Technical Matters Contractual Matters

GEORGIA INSTITUTE OF TECHNOLOGY (21. OFFICE OF RESEARCH ADMINISTRATION u &r.;;---.- •

RESEARCH PROJECT INITIATION

Date: January 9, 1974

Project Title: Infrasonic Wave Propagation in the Atmosphere E-25-638 Project No:

Principal Investigator Dr.. Allan D. Pierce

Sponsor:. USAF Electronic. Systems Division (PPR), AFSC, Bedford, Mass. October 15, 1973 October 14, 1976 (Tech. Performance) Agreement Period: From Until

Type Agreement: ontract No..F19628-74-C-0065 $120,000 (Incrementally funded at $23,500 thru 30 Jun. 1974).. Arn ount:

Quarterly Status Reports; Scientific Reports (at least annually); Reports Required: Final Report

Sponsor Contact Person (s): Technical Matters Contractual Matters

1hiria5 W Colan 0- ,• (thru (RA) ) e) Ms. Patricia A. Murphy (AM) Air Force Cambridge Research. Laboratories Electronic Systems Div. (PPR) Lawrence G. Hans con Field Phone : (617) 861- 4018 Bedforit Mass. 01730 Defense Priority Rating: None Coil ael • Assigned to: School of Mechanical Engineering enifec-e /..(4.t4e • COPIES TO: -60P/Ar00,,,07 adriAattIVII.C.A Principai Investigator Library • School Director Rich Electronic Computer Center • Mien of the College Photographic Laboratory Director, Research Administration Project File Director, Financial Affairs (2) ■.■ SocurltY-Reports-Propersy Off • Patent Coordinator Other

AA-3 (6-71) ' GEORGIA INSTITUTE OF TECHNOLOGY OFFICE OF CONTRACT ADMINISTRATION

SPONSORED PROJECT TERMINATION

Date: November • 1976 Project Title: Infrasonic Wave Propagation in the Atmosphere

Project No: E-25-638

Project Director: Dr. Allan D. Pierce Sponsor: USAF Electronic Systems Division (PPR), AFSC, Bedford, MA 01731

Effective Termination Date: • 3/31/76

Clearance of Accounting Charges: 8/ 31 /76

Grant/Contract Closeout Actions Remaining:

Final Invoice and Closing Documents Final Fiscal Report Final Report of Inventions Govt. Property inventory & Related Certificate Classified Material Certificate Other

Assigned to: Mechanical Engineering (School/Laboratory)

COPIES TO:

Project Director Library, Technical Reports Section Division Chief (EES) Office of Computing Services School/Laboratory Director Director, Physical Plant Dean/Director—EES EES Information Office Accounting Office Project File (OCA) Procurement Office Project Code (GTRI) t-Security Coordinator (OCA) Other Reports Coordinator (OCA)

CA-4 (3/76) E - 2 5-638

GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN THE ATMOSPHERE

Quarterly Status Report No. 1 October 15, 1973 to January 15, 1974

Contract No. F19628-74-C-0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff

Prepared for

Air Force Cambridge Research Laboratories (LWW)

Laurence G. Hanscom Field

Bedford, Massachusetts 01730

"This report is intended only for the internal management use of the contractor and the Air Force". SUMMARY OF OBJECTIVES

Item 0001 of the contract. (Revise multimode theory; develop fully documented computer programs; conduct systematic study of infrasonic data; combine analytic and numerical study.)

INVESTIGATIONS IN PROGRESS

During the first quarter, much of the effort expended on subject contract was largely administrative and educational. Two graduate research assistants have been recruited and are presently being briefed on past accomplishments in the field of infrasonic wave propagation and on techniques applicable to the present study. The computer program developed by Pierce and Posey at M.I.T. is presently being adopted and made operational at the Georgia Tech computation center. Dr. Pierce is.•working on the revision of the multimode theory and has new ideas in this respect which will be elaborated on in future reports. These include the incorporation of casuality (no spurious precursors) in individual modes, the analytical approximation of dispersion curves, and the use of Airy functions (as in the Lamb edge mode theory) in the numerical evaluation of individual model waveforms.

PLANS FOR THE NEXT REPORTING PERIOD

It is planned to go ahead with theoretical developments and computer program modifications simul, aneously, although the computer work will be based on less recent theoretical developments. In particular, a modest goal that has been set for the research assistants as sort of a means of getting them immersed into the problem is the modification of the program to allow for propagation past the antipodes (over half-way around the globe), then running the program for a number of cases corresponding to data that has appeared in the open literature,

APPENDIX A

1. Abstract of paper "Acoustic Gravity Wave Propagation Past the Antipode".

2. Reprint of "Reply" published in Journal of Geophysical Research.

3. Preprint of paper "Theory of Infrasound Generated by Explosions" presented at Infras-Sons meeting in Paris in September 1973. Acoustic Gravity Wave Propagation Past the Antipode. Wayne A. Kinney,

Christopher Y. Kapper, and Allan D. Pierce, School of Mechanical Engineering,

Georgia Institute of Technology, Atlanta, Georgia 30332. - The previous theoretical formulations and numerical computations of pressure waveforms

[such as described by Harkrider, Pierce and Posey, and others] apply only to atmospheric traveling waves which have traveled less than the distance around the earth. In the present paper, a technique resembling that previously introduced by Brune, Nafe, and Alsop [Bull. Seism. Soc. Amer. 51, 247-257

(1961)] for elastic surface waves on the earth is discussed and applied to the acoustic-gravity wave propagation past the antipode problem. The principal modification to the older theory is a shift in phase of r/2 to the Fourier transform of the wave after it has traveled over half way round the globe from the source. The source of the wave is presumed to be a nuclear explosion of given energy E. Numerically synthesized waveforms of antipodal arrivals are exhibited and compared with those for direct arrivals. The necessary modifications to the Lamb mode model theory of Pierce and Posey [Geophys. J. Roy. Astron.

Soc. 26, 341-368 (1971)] are also described. [Research supported by Air Force

Cambridge Research Laboratories.]

Above is abstract of an oral presentation intended for 87th. Meeting, Acoustical Society of America, New York, New York, April 23-26, 1974. W. A. Kinney will present the paper. GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN THE ATMOSPHERE

Quarterly Status Report No: 2

January 15, 1974 to April 15, 1974

Contract No. F19628-74-C-0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff

Prepared for

Air Force Cambridge Research. Laboratories (LWW)

Laurence G. Hanscom Field

Bedford, Massachusetts 01730

"This report is Intended only for the internal management use of the contractor and the Air Force". SUMMARY OF OBJECTIVES

The objective of the study is to develop new analytical and computational

. techniques for the reduction of infrasonic data recorded at moderate to greater

distances from atmospheric explosions. Present research is concerned with Line

Item 0001 of the contract. (Revise multimode theory; develop fully documented

computer programs; conduct systematic study of infrasonic data; combine analytic

and numerical study.)

INVESTIGATIONS IN PROGRESS

During the second quarter, some of the effort expended on the subject

contract was in the adaptation of the Pierce and PoSey multimode program to the

Georgia. Tech computer. This task has been completed and the program is now

working quite as well as previously. In addition, we carried out the modifica

tions discussed in the preceding quarterly on propagation past the antipodes

and did some exploratory calculations (U. S. Standard Atmosphere, no winds).

The results of the computations are summarized in the attached figures (Appendix

A) which were used as slides in a paper presented at the 87th meeting of the

Acoustical Society.

Other computational developments include the modification of the root

searching subroutine RTMI to what appears to be a more logical and satisfactory

form for our present purposes than that of the IBM library version. Incidentally,

the IBM library version had to be modified because it led to numbers too large

for the Georgia Tech computer. Some modifications have also been made in the

subroutine TMPT to enable more logical interfacing with the CALCOMP plotter.

PLANS FOR THE NEXT REPORTING PERIOD

The next quarter should see considerable progress on the contract, as a

larger fraction of the investigators' time will be devoted to it during the

summer months. In particular, Dr. Pierce will be working full time on the con-

tract during the summer rather than quarter-time as during the academic year.

The first priority for the summer will be the extension of the program to

APPENDIX A

The attached are_figures used in an oral presentation at the 87th Meeting of the Acoustical Society by Wayne A. Kinney in New York City on April 26, 1974.

[The abstract of the paper, "Acoustic Gravity Wave Propagation past the

Antipode" is attached to Quarterly Progress Report #1]. 04 ;rawq. 0054, riv. qw1** 4PA Direct versus N u„cito.r 114$ Ant-11,010 Expiosion Arrivals

microbetrolrapil

• 0 I tlretipoole

yvv,v e COUrtiC t4A Vifr Typical Pressure Wave forms

Source 58 Melet.-i- orts at Novaya Zeryll y q USSR Oct. 3 0) 1 9 41

eLt a recorded in su4f..040,n New York by W. L. Donn Ara ColleCtitAeS

Antipodal Az 2 3 3 3 0 k

2nd A ntipodat A3) 1rz----:16 720 al

Time ( 1S minutes between marks Observed waveform

Way formS at eeddei -follow/in 5 a S.

Multi Housatonic ekphsion 50 - mode synthesis 6..ssamed lo M 7-) ill soLem Pacify

157 p. bar Di r et arrivals -50 oniti)

50 Edge mode synthesis

tli p.bars -50 1 1 1

I 285 290 295 300 .305 Time after blast (min) Direct arrt vcd ot.t. r > 500 frrn.

A cou.sti c. pressure

Cicr4. (cot kn r c41) dco

sum over vAiclecs2 nide eometrical intetpi over ;rerencr factor corres. 19n (40) amplikude factor to energy phase C on ser vatiort . kn ( 0) -- velocit".

Pre vious elieor ► .6rea..kS down wAen e /eo ° r ozcv)00 ion

MalytiPicet,tion 0.1° Wave .for QS1 wave 0 ly roa. ekes rd'ipocie

View oP ea'r'th. rz ear a. ntipode.

.1.111•1.01•■ ■ ■ ••••• 10.10••• •••••11101 1...••■ ■■ ■ ■ •••••••• •••• • •••• •••••••• Antipode= poin' on olobe

_ NA/a v e neCtr•

the1tipode _ ..

/el hours r= 17,oao kol

120

>I .fOri;11, 5420.00 5430.00 5540.00 sp oo.00 5660.00 5720.00 5780.00 5840.00 TIME (SEC)

• 14.0 !lours ez ffrer- 18 ,000 ken eA'Flosrov■ f m a V (1-fousal-arsic) - Vot H tip' 5750.00 5p 10.00 5670.00 5p 30.00 5p 9b:00 5p50.00 6p0.00 6170.00 TIME (SEC)

r:r lei 000 krn 14• 4 A dur.1" a I ter e OS ' Oils

1,0114;n1,To i 6p70.00 6,130.00 ,190.00 6250.00 310.00 6370.00 6430.00 6490.00 TIME (SEC) (efter burst) Waves nea.r Antipode

(feat circle distance From an,IL;pode.)

p A cos (cot c,03(,t.k+ 0) V ciet incomin5 wave oatloinl wave

-For 57,2(Z.11 r

p 1) ,70 (k r‘ics, ) cos (cot + m a.tcluncl aLove cf.. r3 tives

(.277.‘0/3 ) =

ove frereny and modes ;piplied)

Phase Siti.ft on Pa ss?ncs +hroksh (Theoreticcal ✓avePorms) r /7) 000 km (before antipode)

‘0300• x10 4 3o r n 6030.00 6090.00 6 ,150.0 6210.00 6270.00 6330.00 6390 TIME (SEC)__

r = 22,0o0 (cit.ker claztipode)

7100 N10 1 550.00 6710.00 6770.00 6830.00 6890.00 6950.00 7010.00

TIME f SEC GIC4"er deo/lea/on

U SSR 52 MT EXPLOSION red)

Farther Dispersion bey ond the AI p o de 1 )1%6 hours o ffcr r = 2/) 0 0 0 krn egiolovio

IiV

Crain -44— xl 0 t 6710.00 5770.00 6830.00 6890.00 spso.01 7010.00 7070.00- 7,130.00 1 TIME [SEC)

22.Z hamr3 I o.:Pti!r exphs464 r= 25d 000 A

JOrnaL x10' 8000.008060.00 8120.00 8180.08 8240.00 8300.00 8360.00 8420.00

TIME (SEC)

3400t) kw*

Iflokins (Hou sa-bon 1 4) s10 1 9600.00 . 960.00 9720.00 9780.00 9840.00 9900.00 9560.00 10020.00 TIME ( SEC 1 62.-ci. er burst) e o ry and Data for

Antipodal igrigi Veils

Tti eory

rime al cr firs t 21.7 hrs. Data

0 im 1'4"

Sotirce: 45-8 Ategoions at A4tfayq Zeml y a, USSR act3oj iciai Data recorded fn suburban New York by W.I.. Dortn and Collea,ues. (r 333601041) GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN THE ATMOSPHERE

Quarterly Status Report No. 3 April 15, 1973 to July 15, 1974

Contract No. F19628 -74 -C -0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff

Prepared for

Air Force Cambridge Research Laboratories (LWW)

Laurence G. Hanscom Field

Bedford, Massachusetts 01730

"This report is intended only for the internal management use of the contractor and the Air Force. SUMMARY OF OBJECTIVES

The objective of the study is to develop new analytical and computational techniques for the reduction of infrasonic data recorded at moderate to greater distances from atmospheric explosions. Present research is concerned with Line Item 0001 of the contract. (Revise multimode theory; develop fully documented computer programs; conduct systematic study of infrasonic data; combine analytic and numerical study).

INVESTIGATIONS IN PROGRESS

During the third quarter, work on subject contract was divided into three parallel tasks. Mr. Kapper with Dr. Pierce's guidance continued work on the extension of the computer program to include leaking modes, while

Mr. Kinney and Dr. Pierce worked on the extension of the program to include higher frequencies in the modal synthesis, In addition, Dr. Pierce has been working on extending the model to include absorption. Further details on these tasks are contained in Appendices A and B.

PLANS FOR THE NEXT REPORTING PERIOD

The work described above will continue. It is anticipated that the work on including leaking modes into the program will be completed by the end of the next quarter and that the results will be available in the form of an internal report (M.S. thesis) written by Mr. Kapper. At that time, the question of issuing an interim technical report will be considered.

PUBLICATIONS

In Appendix A are attached three abstracts of papers which are being prepared for oral presentation at the forthcoming Acoustical Society Meeting in November.

Appendix B gives a brief writeup by Mr. Kinney of the work he is currently doing on the high frequency extension of the theory.

APPENDIX A

Abstracts of papers submitted for oral presentation at the St. Louis meeting of the Acoustical Society of America, November, 1974. Leaky Infrasonic Guided Waves in the Atmosphere. Christopher Y.

Kapper, School of Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332. - Prior theoretical formulations'and computational techniques

for the prediction of pressure waveforms generated by large explosions in

the atmosphere have considered only fully ducted modes. In the present

paper, a technique for including weakly leaking guided modes in concert with fully ducted modes is developed. Modification of previous theory'

includes the extension of the boundary condition at the upper halfspace to

include a complex horizontal wave number. The major alterations to the

computer program Infrasonic Waveforms (as described in report by Pierce

and Posey, 1970) incured consist of the computation of the imaginary part of the newly incorporated complex wave number, extension of the normal mode dispersion function to lower frequencies, and a second order correction

factor to the phase velocity. (Work sponsored by Air Force Cambridge

Research Laboratories). Asymptotic High Frequency Behavior of Guided Infrasonic Modes in the

Atmosphere., Wayne A. Kinney, School of Mechanical Engineering, Georgia

Institute of Technology, Atlanta 30332. - Refinement of previous theoretical

formulations and numerical computations of pressure waveforms as applied to

atmospheric traveling infrasonic waves could include a description of

their asymptotic behavior at high frequencies. In the present paper,

calculations based on the W.K.B.J. approximation and similar to those introduced by Haskell[J. Appl. Phys. 22, 157-168 (1951)] are performed to

describe the asymptotic behavior of infrasonic guided modes as generated by a nuclear explosion in the atmosphere. The results of these calculations

are then matched onto numerical solutions which have been given by

Harkrider, Pierce and Posey, and others. It is demonstrated that the use

of these asymptotic formulae in conjunction with a computer program which

synthesizes infrasonic pressure waveforms has enabled the elimination of

problems associated with high frequency truncation of numerical integration

over frequency. In this way, small spurious high frequency oscillations

in the computer solutions have been avoided. (Work sponsored by Air . Force Cambridge Research. Laboratory). Fundamental Nonlinear Equations of Atmospheric Acoustics: A

Synthesis of Current Physical Models. Allan D. Fierce, School of Mechanical

Engineering, Georgia Institute of Technology, Atlanta 30332. Paper represents

an attempt to give concise but comprehensive set of macroscopic continuum

equations incorporating all physical mechanisms of significance in atmos-

pheric acoustics. While recent literature on sound absorption (e.g., Evans,

Bass, and Sutherland 1972; Calvert, Coffman, and Querfeld 1966) implicitely

contains such information, discussion tends to be centered on the particular (albeit fundamental) case of plane waves of infinitesimal amplitude and

constant frequency in a homogeneous quiescent atmosphere. Two problems

of current interest (attenuation over distances of tens of thousands of

kilometers of ducted acoustic gravity waves and the non-turbulence contribut-

ion to rise times of sonic booms) cannot be discussed in this context.

Resultant set of equations incorporates Swartz, Slawsky, Herzfeld, Tanczos,

Shields, Evans, Bass, Sutherland, Piercy model in which different temperatures

are assigned to different internal vibration modes. Stokes' (1851) original model of electromagnetic heat radiation transfer is incorporated as a

somewhat regretful compromise since Smith's (1957) model is restricted to

plane waves. Value of q is estimated from recent work by Calvert et.al.

(Work sponsored by Air Force Cambridge Research Laboratory). APPENDIX B

Asymptotic High Frequency Behavior of Guided Infrasonic Modes in the Atmosphere

Wayne A. Kinney

One of the present objectives in refining the computer program

Infrasonic Waveforms is to eliminate spurious high frequency oscillations that appear in the computer solution. It is felt that this "numerical noise" as it might be called is associated in part with that portion of the solution which is derived by a numerical integration over frequency.

At present, this integration as it is performed in the program is truncated abruptly at the high frequency end (see equation 2.5.5 on page

38 of reference 1).

It was discovered by trial that merely widening the frequency window used for integration in the program to include a higher frequency upper limit did not eliminate the numerical noise. Thus, an effort is underway to formulate high frequency asymptotic expressions for certain key dependent variables with the intent of matching these expressions onto the numerical 1 solutions that have been adopted to date .

In particular, high frequency asymptotic expressions are being sought for the program variables AMPLTO, PHASQ and VPHSE as defined by equations

2.5.4b, 2.5.4f and 2.5.6 respectively on page 38 of reference 1. It is felt that calculations based on the W.K.B.J. approximation will suffice in obtaining these expressions. In fact, N.A. Haskell performed similar calculations in his discussion on normal modes in sound channel wave pro- 2 pagation .

To date, an appropriate expression for VPHSE has been derived using a calculational procedure similar to that used in describing the asymptotic 3 behavior of the linear harmonic oscillator in quantuum-mechanics „ In addition, it appears that, as was demonstrated by Haskell, the asymptotic form of the Hermite polynomial will be incorporated into the expression for the final solution. Of course, once the asymptotic expressions for AMPLTD,

PHASQ and VPHSE are formulated, it will remain to incorporate them into the computer program along with associated modifications.

REFERENCES

1. Pierce, A. D. and Posey, J, W., Theoretical Predictions of Acoustic Gravity Pressure Waveforms Generated by Large Explosions in the Atmosphere, Report No. AFCRL-70-0134 (1970).

2. Haskell, N. A., Asymptotic Approximation for the Normal Modes in Sound Channel Wave Propagation, J. Appl. Phys 22, 157-168 (1951).

3. Schiff, L. I., Quantum Mechanics (McGraw-Hill Book Co., Inc., New York, 1955), pp. 60-62. E - Co,3

GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN THE ATMOSPHERE

Quarterly Status Report No. 4 July 15, 1974 to October 15, 1974

Contract No. F19628-74-C-0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff Prepared for Air Force Cambridge Research Laboratories (LWW) Laurence G. Hanscom Field Bedford, Massachusetts 01730

"This report is intended only for the internal management use of the contractor and the Air Force. SUMMARY OF OBJECTIVES

The objective of the study is to develop new analytical and computational techniques for the reduction of infrasonic data recorded at moderate to greater distances from atmospheric explosions. Present research is concerned with Line Item 0001 of the contract. (Revise multimode theory; develop fully documented computer programs; conduct systematic study of infrasonic data; combine analytic_a&I numerical study).

INVESTIGATIONS IN PROGRESS

During the fourth quarter, a number of significant developments took place. First, a general computational technique was developed to handle leaking modes and for incorporating this technique into the computer program. The results to date are summarized in the attached draft (Appendix B) by Christopher Kapper. Another development was a technique for finding the higher frequency portions of dispersion curves (phase velocity versus frequency) taking into account the presence of two sound channels in the atmosphere. This study has clarified the previously not well understood features of prior computations of such curves and has suggested that the previous selection of modes to be included in the waveform syntheses may be inappropriate for an accurate depiction of the later arrivals at ground level. Progress has also been made on including realistic absorption mechanisms into the program.

PLANS FOR THE NEXT REPORTING PERIOD

The draft of Scientific Report # 1 is now about 90% complete and the first priority now is to complete it. We are presently working on the details of the inclusion of leaking modes into the synthesis program.

This should not take much time but we are unsure at present whether we

APPENDIX A

Slides of three papers intended for oral presentation only at the November, 1974 meeting of the Acoustical Society'ff America in St. Louis. The 88th Meeting of the Acoustical Society of America

Chase-Park Plaza Hotel • St. Louis, Missouri 4-8 November 1974

TUESDAY, 5 NOVEMBER 1974 CHASE CLUB, 9:30 A.M.

Session A. Physical Acoustics 1: Atmospheric Acoustics

11: 00 A6. Leaky infrasonic waves inthe atmosphere. Christopher Y. Rapper (School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332) Prior theoretical formulations and computational techniques lor the prediction of pressure waveforms generated by large explosions in !he atmosphere have considered only fully ducted modes. In the present paper, a technique for including weakly leaking guided modes in concert with fully ducted modes is de‘eloped. Modification of previous theory includes the extension of the boundary condition at the upper halfspace to include a complex horizontal wavenumber. The major alterations to the computer pre„ ram Infrasonic Waveforms (as described In report by Pierce and Posey, 1970) incurred consist of the computation of the imaginary part of the newly incorporated complex wavenmnber, extension of the normal-mode dispersion function to lower frequencies, and a second-order correction factor to the phase velocity. [Work sponsored by Air Force Cambridge Research Laboratories.

2..

6 ic 1 z 'V:.0—. N. t • ,-. " L. ...._., 6 6 .3cc • GlO I 0 era_ 8 7 e• cc cc 7.1r iii N I/ 4.9 we r '77—te," u• 9 9 0 4. x %., ....• In a P4 H 9 .4 3 4 < ••' U VI Z X 5 C 0 4 7.eC 1:11.1 ^R•11111111

41 Q

CR. 8.o ow.* .--

0 f:C AMMAR SatauCi•C•1. •••••••• CPI 0 U. a ••• X PERK* a KIRIOR1

PHASE VELOCITY VERSUS ANGULAR FREQUENCY M DI

•

*MR. 1.2N•so. INR•l• 111S•••••1,1•,,,, .413•11 .S..••••• ■ • -•-..•-• • -- • 1•1 RS/Inlet 1e4••••• RR. .5 , 115 ...Mite ....yr.. ...

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Int %I So.cloots:c s) 0 root net,.

s t eG =O 2 sp )0

ito root s r's 1- 40-TPiled2eA-43.9_, , 5°1 itwo

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PROTOTYPE EQUATION ROOT SEQUENCE FOR 6)0 AND 6(0

--:;:tr:1: • .. 4 ......

• .2. • v••.• . ..

...... AKR (I/KM) AKR (I/KM) SIGN - OF THE REAL PART OF THE EIGENmODE SIGN OF THE IMAGINARY PART OF THE DISPERSION FUNCTION SHOWING ROOTS ON El GENMODE DISPERSION FUNCT ION SHOWING BRANCH IL ROOTS ON BRANCH

is • '4.1

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• The 88th Meeting of the Acoustical Society of America

!hase—Park Plaza Hotel • St. Louis, Missouri • 4-8 November 1974

'UESDAY, 5 NOVEMBER 1974 CHASE CLUB, 9:30 A.M.

Session A. Physical Acoustics I: Atmospheric Acoustics

10:45 AS. Asymptotic high-frequency behavior of guided infrasonic - modes in the atmosphere. Wayne A. Kinney (School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

Refinement of previous theoretical formulations and numerical computations of pressure waveforms as applied to atmospheric traveling infrasonic waves could include a description of their asymptotic behavior at high frequencies. In the present paper, calculations based on the W. K. B. J. approximation and similar to those introduced by Haskell [J. Appl. Phys. 22, 157-1G7 (1951)) are performed to describe the asymptotic behavior of infrasonic guided modes as generated by a nuclear explosion in the atmosphere. The results of these calculations are then matched onto numerical solutions which have been given by Harkricier, Pierce and Posey, and others. It is demonstrated that the use of these asymptotic formulas in conjunction with a computer program which synthesizes infrasonic pressure waveforms has enabled the elimination of problems associated with high-frequency truncation of numerical integration over frequency. In this way, small spurious high-frequency oscillations in the computer solutions have been avoided. [Work sponsored by Air Force Cambridge Research Laboratory.) OR , GR, GR, SOUND CHANNEL DUCTING 0(3 GAO GR,y

• ' -SQ`—•••,1111 I , S, iv "11 I 5 • • • • I, ',et .” v S2 • ■■ S2 S2 .11 ...... S3 s,.

S4 5, S4

Ss Sy s5 ..••••••■•••••IN ...... N'e _— 75 75 TOTAL j\ArWmftwo141, TOTAL_\1\itikevIg 1114/41414w 1\TURNIND vvv,,y lai TOTALAt ik ! t*104,411,4 g ,tti POINTS If 1 11 -75 -75 UPWIND CROSSWIND DOWNWIND

510 540 570 510 540 570 510 g40 570

TIME AFTER BLAST (min) Vp ACOUSTIC PRESSURE SOUND SPEED PROFILE

DISPERSION CURVES 4

500 MODEL ATMOSPHERE „, ativell

150 ---••-• SUBTROPICAL SUMMER U. S. STANDARD ARCTIC SUMMER ATMOSPHERE ,1962 ARCTIC WINTER

400

100 / 30

t / 0 20 N 300 a. A / r? GR 1 1

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10 5 3 2 1.5 PERIOD IN MINUTES -

0.41 W.K B.J. MODEL U FULL MODEL

P _ ikx (1) - Fce)e- iwt e x DISPERSION CURVES

/- 0.34 , + [Lei ji)) 2 ._ kniF 0 d'F 0 (2) dZ ta

(3) c.27 0 0.2 ANGULAR FREQUENCY (SEC-1 ) + Tr (4) — = vp (A) W.K.B.J. MODEL iluatom DISPERSION

where p. acoustic pressure CURVE SETS = ambient density ta =angular frequency k =horizcntal wave number cml= sound speed as a•function of altitude

VP = phase velocity 0.27 n = 0 0.1 ANGULAR FREQUENCY (SEC-I )

DISPERSION CURVES DO NOT CROSS

W.K.B.J. APP RO XI MATI ON FULL MODEL wu 0.34 to

• 0.32 0.32 F- LOWER CHANNEL

U 0

G l > 0.30 0.30

RESONANT 1 NTFRACTICN Ln UPPER ' CHANNEL BETWEEN ADJACENT MODES

• 0.28 0.28 0.06 0.08 0.10 012 0.06 0.08 0.10 0.12

ANGULAR FREQUENCY ANGULAR FREQUENCY (SEC -1 ) (S EC -1 ) The 88th Meeting of the Acoustical Society of America

lase—Park Plaza Hotel • St. Louis, Missouri • 4-8 November 1974

'HURSDAY, 7 NOVEMBER 1974 REGENCY ROOM, 2:00 P.M.

Session T. Nonlinear Acoustics

(Joint Session by the Technical Committees on Physical, Engineering, and Underwater Acoustics)

4:20 T8. Fundamental nonlinear equations of atmospheric acoustics: a synthesis of current physical models. Allan D. Pierce (School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332) Paper represents an attempt to give concise but comprehensive set of macroscopic continuum equations incorporating all physical mechanisms of significance in atmospheric acoustics. While recent literature on sound absorption [e.g. , Evans, Bass, and Sutherland (1972); Calvert, Coffman, and Querfeld (1966)1 implicitly contains such information, discussion tends to be centered on the particular (albeit fundamental) case of plane waves of infinitesimal amplitude and constant frequency in a homogeneous quiescent atmosphere. Two problems of current interest (attenuation over distances of tens of thousands of kilometers of ducted acoustic gravity waves and the nontur- bulence contribution to rise times of sonic booms) cannot be discussed in this context. The resultant set of equations incorporates Swartz, Slawsky, Herzfeld, Tanvzos, Shields, Evans, Bass, Sutherland, and Piercy models in which different temperatures are assigned to different internal vibration modes. Stokes' s (1951) original model of electromagnetic heat radiation transfer is incorporated as a somewhat regretful compromise since Smith's (1957) model is restricted to plane waves. Value of q is estimated from recent work by Calvert et al. [Work sponsored by Air Force Cambridge Research Laboratory. I Examples o; Problems in which. ,ve rd.? 4.; may dc Important Extended Continuum Concept for Air: and for whi ch nic4d of • plane wares of COrti*Oait fratl•nell 1. Many coexisting "fluids' • with vemi small A mplitude s • in Aemelega sett atmosphere 2. For each deg ree of freedom of ea ch • witA lravdy roes/rate"( me !feu kr type , one in general eonce,ves of a di fferent fluid

Fl uid Properties :

rturnier of carriers per unit volume -

afitragt Diffracti•n Rise Timms of Sonic &•ut average velocity 14 carriers

average energ y of type per carrier Leal. rani. prepasiorme.*C- • aelmistie-yamt, wayes • •• ei = Ni average energy of type i • per unit volume

mass of carriers wlielt carry energy of type •-

a Temperature Concept for Gas pot ••■■•••■•• .. G enera/ Model of Air on. Thermodynamic Eguilihrium

Siappase Ei i s rAe type I *ret4 " '""t1 of per carrier carrv :m 1 such energy 04 Hz 0 C 04 Let g\'i Cr)be the avera5e such energy especrssi per Molecule . Rotational Vibrational earrser were the 543 in Catodet de vets a davit, a Ntermeeirnanwre eludiirisue et -freedom Freedom temperature T

NA 2 (deleorrole) .1 acT) defined as Oz 2 (dezener‘te) 1 Iss, on a f h.ve 3 • Hz ° 3 3 degrees 'a Cr) = trans lofional T CO2. 2 (degenerate) il freedom (.t deienorate) EF festive temperature T for ga s as a. tmerti elf any makede = zlenergl in each tvA • AB le 'Med al COI fairrI dens "ilea Cr. Prordom

Cont rol Volume Formulation e7 Some Reasonable (More or Less) Assumptions of Conservation Concept

1. Fivera3e veloedy of carriers equal to mess- eve 505 v. Flat of some i Vilacit, *11) • at imaotal Z. Stress . tensor minus (/3)64 frice)(1.1ut ieeidt) Net rate of ,n.reese Net rate of proportion./ to rife of s4e.r tensor = S,e ! 1...4tij per mode re/wre n pr anut Via Jae ea colhamas from *tins, cauJaa JP1611 but nit 10'0 neet3soni, . 3 (r)- T() ver y

of utter/ha (rotsseiOrl✓ or riliritInell0 CothsionS transfer: energies pproxolAtely it. (zeirreel &telly moment&on and k /ee;fy un terra Ii led.) anent He.t proportion./ #0 77" Other causes 5ra ✓ ai background electromoretm creld

Pressure dePmed as —69 Irene o0 stress foliar

."'"fSertir tenser) am Alamtntom • Pat tenser o-

1 Concept of Flux Mass and Momentum Esuations J) is flax .2.F. scalar (or vector) physical !manta/ a 0%11 49 (a. .•gdA) rep fa amount flem•ai per land' tarns jet (moss) po,.rive SSNISe across area dA arta • if + v.(0)= normal A.

C Pg. = -ve v.(1s)-1- (2 3 (momentum.) Dt

N afiy mok.ie 3 =. (complete N coral Is Mrsion) P lu.x Theorem for Gases 15t .,1 svitare I(t)= carrier Carrier eRTE, ''pressure M Ana. Alf R=. k on•ers in Air, 1 Ve3CoSatt4

Sij = raft el sheer fres., p ve,.3 a n,,,,,,I.tri kro454 ww.orep &ureic vairof13 = ) •9 earners per • , tr for on. f 'roloot ear star 4 * rt., to ivra = ;4+ v.:yr •.t•t*I" -hole clerovabve. 4- O TH ER TEF.14 5

tr 'V lir Derived form of energy esuation: Value of Stoke's Coefficient'

C to do Dclit4 Mike C * 17: Pt • my has .hlo parr and CO2 peril' p.,irendf + 0 trace (s* S) = 41,4 respective/y to .fraction of air _ Lot) . Note. 3 not Pstat c ,), or are or Co, M.A.% enlIcisMat Calvert et al, conspu.eniLions Jul5ast . e-• enternal trans/47;one/, rat •t tonal, and VI Att-c 1 - en erl, ea per unit Volume 4re 4 Se,O r7 Ra o me/teats/cal __ X 2: (T) thermal conductivity when .. 1'00) = .00; . ___...... _....___. -3,_ -I 1 „d energii oisorded rachotted (Markham. et at estimate ja Ile. ro ,..A see. w_erA e— _ per unit time per sold vo/4one • . — ..a w .r...1 - wall- valtos44, ht meters.) from and to daeki rounA' electromainetfe -fseld

Al Se i- type energy esuation Radiative Term i„4 in Total Imply Eviction

jet.. ). /Yet- aortas. of 1-ton will _ ern, R i t ) • rt. wig tat 7.41i:1 = 2- •e eCit rAk) • 5ria trans‘tions Li 3, (0,t) e Tor, _ • if/pea dna t. = 4614 onfis.wns r tn, •• ik

Se dente...elation enem y per Unit VOIlnkt per unit transition types : ern 4.,.J when em Acid in therms/ .t temperature 7 01...4 4.1r &mina) ,j-type enee51 trans/Ai-moat energy rn.w Spectral isorpi•on coecheient of ern wart in aie per unit our density a+` f'reguene i e em,s4.fr fr.1..... s end .01. coorsdoon.) energy i-type energy tramintional Overly

Volume integration extends over Attie sphere 2-9748 energy deetromouretie energy centered et point x j Rc I 1,1- rrl •

vanishes if 7; al Stoke's assumption 3 Ter 1 vanishes f and 7;-;r T(2)J 3 vanishes e„, (1) = e if 7 7;..t

=4 . • ..„,„ •

Ir 20° evt, augiSi Li near Form of i-type, Enerly riOeff octants at C and sea / 1 prestare

, ply /De = - 1.7 • tol ('l,- e,)+ 4.5 a le (ri Tty)

Dt —4 Yo,i (7; - Tat)+ '42; (irr pli/Dt = ••• LL ?a SOS fi 4:0•0941. 2.7 • 1081a .10 . gyo' 4.10

I t7rrol 14. /.r./04_7(73-.770 /.6 *10407-70 (7; Zee.) J.1 so. 41 ("5- 1,i/0t r — I L P1074 34' 106 Fjj(T3 Tv) # 13. Sa (5/4)F4 apparent -collision freiaenel fe = £0107 (7.; - Te.) dooensioAless collision efficiencies are 074 Lot = .141.16 1 + lc. /0 71;4 U. 'of d(7-7;). o% /elf aomevoliot dependent on total tearrafai ►e • (1;-;) depenilent on con.rtotuent oeraettons

DT/tats - La Ake = .14e • 'vt P total tare derivative I Vo7ft 1(s -- 7.7) 4 C 1 /0 Si

41. 3 .10. C4 * yY.I frort“ta = 2' entfera tare avera3e1 over /Aria volume -74 'Ca l i .) Tart .r awlee.le. DT, /tot -I 7.10 4 .. /4. 61(z-r,r) It 4 a 10‘ ...La are CC altereott (7; 7";?) 40(7: - re) -gr,,,(7;-;)+1, ZY.,;(7, -7;,) fset Bt I Oloomotr.o dretoto) Co A .1 14 fivisLoo Cri (70,-76;) D 7;./Pt. A. beam, .r 00 2. 3 or•Irotra ot 04 3 dill.metNe OreicAml COAr leorLoi ,ot 114 0 tr34 are I t (

Fetrtiter Approximations: Su3vistett Vahr.t,on Relaxation Eligat mat

In rotational energy artettionS , Ignore 20..t eller y transfer to and from v:StAtionci Jraie.1, or to total-Hint/ statt$ of ether maktuks.

rtealtrak Notrevrt On) _Dt/Dt = - L31. A 68 /07,(COA) :1: I.I410 K11,4( .1-_To D7.`'" 7;•tiA Tiv) 4 At .4 toe A

2. 14 voArational en ergy equations forma/Ii consder 0.11 re e .ctista 1 te sweet lore! to nu ,* trag3ktioaai erci fot = -117.ca'ID ef(e01).gs'raf(x,,o)] (T4 -70) temperature. 41 4: 447 104 (To 4-r) Sectroryl a snot ic iSsorptiovo r o diati on Cora r,ketioti t0 to t' 'et la I and Pi bratiorta I enersy etuatiens T,,..te art = -[4.7 a /0` + A0410 1 f(ll,0)](7,,e---7;)". assarned ne9lt114/e., % 6 I AO' (6,- .4) . ("7: - 7;) + 4 );. ZY4 Cr -7- )--ar At a 3 Pr . . Zs the total enersy epo.tion: peve,),/pt [ - rP Dye P3'

-re bro.,) Ajta 44„:4',) ,-. I N.

• oswaRAII.M5.111.4.41. APPENDIX B

Draft of a manuscript on the leaking mode and antipodal arrival theories. It is anticipated that this will be incorporated in a somewhat revised form in Scientific Report No. 1. of 011175KTS

PA%e.

NT S

LIST O li)P4S •

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CHM-

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0;7-- DAcRhso!:\t‘c wA\1 roP.1/5 70 INCLUDE

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corAtiu4.-ktAtml l- E.Arttn svir::je 4 lit e 1.v, ,j.. ../: 11.A

of Lec6c,f , S ,)-6 tOck 1\10(1,;-ric..i0v* zt\kRA6ontc.

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3.1 5 i ire Cov'T Piavle Charter Pale.

,2 1 114n.y.-.; .Cov. Mcd-e •

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4-t To Cl.r.siJevotiitAs for foss Art:lipoid

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AvAsq.--rikt \tia.vei.11/1,“.

\7 AM) RECOMMENIDA7ION 6

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‘Y-1:11_11--( PF.f)GRAAA =OR, DETERMirlA6 THE SiElN OF

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CWEGA 1234-5673 9 ;312345:5799P 1 23'!5-s-7:19912 34 567-393 PHAS E= Ve-OCITY DIR ,":CTICr IS 3 5 .00CD=6"rcS

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++++++ +4+++ ++++++++ +++++++ ■•■■••■■•■,.....M • . 0 01100 12122 +++ T++++ -1-4 ++++4+++++++++++++++++ ++++++ . 0 nNAL9424 - ++++++ .011iLG.573!-: +++++ e-ow------. 061Gu64C 4 1 1E .00n00.1347 4 - .00160a1347 -.Cnoli1J;;4;141 +++ ++++++ t+4...+..,.+Ti++++4+++4++++++++++++4+- ■-■"---.----■■■•*■ --■■■■■ - . 0 4100 0 66735 / -. 0 an012122 - ---• -- .0anuUt4Pln +++++ ++4.1. -1- 4.+++++4++++++.4.++++.4.-144.÷.4.4.11.4. k4----- .0nau017516 -.141;10.;?020. +++ + 4++++4-4- +++,++ -1. 4-4- ++++*+++.4.4.0-..F iti+iti.l. _ +Tsf++++++++++,+++++++.1.+++++++.1.4,4++++.4.+4. - 4-1- + + ++++++ 4-+++.1.+ 4 +4 4 4.4.++++++++4.,T+41-44.+4!- ..- - . 0 (1.14 0 3:1980 -T ■ T+++ ++++++ 44,--1-++++.1.4.0-4.-1.++ + 4 4 4 + .1. 4.4 1,44.1. 4 .--".°014:333673 +++++44-+++++++.4,+++++T ++++++ ▪ ++-0-4+*4-4.+++,+,+++44 +4 +++++++4++.44+,+++ +- --_-_-_ - _ _ _ -4+++++++++++++44++++++ 4.44-4-++++-i.+4++++++-1-4------"'oOng01417571 +++++++++++++--■■■■. -

.....1/11;16° ■ 71U.S +++++ - oliarICALAP37 4++++ +44T+4--1-.4.+ +T-1.++4.1.+++++++++4.4 - . 0 o00 0 -,2531 -- . 00r1; 045,24. +++++ - ■■ +++++ ++ 44.-1-4-Tt+ ■ +++++-1.4 ++++4++4+ 4+4 +++++4+-..---_ ■_---....-.■■ +++++ +++++ ++4+ + + +4 + +++■ 44.* *4 +4 4+++++++++++T-V4+ ■-44-4. 4"■■-■-■..-

1NT:EGA= .n0313.2066 Al:!= .0111300 .0311406 .0311691 .n311866 .0312081 ;------.C 3 1P2 77 .0312472 .15312-67 .0112862 .n313058 .0 3 13253 •0313946 .r.313.,94 .0313839 .03i4034. 70314236 - -.0314425 - - .0314816 .0315011 -03152 0A .0315462 .6315-;97 .0315742 .03%5988 ------0316163 -- •031637S .6315573 .031769 .0316464 _ .03171 59 "- .C31731-.5 "- .1.317050 .03177.5 .0317941 .0316331 •0318722 .0318917 .031q11 ■ .0319306 .u319 .r,03 .0319696 .031 9 94 .032no64 .03k1;264 .032046p .03'41675 .03,0A76 r .0321201 .03214.5i, .0371652 .03,1847 .8372041 .0322237 •4.3;'2.33 .03 , 2628 .03,2823 -03734.1 .. .032321 4 .1.3234D9 .0323605 .0323806------

1-; q%,7-t, . t Sip *tile Refit Pckv-t kvte. Fi8eAvit o ctt Disp-Q'Sioe) cici on. sroowlvt3 gzsts 17-

PAR, DF - 14-4.nFN

. 0noilis , 5224

_ _ - - _ _ _ _ .0na 0+1 755 - ------ -+++++ +

. 3(1 :) u0 .5 3 6 73 •••• + -4-1. +++++++++++++ -.08o(4 63n g80 - - +++ - ------- +++++ .0n5p0p326n . 0 m-. 6 0 :;5592 _ +++++ 4 0 a36 0 ;4 2A9p-■ - +++++++++4.4.4. +++++ +++++ . 0 .116 0 1791u ------,.++++++++++++++++++++++++++++++++ +++++ +++ 4.4+ ++++++ +++++++++++++ .0n(160/2122 ______+++++ ++++++ ++++++++++++++++++++++++ .urenLAti9.429 ---- - - -+++++++++++++++++ +++++ +4+4.4+4++++ +++++ ++++ ++++++++++++++++ ++++++ +++++++ ++++

.6:100.1347 -.0836001347 - + + + + + + ++++ + +++.- - - -

- .0n1tA.04841 ++4 + ++,

-.4.11inu0A;429 +++ +++++ + ++++++

--.COICul2122 ___-___ ++++++,.+++++++,++++++++++++++ ++++++ ++++++ + +,+++++ ++ +++ 4. 4.„0._ ..-.(4,10 ; 321), ++++++++++++++.,+r+++++.+++++++++.+F 44-4.+4+++,4.1.4,f,i++.4-4.4,++++.1.t+++ + +++ 4 ++.1.4. 741. 4. 44. *+++.4.4++.04.44-4.44++++++++++.4. ■ ++++++ + .1.4.4. 1 - -.0apu 0:)a 50 .4. ...4. 4.+++,++ 4 ++++++++4.4. 4.4-*+++++44.1 ++++++ +++++ +++++++++++ ++++++++ ....Onn0036:467 +++++ ++++++ .-.0,1 "0390 6 1 --++++++++++++++,+ +++++++++++++ +++++ + +++++ ++++ +++++ ++++ +++++++++ +++++++++++ +++++ +++++++,+ +,++++++++++++------.01111c.0+444g - ++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++ ++---- -.0n1 b i.4.7143 ++++++++++++,+++++++++++++++ + +4.+++ ++ +°~~ +++++++ - .04100 - 9z:37 ~~~-~++~~+~~~~ - ~+~~~~~ ++++++ + + +++++++++++++++++++++++++++++++ - .31 ,10-'2531 + +++++++ +++++++++++++++++ ++++++ ++4 ++++++++ +++++++ +++++++ +++++ ++++++ .4.44.4-4+++ -.0an„:),791,% ++++++++++++++ + ++++4+4 ++++++ +++4+++++4+++4-++++++++++++++ - :01116 6.7:0Al2 ---++++++++++++++4:+++++4.+++++++ +++++ +++ +++++ ++++++ +++++ ++ + +++++ +++++ -.can u o"330, ++++++++++++,+.+++++++ +++++++++++ +++++++++ +++++ +4.4+++++++++++++++ +++++++ ++++++++ +++++++++*+++++++++++++++++ +++++ +4++++++++++++++++ _ _ .on7s42366 AKR= .03113n0 .03111,05 .0311691 .(1311886 .0312081 _ 7.633p27-7 .0312472 .312,,.67 .0Al2862 .0313056 -0311,53 .0315446 .o313^44 .0313839 .0314n34 4014-230-- -;0314425 .,,314h2.- - .0114816 *.41315n1f - - •- .03 152 0 - .031h4n2 .315597 .0115792 .031595h - • -.0316176 .0315573 .0316769 .0,6q64 .,i31755(1 . .0A17745 .- .n317941 , _ _ _ +r 3 la1 3,, .0316331 .318927 .031,1722 .n318917 --- .031911: .031930 .0319`u3 .0119698 .0304,94 . _..03 20 4,.4 .U320204 .320‘.69 .0320675 .8320R7o -.0321664, .0321261 .03214i, .0321652 .0371A47 .05222 7 .0322453 .0322628 .0372823 .65p361.4 .03232i4 -0323409 .032.i605 .0323A0n - AKR 0/ (4 04)

8 t tr.R.. 2. ci oiti, fri 411. it., ■:%. t4 ( Pi a =:„. 7 44)_:A ._...7- , ,,„,,.., ----' .1--n-% ' .- 4.7. - 0 eittil I° _ ..1 ■ e. o ,_ . _ . . D i 5p..(1,-c,i or. 4- tiv%Cil 0 l''‘ _1:-..-.:1.-_-•-1 %-ei'-; R.oci,S OA 5i-a;At'll 1 1 i N s ik,'0.1.e. Rook: PctizIA a AA a a ▪ ▪ •

e cc on is ertf:t(y rip or LaVIV.S • at-,1;,%, 0 C. ea le Y e o,r ci9€ ts(Li„,, r VA A t fifz, "fisi . 5

pireviousfy p(*Nizect OUP by PiQtrce. Po e\/ ,

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vlo!ii..6t LC) Clillrhy VA ai-JA i C alt\1' \iv . -OA; is 56

k 7 (Y.A4--, ICJ" e e 'sr C't vuetA , ptrec..e.!:Xlvy:3 .Lec-Lc,o,v\. D.Af 6.3 L- 1 -1•' 111.

tss ;. . t eY,t- ti- '''' - i 1 •:z-..',',', 1., t \N ! '“,"A ',.. ',..5 1, Cli... (3- Ii)

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1 , si; . C KL V e vA fly :-.1. -e, 1, kt,... -.;• 41:3, %. IS 5 c yv-p r t s trcA `,c,

• . „ . . Lt = A ∎ Y'• \ cx-t TT\ 4..v‘ At. 7A C a OA Etv- -st

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e c f ss q,ry t kVie wrixivAattavt 6.1

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• Olt 1%.U% LLOrA

s ( + ( c- - 2D5)S.--0 Sp - 0

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2 / b i4 4 e /27 3.2_0 •

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a = LsD2- 6 07--2Dsp - -61-5 3,114] f 3b c3 s.2.72_ 0,)

3 f [1(ZD 11) r)7 Ds X:Lr) -D1L, - Iptie*-1 /21 a (3.12i

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II 1, e if...0 Vv", e

(MD 4 1 ) 3D = D p

evk ty b e t ev LiA Vi■ .r7,2 Jae

D s ,3AT-n 3, 1t e ,c i s .1‘- :sa. ckt t''et

ck - , -213) D)

3 „ -Az1/7-9) 5f- /D

(.(6 \iv aukck

61/4--+ ca(1-ri ff/(q)C,v1)D

\i..!)Art bbe erit tO!:\ 3 45 5Z-0 - PO\-"E\len ts•S eGi-Stty \r4V\ ScA.0 tt/tC\I:. Ov vs)09ta

oukr 7f.)Ir c. 51:"Apl eX 1r oJt k tA us is e•kp e f-f- D rr •• • • ES 1,A.Trac arAA E ei/cy sv\Aca( ) owe wwka. I iV.4 c.trie■ :::ler. root 5,

LA so vr as tt-, c.ovA.p Tv\ Ef-'4 i3A ti\ C ftA c3jAct-1!)-A-c.ok --EVI A is ti Dri C j *1 tVIC_Otrp \N3 -73 yrr 7.,e.tr ioA eis) 0 S

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:AC:\ 15.

CHAPTER i

E.Y,TPASV:TI OP INFRASOt\itC, WAVgFORMS TO INCLUDE DI5TANCrS f3\

'Ae.orectico.i cocnsidevatiovIss incorporated W) iUie digital computev- prosv-am INPRASON‘C.

\tiA\PORNiS syn -Ehesls to waves gnat ii■ act kc, s +,1/10,r, kcd-e, 4 t AistatAce corown4 -One

r Irttel p (3.5. C, tyi +Nis clAavity- is to e,•,( vAph1-.1

tzcl.wAll t evroA)t -e. p ut,4z,tr s,„,f tr,tinesi s acou stic, c-ressove \Nek\m-Qn-tvr‘s po;vNts Whose CstaInc..e.5

qy-e ire r‘ +.1;\ op, IA 04, \N 4v-oov‘a tl/Ne \NOVI 4 VOWS q

■ n 0 CI f-7°Y0P '2 '‹ Pt S tA, E x{ etns .0v, it pvtov to 0! ( $IJNOV-)%

.Cr Ir VZ4 e EVA-t iov1 ri ct ptfN0 t 01A Q t tr ecd, ctistcmce_ prN4,vt. of --1etov■ a-L;o1A (Le.tine ckiA4,:,:pocte),(1

,14 Ti /1. tracV‘a..,,s o tlAe F 0 WI' i C r 0 V Vt'N S of 1 eactrk mottl- 1 NAbcki-ica-Llov) 4,o ite precvravv. esec, ,',..64,_cktes -He irelln. -ev-pretcki.lo!A -Lt-t€ eat

c_Ivr_te (A.1-•4.rincc r fy( tine 1112 ple.“4Le sni

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^ ) " • C-- ti -0 3 IA 5 OY pv-Reira,e61 . -Cocr pre n "/"\ .L St

(kV' 1/4! f) 0 rinS•

.53

4.1TVs• 4: tc_€:‘ Lost C ov■ si a ev- ct,, ■ ov\S ,

(-) r)c.,-11, A v\t'x p D ;,-k, \Ni ClArec blirnS

qc °WY.- • C tra,`/Ity k.AJcJC!, ',A.-\ckt have

Past be yorttt ' C 1 air() Spec“ - iC aEe•it.,:vt,,Ov‘5 401f \Yet tti (DVS r 1•% NI( s t Mckopt-QA„ am tlb Sr.— C'r (:) (-11 4 D ■rs k.V∎ t tr f ace Ck co.( ecortiA

t(fle, av\ot it-Ne cuAtA p c}ct,e, .-Tfr

rp/ 7r, tin s Et- cc gr 4,' ,;k av‘vix..Pal )1r- A t

‘r. el ‘rte:p4 ir,c13 ir os\.) tite. 4,1Ae c-. 1A0v-t,c14:, Tr e

n r 4,1Ae TIA

Otr r t 1,, \I rAt k' C!f l r•" r 6 •.“:(1 C C ; (V Dv\ poi

t1,1 e. A ctv,i; 00,!&['V 1\101. 1kt cA'Pv20,./ 0.1 1.110s

t v. 0„, v1r a kA e. ck'‘i..ciLtAce cy.p16:torp, otirct- 1-etuwet --LtAtrou3trI *)AC mAtIpod<

v-4,4%A tteq. Opser.i.1 1:to.^ 'rAt. r.vrtp ■ VO,t IS 14\e

pyes‘vr-f wAsit24 t-rtPA 4,irckv€'1cci. covAlAd,C(\ rAti-our ,.61

• 114\ •Y\ 4 oinsev-v (A ■ ov■

pckcse. d gAv000ln t o (A)-sek., y At', 0 v., 1)o ‘,\+.,, 3 *IA ( 1, at ,kt ovittt i 0h iptiv,t dn;1. tootr 1. t t'llf rovOnl.v.- -e avt v\ot (-,o,Asir.hrea IACtre., Tire Aisc,„c if. VA erk5cpr-tot ow, t..-t- e s tt ,x ct i5 -uAe. Trea-t,

o,-ctuv, p oli,14. r- ` Ir c-1` I1A. 91 6uv- tct -cychm, ki4e aetnv po ■ Int, skow s s 0 kAA e

Fir tr?'":-,q1:2 \rv.‘‘.1tf 0,1 MS v ec_ottc.ted 1v1 suburban N e'vg

\ tr- ‘4, AN\ R ‘JSS 'iatiN t•iPt CM. 0+ EB imeloCtotA% at sit

Direct Arrival A1

r= 6)630 km

350 pbars

Antipodal Arrival A2

L r.-= 33,360 km -0

4VVNiNg\I\PrNy\ d r\II ,FkrINJAr

240 pbars

2nd Antipodal Arrival A3

r= 46,720 km

190 pbars

Time (15 minutes between marks

5 tf Rtt'Ovct-tCik tvk SUburr IA4V1 1-"rt!L“-Pre Ni\b"If4131,12v1-. -Co r tx13 at 0 vl 5.; Oxi os

lq(01. 141. bin kl _A 'VA 3°

1\\4Vf&Nit tv1A 1 t'N. 1 (-) (tOtle 1 1 (Pi•

e try e. s es oi cou:.,!: - y

1./Otiv OV w0 t Ova ' cot., ( Etetreet Ai vc `J :tts. VtAc 11,‘ ‘ r vr, 04- )1r/ iv\ clutE ci ,)1t--/A p tr- )hf\ f Ilv- qtrvE. ■ pa CAI clEscvitc-cd r)etre. AtA ef't 1*- r)IA

11 :0.1 V- fl 0{- !..14‘e e\ . s-.Prt."- I.V.,e qvvti 44e. Ar,14 tve iot! t-V\ cory br (low(A Yi et,ts.

( '!"1 \.1‘) -V) 07- v rJr C,CA 1r `_ir 4- IA rAt et/. 0_101e \Ai v ov t/A

t..‘ s cv.t. ei.e■ 1 t7 %,-1 .,1 s et(Attporit .

tutkv,,,) : rt. A.t r -110v\ svivxttev- %In aired tVatA say

t Jtr)O 4:7 1,r■ of Jar .e!vtilf\'; avec4 q ‘1.1 (lvt cwAsi,,tetr

td Tlyvt,„, cr) ,A;.: iAv\ lovervNiv,3

(fin tc 0h of' any - vrettl'iAcy 1VN yft vA 611 €. tA el vs '.ire

•1,"-;(,) y ; ;kv- f.a.( Ai., -

( /rA ) V,•: V1,1) )7- (4.0

v1/417?, t'tf 1,1 •vuovict v ftrir r 6ttAcA t eta ttA c:A*. -p ckv-*:

V- V1 t { s^ y b1 t Uy e ..s1 = 1 C < DV\ • , f p lirttr, 37 ,4 t.,6 4A 0 %Kir- f-v :it my NW/Ale-COI/VA . 3 414A Er e V) r \ poelk tneve.)

14) GI VIE y tvi c-,,,,respo -TIAE 751) k ..picv)t t e is

etAt. C 1. rsEl \ ktA P au\ blut `•/ A 4-- 1 \J ,e_ Tf r e#12/1,:!•; otA. Tt/ve A f. r, isss (IA e SU tr c rt

to5i ovt 'r YIN ctint.p(sae Tlrve sory-J4z.iot•A -1;5

ot _of vreSsoTe

clif\- tr.& tr c (AS

7.` co s( c-t) t

!: e .11;, c V. 1r err CI, S •) ft. Clk1 C

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tAtle \ton Irt rk solo (,)tt‘ -fotr Une '7,Dt(tt eitousik(.- it cf,f,utte

itA for covI si pv cot (15 r_k s(Jvf\ n4- ivolo‘tel kt,‘ Tit )t•/%3 trf c r 't ikt re)i )'A . TL e

at'`-rAp-C. atlef. 4.1\'x1 . 1 q' cf-3 C krf, cats e cacti t CIA +V- t its

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16 -O ki ktfii“ 61 Luta I.!) To\ olyr0 cL CS -.Ler) \ivo.‘„Je

civApi ; 1 (1,A:k *Iv\ is r ,r P. T_Are IS `Awity1,tc5 pi; 1.-; Tr\ 1 rt ,r\ti pv Ike.

.‘k , •"<"/ Ll teen o ►n (AO ckear

;up+, Ipod eadnect \hitt\ tin r 24,0 0 0 ktiek

4 AAcivit., It-vA C4.3) 5\nou ick tr ep.tre_smi. 11--A ,J,Je Cqh V\ Oki/ e.,I1o.4, i r r cl.SC c1 Qv- ecAce, \Nrkv L ovvis

lrf 'tNe 0:4\t; pOrrte... rokr- katre rA -0A.e

.Bc.ss :Iv\ be cc( 'b.1 its

EN{ tIC fl. v 1 )4" ; O (.4..7..)

D(7-11, V) CV) ( kVA- — T/4) CC1S w C) (A.4) A

61, Oct\ .04 (09+4 3 C. _ t ‘CI(Itkf..QCS IS

1)(,zi-rsv-A Q [LDS Luxt-t-E -i4A It/4) C4-.5) CTs C cot krA i x/4)1

L4,3) -1, co U0'4- e

A ---- B D /(2-TE 1/2 5 8

.15.1 Hours After rz17,000 km Detonation

x10 1 5 4 20.00 5480.00 5543.00 5600.00 5660.00 5720.00 5760.00 - 1 1 Time (sec), after detonation tfI 1- .18,000 km rai-- 16.0 Hours After sa Detonation

L \ig

140 pbars 41D' 5750.00 5810.00 5670.00 5930.00 5990.00 60513.00 6110.00 Time (sec), after detonation r=19,000 km tri 16.9 Hours After Detonat ion

190 pbars 11 10 1 6070.00 6130.00 150.00 6250.00 310.00 6370.00 6430.00 1 Time (sec), after detonation

.c -e W 4 :14 0.1. A.A law\ p l;41(0,tir_m 13 ,P.fov A p S ,) Ur entvf) toy\ g -fTIOA

out

Tine sito• i ss tijv)•k, v\icwei-ov-vey ov‘ciev-c.Ao es IA pcicase

-5V)14-k.. oc 10 de 3 vees upon pck sb 'my -10,1 v-ou 3 fkint2ipoie. tines -One A avtr-ival sifoolck be out or , „ VI. e -e A 1 (Orli:R;(4.1 /0 ,4 ctelreeS.BaSeA

-4_17‘14,s 1,,,?,,r±‘/J s (t)-e -UP Q c.ovAp ut..€ kr- pip °Trawl has been ctttetetct s T .-I.4esize coressvtre wmjeCovvf\s -cocr tth € A., oky v-1 NI rAk 4..\A ss eS k..■Airou(5 1A tie avv.1 poote.

4 Z. Mo4-14.0cLoms to IRFRASoNtC wAvEPoRm 5 -cot. Pps-t. Av\t1 poact( Wave forms

WctvifctlevA syvvOstS 15. ti- OVA •C' o pt) Q tt_cessi-Latos v ,Niy vin1 vAo t1 rr ts e ((WA p st prolvdrivN ., 5-‘ tt,i5vvItAvv-t v, i 421, •;?, r tt 0 p/r..`",',

"0+ brUtAl , Mct e GI 41•0 A\ Of 6. 1V01 ) f) tr ^ 'r^rc S e --1:1•"3 it vu.(u. es qv tle svfitle-st cu.( S' ) a e re. ■ I z 6394. kw . -A)v- 7-tip) -Live 0.vv p I t ek.,2 coy I Oirl

f` Op-ki n ( covv44,,v v ArC s cu.( e olD F -eav l tryi in sob', oUttti e 1 ) is 0.1tered 4'ov-- .1)3 7c.E.21 F 0.0.1

uoav e-C rivv-r,s L.! e v‘3,--eArw, 4 • Li v\ / r IDy . , ra1 50110t-V. rekrt +-‘) \tvIntyc Is Int,eirptr . :.-ZrA

(90

AS e #.:ki tAvs€ ctEe)t. stA E.' V .T.): 1.A 0 S e d 4 vow, point, ot- (iketohNcioci-,. reor p :1'1C71:k -"„t tr trl V OA

v„fct 10, e v r viwJH b -Q be .Lw.Qe-P, IT re co,\ck z 2,, IT re (KU 0 tx\ et er 719‘ g ar C 71A. 9 (_> "E: .-)v)

{--4. ck.otr n sad rout ik\A 'PT OIDIDA'r:yncj nt

CF 4, ,((0314, Pt ) ) ) 11: • (4.8)

ir eV, •fott- r./'..\ -7,‘ pa \iv c 7. 'ft\ a J \it

CF (I. ro314. is .5 (SI N 0>AD ) ) ) •A?- (

•r P (; =

RAD = Ko53 / G394. (4.to

Ti accailv\w‘o dose. c(AckvIc W plro.:,.c. C4,5 ‘'11C

PaSS t " kirk f 'Ast:tF • -(-APJ Co cow. ?,) Ccu,-(A

(-0 tVel fR2-= Pc4Z, *\. 5107 vo (4.1 t)

qv( ;11S-Ake4 .TWPT ctittp- - tiAts ckyd

Ake v- iv‘c•tite 1,TAC ) 4'_ (• SOlbeeteteitel#7. T rit v. F., v- ayr ri..,7\ • -CV\ - 4•\ k ■--L.r,:.A .,v. s e v av:\ yr c_ Ai.

Cg I

0,A t k•6rofts , 0 51 (1 t ine S ovie t s'vxo'c 9 -4 30- Uri cf Fir Ick ,t

•rrk-c, 1,•.(/) SOU'r Ce r tl VinAS -r shits rtri k•ve

0 i5 tYtA itO?t .',7-:,,k, IC?I. .-acv- obz,crvati(yA. c• Syrkf), Ft/ e vg Foy- ±I r

11)77)0 V\ - ,Vve

17 ,1(► ov1/410 ttA Fkt3 s I ! .Cos- bony' trl •Ki-jto.i oz. A soy)

rkit \la\ s -c 0Y Ck COtiKputQV SVA e size d, pt t ssuv-t

\Nr.kv ,,z, OVVV) '1'.A.0% Mi Ctre. brcintraplA r tcorol-e4 6y Do v' alneit S i vy 3f.tif\ \NI .cvk tlne 5 16 lineratotn soviet -Lest is F kr..eze,L11,..p_r_t P,c:j . 1g . Clz.v‘sitIv(i)A3tine scattevl'v,5(

c.av, occ..utr. at suclA curg,e, av ► ca clistuv► ues

11,-, vo 4,., -wAv- ectco0vialote 4.0 say - Cmkt ti/ve avv\pVt tucle.s atAct { Y Fi r) . ‘121 OCk D-c tint vv p‘ots Gur e. Ur■ e 5cupne Mriteti Int,01144:,UCt Expected First .Before Antipode Peak r= 19)000 km Arrival \1000 Mbars

0 L cO 1 0 3 , 6030.00 5090.00 6150.04 5210-.00 6270.00 6330 -.00 6390 . Time (sec), after detonation D. ovl Expected First After. Antipode 1- Peak a_ Arrival r= 21,000 km 1100 1.4 bars

*1 ..0 t .6650,00 6710.00 6770.00 6830.00 6850.00 6-95 0.. 00 7010. -00 Time (sec ), after detonation

164

. "'t `1 (03

18.6 Hours After 1- .21,000 km Detonation

L 200 pbars 0 *1 0 6?10.00 6770.00 6830.00 6890.00 6950.011 7010.00 Time (sec), after detonation -) 222 Hours After r= 25,000 km Detonation

Q1 L 0

85 fri bars

I0' 8000.00 8060.00 8120.00 8180.00 8240.00 8300.00 Time (sec), after detonation 26.7 Hours After Detonation r = 30,000 km

70 pbars

*10 1 9600.00 9660.00 9720-00 9780.00 9040.00 9900.00 Time (sec), after detonation

f y:r: \j\i tt.N.3 ( f Vb D 5.4( NA '7_, e yu A va. : yA-C

11 e o toy. t 14)1 oc:, i r,'%

Theory r=33,360 km 380 pbars

U) ro

29.7 Hours After Detonation Data r.33,360 km Ui 0

Time After Detonation (15 minutes between marks)

1c3‘ D p 0,11k t A v VC1kS. i frilfCC IC> ■ fAir. c \ c47- k\k r) =" vt ..(

L vs" r yik-.). tprIzaLv‘ N ew 4\(ovl.c., b y w. DoNw aq. r c_olt ,eckT,es AvAplikoac scut -Es te, 6-1 stri

C‘1APTER

A'AL)r. . a co tv \E 110 AT ON5

use iocts'ks af v 413 Cti1 (c"k akv-a\tiv-t fti te§ tl»e- e.Y.4,Q:ALUrt. UnQ r pvalif(kok 11\1 7:7 \NAV M,5 inctuae,

%%ICI 90 a al Ct Girl" (k k"(1V•it 'm 0 ,1C

tkkb eVil"y 14y A fev‘i CIL v S 5CibtrOUt t ice -ThAPT

etvik- cpcfokttl catn 64? 0,cc.4v-Avv\rvi,-tt.,Q4 1v-1(1'4 JAL q 614 4c3vT .Q.

5141,-.t. cot,- TQUfG p tnrov3111 c,ect„1 ..Ice , , vAmestlloczo-A 4ov {:one (it C. 1 0 Si.1";\ \meti0,-\( te*.i ,A3

*No $ 411 at k ip incotryv6 ')it < J.+ s p ev s 01(..f.2171 (*.voile fl.teKarA ‘1.6.;tc) kaki( vvrode

Z it\ kw/t). tt, ftp A ori 60s;f: 11-1(-1- tAp LKe c P., q,);:-., 1', avt .c(^Q

6 wAr) ef OVA 7-0 Wt-°4. C4Ate U e 4.,1A 111 e f.: ; s di} e_ { 13. 41"1 tkAk' 0 4 S Stl \x, 11 IA GeA — 1 t ctkit.\( ►

Arovic 3 " 61/' jt P° Ode& u, e-1,\.A\ 9 !A "kirve. CCM/VA Vol" (VA

Vt.) klUir riMbrz.1 a COON' 1.1 611C:A Sive. C LtA v\ kJ (1 ivil());PFOY- 0,t;

1 1.).1.1i f.:53 V\ -A, vv0.VE OT TArytet

Et-tv,,,IspiAf_li-rs two- nlitli'Aef,L, It is wictic:tpo,t;eil -Uro+; -rcArle st -c2 (.1t-.1 1 -c.). 6\+- 011 e (innq C. \iv► k\ \De 4:_tAe- iyeo.r utAo- A FP E D I

UAbrkP, Ok'\DcM:k i ANC) Li, D(kP.

cotnta.4S tiAt ec.k stitA15 tr; 59f4ttioAtAtS

DA r30,P, I t) AA 00,P, R t e s also 1.(A ctud. . q sitort I tAtv o ,.-A0 o tp, 61, rie.,sc,v-ivtlatn Of tine t‘ inrot avid, output

.(3v- • Q.C.AA StACOPY1.-iV‘c , 1* C 0A-l3R (sUBROJTINE) MoOIFIED_7/11/74 LASTCARD.IN DECK IS NO. 2* C 3* C 4* C ------ABSTRACT-- - - 5* C 6* C TITLE - DADOR 7* C TH,7 FjVCTI0V OF THIS SjRROJTINE IS TO COMPUTE THE COMPONENTS 8* C 3= T -I= MATRICES DADOMPJA3KX. A4) DAJKY ,WHICH REPRESENT THE 9* C PARTIAL DERIVATIVES OF THE MATRIX A WHICH WOULD BE 10* C COmPJTEJ BY SUBROUTTVE AAAA. 11* C DAJOm IS THE PaRTIAL DERIVATIvE MATRIX OF A wRT OMEGA 12* C DA)AX IS THE PARTIL DERIVATIvE m4TRIX OF A wRT AKX 13* C D 4 JiY IS THE PARTIAL DERIVATIuE MATRIX OF A wRT AAY 14* C LIAE Ar ALL ARE 2-3y-2 MATRICES. 15* C 16 4! C LANSJk;E - FORTRAN V (UNIVAC 1108.REFERENcE MAOAL UP-7536 REV. 14 17* C AJTHOR; - ALLAA O. PIERCE. CH ,RISTOPiER KAPPER• G.I.T.. JULY. 1974 18* C 19* C ----CALLIvG SEDUENCE---- 20* C 21* C SEE SURROJTINE COIAK 22* C DImEVSION D(2.2).DADOM(202).DAD

• ▪

411

63* 64* C _ DlnOM(I , J) ELEAEVT 0= MAT9IX_DRD 04 65* OR -r(X(1,J) ={1,J)-TH =LE .15.:47 0= MATRIX DDKX 66* DR1

65* D O 15 1=1,2 -- D O 15j1.2 87* 15 0P,(1,J)=RP'(/...1) SS* DO 100 J=171-mAX 69* 1=1mAx+1-.) --- 30* --- C 1(11 91* 0.7.1/X11I) 92* VT=VY/(1) 93* 44* tA L L )4554(D1L,A.A

--- 45* A lvT(2,1)=JD , (2 , 1)*Em(;..2i-)PD(2.21*EA(2,1) 99* AIyTi2,21=-D ,, (2.1)*EmIl.2)(2 , 2)*Em(1 , 1) DD 20 11=162 101. DD 20 J.J=1 , 2 --102* 20 DPatII.jJ)=AI4i(IIT7.1) - 103* DO 30I11.2 --104* ---- — •DD 30 J.J=1.2 105. 0 0 30 (.,(=1,2 DV-30 LL=1 , 2 107* DR13MCII,JJ)=DRDOM(II , JJ)s)P(II , KK)*)m00m(KK.LL 1 *UP 6( LL.J.1) ,, K0s:lmD4X(KK.LL)*UPPILLyjj) 3R1tXtil.JJ)=DRDKX(II.JJ)+1F(11 30 DRliT(II.JJ)=3RDKY(II.JJ1+ -PP(110(0*)MDKTIKK.LL)*up 6( LL.JJ ) DO 40 ii=102 30 40 Ji=1e2 40 Alvilli.J3T—LACII ,A ) *J2P117,=0*III/O*UPP(2*JJ) 113* DD 50 11=1 , 2

125* 50 uP(II,JJ)=AlyTtil.JJ) 7-116* 100 COvTI ,JUE 117* RETJRN4 116* ---Eg1

• .•• ■■.-"e'

14 Dmo2R_.!SU3i0JTINE , MODIFIZ1 7/11/14__L4ST_CARD_IN_DE_CK_IS__NO& 2• 3* 4* ----ABSTRACT-- - - 5* C 6* C TITLE - DmOOR 7* TH= FuNCTI01 0= THIS SJ3R3JTINE IS TO COMPUTE THE COMPoNENTS 8* OF rAr AATPICES DmDom.DmDKX. AND OADKY WHICH REPRESENT THE 9* C zA=7rIAL DERIVATIVES OF THE EM MATRIX WHICH WOULD BE 10* C Co4,pJTEJ OY SU9R3UTINE mMAM. 11* 71DDM IS THE Pi.RTIAL DERIVATIVE _ MATRIX OF -_ EM WRT omEGA 12* C pmJ0( IS THE PARTIAL DERIVATIVE MATRIX OF Em WRT AKX 13* C DmDKY IS THE PARTIAL DERIVATIvE mATRIX oF EM ORT AKY MATRIX Em IS ALSO ComPiTED IN THIS SUBROUTINE. 15* 16* C LAN5J 4 3= ?-oRTRAN V (JNIVAC 110B.8EFERENECE MANUAL UP-7536 REV.1) 17* C 18$ C AJTIORc 7:74-41.1A4 0 ---OTERCE , C=RISTOPlER KAPPER. JULY. 1974 19* 20* C S?aUENCE --- - 21* - - 22* C SEE 5.44DAlf■ift0mP‹ 23* C 7IgET;1D•4 A(2*2 )1E .4 (2.?).0M)X(2 , 2 ) DMD1M(2 , 2 1 tOMDKX ( 2 ,2) 2 4 * DIA = 'iSiDV ..01)

•

.63* D(1•1)=.0099 64* ot1•2)=-c53 55* p(?•1)=(96.04=-6)/CSO 65* DtPt2)= - .00 9 8 67* BOm:0vE5A - A HE DERIVATIVE .ITA RESPECT TO OME5A OF 90MSO/CSO 81* .DA10v(2,1)=D4DOM(211)-xAT 82* DAliX(2 , 11=D4DCX(2 , 1)+x4TtVX 83* DA1(Y(2.1)=DA3(Y(2.1)+xAT*Ia 84* RETUR'l E 4 1

1* DR)OR 15,I3ROLITNEI_ 4091.71E7_7/11/74 LAST CARD IN OEC_IS NO. —2* 3* C - 4* --T. --- -A3ST4Ta:T-- -- 5* C ---- '6* - C TITLE ..: DRaOR 7* C TH , I.JRPOS..._ OF T4IS SL.11DUTIVE IS TO COmDUTE THE COADONENTS ---- 13* --. C . OF f -1 4ATRICrS'DR3M.DRDKX. AND DRJKY nlICH REPRESENT THE 9*C DAITIAL DERIVATIVES O= THE /DD MATRIX ii-IICH .1DJLD 3E ___ru, : eiAPJT , T-3C-5J3ROUTIN:- ///. 11*. DRDOM IS THE DA/TiAL Dr4IVATIvE MATRIX OF RPP WRT OMEGA 3:RIVATIvE MATRIX - OF RPP W1TAXX - - 12. - C ------ORD ---- .2E2 7i ---- C - 28' C ----ExTER4AL SJR/OjTINES RE0ULRED--- e9* C 30* C DmlaR (DMD.11 CALLS DADnR•CAI.SAI) 31* C 32* C ----AR3Um7NT LIST- --- 33* _ _34* C 0 4 73A R*4 ND _IN' 35* 6 AKx R*4 ND PP 36* C AKy R*4 ND / .0 37* 6 FeP R*4 2-9Y -2 14 2 -- 38* C _ A 'R*4 2 - 3T- 2 14 2 39* C DRUM R*4 2-3Y-2 OUT ______40*_____ C 7:01AX_ 4.4 _273Y-2_DiT 41* C OBIT R*4 2 -3Y-2 OUT • 42* C 43* C 70m40 4 STORAGE USE) _44*__ C C0A'40•.LIMAX._C100(1.1.,YI,41 45.' C ____A 54 _____C_ ____IMAX ' ___ I*4 No 1ND 47* C CI R*4 100 IV' 45* C vXT R*4 100 1ND 49* C VIT R*4 100 IND 50* C H I R*4 100 /NO - 1,4, C- 52* C --.--INPUTS---- 53* - C 54* C OM -'GA =ANSuLAR FpE0dricl RAD/SEC 5* 6C AXx =X C0 42 O'JE'T LI= HDRIZOWAL WAVE NUMBER VECTOR IN 1/01 55* C WC., =Y COMDONrNT ;,)= .toRIZOTAL WAVE 4,PA3ERVECTOR_IN 1_/K. 4 — 57* C R D ' = 2 - 3Y - 2 TRAYS==/ MATRIX W-ITCH CONNECTS SOLUTIONS 53*. C 0= THE_REsIDJAL Ez...JATIoN,, AT THE 3oTTOm OF _TIE - 69* C • UDD , R HALrSPA:E To SOLUTIONS AT THE OROUND• 60* C A =MATRIX A lr CDEFrICE41S ol* 62* ----OJT7uTs_7__ ..

-----63i- C TO THOSE AT f4EMOTTONI OF THE LAYER. 54* C D''h04(11...) =(I.j)-TH zLcv.E4T 0 7 mATRix,DmOOM 55* C . DM -M(1 , J) =(I+,)-TH FLEv. 7 4T 0 7 MATRIX 0 ,13 ,(X 65* C Omm(Y(1.J) :(1+00-T4 =LE ,1ENT 0 7 MATRIX DIDKY 67i ______

65* C . __. ----PR0GR4m =OLL065 BEL04- - - - — 594 C 70* SUROJINE Dm)R(OMEGA.A.tx,AKY.C.VX , VY.H.A.Em+DmD04 .0 4 DKX.DMOKY )

— 71* DimENsID4 Em(2.2}+O -0•DX(2.2).74D0%02+2)+DmO

- 7 77* 78* 17)% 7(n*SAA . ---79* Y-.135(4)

1$0* /F(y-1.0r-2) 1.3.4 ,50.0 4- X**2/1650.0*X**3/90720.0 __ bl* 5 D5a1Xz1•0/6.0+x 82* . G0 TO 5 .._ b3* 4 DSAIX=0.5*(CA-SA)/X 84* 5 GE4=q+aSAIX _ 85* DO 20 1=1+2 55* Du 20 J=1•e 87* 20 DA1X(i.J)= - GE1.! 1(1 " ) 88* Da 30 I=1•e _ 89* 30 - 041X(/.1.)=D4DxlItI)+DCAIX Dx-om={2.0*A(1.2)*DAD3m(1.114A(1.2)*DaD04(2 , 1/*Al2.1/*DADOM ( 1.2 1) * 90* . , t_ __91*, 1+.5r.. 92* Ox-CX:(2.0*A(1.1)*DAO.Cx(1.1)+A(1+2)*DADKX ( 2 , 1)+A( 2 .1 ) *DAD

94* DX1Ky=(2.0*4(1.1)*DADAy(1.1)+A(1.2)*D.O.CY ( 2.1)+A(2.1)*DADKY ( 1. 2)1* 954 111 • ------— - - - - -05* T = H*SA _ 97* DO 90 1=1 , 2 18* DO 90 J=1.2 _____..719 * D%4 104(1!J=DYIDKII!J ) *DxD04=T*DADOM(I: 0 100* D 41.(X(I.J)=D4Dx(1+J)*DxDKX-T*DADKX(I.J) 101*. _ Dm14Y(1+J)=OmDXtl+J )_*DOKY-!!..T*DADISTAIrJ) 102* 90 E 4 (14,1)=T*AtI+J) 2 _1034_ -0 0 . 1 90 1;1. 104* 190 EM(1+1)=E4lIpI)+CA 105* 106* EN1 •

7 r:

M.' PE NIDY1:• B

k-11- 1 1._11 DGTE:RtAkPlitM.-1 SV1N

)F" Tc=e ItiE1•11\t\OPE

bv.re.12,S,tor4 m ON(

-46% eµo trQ 11.fl ckro';'; ei 4 A€ ci) ,A q( yeAt tulci

1Yvkltfirtro-1 forts u t-14e e i vAodc • 4 61A

fair ct sip et e 3 r 0( CYC" 4111 12 vi- Qct‘ y

faritt tie co"pley fAovizolAVA\ 7\9tvA Cr. i (tie cLu c5

Frevlooc,-.1\f. _tv\ Uv•e 3 f,)Ao 3 cti) C. ovc ck 5 ,te

Ii3tiv- 41 al L otA tA e t cf,v;AF,Irk, eir VCIV'0, 10(e. v•kev-s to -One

10-0.7t0.- t Q^ ttr A Alpo 4 ,/6 ockNAOSp rAelft v•l\ikt a U Aesi To\ e.s t +qv h isf ckc TA;.1. ptral•p*v.'n p -trcOVVAS it3

S 1 cDtm4plelc CAI WICLk ° vI 1" okottl)re pr-ci:-ANA (Ir► vtktr i tt(nif nokcitcotn at/kr- R.0. C iAlikiru) . V. v t ti.n .e. avyto,N 1 fccsitt o.As I qvvi 42, repre5evnt conci imi,3itnittry

p&rtS ck-t tittrft.t.ive rhe pProe3C di • EtImptnttnol .si`cyy ti ea( (tw,ok 1)fi•-1,G 0c -LAVe_

ttrSiOrt •(U V“ {,Vk uat10)•7■ cttivieck ∎ 3 1.

1- 1 iS the_ J■ istctioce LT) &int ot fot ,r

sc T rAe. 4 A rt. ,acre ciec tvAta kV-%

S-qc;,,Zov, v-, ETUkut3 -IOU ,A 0 IA cpl -,0fAtal _ r RrocAA ett-c-e ‘,/,) -vt\ cotAci.-; t' 3v‘s arA ocz t vIS

CAI (DO = c...05V1 (x)

S (.x.) = si:p!A( x) /C

it = cA-ssc)()-- (H) sAIcx)

(-i-1 )(A rz_ l_)s Ai bc) (A-4.)

- t ■ 7“.) t v y t- A A3 L si v\

det,' because .1,1)-e u ∎ c-i--2t ovts SAT are COVNStY1)(..tea blz)-LIA tIyati ,*5

WACk k!Ae OU 'P a:i'V . Litt t co" pu-V!tr- t--:troa raVA

CANPAffittS idAe 4c,'"1t fr€At COA4 payt 1/1 .011:A"t..■1 rAoeve fAisp€rzit:v1 tks 6. (z_ko) v‘A 1' • jitve IAptik..S Ftr0.)tratAl air•• +:14-kt spe•e ct sj

firt 13vaett 43.(pA er Lit,v,A 0 c.):11 ckepi a v- ;

CL A cu. A mku k ratoiAtA a • mt. 3,1 alia ,f a..tt C. 1/% -e.

NR AAA 1 11/1e itApe's-S atsuf-lAe 4tvve4 krw,) ,,,Ar cv„,A t_rppev-- -.Levrcir,

z.7-4 a:A(1 t vAal 4.1 k_ • I ,1f tneialr.* H (S 11S

t f-O'r I.Aev) -On e e

1 , p t j -1/4.10 0 . iv\ I f,) ere • I

5 b11 0:I'lle IPT Aki-e rr OWE CIA / 4V,R

'A I GA17\ \ ; LB 4 S

7 ! • s 4.. • •-•-= crie`A ';',1 0 1-1. p 's 3V; cu t^ c. !,,A e p e

)t) acri•I AC)W\li.) telt' rry■."4 - t. ai r -r? ,

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-e7111i. 1:,nrcx\I , Dv . fA3 ApoA . bv-a.,...,CFA StItAS bi GU CI) a v13, G U C2..) c..cft_zr V'A e .tiA cmt\ADN e-K r, 11 37—eck. 1* C MAIN PROGRAM 2* 01-,Z(.5;10N DORN(100).5To(1(1n0) 3* NAmELIsT /NAM1/ CL.CD.OMEGA•NR.NI.AKRL.AWRN.AKIL 0 AKIU.H.IOPT 4* DATA 02/1104 5* DATA o3/1H-/ 6* RFAD (5.NAm1) 7* DFITR=(AKRU - AKRL)/(NR-1) Bs DFLTI=IAKIU -AKIL1/(NI-11 9* AKI=AKIU 10* WRITE (6,12) 11* 12 FokmAT (1H1P5X) 12* Do 100 1=1.N/ 13* DO 50 J=11(4R 14* AwR=Ay.RL+1J - 1)*DELIR 15* CALL tW4DRN(OmFGA.AKR.AKI.cL.CW.H.RNmD.AINmD ) 16* IF (ICPT .GE. 0) GO TO 29 17* WRITE (6.23) OmEoA.AKR.AKI.RNmD•AINmD 18* 23 FOOIAT (1H •6HOMEGA=1F13.8.3X.4HAKR=PF13.8.3X.4HAKI=.F13.8r3X.6HNM 19* 1OrN=0613.A.3X.2H+1.G13.8) 20* 29 CONTINUE 21* IF (RNmD) 31.31.33 22* 31 DoRN(J)=03 23* Go TO 35 24* 33 DokN(J)=02 25* 35 CONTINUE 26* 5n CONTINUE 27* wriTE (6.71) AK/•(DORN(J)*J=IoNR) 28* 71 FOuMAT (1m .F14.10.3X.100111) 29* AKI=AKI -DELTI 30* 100 CONTINUE 31* Do 110 I=1.NR 32* 110 STORIII=AKRL+(1)*DELTR 33* WRITE (6.115) nMEGA.(STOR(.)14=1.NR) 34* 115 Fol4MAT •6M0mEGA=.F13.9.3x.4MAKR=.(5F11.7)) 35* WRITE (6.1181 36* 118 rOk•AT (1H1 , 15HSIGN OF THE IMAGINARY PART oF. NMDFN) 37*. AKt=AKIU 38* DO 200 17..1.N1 39* Do 150 J=1.N4 40* A1(.74=AKHL -1.(J- 1)*DELTR 41* CALL CNmprN(OmFGA.AWR.AKI.cl.CU.11.RNmO.AINmD) 42* IF (roPT n) GO TO 129 43* WRITE (6.23) OmEGA.AKR.AKI.RNMO•AINMD 44* 129 CotaIwE 45* IF (AIIM0) 131.131.133 46* 131 DokN(J)=03 47* GO TO 135 48* 133 Do4N(J)=o2 49* 135 CONTINUE 50* 150 UNTINuE 51* WRITE (6.171) AKI.(DORN(J),J=1.NR) 52* 1 7 1 FORMAT (1H , F14.10.3X•100A1) 53* AKI=AKTDFLTI 54* 200 CONTINUE 55* Do 210 I=1.NR 56* 210 STOR(I)=AKRO.(T-1)*DELTR 57* WRITE (6015) 0MEGA.(STOR(T),I=1.NR) 58* 215 Fo-MAT (IN .6HOMEGA=.F13.9,3)(.4HAKR=.15F11.7)) 59* ENO

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4.1C TPAN.01 tir e, 3 t-.01/4-1). Pin\f5. 343,35R, (VIVI), GEORGIA INSTITUTI, or TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN TEE ATMOSPHERE

Quarterly Status Report No. 5 - October 15, 1974 to January I , 1975

Contract No. F19628-74-C•0065

Project No. 7639

Contract Monitor, Elisabeth F. Tliff

Prepared for

Air Force Cambridge Research Laboratories (LUW)

Laurence G. Hans cm Field

Bedford, -Massachusetts 01730

This report is intended o.-1y for the. internal Tiona.tement use of the contractcr and the Air Force. SUMVRY OF OBJECTIVES

The objectives of the study are to develop new analytical and computational techniques for the reduction of infrasonic data recorded at moderate to greater distances from atmospheric explosions. Present research is con- concerned with Line Item 0001 of the contract and in particular with the revision of the multimode synthesis program.

INVESTIGATIONS IN PROGRESS

The bulk of the research during this reporting period has been on the incorporation of various modifications discussed in G. Y. Rapper's MS thesis into the computer program. (A draft of this thesis was included with the )revious quarterly and a copy of the final version was sent to the contract lanitor.) Tentatively, it had been intended to issue this thesis with minor codifications as Scientific Report No. 1. However, insofar as the examples reated in the thesis were only for the special (and unrealistic, although xemplary) case of a two layer atmosphere, it was decided that the issuing f the report be delayed until the N-layer atmosphere program was suitably evised and debugged. Unfortunately, this has taken longer than anticipated. lrt of the problem is that Mr. Kapper left Georgia Tech in December after 3ceiving his M.S. and it has taken some time for Mr. Kinney to become fully :quainted with the computational theory. Also, the modified program was )und to contain a number (perhaps inevitable) of bugs. The current version the program is included here as Appendix A, even though it is not opera- onal. Its inclusion is partly for our own benefit Since the piecewise revision, subroutine by subroutine, had caused us to lose track of a single comprehensive listing of the current version of the program. It also should indicate the nature of the modifications which are being incorporated into the program. New subroutines include CUBIC, PDIFR, RCHEK, CROOT, DRDQR, DMDQR, DFDQR, AND DADQR. All of the subroutines just mentioned have been compiled and the program does run. The problem at present is that the results have a certain erratic nature which suggests there is an error somewhere. The work on finding this bug had slackened off during the last month or so because of certain academic interruptions (such as the doctoral exams) but is now being pursued vigorously. It should be mentioned that we now expect the modified program, once it is running properly, to give much better early portions of waveforms. The spurious low frequency precursors are expected to disappear and we expect a much better agreement between the multimodal theory and the Lamb node theory. This is not so much because we have accounted for leakage Df energy out of the lower atmosphere, but rather because we have circum- rented the artificial low frequency truncation in the synthesis of modal waveforms. Other lines of investigation include the asymptotic theory under levelopment by Mr. Kinney to remove the high frequency truncation. A :opy of a paper on this subject was recently sent to the contract monitor end: is included here as Appendix B. Dr. Pierce has also done some additional mg( on absorption of infrasound. However, both of these studies have been emporarily discontinued in order that fuller attention may be devoted to the satisfactory completion of the leaking mode modifications to the program.

PLANS FOR THE NEXT REPORTING PERIOD

Needless to say, the work described above will continue. We feel badly that Scientific Report #1 is overdue and want to get this issued as soon as humanly possible, although we are hesitant to do so until we have the program working satisfactorily and have some specific numerical examples of a realistic nature to include.

PUBLICATIONS

Three papers were presented at the St. Louis Meeting of the Acoustical Society of America in November. The titles and abstracts are as given in quarterly Progress Report #3. Also, a Georgia Tech MS Thesis "Computational Techniques in Infrasonic Waveform Synthesis" by Christopher Y. Kapper, was issued during the past quarter. The abstract is the same as included in Liarterly Progress Report #4.

'ERSOMEL

As mentioned above, Mr. Kap Aler has left Georgia Tech and is no longer corking on the project. Mr. Kinney, who recently successfully completed ds doctoral exams, and Dr. Pierce continue to work on the project. Due o the limitations of funds, no additional personnel will be added to eplace Mr. Kapper.

APPENDIX A

3 4 • o***********P***********************************************************- s C 6 C • PROIM'iM TO SYMTHESTZE PRESSUR! wAvEFORHS OF ACOUSTIC C GAAVITY WAVES GENERATED H Y NUCLEAR EXPLOSIONS IN THE si C AT1OSPHERL q C 10 cs*******************************************4.************************ 11 C 12 C ----AfisTRACT---7 13. C 14 C TITLE - MAIH.PRoGRA'A 15 C GE•ER ,J. PURPOSE PROsRAm FOR STUDYING THE PROPAGATION OF NUCLEAR 16 C EXPLOSION 6FNEPATED ACOUSTIC GRAVITY WAVES IN THE ATMOSPHERE. 17 p.,. • 11, C TH7 ATMOSPHERE IS APpROXTMtTED Hy A mHLTILAYER ATMOSPHERE 1• C wlIo CONSTANT WINE) VELOCTTY AND TFmnERATuRE IN EACH LAYER. PO C THE Nu•iiER OF LAYERS ► wioTHS OF LAYERS ► AND PROPERTIES OF LI C - LAYERS •AY BE SELECTED HY THE USFR, THE GROUND AT 27.0 IS 22 C. ASs.u, iD FLAT AND RIGID, THE uPPEPmOST LAYER OF THE ;IL ..' AT!!ospH!:RE Is asSUALD TO FE UNPOWIDEO FROM ABOVE. L4 C L5 C THE SOURCE IS SPECIf-IED FIT ITS HEIGHT OF hURsT AND ENERGY P5 C YIFLO, IT IS ApPROYIMATTO As A POINT ENERGY SOURCE WITH i -t C 11 , •E ;1EpENIDENcE CON}- ORMIN To C1107 ROOT (HYDRODYNAMIC) '26 C SCALING r,FRIVE:0 PROm THE EFEFCTS oF NUCLEAR WEAPONS Lv C (U.S. GuVE •NyENT PRINTING OFFICE. )9(,2). 9 C .1 C THr oriszPvn LOCATION MAY RE SPECIFIED ARBITRARILY. 3.! . C HOt!?EvER ► THE covPUT ,, TION INCLUDES ONLY CONtRIHUTIONS FROM 33 C FULLY DuCTcD GUIDED MODES AND ACCnRRIMGLY GIVES A SOLUTION 34 C VALID (AT NEST) ONLY AT LARGE HOO T 7(.‘rTAL DISTANCES. J'S C ALsor THE pRoGRAMING IS dASFD OM THE PREMISE THAT ONLY 36 C PORTIONS Or moDES WITH PHASE VELOCITIES GREATER THAN THE 37 C 14AXIgUM WIND SPEED ARE TO RE INCLUDED INTO THE COMPUTATION , .!, ■) C THE PROGRAm CANNOT THEREFORE BE APPLIED TO THE STUDY OF ,S9 C CRITIC AL LAYER EFFECTS. 4.0 C 41 C LANGUAGE -. F0qTRAN IV (360 ► REFERENCE . MANUAL C28-6515-4) 42 C. 43 C AUTHORS - A.O.PIERCE ANO 4.P05EY, M.I.T. , JuNE,19811 44 C 4 ti C —..".—USAO;:.• --- 4L, C 47 C ILL DATA 1, INPUT IN THE NAMFLIST FORMAT. EACH SEOUENCE OF DATA 4h C RUST 10CLuDE A NAM1 GROUP AT THE REGINNTNG, . to 9 C NPNoH= Su CAN/WI HSTART= ► NRRNT: ► REND t>1 C ',;-: C • THE RE1AIN•EU OF THE DATA TO 13E SUPPLIED nFPFNDs ON THE VALUE s3 C OF NsTART. 54 L '

t,b, pp., 56 cgNA:42 LANGLE=. . ImAX= ► Tx. ► ► VKNTxr. ► ETC. RENO

09 CiNA•ilo Tr:' Il ( f 1.7. , TOD= p DELTI: , RO1S= 1 TOP% REND 1,1 C 62 C *«ttN.,rAkT=:, *ii* (0 c4Nk.3 1•AX:: e CI= , 0..1 \NI= 1 , 0, 0 ETC, &END o4 CNAt4 T WTK= , VI= ► V.= ► omit: r ECT. REND c.,.. C&NA(.3 TJCRL 7 : , 1.0r:R= 0(F Mn hr., C kNAI 11 1, 11-J.• .7 REND t,7 CuNAN,10 frIRSIt , TEND= , nELTT= Rorls= 0 topT= REND • (.3 C ► t.) C , **N5rAkT=.7c**** , cp.: , .., 7C C&NAyb rihx= ► VxI.T. ► I ► ..t ETC. BEND 71 ,.:1NA; , u zSCLi7:-- , Z.VS:1 &END 7.2 CINA!J1 y1(.71-D7 - rIEHO • 7,5 Cr .NAIC TF1q7 .7. I TEN% , DELT1= , 1tO1S: WI: REND 74 C ► 75 •C ***44• ,,rART=4**** . 73 C ,rt,NA ,.7 ,,,, VP 11 01 7. r ETC. OT'',0 ; r ► ► ► ..•, 4DPNO= ► 77 C.,u4A!,3 yli:LD-.; REND 7.1 C&NA ,•i .,) T 7 N(ST= , Tt. Ht): , DELTT: , Rols= , IOpT= REND 79 C t.0 C ****ITS, TAT=ctt.. N1 C4NAy9 wX7 N)-7 , KST= 1 Pe" Krim= 1 ft*, WIMOD= ► ► ETC. 6,-! C.it4A Ti: 1ST= p h ►Inl, I DELTI: t /4 01S= , TOP% RENO a..5 C 0. i C *t*.W3r ,v,iT".:A***•• L..) C(No Ab)IT/OdAL DATA 4G N r.:1:0ED. CNPUrATION TCRINATES.) il.r.; C 57 C FOk A C.0:-, L7TE LIST O VA!ZIA .,LES THAT ARE TNCLUDED IN A GIVEN , i_i C ,,(011. c.PouP ► SPE NA 4aIST STATEmENTS IN PRY AM. NOTE THAT D4TA v,i, Ibi7AD(5 -) C LJT flY REI,P0ipriA:011) , ► Al2) ► 7Tr.,, NEED NOT INCLUDE ,,, 0 C vALuTy., o'r, ALL vARTA!iEf, S- IN THT: CORRrSpONDI'IG NAW.LIST GROUP. ONE ?I. C iE':Cr) ;•:LT INPUT THOSE VALUES, NEEBEO Fon Ti: CALCULATION AND WHICH ,:),! C rwE NOT io,..ro7 ADY Ir STORAGE. C3 C <.: ; C DATA ASSCIATFD WITH NA13 p Milm5p IstAy7 p ANO MAMq SHOULD IN GENERAL 9:) C NOT or_ :-.,6; ,PLI7n ARITRARILY , Hui. mAY HE !"flTAINED FROM PREVIOUS ).:, C kW19 OF (HF Pr;OGRAYq. IF NSTART=1 ► NP!1CHzt, rATA CARDS FOR NA•3t 97 C .r11"15, •!A'•,7t AND Ni •1') ART AUT(NATICALLY PIII,(:Hir.O. IF HSTART=2t 9.3 C NP\SCH=IF DATA CARDS FOR NAM, NA,,17, AND NA"!) ARE PUNCHED. IF 99 C NSTr.RT ..751 t•2NCH=1, IIATA CA: 10 r, FOR MAM7 AND NAmq ARE PUNCHED. IF 1 0 0 C NSTART=q ► HPNCH=1 ► DATA CARDS FOR NAMS ARE PUNCHED. 1M. C IL?. C THE NEXT lATCH OF DATA AFTER OM10 SHOULD !"?F. NAM'. THE LAST DATA 1(, ..i C CARD SHOULD 5;.-.. NV1 WITH NSTART=6. 1"q C 1LJ'i C ----PXTERNAL SW'ROUTINES REOUIRED--7- 1 0 C I r7 C SI"POUTINE: TYPE CALLED BY 1 1:3 C 1 ■ '%1 C A AAA FLTNT,MMAM , NA,OPDE,NNPFN 11 C S: Urr1J T"PT 111 C SUB 11 .! C 1:1:5, 1 T SUB 113 C ANC'S SUB ! :iN°E T"PT (M.I•T. CALCnMP ROUTINE) A . v LAL,TI SLJO I11.!C f.L1WT sul TOTTNT Tt-PT (M,I,T. CALCr)NP ROUTINE) IP) C L:Hi ,oLr OF 11.1 •C Ft"lonl .UC mmcin (c)

c-w WW1'. MODE IS STOREU 22, 1 . C FOR M DETWEEN KSTINMODE) AND KFIN(NMOOE). , 2.2. C P33 - C NAVA '.!' Ni;gLI4ST GROUP 6 , :14 C • 23!, - C ISCRCE ' =HEIGHT Im KM OF nuRsT ADoVE. coroNNO 2.56 C - . zO8s =HEIGHT IN KM OF OBSEPNFR AROVE GR)UNO ;, 37 C 2,51 C NAO. -- NAMELIST GROUP 7 23-) C . 24o C ONMOD(N) =ARRAY STORINn ANGULAR FIEW:NCY ORDINATE (RAD/SEC) OF POI C . POINTS ON UIcPEUSTON CURVES• TUE WIDE MODE IS STORED 24.! C FOR v RE_TwEEN KST(NmODE) Am') vFIN(W40DE). :•43 C vPmco(q) =ARRAY STORING PNAF.E VELOCITY ORDINATE (Km/SEC) OP p 10) C POINTS ON DISPEPsION CURVES. THr NMODE MODE IS STORED w., C rnR V DETwEEN KST(NMDDr) AO yrTN(NVC)E) Z,1-16 C MUFND =NUM3ER OF NoRMAL NonES FoUmo ;,147 C KST(N) =INn Ex OF FIRST TABNLATFD PoI,IT TN N-TH VODE li4 L KFIM(N) =INDZY OF LAST TABULATED POINT IN N-TH NonE. IN 49 . C - GENF:FAL/ KFIN(N)FKST(O+1)-1. 2.511 C JW(J) =AMPL1Tu0F. FACTOR FOR GMTKO l!qvrr EXCITED DY POINT 21=1 C ENERGY SOURCE. UNIT; APE K 1:1 4-1, (-1). THE J-TH ELEMENT 2b2 C CORRESPONDS TO ANGULAR FRrOHFNCy ONMOD(J) AND PHASE 2:, j C VELOCITY VFMOD(J). THE AMPLITUDE FACTOR IS APPROPRIATE ;.,.4 C TO THE 1•10!)E-TN VODE IF J .OF. KsT(r•oDE) ANn J .I.E. 255 C KFIN(NMODE). A DETAILED EEFI-ITTTON CF AMP(J) IS GIVEN bb C IN THE LISTING OF SURRotrylNE DIANPOE. • P 5 7 C ALAM =A SULINC, FACTOR (I f:PP:Mt:TNT OM NET HT OF PURST, EOUAL 2!) , C TO CORE ROOT OF (PREsSHRF AT 'MOHOD)/(PRESSURE AT iS9 C HURST HEIG)1T) TIMES (SOUND SPEED AT GROUND)/(SOUND pu0 C SPEED AT BURST )IEIGHT). SEE SUDRPuTINE pAMPDE. PGI C FACT =A GENERAL AmPL/TUDE FACTOR DEnEMDENT ON DURST HEIGHT P 6 2 C AND nRsERvER HEIGHT. A PRECISE DEFINITION Is GIVEN l.„3 C IN THE LISTVG VF SUDROUTTNE OAMPOE. 2b4 C ;55 C NAO - -- NAMCLIST GROUP 9 ;'6u C 267 C YIELD ' - =ENERGY YIELD OF EXPLOSION IN ITOUIVALENT KILOTONS (KT) 2w3 C OF TNT. 1 KT = 4.2X( 1 0)01 9 ERGS. P6 9 C. P7U C NAm9 -- NAMELIST. GROUP n P7) C 272 C .=NuMvER OF NormAL MODES FCUHD 273 C r:-TI(it°,1) - =INDEX oF FIPST TAPUL•TEO POINT TN N-TH MODE 274 C KElm(N) =ImDEX iF LAST TIRULATO POINT IN N-TH MODE. IN 7t, c GENEPAL, KFIN(N)=KST(M4J)-I ; 7 5 C OMmeD(N) =ARRAY STORING ANGULAR FREQUENCY ORDINATE (RAD/SEC) OF 2 7 7 C POINTS ON DISPEI( sION CURVES• THE MNODE MODE IS STORED 278 C FOR r GETwEFN KST(NMOOF) AND KEIN(NNODE). - 279 C WIMOD(0) =ARRAY STORING PHASE VELOCITY oRDIMATE (KM/SEC) OF 280. C POINTS ON DJSPERsION CURVES. THE mNo0E MODE IS STORED 2(11 C FOR N 9ETVEEN KST(NMODE) AND 147FIN(NNODE). 2 8 2 C ANHLTD(N) .7i•IPLITmDE FACTOR REPRESENTING TOTAL MAGNITUDE OF 2 0 3 C FOURIER TRANSFORM OF WAVEFORM CnNTRIDuTIoN OF SINGLE pl,14 C GUIDED MODE AT FREouENCY OmMnD(N). IT REPRESENTS THE -, . ..ic HHvut moor. WHL-N H or.lwFtm KsT(mmoo•) Aro Kplm(nmOnE), RESPECTIVELY, 2 ,./0 C, THE INTaRAO tc. UMDFR ST0CD To HAVE THE FORM P°1 C AmPLIO*CO3(OMMOD*(TNE-DISTANCE/VP110D)+VHASO). FOR A P9,1 C PPEC1Sr; DEFINITION O PHASO, SFE SUnROUTINES TMPT pO3 C AND PPAMP. • 94 C q'J C NA•IJ -- HV•LLIST SROHP 10 ?9.) C 2 91 C TFIRsr =FIRST TI•ir. RELATIVE TO TIME OF DETONATION FOR WHICH ,i 1 C wAV;i1, 081M TS CIMHUTED. LIMITS ARr TN 5 •cONOS. ;-,,c,9 C TEND =APPROXIMATE TIMr- VALHE COPREczOomDING TO LAST POINT 30„) C TAOULATEO FOR 4AvFFOR'A (RELATTVr To TINE OF DETONATION. 301 C FOR DRECM: DEFINITION, SEE r.HT1RoNTINE TMPT. ,.. 3,.! C DELTT r.:INC•7MENT OF TI"E VALUES In r,i,:CorIns. FOR WHICH SUCCES• 30:5 c SIvE wAV7FoRm POINTS ARE TA3HLATED4 304 C RODS -v1A3NITHOE OF HORIZONTAL DISTANCE IN KM OETWEEN SOURCE 7,0:i C AmO DilcERVER. :“..) C 10P7 =INTEcFR CO•TriOLLINO NHICHimOoFS ARE INCLUDED IN THE 7.07 C Cn•PvTFO wAvEFOFm, FOR PRECTF-',E DEFINITION, SEE 3 1:1 C SU3RFUTINE 1"IJT. :09 C 310 C :Ill C ----DRoGRAM FOLLOWS PELOWC -....- ,31,? C 313 C .14- C DII, ,CNSION SI/crE1ENTS 15 '01.'iENSION CT(10°),VXI(100) , 07(100 ) .HI(1001,AMP(1000),AMPLTD(1000). 31 oUIENSI.)N T(100)01KmTX( 100) ,VKNTy(100) f ZI (100) WHAS .3 (1000) 117 DIENSION t,IC:LE(100).ANDY( 1 00),AKI(1000),PHAmPI100(1) • 313 D.PET_;ION 01(130),VP(100) , IN"ODE(1 0 000) 7.19 AMEN ..11QN KST(10).KFIN(10) , OMMOD(1 0 00) , VP'01D(1000),IDUF(1400) C Z21 C ALOCAlION ,Tr: VAnTAPLES TO cC!Y1ON STORAGE 2. ,[2 C(Y4N0A P':AX,CIFVX.I•VYItHI 323 C :34 C NArELIST SrAlEENTS : 4 1,7Lrj /NvAl/ HST ,MT,NPIMT.NrucH 3',.:6 Ni-IFLLST /N ,1 2/ LA•oLE,IMAX,I,VKNTxtVKNTY,'IINDY,WANGLEF2I 327 ;;ArrLisT /m3/ IN'Ax,CItvXIIVy1,HI • 32:., . HELIST /NAm4/ THETKO I VI ► V2P0141,0M2,NOmI,mVPI,MAXMOD 3; . .) Ht.;.4LISt /NAME;/ Is7Ax,CIFV):I.VyI,HT , THETKD,0FNO,KST,KFINIOMMOD , 3:5,-; 1 VP , o', 3.:, 1 N4•ELiST /NAW1/ ZSCPCE,ZOAS 33:1 HANELIST /NAw7/ Om.moO,VPMDO.voFNOrKsT,KFIN,AmPtALAM , FACT 3.50 fit' LIST /N.!..NS/ YIELD . 334 NAMELIST /NA19/ A1,FIQ,K3T , KFINtOMOD,Vr1On,A1PLTDIPHASO 33.i NAMELIST /NAM10/ TFIRST,TENOIOELTT , ROBS.II,PT - 33o C 'nz, 7 C 3:, 3 C dEFORL ANY i!TA IS READ IN, ALL NAVELIST VALUES ARE PRESET TO ZERO. 33') C THIS IS DoHE sI'lpLy TO MAKE NA:4ELI ST PRINTOUT EASIER TO READ. 3 4 0 1.STAI(t=0 34 1 f.PUT=0

.--... 3 1 =i Tball0=04, 0 346 V1=0.0 341 Vk=0•0 -.1i3 Mil=0.1) Ii49 0"2=0.0 3!)U hOJI=0 3!'l I;VP1=0 35: OAXmo9=0 353 rT,Fri!).7,! 75t ZSCRCr:=0.0

3'36 i=(I,'Z :- _;7 FeC1.7.0..) z'..1 yIELu=i.;:' 35, TFIRsf=0.0 306 T:Na=loso : , t 0ELTrzo.o. ,o, P! 1,1 5=0.0 n3 lOPT=C 3E14 C•0 pi iPii=1p10o 765 CI(1p,Ozid,p- 36.6 ‘01(1PR)r.n.0* 3'.$7 VVICIWO=0.0 31..,'.i HIt/piirg.,.0 h9 T(ipR)::0:0 37u VOTO(Ph1=n.0 371 WiTy(TPR)=1,0 3 7 2 2"f(IPit):e.0 373 L%A14 61..:AIi- R)=1).0 . q 4 :;IN0y(T:)1%)=0.0 375 W•:(1rii).1%,.0 ZI'v 21 VP(IPR)=U,C . 377 00 31 IPR.7.1,10 373 KV(TPR)=0 3,7g 31 KFIN(IPII)=0 3F,U 00 41 I 11 P=1,1000 :=Iii AmPCIP).7.‹).0 . 3::2 AKI(IP;1) = n.n 3'23 PrietYp(Pk) = .0.0

- .34 w!.PLT';(100)=0.0 1:, P011SOTPrirztl.11 .7b6 0:1`10D(JPN)=0.0 'I.7 bkI(/PQ)L-0.,0 3•3 41 VM•00(IP10 7-1.0 ,n C rl C START OF EAECI,TARCE POPT1ON OF pRosgAm 31;2 'c :193 L tlEvFLT IS A cacorP sunRouTrmE wHicH INITTATEs THE CALCOMP PLOTTER 3w; C inrC FILE. 1✓ 64n TS NE }4.I .T. CO1PUTITTON C&•TEP FROPLEM NO, 5923 IS :,11-i C ilci PkO6pAll fin, AinAnii PAPIA wITH 9 1 •ACK I t's( lc QEOUESTED. ZYL, CtLL ► LOIc(VUTPI I'nnt 2 .a 001130) 3 9./ :5Y0 j IcLAD tri.,Nro41) . - ' ///// e/MIHA•1 rIAS OUST liFFN REAn 403 (6 , )1Aml) 01 .4 05 C CUrRENr VLUL Or NcTAPT CONTROLS THE STAGE AT NH/CH COMPUTATION BEGINS. Wori C SINCC ComPufLD no TO STATEMENTS somETImEs no HIT ComPILE CORRECTLY IF 407 C 11JoEx Is NoT .)(7LICITLY DEFINED, WE PLAY IT SAFE WITH REDUNDANT 403 C STfTi-YENT. 409 HST4R1=NSTART 413 4;1 00 To (200,300 ■ 40°1500,000,9 99)PNSTART 412 C ARNIV L: ;IEfq7 IF NSTART=1 414 20n (5,NAN.2) 415 416 (6r37) 417 237 FOINAT(1+( ///// 27H NAM2 HAS JUST BEEN READ IN) 413 (6,Novic>) 41') C CO!IVD-C1 AT,I 0 PHERIC DATA TO STAWARD FORM 421 CALL AT mo'7, (T,VKNTX,VKNTY ,71, ANGLE , WINOY,LNGI.F.:) 4y'2 N1"LCl .LE. 0) RO TO 2 7 0 423 ✓ 4;::4 C PRINT ATiloSPfi nic PROC:ILE IF N ►'RMT .GT• 0 425 C1LL PRA1m0

427 27u IF( Np!IcH .LE. 0) GO TO 303 43 C PW;CH 1.1r1.; DATA IF NPv!CA ,GT. 0 :RITE, (7,271) 431 271 FO;WAT ( 7H kNhM3 ) 432 INHs = 1.1X + 433 kITC (7,27p) TWO;,(CI(I) , I= 1 ,IUNc) 434 272 F0141AT ( 10H Il';Px = ,I3,1H, / PH CI = / 1!3 .5 1 ( uX,(J1r;.1,1-■ ,,015.A,IN ,, G 1 5.811m,,s15.({,IH, ) ) 'AIUTE(IPC/4.) (VXI(I),T=1,IO)iE:9 274 POIOIAT(H VXI = / 4.s3 1 ( 67;, 0 6 1r,,nelti ► ,(2.15.11 ► 1H ► 1 5.8,1H,,015.F1,1H, ) ) WRITE(7 , i:76) (VYI(I) , I=IrIUHC) 441 276 FuRmAr(911 vYI = / 441 1 ( 6x,:515.1,1H,pr15.8,1H.ls 1 5.8,1H,1015.somp ) ) 442 Ii-, ITE(7•27))) HI(I)pI=1,Ims) 443 276 FORAT ( :1H HI = 1 ( 6.0 6 1';.P,1H,)(4 1508,1H ► 015,8,0,,G15.11,1H, ) ) 4o5 ,', RITE (7 , 779) 40 27t i;olzMr ( f-H &ENO ) 447 (0,985) 10 , 1 ITE (ot,:7, 7)) 449 (607:3.) IMAYt(CI .(I)PI= 1 ,IUHS) (V)(I(I)ri=1,IUHS) 01 (VYI(I)rt=1, 1 UMS) 'ARITE(6,27:t) ( HI(I),I=1,IUHS) 4!).5 ( brP7n) 454 28 CA To 315

4.... ' f r. t4,5jq) t1L0) 304 1-0ATtit, //j// 2711 N/ 43 HAS JUST BEEN REA1 /H) 460 ANITE'(6 , I1A(•3) 4 101 IF ( WWII .LE. 01 Cr} To 305 Fic),! C FRINT ATMOSPO7RTC morns: IF flP1NT ,0T• 0 40i CALL idi“00 46.C6. (,f1!) C cONTINUTN:; 1-)tom 270r 2(1e 302' OH 303 4 66 305 PEAD (500%14) 04/ ■,01ITE 0.'4307) , . s.'f.'( 407 1. 0vATI1 1.10 ///// 27H NA ►14 1-145 JIJST BEEN PEAT) TN) 4u9 diITE (b.rjA,44) . 477 C 471 c coy.wRT Tiv_TK•, rR(-1 !)rt74qEEI!..; TO 0111A1`IS 472 THf:Til. '..; (3,14159) * THi7tKr / 1E30.0 . L70 1:31 = (Ry.,,, 474 (.VP = 1VP1 . L75 r . 4 7 -, C colsTmIcr rAtiLr OF Ii4"OoE VALUES n77 CALL VOILFW11 7 0•!eVI,V2v ► O• , •VP,THrTKPOMOPONMODEWPRNT) ;1 7 3 C ti i'l C CO:.PUTE 01:PLPSTON CUtlVES OF 0UrED MOVES . 479 CALL .ALL?,, nr.'w1v0PN0Y11 ,411)r.10 ,7e trioFNotomt.VP.K.ST,KFINFommor), VPFIODI 401 1 IHM()EITHrjhOWT) t, ? C. 4(x.3 C CHECl ro Sr.E IF AMY ;,10D 7S WERE Four() 01 I!:( OOP .n7. 0) oo TO 320 14Z, :i C c, ,.:6 C EXIT IF K111 ,LT. .0 /0,7 4:r, ITE (0,?)111 K4or 10, i 311 F••, •:•AT(l'i , SHK•io ('t:* 13) tit.) CALL E(11 4 r o C ii c.'.1. C CO;,T1W1•6 WITH Kvorl .5T. 0 FPON 308 492 120 CALL 170LsT(mo=N),(10 , 0v10 0 1KST.K=D1FAKT) 4':,5 CALL CO(,TOFNOrKST,KFIN , 0,01 0n,VPM0n,THErWKI) C.''i IF ( H1'1 1-11 01. 0) 01 TO 3 5 0 49•...) C .. 0 j C PRINT N2RAAL 90o7 PrStlEISTON coves tic)/ ci_s_ l.oL!w(murft),o:. ,..,Arl,emon,gsT,KFIN , AKT) 49 3 (: 4c.') C c0;;TINU1N(i Fi(nM 320 0!( 121. 5•, 0' 350 1r( NPNCH ,LE. 0) GI To 3A0 YJI C. e!.1 ;.! C PUVCH NATi UhrA /17 .NPrC!1 .GT, 0 ,,J..5 :.CITE (7,3 1it) r.)04 351 rOT°AT ( 711 :iMA15 1 . by ,..) 10Hq = MAX + 1 5 1, 6 ., :a Tr ( 7, 1 7 ;0 IWO. (CI ( I) / T=I tIUHS ) !%0 / ei(Iir,(/t:74) (VX1(I),I=1,IUHS) `o1 :.'(It":(7•i:.76) (VY{(1)PI=leIVH 5 , j ,I1 ! :(7p,-,/ , ) ( 11(I)ti=1,TUHS) F•19 • v(1•1 (tt. ■ !, , ) TH:r;(*),Ar:(.7N9 1( v c,T(T) , /=1 , Mf)rHO) .gill 3.x. )&1 ,1r (I1H Til.N) :.:,516.3 , 1H,/13H MOr'HO =,I3F11-4/8H KST =/ 51.! 1 ( 6,0:ii!,.n t 1H,,'0,5.ht1H , Fro5.0,1H,Fn15.n,11 4, 7 )

...., P. `“ 1 " Af.1 11(AN-AD) 517 . 1:h(ITI: ( 7 t ,!c57) (0WM00(I) ► 1=1 , KLAST) 513 •57 ro:zAr ( 1.1H 0;4v01 = / ,, 3 51J 1 ( 6X,O1'1,nr1Hr ► nlq.3,1H 1 5.A ► 1HI ► G 1 5.r1,1(l ► ) ) =.',:u ,,iiITE(71J!';9) (\-w.)(I),I=1,KLAST) f,21 359 FoRNIAT ( 1141 Vf-m01 = / h22 1 ( 6X,G1,1,1)0,G19.5,1Ht,G 1 5,8,1H,,G15,A,IHr ) ) 523 WIATr ( 7 '271) L3 ::1(11- r (,5('3) 5;2'3 ,,,: ATF, (6, -0%1) 52.3 ;I:ATE: (up:-, 7;1) IAA4r(CI(I)tI=IFIUHS) 51'7 'R[1. 7(6r;.-74) (0(I(T),1=1,IU 05 ) F:L3 ..-;kITE(6 , 06) (VYI(I)II=1,IIYI`r) 5.2 -) V-■ IT,:(ris2/S) ( 11I(I),T=1,Ii 5.3i1 Alari: (605?) THETKr) , FW , (KST(I)tI=1 , W)FtIO) 5!I :1H1Tr(;r655) (KFP!(I)IT=1r'4W*NN 5 3 :.! OITE (0 , .;57) (0 ."1)0(t), 1 =1 ► KLAST) .5 .!3. :;111T7(r),3591 (VNI0O(I)II= 1, KLAGT) 534 '.1 MTE (6 ► 27q) 3'j C .):,-) C cO ■jir.,JIA;:, FR0 ,,,I 7,50 or 351 S:!, 7 36c GO TO 'lib S!-J C ..!) C $4 :1 C dE AkkIVE l iCi": IF NSI7RT:-.3 f..Y.1- ? '.!aTE ( 6, 4:7,) .5 4.5 403 . FOR”AT(ld ///// 27H NA15 HAS JUST BEEN RE 10 TIC) 1-1:∎ 4 . NITE (utNA , ;(,) 5 4 5 C 5 1 b C C13:,IVERT 1-01.K!) PRW1 0:70..REES TO R"DIANS 547 • PIET< = (3.14159) * THETKO / 1 6 0 ..0 5q• C 1.:- (1-4 C COrlI!'!UIN6 FIN0N; 363 OR '102 55C 415 (LAD (INAPAr,) 5 5 1 7.: :;. 1 r:". (el'to 17) 5 ,:: 417 rO;(4A((ITI ///// 27H NAM HAS JUST HEEN REPO IN) s53 • ihiTi:r. (5,NA•.1) 554 C 5"5 C CO, PUTE YIELD IfnrPENnENT AMPLITUoE FACTORS FOZ (UIDF.D MODES CALL i'AAR)E(2scRcE,zos oloFflu,KsT,KPINoAmnn,Vmon, 557 1 AKTOMPWHA+4POLAM,FACT,THVONPRNT) 5:) ,:s C WIM 450 IF( NPACri .LE. 0) GO TO 460 !.;')0 C 561 C PUNCH 'ii7 OPTA IF HPNCH .GT. 0 5b ,1 KLAST = KFIH(MDFNn) 56,5 ',,i'?IT: (7,451)(AMP(I) ► 1=1,KLAST) !'4, 4 451 FOR AT (.7h r?r,A,A7 / (.) AMP = / .)°,) 1 ( 6.<161 .i.'it1H , t (150t1H , IG 1 5.3#1 11, g7,1 50,111, ) ) rj Uo OITE ( 7, 45'79 ALA' ,FACT 567 '452 FOR•ii■ T ( 10H AL/Y4 = 1516•8f1H, / 10H FACT = ,G16.Ae1Ho ) f, 03 4.itITE. ( 7r /07)9) ",UPNOs (KCT( 1) t I = 1 t'OF MD) 5 6 9 455 FONmAT ( • • 1 0 H M)FND =rI3r1H,i8H KST =./

.0Ilt: (.w-41•) n76 'e;RiTE (6045)1AANNI) , I=10( LAST) • s71 ,,IIITE (6445?) AL!011, FACT !)7i WP1TE (6.45c) MO=NO.(KST(I) ,1 =1.MD5M0 ) 579. vNITr(i455) (Y.FIN(I),I=1 , MDFNO) 5ro 7:RITE (6?357) (0M1D(I),I=1 , KLAST) !.-Al WnITE(6 ► 359) (vP90n(1),I=1,KLAST) si, a • 459 i,dIrE ( 6, 279) 51.1 C .t;&+4 C CODTINUIN3 FgON 450 OR 459 5 3 3 460 GO TO 515 c) 60 C n67 C N(-1 C WE ARRIVE 1V i.57. IF NSTAPT=4 50n ROD (tio4•7) !' 4 4 .) fi i.P, (TE'tot.))1.1 S91 501 FOR!A4T(111 ///// P7H NAM7 HAS JUST BEEN Rc.7. AD IN) 5 9 I 50;' 'ARLIE (NPNAN7) !-,9.!, C !,r., 4 C COLTIt ,J/N; FliDm 460 OR, 502 F,9 6 51,-, cii.:40 (5NAMI,) 596 ;!NITE (6P51.r.) 5'17 516 FORMAT( 1,--1 ///// 27N NAG HAS JUST PEEN Rrym IN) 9 3 517 ',MITE (601A ,,,6) t,':,,) C OU C CO:.PUiE yIrLD DEPENOENT AmnLTTuots ANn PHASE T7RMS OF GUIDED MODES 0 1 1 CALL R2ANP(yIrLOp'.T11:Nnp

OPRT ► S 1.SU5b5 1 IS In EXCLUSIVE USE HY AUOTHFR RUN FAC STATUS: 440001300000 ERR 000 RLENT ADDR:040241 r1 D1:2000e6

A 003001 0 60277 000000 0 00000 00000 0 000000 000000 040342 000000 001251 000000 000000 00110n0 000017 001)005 040204 000001 001602 000001 0 00014 00000 1 002234 000000 000000 350505 050505 000000 0 01 250 000000 000001

A 00U000 000000 350600 050505 000000 001250 000000 000000 000 000 000000 050505 050561 000000 000000 000000 000000 000000 000000 000001 010000 000000 0 30000 000000 000000 000000 000000 00000'0 0 0{)0 0 0 000000 000000 000000 000036 000000 000000 000030 0000n0

Oev0On 000000 141512 2 02211 000000 000000 000000 000000 000000 000000 000000 procoo 000000 000000 006000 000000 00 1)000 000000 000000 000000 000000 030000 00 00 00 000 00 0 000000 000000 000000 000000 440001 300000 RUNsTREAM ANALYSIS TERMINAIED

RUN10: 106.11 ACCT: 061A1019 PROJECT: INFRASOUND IT.T1Ou I'VIVFFnPA 'Lb TwcIt.VINcT.TLII 1YRO0S,MOFNO,KcjoKFINIOMOn,VPMUD# r • , L c itLptAT ui-Xf WAwrl- •0 *fk,

G AuR;V1 1F . 145TtNT = A. • C LOWIni iriwINi.TFS THE CALEOMP TAPF FILE. 9 c4() LA; L LY1; 0116.1 6,7*

C.4) Or LUO'IL.TIVIJ: 140 nIAGOSTiCS. 04,6, P 44- . 44' 9 SOnCu *48 ♦ f* 1.!-)1.000ISOISCO IQ" sliP.1,/10/74'0:03:57 (64) .

!,01)011TINE 4A6LF rNIRY 1-'GIN r 000046

Sloarto,L CoUtil) 0,10537; DATA(n) 000437; MIANK COromON(2) 0110621

EATIPIA. PErCkEuCLS (111.,4K. WAVE)

410,13 %TOO 0114 SuSI'CI “Oh'', LU61Hu 0006 WjOcr4 0007 14 ,,Jou1, 11011 N104$ 11011 NT0i$ 0 0 17 NrR;i3.1.

SIOuA4 AcsiGu4.141 (HL.CK, TYPL ► RFLATAVF IOCATION, NAME)

001 On0u3:e. 101. 0 1 01 0001/.7 10uL 0001 00005 i21L 0000 Onn392 122E 0001 000261 123L 00111 001).)211iniL 0001 UnO1P4 1261_ lino1 non.5P6 1271_ n000 0nn3(,3 130F 0001 000043 20L 000n n(10,6i 0b0:.)7r, 213G nnol Ooo691 2160 0001 0003)7 2511) 0001 000352 262G ■ 10111 Ou0A6/ /70; 000a-0 30L 0001. 000421 :ff76 0001 00(1064 41L 0000 000373 413F 0111 Or0u2t) 111‘ (Jowl 000,105 :03F 0001 000072 csUL 0000 000332 61F 0000 000407 613F Whit uu010/ (4.11. 110 0;1 1.1 O0:02 0:5F nnnl 000174 7Ak. 0001 000137 75L 0001 000151 80L 1111 0 1 1190i6p A5L • 1i01t12 000;01 CI nnon R 00oonn nORN 0002 00045 HI 0000 I 000.321 I 00,in I 000.5111 iFIA6 hrirw . nnnunn 1VAX 0000 000417 ilIUI'$ nOOn 1 000314 ISUS 0000 I 000322 J v011n I 000.530 J1.0 11006 1 000.7:i ,ANA 0000 ] 000374 ja9 0000 I 000144 KORN 0000 I 000313 M 00,10 I 000An Mi uncle) I 000 11 0 nnon I 000320 NOMP 0000 1000311 NOPER 0000 I 000317 NVPP 0010 I OCIOJIt) ut 110110 R 0(10 A25 0000 R 00u326 n2 noon R 000327 03 0002 000145 VXI 002 0n0J1i vY. onto It 00001 A _. ..,_ .., L uutOu 4* C -7*--AuSTRALT---- L,010() `Jr C (JO,00 6* C TITLE - 1A;ti.). 00100 7* C (j)v;,A1ION nF SUSPICIONIESS TOLE OF NORMAL MODE DISPERSION uutOu 8* L FuNCtiON SI;;US . uviou 9. C uiliaw 10* C TAFLE CALLS SHPRuHTI•F MPoUT TO CoNSTROCT THE MATRIX OF uU00 ,1* C HuRto ,..00E DISPE.*SIOu FUNCTioN SIGNS /NMODE (STORED IN (JOkuti 12. L VtCTOp FO.%N CoLIO,U AFTER COLHAN) FOR REGION IN FREQUENCY - ud100 *3* C PHASE ITLoOIT) PLANE (OHI.1.E.WEGA.LE.0M2.AND.VI.LE.VP,LE 0,11011 ,4* C .v2). sUuROUATNE SUSPCT IS CALLED TO.FVALUATE THE SHSPI" uOlop )5* C CON TLDEA rIc.,11S. OF EACH IN ELEMENT IN THE MATRIX u(.1400 , 6* C fL rklItlimG ii?Om LEFT TO RIGHT* TOP 10 UUTTOM. IF ISUS ,NF. uuLoo ,7* L 0 r 4t•oDIE IS ALTFPVO AS FOLLoWS. u0A.U0 16+- C • ycnszi Ri'w Aranw An nvE SU S PICIOUS ELEMENT AND COLUMN UUOU J(:;* C 'ill,r1FD To ITS LEFT 00,0u „O• C =2 CoLUMH AONED TO RISHT OF SUSPICIOUS ELEMENT UOLni) el4 C Ai 4) POW AHPLD AHOVE IT i,VIO6 ,.2t C. :.;.3 Row AsOrn HVLnW SuSPIcIOUS ELEMENT ANN COLUMN uulou , 3* ca AHOED To ITS RIGHT u01(!0 e lio, f.. .t4 C ■ )111: ADI)E0 TO LEFT nF SUSPICIOUS ELEMENT uulf:u ,5* c, A,A1 P.lw A1,DED nELow TT bull)); ,n1 C • lq).:J- Vrr , . NFTTHER •HF i,UMPR OF ROwS NVP NOR THE NUMRER OF uOtOu , ,7• C •CoLut. ii:, WILL. NE /NCREASEj OFY(IND 100. IF ISUS CALLS ut,Piu , 0t L. FI.I1 1.,1 AO,JITIoNAL RuW WHEN NVP = 100 r THE MESSAGE uULUu ,r4.- C (,:VP 7.: 100 N = XX M = XX) WILL NE PRINTED. uOlou ,0* •C H IS w(l'.4 Jo. or SHSPICIOUS ELEMENT. M IS COLUMN NO. IF uulOu ,I. C IUS rALLS m ADIII1ION Or A COLUmN aHFN NOM = 100 ► THE uOlOu .-,2* C ",:';Star- (NO!A .7. lOn N = YX M = XX) IS PRIMTED. u0Lud ,3 4 C ViiEri TY'!OL,E Iii.c, D;-Fm EXPANDED) SCANNING IS RESUMED AT THE UU106 .,4* C . FLEMLIT Li IlLo MAMTX WITH SAi.r. Row ANn COLUMN NOS• AS uULUG ,,Sr L Trio!7,u or '..;u$PIcioliS ELEMENT IN an MATRIX. IF NOPT TS ,10100 :,6* c PwATivr IN•OHE WiL) hr PRINTED As IT IS RETURNED FROM uLliOu ,7* I: 'Ir,OHT t,ND IN ITS SINAI.. FORM. uUiDu 6* C t,010u .f (lir C LANGoAnr - FuRT2A!1 IV (3,,n, I:I:FP- HENCE MANUAL = C26-6515=4) UU 1 !III ..01, L 0.ul0d •.1* T. AUTHoR - J.W.PnSFY, M.t.T.. 1UNE*1966 tAllou ,2 k C uQlou .3* C oulOu .4* t„ --- - uSAGi----- uOluu •.!0, C ,64, viii0J .(-: c,UA RoHTI).Th mPnUT*SUSPLT ► LNGTHNIWIDEN*NMDFN ARE CALLED TN TAHLE. uulOu •.7* C uL110u •.8. C EORTkA, W;A:JI; (A.400 ...9. C (101. TAI0.F.(Onl.nm2pVlftiP.*Nnm,NVPITHETKtom ► v*INMODEINoPT) uOluo A* C UUltiu :,1* C ItipHIS uUlOu '12* C uu 1.1h■ :,3* L. 0m1 IIINIPH*• VALUE OF pRi=noENCy•To RE CONSIDERED. uulOu .,4. C R.4 ch.11.0u •15* C O.2 m„x/ril - VALUE OF •RFOOENY TO OE CnNSIDFRED u04.0t, .16* C R.4 • uL LVH11)LP(LO uulutl •.1,1* C IJUIOU .1* C t40: 4 114111AL NV* OM FRFOUCNCIES TU nF, rON:JIDERED C 1.4 t1U,00 ,,_)'l C ,HuP IIr11IAl NO. 01-' PHASE VELOCITIES To LIL CONSIDLRED 00100 .4* is 1.4 uUiUu "5* C ToEIK hiASr VFLuCITf OwECTION (RAuIANS) UU1Pu „6. C R.4 uUttJu "7* C NuPt PRINT OOT OPTION. IF NOPT = -Ir m0 PRINT. IF NOPT = It u0)00 .6* C 1.4 1.i1.1OC/r IS PRINTED IN ITS INITIAL FORM (GENERATED BY MPOUT) "9* C- AND IN 1Th FINAL PORM'. LU10u .0* 1.

uOtOu $1* LC OTAIT‘ uulOu a* uO.Ou .3* C Nu• TuTAL NO. OF FRrONENCIES CONSIDFRFD uulOu C 1.4 u0100 i5, C HvP 1uTAL to. OF pHrkSv vELOCIIIES CoN,IDERED. LU1O 0 ib , C 1.4 u011.1u '7* • 0+4 VCT01., WHuSC FLEMi:NTS ARE THE VALUES or ANGULAR FREQUENCY ut)100 'It C R.410) CNRPEsPONJING TO THE COLUMNS OF THE 1NmODE MATRIX '9. C LiOlOu "n* C V . VtCTOu WHoSF .FlovNTS ARE THE VALUES OF PHASE VELOCITY tAliOu R.401 C.REcrONLING TO THE ROWS OF IHF TNMOUF MATRIX tiUtUu .2* C "3* • Iba.nr LAcil rtFMENT pF TNIS MATRIX CORREsPONOS TO A POINT IN THE

u0i00 .4* 1. I.4tD) 1-1EQUpt471 (0M) - PHASE VFLOC1TY (v) PLANE. IF ThE NORMAL uU,Ou .5* ouPF hIsPEPSIoN.FIINCTION (FPP) TS POSITIVE AT THAT POINT' uullu .0* E, FNFliT Is +1. IF FPP TS NFGATIVL, THL ELEMENT IS -1' u0,00 .7* FPP' DOLS NoT FyIST, THE ELEMENT 15 5, INMODE HAS NVP C Qur, 71"D NOM cOL,OoNS. MATRIX IS STORED AS A VECTOR, C COL' 1M AF1ER cOLUmN. o0i0u •$0* ut,,U0 .51 4 u010 -0 uUlOu u0 ► 0u C cLT IN•ONE -11ritilt-1.-Ir..1.1,1, - 1 , -Irl,1,1 , 1 uUL00 • WIT, NON z NVP 7. 4 u00.10 • AND ON z 1.u 0 1.go? , 1, 1 2.fl THEN = 3.14159 uUiOu v7. C V = 1.012. 11..5.00.0 uulOu NOT CouRUCT, FuR IILUSTRATION ONLY C uUlflu •-i9$ uOlOo 1"0* C OWN TAigE viILI BE PRINTFD AS FOLLOWS. L.010u 1..1* uulOu 1"2. C VP1,4,-4. &WMAL POOL DIsPERsION FUNCTION SI(-44 oUluu' 1,3* c 1,011o"o • -+4 ,1. cULOo 1.4* c 0010u 1"5* C' 3.1)1J0.,n A-wr u010u LA* C 4.oun"ii A --t uVLOU 1.7* u04,10 1-„8* C 0-r;7A 1p34 WILL1 1.9* VpLoCiTy NIRErTION IS 9o.nOnuEGREES uUiOu lilt C uu100 lil* c 0MFGA uOlOO li2* • 0.100uOr ul 0.150u0E 01 0.20000E - 01 0.25000E 01 Uw100 1,3* .....-...*r. J u L fAruM , V4 I NMO D F*N O PT) ..... t. )0.0.) 1,0* 0I,.I.NSTON om(In0) , 4(100/I41M0O•(10000),DoRN(100)*KORN(100) . ,0104. 1,1* CO-VW! li,Ax ► CI(100)*VX1(1011),VYI(1001 ► H/(1n0) ,L1Q 44 I,P* C Ju /04 1,3* C *,PnUl tS CALtEo TO nnOdUCE IN0o0F M4TRIX Awn Om AND V VECTORS. itil(lp 1e4* LAIC WoUT(UM1.0M2 , V1:04NuMoN041714 4 0a.:40M4V 4 IFIFTK) JOInS 1,5 4 C J01.0!) 1,6* C IFIAb ::. 1 T..OICATEL; FIliST TIME THROUGH WR ITE' PROCLOURE ,O10,, 1,7* IFIAu = 1 ,O.00 1,6+ C will, 1,9* C lumOuE IS PINTED Tr NoPT IS PoSTTIVE ;010/ 1,0* is (NoPF.nt ..0) (40 JO 123 ■ 01.11 1,1+ '. IF, At, • :: o M112 1,2+ 0.1ER=0 6.J.1,-? 1,3* C 10PFIR 1; Tiv: uoAvEo or EXP,INSIoU OPERATIOAS PERFORMrD IN THF PRESFNT , ni12 1 • 4* 'C SCAN 0.. IHF *IAIRIX. TON. HOPri) IS 'W NOMI'AFR OF SUSPICIOUS POINTS ill i 1;! 1•5* C rOut-.0 t', To. P(ESEmt S4A4. wile! 1.,6* C 10(12 1,7* C 11:::M0 sCAN1.4 .1,6 OF ptTEOIOR ELEACHTS OF INMOnE TN UPPER LFFT CORNER W11-i 1.1 ,1r I. = 2

1) 114 1,g* 1.1 f . U11.) 1-or In (,AI t_ SuSPCr(tumpuVP , M.MDE.ISUS) - ullt. 1.1* C u115 1_2* c PoT:.1 ,N,) TI7, SuSoICIuUS IF IczO5.NE.0 (( 1111 1,3r IF4ISUS4!1F.0) 0 To 60 0110 1..4* C . oil), 1-5* C CrirCN r.OR p-,,0 or Rn'4 WL2o 1..6* 2n IF (m.IT.t.,0m - 11) 60 To 30 0120 - 147* C t1.2.0 1,13* C CIFCK (Ori IAST ROW 0127. 1-9* IF 0,i0 T. (AP-1)1 60 Ti) 40• 0.24 1,0* 60 10 121 0124 1:w1. 4 C 0124 I.”14 C MOVE 0,..E CoL11+,. TO PIGHT 012n 1,3* Si, In '7 ivr} 1 ti.2., 1,4* oo To 1 0 • 0.20 1,5* C. 1120 1•,6* C Apv.,\I.C,• our Rna ANn START AT CnCUM TWO )12/ 1•7* 4n . I. .-: 1 4+1 1130 1:,6* 1.1 - ;., 101 1 ,94 i40 TO - 10 •131 1,04 C '41. .1..14 L LHFCK ,-no kotxI•um w*LUt. OF NV(' L'',.! 1,2r onIF,HvP.(.T.100) nn 10 62 134 1,3* ot r04;.titi (plot MVo .-.: 100 N ;0I3011( M P*I3) 13:. 1„4* .,RITE (,, , ,1) td. ,, 01 1-5* (;0 To pn 02 1,6* Op IF,No,1 .1 T. 10n) Gu TO 70 04 1,7* 0:i I-n-mATt,J.10-ii.OM.-.:. - 10u N=rI3r AN m=.13) 0!) 1,A* 64 AotTr(o,63) 1.1*m 151 1,9* oo To 2.1 152 1,0+ 7n IF,ISUs .hi::. 1) GO TO 75

DI54 1,5* C Ann A COLUMN TO LEFT OF SocPIrIntrs POINT ul9t) 1,64, ("1=41-1 015o 1,7* . t,0 To fou j14/ 1,8* 7 '1 IF,IStJc 01•. 21 GO TO *in )157 1,94- c, J15/ 1.41. C Ant) A CuLw1,4 TO RTrolT OF cUSPTCTOuS POINT 3.0.,t 1#.1* Mlel Jitil 1„24, t; 1161 1“3* c ADD Rt%W ADj0.1 SUSpTCIOUS kOINr .40.4 LAt . NI -ttl- 1 . 4466 1.05* 60 Tu 100 )1h4 1,.6* tin IF(TSUs .PI•. 3) GO TO 89 JO ,t 1,,l+ (; )164 L•* C /WI) A COLO•N 10 WO1HT OF cUSPICTOOS POINT 460 1"9* 1.1..--.M )1,60 140* Jlfiti . 1 '414 C AnU il,w OrLoW :.3USPTCIOUS POINT Jib/ 1"24 1.11:::N iI7u .143• (70 Tu lull ji7L, 144* C )17v .145* C Al R.e., PFLO'fi SUSPICIOUS PO/Nr Jill 146* 1st, 141=N J171 1,7* C i171 1 .S. C Ant) A Ckilm,o4 f0 Lt-FT OF S0SPIrIOUS POINT )171 149* Na.--_-1 ,1.7.5 2"04 10o CO.,TINHC J174 2"14 (:AI I. LtJ,1!144(0M,VrINMOD1.:WO:JrNVP.NVPPPN1414THETK) JIM 2.2* 0111 W6FN(0444),,INI10DE.NOm.NOMP ► NVPP,M141•THETK) )17t, 2.3# NV - =14VP; ) )17 -/ 2.4* m0":7.20m1:4 J...0u 2"5* NO0F4=mu ► EO1 Je.01 . 2"6* 1,0 Ti, 10 leO2 2..7* 121 L0•Tli:0E 1 e03 2„t1+ 1F(NuP•R ;w .f. n ,AND. NW) ,LT. 100 .ANu. NnM .LT. 100) GO TO 5 le:U6 2„9* C e03 2,o* C DO NuT PRIu T IN•or.A- IF I'JOPT IS NFGATIVF ,..!0'..) 211* IFtHuPT .Li • 01 KEf•RPJ 0 ..i. 2,2* C r. n5 e,3* C LAoF1I.43 cu/ 2,4* 122 i-0•'"HT (011VPIK7.6X ► 36HNOR!JAL MOUE DISPERSTON FUNCTION SIGN/) r!In 2,5* 12A wPrTr• (6.1,[2) r.1, •2,6* 00 133 1=1,11V1 In e,7* 00 11 01 its ,!2d 2,8* .01.,=1,1-11 1 1.VP+I .21 2 ,c)* .J1?u=111(4014.I.JAII)-1 :22 2,o* . IF (o!10) 4260125e1Z 4 •2h 2.14 124 L0,1100t. 2'„, e,2* C :2t) 2,3* C IF I(iM“DE .1' 1) DORm = 1NX. Pil 2,4* DAIA 61/10A/► )u 2,5* 110(J1 al 31 2,6* 60 Tu 1;"/ 3• 2,74 12g, .L0.,T1NNr

,s+.1 4,414 HOwnIJ) = 142 u0e30 2.)2* . 60 To 1'e7 t,Q,-./ 2„34 12F, (.0.,TINIIi:. uU,3/ 2.-44 C L0,37 ',.r.i* . C IF INN1.ir+. : -It ilOptl = 11(- u 0equ 2,6* 1)ArA 01/111-/ bu,41 2•74 ii0.N(.11 = 63 ub,:if 2,8* 127 CO-iT1M14. UU.44 2,94 12A CO.ITIN01: 1,0,: 114 2..0.4 C L,(144 4+ 2-14- C )-111N1 ,,ni.., 1 a rAnIr u(;e46 2 .. 2 + 'i,RITt tf),1JC)1/(I)r((IOPi..AJ). J=1,N0m) u0e!)!, 2.3* )30 1-0•NIAT(11-1 trii.'703X,1110;0) i.A.JeLI 2.4* i33 Lo•isntv.:: uu;,:4,0 2.5* 011i = 10 UL461 2.-64 00 1!i0 J=10ION ude,.61 2,7* . C U0,01 2-.64 t.,UI. - r..lir k ail 1;.,1;1, uLle:64 2-9* .CJII 1.0_,N(J) = NO0(j, au) bu, 2,04 ,A, APITF (,,L13) 1 , 0P.4(sA ► ,J=irki0m, b0,--:74 2,1* k 1 3 FO•,:vilT (tdloOloi(:' ► 6X410oT1) u ■),:74 2,2* C Wie74 2.,3+ C V H 1Ni-- TA FROM PAPIANS TO nEGPFFS 1,0,7n 2,qr x Wt1,41/3(1/1,.1 440.9 uue7c. 2.51, v.P. Tr' (or i i 1 3 ) y

u3..,91 2.,6* 41;. 1-0•*:.,i1 (iii P1Iv,2711PHASE VwLoCITY OTRCCTION IS ► F9.3 ► LU,V11 2,7 4 1 Alinvur:Er.S ) Li...Ak 2,6* a. -211.1. (..:151.70 uu,n4 4:,qt t,131-0.,!Ao ( 3,1t.inrAr(:A :-.) 11u:J.14 2,0t C 60.)(.14 2,1* C LIcT V.11.07c oF 6N , F44A WoICH CoRwESPoN0 TO COLUMNS OF 'TAUS. uuoii',) 2„21 v,1c1Tr Co,i,14) irM(1),I=1rrinm) uu...1:1 2,31, 613 FO,:-"AT ( Ill r5;- 14.:)) u 0 .:, la 2,4* C uk);,1.', 2"5*• c IF S1)5,IcIn;• ELIMI•TIvN HAS WIT (?FEN PERFORMEnr bEGIN IT AT THIS TIME L,0..)14 2„6* IF(Ir.Et.i•I.) CO 10 5 u0,:)16 2,,7* I-4E10,N • UU,17 2„8* rN

F,L1 or COmPILkT100: 1.10 nIA0i;OSTICS. 4*4**4 sUl iC1 1.S,1µ(J/SoL4LI iOr !,11r.-1,/11:1/74 -'13:0';:67 (IP)

FN1RY rOn125

510‘,A6C USCa: CopE(1) 0,.015,1; D,,TA(n) 0un07!i; RIANK COMmON(2) On0000

ExTEnp.;A. PEi-ErIENCriS (hL.ICK ► tvir 1W"/ 11

51 0100:: ASSIONPENI (141..ICK. P$1.14. WATIVE IOCATIOn. NAME) 000 0o0u0• 60o1 UP0101 112G 0001 000065 1236 0000 000016 21F 0000 000033 51F 000 I 00000 0000 000043 IMP% 0000 I 000003 .1 '0000 I 000001 K1 0000 I 000002 K2

0000 1* C. 1A.$PkT tr.UHRO(JTTNE) 7/.31/68 LAST CARD IN DECK IS NO. c0100 2* C uolUu 3* c , ----AhSTPACT -,-- u010u 4* L - 1,0i0u 5* C II1LF - lAisVicT uOlOu 6t L . . P$4061ZAM TO 1..11:INI OUT LIT of' FRE0NENCY. PHASE‘ VELOCITY. u0100 7* C. • &PI Whit.. WO PHASF F06. EACH GUTDEO mOnE EXCITED DY A NUCLEAR ,0C,i00 * C ExPIt,S1.0 1 Or r,IVE• yIELn. THE STMuETAHi:OUS LISTING OF FREOUFNCY tailDo 9* C AhO olthE VFLoCITY kEPRwSENTS TiF nr,PERSIDN CURVE FOR THE IJOLO0 $0+ C 0o1hfo .40I17. THE OHANTITIrs AWPLTI) AND PHASE DEPEND ON SOURCE LO IOU , 1* C A•D oP5RVF. ►4 HEIGHTF, AS WFLL AS THE MOUFL ATMOSPHERE. HOWEVER' A.,0 4 01, 12* C ThE LAT1ER 7HFORMATIOn IS NOT LISTED nY TAOPPT AND IS PRESUMED ulliGu 13* C To $0_ L1STEn HY ArminrR SUPROUTIME. THE SUBROUTINE TADPRT uulN, ,4* C • ShODLO NOT kE CALLEo OHTIl ALL THE UHANTITIEs TO RE LISTED 1,0 1 0 o ,5* C HAVF WIA.N CWiPUllb Ann cToRED IN THE mArnIhE. NORMALLY. 00100 ,6* C AIM ► •,, IAPLF. ALLMOop ro-ApnE, AND PPAmP VOULD 13E CALLED BEFORE I./LW-1U ,7* C . TAPP; ■ T. UULOU i t C oUlOu ,9i C LANGoA..r - FuPTRAu IV (3h0• MEFFkENCE MANUAL C28-61515-4) LUIDU .0* C kUTHtiR4.. - A.D.PCTRCE ANN J.POSEY , M.I.T. ► JuLY•I 066 ,Alms _It C uuL00 .2* C ----CmLLING ScONENCE ---- uOlOu ,3* c U6i0u ...4* C u/..i7NSION AST(i).KFIN(1),0010D(1) , VPMua(1),AMPLTD(1),PHASOCIA UOIOu r5* C IIIF !..,1Ji:PoUT11. 17 USEc vARIADIE nymEnsIONTN6. THE TRUE DIMENSIONS MUST

$,0100 , (ii C hE GIV.fl I,., THE PPnr,RA.:, WHICH r.EFINES THESE QUANTITIES. SEE THE uu$0o ,7* C t).4471SION cTi,11:.HEHIS Ii THE MAIN PkOORAM. uU100 ,A* C . CA(L TAI.Pftl(YIFLDrwOrMO,KSTPKFIMIOMVOD , VPMnDfAMPLTD , PHASO) u0i00 r9* C 001D0 .X1 C HO EXTFPNAI SUHROUTIMES ARE REnUIRFO uulDU .,1• C uViOt, :%2* C " --ANGUPF NT lIsT--- - uulpy ,3* C L/0401., “4* C II.I II IMP uUJ.0.0 C $..n...ho V*:11 IN:, 0010u 2.ii: L A Sr 1*4 fiiJAR■in MP uOillo .-.7* CI*4AFII: VAR IN' , uuLN, .,111. .. om .0o V 4 4 VAR IMp Oitati C JP ..01, • Rill VAR Mu 00100 "0* CP.utit,:i 10 11 VAR INv wil 00 ...I* (. rn.co I(+4 VAR IMP oul00 .,2* c yulOo -.3* C NO Cull .oN cTor.OE ABED u010.u . 4.4* c

LiN, pUW4VALLNI KILOTONS (KT) vutoo • dr TNI• I KT = 4.2X(10)*+19 FkGS. u6t00 »9* 01 . =0Ni - 1Ek OF MOR...M mODES rourm uOtoo .,Oa INS, (r+) ./=ImPFX OF FIRST TAIXLATED ► OTNT IN N-TH.MODE uUiOu 1 1 KFiM('I) =Im(`rx OF Insr TAMILATED MINT IN N-TH MODE. IN 1401.00 C GOT:RAL. kFT•cN)=KST(N+1)-1. 1, 010o .u41.,1t0(N) AorAY STOuTMG AwOuLAR FRE,JHENcY OROWATE (RAu/SCC) OF tAtOu ,4• • -PnINTs ON OISL4F445110N CHINES. THE NMOUE mOnE IS STORED C F(12 Fl Hr.T.FFM KST(NNO0E) KFIN(NMODE), E.,6.10o C VP400(N) 7:AuNAY STOKING PA %F. VFLOCITY oRDINATE (KM/SEC) OF • EJUI3o :,7* • PromTs Up niSt.Lksloil CHR-JCS. TILL MMOUE MODE IS STORED uuino Fnr N OET,FEN IT(MMOHE) ANn KFINtilmUDE). C AM ,LID(N) =AmPLIIUME FACTOR kEPRESENIInG TOTAL •AGNITUDE_ OF "O. Fr-WRIER LOM5,,Oiel OF THE CONTpMUTION TO THE WAVFFORM uULOu ,1* Fr) vi A SILGLE MIMED mon: AT PREUENCY OMMOD(N). u010;, ula C MITS cmOHLo hE (oYHLS/04.•2)* (KM*.*(1/2))*SFC. UtitOu C IT PFPNrSI-NTS THE. AMPLITUifF OF Ni OnE -TH MODE IF N TS 110100 “4* litiT(M.,COF) ANo KFIN(HmOnE). INCLUSIVE. FOR 0010u C ; Ppl- CLX WFTHtTION. SFE SUIPOHTINE PNAMP. C r=11,S0(1.1) =r,1 ,1=7,E LAG TM pAMTANS AT FkEoUFNCY oMMUD(N) FUR THE Nwmr - TI: ,,nnt: WEN N IS liLTio,Ert, KST(NMOnE) ANO uUlao C KpTtikAT11-). rNcLuSIVF.• I HF TNTEGRAND IS UNDERSTOOD 1.010o HAvr. Thr rto.m AmPLTO*CuS(CmmOU*(TImE-DISTANCE/VPMOD) LAJLOE, +pyASu). FOR A NECI ,;E DEFINTTION. SEE SUOROUTINE uu1Ou ,la Pwi•AP. oulOu ,2* C

uo i.nu ,3i - ---OuTPUrS---- u0,0, Or C ujIG0 ,5* C PRI11I01.T. THE ')NLy FMCTIurI TAIIPRT IS TO PAINT OUT RESULTS. u0106 4 uNiCti ,7* . uuiliu ,8* L uolOo ,94 C 1HF OUIPUT Fo.AT Tr; li_LUSIOATO RELow. 041Uo "0* vOlOb monE TAkULATION FOR Y.= 100.00. KILOTONS uOlOu ut410k, -3* .1)01.0t , 1.:uif)u C uLiOu nh* 1 u010u - 7i AMPLIn PHASE Luluu 41,100 0.334;,6 -7.013420 70 »0* - .00200 0.243^/2 -8.02394E 20 : =71 1010u c udi0o ,24 u01 ,10 C J 4* C 1001.E (tUttth , ,r)t L,OiOu •'u• CrA VPHSE AMPLID PHASE u0100 vu J.00. C .0o1(10 0.552=48 -7.95321E 10 -2.4n798 QUiOu C .0u2n0 0.48391 -;1.2.1106E 11 -2.30524 11,04 uUiOU LI* C • F Tr. •

--.. L DU101 1"n* C AIARIAnir PP•,111%IerWin IS UcED uU106 1.9* oI-riAltifi KST(i/PKVIN(1),010MOD(1)+VRMOU(1)/AMPLTD(1)1PHASG(1) uU104 1,0* - 11i.....1714,TuN hKI(In00) uu10 ► 111* C uUlu!) 1t24 laRTTL (6,11.) Yin.° I, .5•- ii )•04 010( ini ,111 t25X,2211.,10147 TABULATION FOR Yr. , F9.209H KILOTONS) uu ► lu - u01 16 1,h+ C !Atli, lib* C STARi .1F OffirP DA 10nP . twill 1,6* • 1)0 5u 11=1•P► ND uulli 117* ( 0).1.14 1,64. vaiiTti (6,21) IT 0)117 1,9+ 2i 10..IIAT(Iti /// , 11 , .04X, ,.:1410nE +13//4 111 ► 9X4SHOMFGAr9 ► 5HVPHSEr9X. t,uJ17 1,l1 1- 1 6,W1 110•64(,5147HALIE/ ) Oil/ 1,1*- uVi2U 1,2i K1_KS,T(11) uU121 1,3+ K2:.:1(1.IN1II) %,0 ► ,?1 1,4+ C u0,211,5+ C STAR) ,P Ir.itir PO tOoP L00;.' 1,.0 lio nu J=KirK2 ' , utitP;? 1,7+ C 7 uUt2n 1,11i to ,vilor (,, ► !41 ow.oD(J),,,nmon(J),AmPLTo(J),PHAso(J) uu11:4 1,9* 51 1-0..PAT( IH OX,F14$5.F14.5.1P014.501PF14.51 0005 A-1 0 * C ENn or LuOP!., u013, 1,1+ C u0i3u 1,2* 4,E ► UMI . uui:Ar 1.,3* Fn.,

FAA.) OF L0,-PILATION:- nIAGuOSTICS. 1.FoR4S 1.S011C24SMC2 ' fo" 1,11p7-1,/lU/74 - 18:04A (II+) NO CuU.CF CANn 1CHOurn. I.S"HC3.Su1IC4 FOK SI1rwl,/10/74-1n4Ut,:?2 (19•)

Sn0,400,HF WO" FwtRY pottil 001262

siokA4. USW: CoDF(i) 061:517; 0.1A(n) 0.'17041 ninNx CO1INON(2) On000A

Lx4A7NA, pEruiLNcis (IrLe !CKF uArt) 0 1 '13 AKI u0u4 • PLOI ufh i r", SrALC uu.4 LIMG u067 ' thIV,,E14

hui3 N1o4$ 0014 STN 6015 0 0 16 EXP lt 7 COS 00/0 N101$ 60/1 NrR1.3:1,

SloHA0L OT L. WATIVF. IOCAII0N. NAmE1

1,001 0111141 101L tiOnu 0311,34 102F 0000 03154 1041; 0001 001170 1071, 0001 000141 11L 00,11 000105 13,,0 0001 0001 0 7 134G 0001 000115 1426 flout 000234 156E 0001 000350 1746 0000 U3101? /F • (W01 u01234 4.00L h001 000024 2300 0001 000526 2330 0001 000570 256G 0010 0000,5 274(.1 tiO0u 0311,P1•3F 0001 000016 3036 0001 000754 332G 0001 000771 335G hold. 00102/ 351.6 000) 0000.43 4L . nOnl n01157 40E0 n001 001200 412G 0000 031462 54F 00.11 '000L)4/ h5L 0000 031.416 L,7F nnol 000074 Fl9L. 0000 031503 61F 0001 000657 65L 0001 On005/ r.61. 0001 00103 70L 0001 001032 75L 0001 031441 753E 0000 R 031411 AIIC 011 .,0 031404 AK11i0T 1,100 f, 1,31374 Al 0000 R 031363 a n000 R 031361 CF 0000 R 031376 CTR/G1 0)00 ii 031J71 LTRIc,2 00n0 k 031403 LJELF.1 n000 R 031367 n1DULF 0000 R 031407 DY 0000 0311.27 1NJp$ 0L),,0 1 0751„;5:1 IT 0000 I 03I011 J 0000 I 031362 J26 0000 I 031353 K 0000 I 031351 L 11n(,0 I 07,110!) .44 00(10 I 031',,c:4 (4 0000 I 031364 HHVIR 0000 1031357 NOFN 0000 1 031373 NOM 00,, 1l 1 071,J5n NOT 0000 rt 031ur1. OPIEGI 0000 R 031402 nMEG2 0000 R 031375 PHI . 0000 R 031370 PH2 0C;:,(1 N 031.,60 0000 f , .031152 nfl00 R 031666 cLOW 0000 R 031377 STRIGI 0000 R 031372 STRIG2 0000 N 031400 til u0n0 R' 31365 S2 0000 R 000000 T n000 R 003722 TNINT 0000 R 001751 TOTINT ouhn R 03140,1 (wilt) 0000 P (27354 Y 0000 D 031125 yAX •

00100 1* C IML.T (comROUTIHN 7/19/68 00i0u 2* 3* 4* 00100 5+ C uhlOu 6* C 1ITLF oulOu 7* C ChLrOtA1ION .m, 1") PLOTTINc: oF FAR-FIELD TL ANSIFNT RESPONSE TO A hOlOu 8* C rhrcv.,upi;. SOHRcE IN 711F ATI.:OsPHERF. 00io0 9* C u0100 .0* 111E. Rpf:;PONSF NE MIN- 14 IS FOUND BY INTEGRATING (AMPLTD * 00.0'10 .1* CuO( nflEGA * IT - R/VP) < PHAS(1) nVEk OMEGA FROM OMMOD LUInu .2* (hST(N)) (0 014m0D(KFIN(N)) ANU OIvID/NG DY SORT1R). VP, 001,00 13* PNASO. AtiU AMPLTD ARF FUNCTIONS OF 1301H N AND OMEGA. THE 0010u ,4* TNTAL wEStiCNS• IS THE SUM OF INF 1000AL RESPONSES. THE uuind PSPOPIrF IS CtICUiATE0 FOR Tirr TFIRS1 AND AT INTERVALS L,0100 if)* ON PE. TT INFRI-AFTr.R UNTIL TEND TS REACHED. THE VALUE OF 00100 .7* 1uPT nFTE.

, 010U e4* CAL Pc.FsSuRF PERTNR8ATIONS INCLUDF THE EARTH CURVATURE

uulvAJ .0* C AUTHOR ■ LI.W.PnrTY, M.T=T.. J1 E119h3 'uU,OI, ,9* C tiUlOu "0* t: • idUlOu "1* C .. ---0SAOP---- uUluo ..!* C UU1OU •3* C FO.ThAn SU6ROUTINE AKI IS CAILFD uOtOu 04* C (JutOti .-.5* C CAICUPP PLOTTUT SU0HOUriNEc: PLOTI ► AXIS1, NUMUR1, SYMBL5 ► AND uulOu .,6* C SC1GPH WIC CALIF!) TO W ► ITE THE CALCOMP• TAPE. SUUROUTINE NEWPLT. ' 6010u ., -/T 4 •iU•.T 1, 414 nELN CALLED pRlOw TO CALLING TMPT ► Arm ENDPLT MuST 5E u1 LOu 084 C c.AlLt0 AFTtR RrTHRNING FRnM TmPT. (SEE MAIN PROGRAM) u0t00 .19* C uU.00 ...O.T C I-UkTRA, USAGE tiutOu ..1* C ' CAtL TioPT(iFIROrTENDIHELTItROHS,MDFNOIKST,KFINpOMMOD , VPMODFAMPLTO CU100. • "Pi, •C 1 ,rHitcNfluPT) UulOu ..3. ' C uUi00 »04 c 4NpHIS U 0 100 »54 L uv100 ..6., . C IFI.SI Twr AT WHICII TA9HLATION AND PLOTTING OF RESPONSE IS TO U011.1u ..7.6. C14.4 IAEGIN (St C) UOIOG 484 C uUlOu *9* C TEN,, .iFTi AT WHICH TAPuLATION AND PLOTTING OF RESPONSE IS TO uU10o •0*. . 0 R44 E.0 (.LE,(TFTPST#510n.)) (SEC) uulOu ,l.s L 0010u !b2. C rWL1T Ti" t: T!!TERVAL RET•EFN SUCCESSIVE CALCULATIONS OF THE uOlou •,3* C p.4 kL5P0mSE (.GE.I(TwNn—IFIRST)/1000)) (SEC) u0.1.00 : , 44 e uulub r.5* • C PuR.: CASTAmCE uF ThE (); .:SFRVOR FROM THE SOURCE OF THE UISTUR- uuluu -ao, C R.. 4- P ,LICE (KM) uulOU !,7* C uut Ou ,F1* C mliF.,0 NoM1.1Ep OF MOD1;S EoUNO (.LE.10) • u0106 •.9* L 1.4 6000 HO* C uGiOu ,.I.* C. • KST ELEmEmT N OF IHIS VECTOR IS NUMDER•OF OMMOD ELEMENT WHICH 031st, ..P* C I.41(7, ) I..-, FIRST FREOuENCT CONSIDERED FOR MOUE N ui,110u ,,3* C UUlou n4* C KFIH CLEmENT N OF THIS VECTOR IS NU"RFP OF OMMOD ELEMENT WHICH 60100 n5* C I.4(D) I!, LAc.T FmEOUENCY CONSIDEliED FOR mODLN • •u0t00 C untOu . ,,7* C 0°4“D4 lik.D'EmTS uF THIS ACTOR NUMBERED KST(N) THROUGH KFIN(N) uu(Ou ,.At C 11.4(D) 'ARE THE VALUE ,' OF FUEDUENCY (IN INCREASING ORDER) FOR uUtOo ,ot C • v:ItICH THE CORaF$RoNDING MODE N PHASE VELOCITIES HAVE BEEN LulOu it)* C OLTE.I4OHEI) t.0100 tl* C uo ► Ou /2. • C VI,Mno • Vr.CTOL4 nF PHAcF. VpLoCITIEk, WHICH CORRESPOND TO THE FRE- 60too ,3t c R.4(0) outAcirs uF VFCTOI, OMNIOD todlOo i iii C: 03100 /53 C AMPITD `.+ALIIEc OF AMPiTTUnE FUNCTION IN Awl INTEGRAL (ELEMENTS uolOu ,6. t; . .R.4(D4 CAREsPOND DIRFCTIY TO ELEMENTS OF OMMOD) (DYNES/CM**2) 60Inli ,7 4- C oL)10u : t)1:. i. PHA0.: 10m TN A. ,(GUMI-MT nF CuS IN AKI INTEGRAL WHICH IS INDFPEN— LJOAnu C R.', In) NAT nr TIME AND nISTANCE (Ru0S) 00106 n ot C utliOu n1. 4 c TuPr CuMPUrATION ANn PwINT OPTION INDICATOR

- -4(;IAL. Rt.c.Pu(ic.E 61)4.0t,-54 c = 19 CALCULATF AIL MODES. PRINT AND PLOT TOTAL RESPONSE 0.1tOu "64 C ' rv:LY 4,UON "7* C (1 4 1.10.0u.- C oPTPuT4,.. uOitio ,,q4 C u0100 404 C THE ONLY OU(PUTc ARC THI: PRTNTOuTS AND PLOTS CALLED FOR fly ion. uUlgo ..,1* C .ALL GKAPHS AqF n/AWN TO THE SAmF SCALE. THE PRESSURE SCALE IS 41010U .-,2 4 C ne.T,:amIINiEn dY THr MAXIMum A'..1.4.IfUDE OF THE TOTAL PRESSURE, ANn THE utliOu L,34, C TIM,: SCALF IS bon SmC ('FR ImCH. PRFSSURE. Is EXPRESSED IN DYNES/CM**2. uutOU ,4.1 C utlinu ,.5* C buIOU •,6* L ----PhuGHAM 4)1.1.0W5 HELU6-- -- 0ULOu ,./74 C wi10t, -.iii, C u1101 / 94 sU-R0HTINE Tvil T(TFIRST,TENo4OFLTT.ROBS. . uU101 1„04 IN.D..w.KST.kFIN.0 ,,'MuU.Vpm0D.CKI.AMPLTD.PHASotIOPT) uu101 1.,1* C u0;01 1,2* C. 6L1.0.5 1,,34 Hi.rNcTuN KST(IN4KFIr(10).0m: ,, oD(1000),VPMn0(1000)4AmPLTD(1000). uutu:, 1.44 1 .-.11A(104.0)sT(10u 1 )410TINT(11)0111TNINT(1n'1001)0(1001) UU,04 1.54 dI-Fr.SION L\itinnn) uU104 1„64 C Lm1n4 1,74 C YAlt IS Vi..CTOr! k,F L.TFRAL CoNSTANTS. ELEMENT N IS THE EIGHT SPACE LABEL uuiO4 1.14 C FOP IH„ i-,4,ry-ohd: AyTs (JH ThF GIJAPH OF IHE oriODE N RESPONSE. LOIsb 1,9# DO.TiA.t. PJO- LISin - YAY(16) U0104, 1,04 nillA YAA/6"•mOdr• 1 PIAH monp P rSH MODE 3 fsH MODE 4 , bH MODE 5 . Lutot, 1,1‘ 1 ;ii , !WT n tAt.4 mODE 7 t8H mon': 8 18H MODE 9 thIl MODE 10/ t,Utlu 1,24 IF,IuPT .N.-..'11) u0 To 4 4,U11i) 1.34 1..R4Tr (6 ► ?) u0114 1,4* 9 i-0,.mb1 (4dir 4nY.2.1HTAI,ULATIoN OF RESPUNSEc//) u0115 115* .WiTt 16t3i UU117 1,6r 3 1-0..1- lid rPOvr4HiIl".12Xr!iHIDTALt11.(,6HMODE IrlUXtfiHMODE 2,10X , OUL17 1,7* 1 :,H,.011! 11.101(.6HMoDE 41101tAHmODE 5/I uu12u 1,8* u IFtIt,PT.FO.12) .mtlE(6.753) ou123 1,94 753 FO,MmT (1H1 1 45x.hOHTRE0LATION OF ACOUSTIC PRESSURE RFSPONSE///1H , • .712..) 1,0* I .AXI1nHIIME (CFC)•9Y115)P (DYNES/CM4*2)//) AAP., 1,1. C d012.3 1 ■ 2# C L IS NiOltiCd t, 17 TTmp.s Al WHICH RFSPONSF IS TO kE CALCULATED )1304 1,34 . J. r Ilrm -IFIPST) / CVLTT + I ,U12b 1,4* ,uLan 1...)4 C , U1lh 1,6* C SI,E 14. THF 1E.JUTH OF THE TIME AXIS IN INCHES Wi2t, 1,7* SIIE = ( .N..NO - TF1kST ) /•600.0 0,2c, 1,ii. C Ul:'n 1,9. C 1,44- S.-.7 AIL urSeONSp- VALUES To.n.0 017 1.04. 1,0 7 AzIrL U137 1"11 101INT(s) ;; n.n 006 1.121. DO 7 m=1.10 ut3t, 1.,.. 7 INJNI(HrK) = 0.0 013n 1,14. C 01.3t, 1,,5* C SET OP TAPrr Or.; TIMES HEGIwNINA AT TFIRST AND TAKING VALUES OF TIME AT Ul3n 1..64 C IHTE,iV.LS or: OtLTT ONTIL TEND TS REACHED 1141 1..,74 Q 1,0 It, TT=14L j44• 1"84 IA T(C() r. IFIRST + (1T-1).DFATT

Ulle22 2.0* C iNcRrAcr YnIA- NL•PPr BY OW uu,.24 2"1* N .7.Ntl u0,:24 2.2$ C uU,.24 2,3* C lr N I.. bRFAT;:ii . 1. 11AI' PAJF-NO ► AL, MODAL RESPONSES HAVE BEEN OETERMINEI) u0,f2b 2"4*. IF (N.11..•luFNO) GO TO i1 u022:-., ,_,,,,,,* C tAle:.n) 2,1')* C I.Oir t.Arli Tr:41- IN T !'4- T TOTM. PoEsSuRE EQUAL TO SUM OF MODAL PHESURES u0,:27 2.7* u0 Ail Tl=lit. uUe.37 2,8 ► HO 5,) N L. ir;,OrHr% . uil.t, 2.9* t).1 10,T1.,T(11) = TnTINI(IT) + THTNT (NI IT) uu.3/210* IF(ToPT.O. ).11 AO To .19 ,u..4.57 c, 1+ C L,037 2,2* C IANITF •II•E hHU conprsruNDING T6TAL ACOUSTIC RESPONSE (OYNES/CM**2) u0e4i 2,3* w.RITi- (0,4) T(TT),TOTIHT(TT) . LO,4!) 2. 14* 54 1-0-;iAT WI r49,,,F9.1,111Y,F1242) 0,41) 4.15t C ii/J/1:, 2111* C y.liPM Ifirl*r—ll MY TJTAL nrSpOwsr IS PRINTED w o c oo 2,7* IF (1wi.Lo.1) 00 TO tin uti,41, 2,R* L (,(4 ,i0 2,9* C i.ili-il LiPi.F6.1i ALI 0,.,AL riFSPoNSES ARK ALSO PRIN(LD LO,;,il 2,0* 5t, Iv = Yi:of:•UFNH,S) 1.,j,:;1 2,1* t.rt1Tr (i,,S/) ITFT(iT),TOTImT(Il)s(TNINTINfilltN=1 , mM) u0e b e. 2,2* 57 t, O.,J-T (ii! t3X.14 , 10X.1.5x,K12,40+.0F12.4 , 4X,F12.4*4X,F12.4, uu,_' i' r 4,3i, 1 -'irF12.0,4Yro7.4) o0,on 2,4* bq t:OTilur „.0,f64 2,5* 61.1 co"TiNtw t,o,,. 2-.5. IF (1 ,4-Ji4I .LE• 5 •uR• )OPT .NF. 11. ) GO TO 55 LIO?7u 2,7* 01Tr (16*) ut),-7,-! 2;1* 61 1-0•.vio. (1)9,20A, t4HTIE,12X,SHTOTAL,11Xt6HmnDE 6,10kr6HMOUE 7tIOXP u0e72 2,9* 1 -rw:Inc: 610X,C')HMuDF. tqp10r•7HmODE 10/) i,u,, 73 2,0* HO 6.i 11=1;1.

uge76 2,1* U v.RITt• ( ► b/) TT,TtIT),TOTTNT(TT) ► (TNINT(N,IT),N=6,MDFND) C0470 2,2* C u0310 2.,3k 6r, L0.,T4,0L uno11 2,4* 6 0 LAIL Pi ,!ft,....11-3) u0.,11 2,5 ► C sI/i- IS TH'. liUm . TR OF SFCO3ADS PER INCH IN THE PLOT uu..,1,! 2,6* cuI/E = I1(L)-T(-I))/600. L/Uo16 2..7$ 1E (icip1.LL.10) 00 10 1n7 u0,15 2,6* CAI L crm E.(ToTT'ITP:)•OtI 11) uUolh g:. ■ ,3* C AF1Ek SCAU. RETURN:2P ToTINT(l +1) IS THE MINIMUM VALUE_ OF THE ou.ilb 2.0* C FI-S1 i vALULS. c0015 2.1* C 10iIt4T(C*21 IS (AAX-MIw)/3.0 OF THE. FIRST I VALUES OF TOTINT , LIO.:,1!-, 2,2k C. oNliu I',--, :IAA - vIm nF %TINT utioIo 2-3* will;=ToTI , ;01. 4 p)*3.0 uLl.)17 2-4t o•r'; = AINI(W,:T6/2:1) * 25.n uui2u 2,5* 1,M;=A..!AYIII(?Ird , z)5.) uu,2o 2,6* C AT 1111 ,..; PLANT ICY I4 THE TOTAL RANGE IN TOTTNT MOD25 i40.):1 2.7* MY = Aro,(WIH) uo.',22 2.11* 101INT(L+2)=DY/3.0 u0.326 2-9 ► 1Jy71-U)/.n tau.,2.1 2•,0* . C u0,23 2•,1* C IF Iu1'I.E0.1? PLOT ntILY THi-- TOTAI_ ACOUSTIC RESPONSE . uu.)24 2.2* IF (IOPT*Etv•)2) GO TO 70 uLIA4u 1.3s C IF 114)1.1E.1n CALCNLATE ONIY MnOF !OPT RESPONSF LAJ.147 LA. .IF (10PI*Lt. t101 N = IOPT 60031 1..St 11 NO,I = 1•.ST(N) + 1 140...v. 1..6• htUM = 1.F11,1(N) 01152 147* C Uul'i2 1.(1*. C OLTEi?MiNl THF- t•ARTw CURVATURE rORRFCTION FACTOR TIMES , RONS**(-0.5). bUt5J 1..9* RA,I = Ri_l tS / 6 ,0 4. uUtS4 1:,0* CF = (1./()374.*AO5(SIti(RAn))))**0.5 011'.i4 1,1t C" 01154 1•,2* t. IHF if, 011F M ,ZPSIJONSF IS Fou"n Foil ALL VALUES OF T BEFORE NEXT MODE IS L0i54 1•,3* C LCNISID).Qtrl uui5b 1•4* WI L•I TT=1,L u015t) 1,54 C uU15b1!,6* c bET A2,PHP toUAL To VALUES FOR AltPH1 IN FIRST INTEGRATION INTERVAL 4 u0i6u 1•)7 02 - 74.(10 ) (A/161 1',3* A2 = AWLr0J2k)

rND or Lo..;0 4 1.“TyoN: N O OT A GNOS TI CS. 1.110 (ito SUHC4 *4#•$4 FuR •11P-1,/)0/74 - 186 Olt)

SuROJT.NE IOTIr.tT 1-NiOT PAN) 00n174

51001',-,L USED: Co(101) Ott• DATA(n) 0o003!1) 12 1ANK COMMON(2) On0621

rEt: ERCr.:-; (Ht....Cho :WY) 110113 (.1,;r;,5 60 1 ,4 ,PLAT 0 0 11 r; 00nE, NCRR3J,

STORAt,E A`ISAGNfit-.NV tyPI:# R::LATIVE IOCATIoN. ,;(1 111 0!lOc711 ton ,) R 0O0nOn AINT 60n0 R 000005 Al 0000 R 000006 A2 0000 R O000n7 A3 1c.i,,0 It oLJOLJU..) 00110 H 000010 Fl 0000 H 000011 F2 0000 R 00001 !Inn:. !I 0001191 CI 4 H u'3112 R 0004:3S III hOnd I 00Om3 1 nOOP I 000000 TMAX 0000 00001& INJPS 0000 1 000001 0060 k Donut? oil4r finn0 P 000003 VX 11002 R 000145 vxl. noon H 000004 VY 0002 R 000311 VYI

uOuOil thI,..GNOaTIC* ThE N.4fti. F.• AHPEApc. IN A HTME(ASIOIA OR TYPE STATENENT OUT IS NEVER REFEREPCFO. buino 14 C 101 Ti (cul,POUTTkJE) 7/27/68 LAST CARD TN DECK TS NO. u0100 P.* C uUlOu . 5* C ----AdSTRACT---- utli00 4* C UUiQU 5t C 1I1LE — WT10 100 Gs' C -Has stLIROUTINE COMpuTEc THE TOTAL INITEGRAL . LOA0o 7t C 11.00 ist C YIAT = IWGRAL OVER 1 FROM 0 TO INFINITY OF uUtOu Yt. l: UOICW ■ 0* C A3(7)*(ik1(7)147 1(Z) + A2(7)*F2AZ))*v2 (I) uolOu ilt C LuLnd 12* C ThE ATI'o5PHpor IS AsSUmrD TO UE REPRESENTED TN A MULTILAYFR FORM uOLOu 13i C 1.IT11 ;.1,APF AND A3 CMISTANT IN 11ACH LAYFR. THE INTEGRAL IS uulOu ,4* C FvAI UATtlf) Ac A :',UM of INTEGRALS OVER INolVIDUAL LAYERS. iLiUu t5 4 C LO,Ou . 1.64- C Tnr ENNCTIONS Fi(Z) ANO FP(/) . ARE CONTINUOUS ACROSS LAYER- u0k00 1 71 C isoU,INAIE 4N1) SATISFY THE RESIDHAL EOUATIONS ut)10u ,Si- C 1 9+ (2A) uulOu C 1)C ► (4)/D7 =•A(1 ► i)*F1(7) + A(1,2)fp2(Z) u0Atiu .0+ C ;)F2(1)/1169(1)/117= A(21i)*F1(7) + A(2,2)*F2(2) 42R)

.H:-.. r.unci/OmS FL(Z) AND F,, (7) ARE ASSUMED TO SATISFY THE UPPER 1..010u ,6* C Pu► HUART COLOTTION THAT )IUTH DECREASE EXPONENTIALLY WITH INF UPPER HALFSPACF• THE NORMALIZATION- cUl ► u ,7* c INCwE.4SIdo ►rIGHT T ► LOIOu 'di' C 01- THE OJNCTIONS IS SUC1* THAT AT•THL LOWER BOUNUARY ZO OF THE .uutOu . ,(J* C UPPER HALESL.0CE • 0.401, .,Os C . uL)40.L1 A* c F1(70)= -SwRI(0)*A(1.2) (3A) U010,1 .J2 4. C . W iA): c,JZ T(G)*(G+A(1.1)) (38) uulOa 03* C 00100 ,4* c w1To uutOk) 5* C uOino 06* C. G = slqT(A(1,1)**2 + A(1,2)*A(2,1)) (4) uOLUu 07* C uulOu At C rft!: ci.r,..ENTc A(IPJ) IN v(INS. (3) ANO (4) ARE THOSE APPROPRIATE uUlti,1 .•LJ* C Iu THF oPPEo HALFSPACE. IF G**P Is NEGATIVE. THE PROGRAM uoloo .0* - C la..TH,06 L= - 1. oTNERWISF IT RETURNS L=1. oulgo .1* C • L.UiOu ...2. C eRnONA., NOTr.LS ouittu , 3 * c 4.X,0u • ..44 C THE I ► T7GRATIoN OVER THE UPPER HAIESPACE IS PERFORMED RY

ut.)161. ‘,5* t. CALLING UHIHT. THE INTEGRATIONS nVER THE LAYERS OF FINITE LJUIOu .(1* C THICKNESS APE PERI-ORMFD DY CALLING ELINT. uOIOU ,7*. C uuL0u ..81 C THE PAPAVETERs Al.AP.1■ 3 WEND TN GENERAL ON ANGULAR 0)100 .9. C FltEoUp- CY ONIrcat, $101?TZONTAL .4AVENHMPLR COMPONENTS AKX uUlOu •4t C 11140 AWY, SOONO.SrFEn C , AND 'HIND VELOCITY COMPONENTS VX uOLOu :,li C AW) VI'. INF I-ORMHLAS - USED A. CONTROLLED fly THE INPUT onuiu :,72* Q PmRAMrTER IT v. ► ICH IS TRANSMITTED TO SUBROUTINE USEAS. uulOu '.3* C uOlOu :,4; C LIE PnvAMLTERL: DE.:INING THE MULTriAYLR ATMOSPHERE ARE (JULNI •5+ C PNEcU.AED STORED IN COMMON UO LOU :alt C uUl(lu !a* C LArJGt.A,s • - FoRTRA1 Iv (36n. pEFERENCE MANUAL C28-6515-41 LJui(lu ,:iit c oulou ,q •C AUTHon - A.D.P*rRCLt .T.T..M JULY') 36u uulOu ..04. C uOLOu ..1* C -...—CALLING SFOHENCE ---- uulOu ..2* C UtI ► iki 4,3i C Srf. LNUWJWITINC NA'LlorsE utikOu "4* C III•ENS(Ori c1(1nr.),VXI(1(10).VYI(100)•NI(100)/PH11(100),PHI2(100) uulOu ..5* C CO.awN IMAA,CI.VXI,VYI,HI (TiiESE MUST. N. IN COMMON) u 1, 40u ..64 C CAI, ToTIAI(OMp-7 A.AKx.i.KY,TT.L.XINT.PNII.PkI2) oUtOu ,7. C oolnu ,01* c - _--ExTrR N AL cunnoUTINES REONTRED---- hutOu ,,qf ' C tgitnn /Os C AAAA,PW4'.1,CATtq4I , USTAtitUPINT.FLINTO)6131 pil i Ou O.* C 6V1OW a• •C AAA Au.) 61 .3" A"r cAuLi:n nY FLINT. outtio .3* C CA, Win SAL ARF CALLED OY fifilili. u0101) P. C till I pa 1:0 C ---.AJOUMI-NT ► IsT---- (4LOu .6* C UU t 00 Os t: .uM6 Oh • N*4 NI) INN „1 C 1. . T*4. ND OUT ut.,100 .•2* C xI•T I*4 ND OUT u0Inj ,.3s C PHrl. - R*4 Inn 'No uJtilu .4* C P1112 IVO 100 IN 0.'0)0 .5* C uutOU „Lk C COMLION SiOdAa. USEr) uUlOu .%7 4, C co own 1mAA•CI.VXI*VYI*HI utilOu ,421* C onina "nt C IN.X 14 4 . ND iN uniOu -,ii* • c Cl• Rt4 Inn I•0 u010,) -di C VX1 R*4 100 IN0 ullL0u -,21, C v Y T R*4 100 !Nu uOlOu •,3* C HI Rs4. 100 IN-1 Ou100 ,4* C U0400 .,5* C ----Ii.PUTS---- uOitiu • .!.14, C uUinu ,7, •C -Ant `.I FIJ.:014.1•CY TN RADIANS/cEC ullio, ,0*. C V II 7-X roMPOUFIJ DP WAvE NUrillEa VECTOR TN KM**(-1) unIlio •g* C AK, = se rOv.i•UNENT OP wAvf NuMBLI1 VErTOR TN KM**(-1) 00100 1,0* C IT =1, 11 0 AmETFR TRANSmITTED TO USFAc DLETNING FUNCTIONAL tiOlnu 1"1 ,* u Dr:47 NOLNCi- OF A11k2.A6 CO,,..,PUTFU UY USEAS. uO i nu 1,2• C iii 1(I) =VAI.HE OF f-1 Ar 1401TOM OF LAYER I owinu 1,3t C POt2(I) =VA•ULT OF FP Ar KOITON OF LAYER I 1,010u 1..4. L 1•,X :-.1111-nEN of:” ATMN';PH1;RIC LAYLRS wITU FINITE THICKNESS LA.;1(16 1.r.)* C LI11) ...... ,INND -4-, E,I) (gM/SEC) IN I-TH !AYER LulOil 1.6* C vX1(1) ..L.x rnt11 , oHEi,T Oi.: ,iIHn VELOCITY (KM/SEC) IN I-TH LAYER uu4tio 1,7* C ■,Y1(1) :-... Y r.W.IPOUENOrT O wINO VELOCITY (KM/SFC) IN I-TH LAYER Colnu 1"10- L . - HI,I) =TNYCKhESS IN i,-” OF I - TH LAYER u0101, 1"q* C - uulnu 1,0* C __-001PUIS--- - unino 1,1+ C Lo..1tOti I ■ 2* C 1. =1 er2 —1 DrPENNIHG ON wHETHF14 NPPLR ► OuNnARY CONDITION uuL0. 1,3* c Cet OR CAk,noT NE' SATISFIED. cEL SUHROUTINE UPINT ui;Lnu 1,4t C AI•AT ::INTEGRAL NVER 7 FROM 0 TO TNIFTNITY AS DEFINED IN THE u010u 1,5* C Anc,TRi,C1. (,01n0 1,6* C oti ► 0,. 1,7. C uu 1. 00 1 C -_-_pRonR.4m EnL1OwS DELOw--,.- uOtOo 1 , [4:, C l,0 LOU 1.0* C L'U101 1,1, ,-,U.RNUrIuE TOTTUTLAMEGA,AK•fAKY/IT,L,XINTIPHII,PHI2) uu101 1,2* C u0 1 01. 1.3* C hIr•E14STON ANN CO!,:!,IoN STAMATNTt: uuloo. 1.4, III .F.NSrou cI(Inn)IvX1(1110).VYT(100)*H1(100) , EM(2,2) U (1 1 0,, 1.5s N1..114Sr,,u i•IiII. ( 100) /1 , 111P ( ion) •, u010 1,6* LO•• , oN IkAA*C.I.W.(IIVYI,HI u01.11, 1.7 , C uu ,..0!, 1, C LOmPuT.Tior4 or CONTTRuTION ERNH UPPER HALFSPArE 1010h 1,9* ,1=IHJ.Y4-1 6k.l1t.)7 1,0* L=rTtJ) u0k1t, 1,1s vX:...o.I(J) uOLli 1•..?.• vY .:VYI(JI. uUilk 1,3+ CA II IJS1 .5(0mEnA,AKX/AKy.C.Vx,vYFIT,A1 , A2 , A3) u0113 1,4, LAIL UILINT(('vEnA,ANX,AKY,C.Vx,vYrAl'AiIrL,E1,E2ruINT) 1.U114 1.19' C a DI-NoTE THI- CONUTBUTION A3*HINT fly xIHT. Ac THE COMPUTATION CON- u0114 140• c Tri.:1 11-5. ;ilia WILL CIICCtSSIVFLY HITIcESENT Tuft VARIOUS SUBTOTALS UNTIL Uuil4 1-.1* C CONTuI..ialo"S M-HOM PLL THIS LAYRS HAVE BEEO ADnEU IN. UUllo 142* AIwt=ABto/i.T 011.11, 143* C uutlh 144* • C START tiF PO 1 CuP tiUill 1.51, HO ' H t T=IrlrAX uUt2e,_ 146* ozi 1onXi1-.1 LUI2i- 147* C. triii2.., 1,n* L LON,PoThilOn or CONTPTBOTION FPnM J-TH LAYER OF' FINITE THICKNESS. L612e 149*. L 1;4: LU.O, NT VALUES. rl ANb Ip Hi-PRESENT F1(Z) AwD F2(Z) AT TOP OF u01.2._ 1•,ni. c J- TH LI.YFH. 0/1.2:S 1:,1 t C=rI(JI 0)124 17,2* ',11,.z.VXI(o) uOi.:".:) 1,5t VT..:VII(J) 00121,1"44 1,=“TIJ) tit/07 1•,5* LAiL UcLASOMEO,AKX,AKYFC.VxlVY,ITPAI,A2,0) 1,0130 1,6* CAJL CI I HT(ONEGA,PKX.AAy.C.Vx.vYtH.FlrF2,A1.A2FAINT) ut1L31 •1",1* A I.,,T=X f I I +• ■ INT*113 tiu131 1..n. ‘ uul:51 1.,9#, L (.O4.,P4,1.,TION (A FI AIM F2 ApPPOPP1ATE TO TUP of (J -1)-TH LAYER ut113, 1,0+ rl = PHII(J) 1,013.1 1"1. 9n FP .7: PHI:J(0) uu 1.30 1"2* C Friti IW no I onp u013j 1,.3+ C 1.01311 i"44 HEIHRH u013t, LS. Fn"

F"...0 or ‘00PiLi1TrOH: 1 hTAOOST1CS. 104 0 G,p *4.4,4* sUI;Cb $44.** WE vt ;r5 1.S,1%.5,SuhC5 VUR s1IF-1,/10/74^1A0W43 (tit)

SoHRIoTtHE vOINI POIMI nOnn70

SlC]KA0L USEo:'OODI(1) 0..012A; DATA(n) 000021; niANK COMmON(2) On0000

ExTrJb.itt rUERE,-(Cf-S (hL0Crt. HA"E) 00113 AAft, ■ unan SORI • tiOu5 ta..) ,:"3*

S1OJ(A.L Ae,S*ON(4HT (hL"CK. TyPt. RrLAT1VE iOCATIoN. NAME) 001 0004121 3L lalno P UnOn00 A nOu0 H n00005 r, n000 R 000006 GRT 0000 000010 INJP$ iNuu t( 0014104 X u0101, 3* C -- - -AoSTP4,CT -- - !- uUlnu 4* C 6010(; Eit L lITLI, - uPfNT 0110u 6* C THN cUHR0HTTrIF COMPUTE::. AN INTFGRAL OF THE FORM LiOLou 7* C uulDu A* C H1NT = INTFOKAL OVER Z FROM 70 TO INFINITY a tAi00 9* C UUiOU 10* C (A14F1(1) + A7*F2(7))**2 (1) uulOu ,If C u0IL0 ,2* c • THE 1•HHLTIOroc, F1(7) ANN FP(/) ARE Tir SnLUTIONS OF THE COUPLED Loin() 13* C oRDT-ARf NIprERL.NTIAL EAUATIONS 60t0o / 44 L uLleo ,5* C n1 1/t.1 = A11•F1 4 Al2*F2 (2A) u(1 10o it* C (1t-P/„1 = P21 ► F1 + A2)*F2 (2A) u0,(10 ,7 ► C ublOo .;:s* C khrPc. IiiC EIPTNTS nF.THL MATRIX A ARF INDLPENDENT' OF 7. THE uOLOo i wt C Ft,rilC„IS F1(7) AND . F2(,) AkE SNBJC.CT Tn THE UPPER BOUNDARY OJLOo ,0* C CONninW.4 TWT BOTH rECuFASE EXPONENTIAILY •dITH INCRFASING LJOiOu ,1* L t1 T1llg%1,_. c T NCL TN MAT R I Y. A IS CLAPHTFD BY AAAA, INSURING colOu , 2. C Al2.2)::-4(1.1), HOTN . SH,,UIT) VARY WITH HFIGHT AS EXP( - G*(7..-70)) uukuu '3* C WHEhL otlint, -4* C LA)100 .,-* L. 6 :.; :.0AT(p(lt1)**24A(10)*A(2,1)) (3) dUtOu , iii C a0100 ,7+ C Il i, AS!T.WFP C**2 IS PcISITIVE. OTHERWTSE L=-I IS RETURNM. JUlo. ,At C JOLUU .9+ C • THE i_X;'LICIT FORMS ADOPTED FOR Fl AND F2 WHICH SATISFY (2) ARE JOtOu .10* C o0i0D 01+ C FI =-5.,1g(f1)*/) (1.2)*i:XP(-G*(1 - 7C1)) (4A) ,6106 02* C F;"! = SORT(r0*(G+a(lti))*FX0-G*(4 -70)) (4A) doloo ,3* L dOIOU .,4* C ThUg t11141 1c GIVEN kY ,j10w 051 C ,o1Co 06* C HINT =((-At*A(IrA+0*(G4A(1,1))**2)/2.0 (5) dUlOu ,7* C ,u,Osi ..,f1* C LAmSNA.:.r .= FuRTPam IV (,71,Or 44EFERENCF MANUAL C22-6515-4) ,UL0t, 09* c AUTHO - A.D,RrERCE , M.I.T. , JULY , 1966 ,u.L6L, -o* C ,010u "1* C ----CALLING S;:UHENCE - --- .01i16 ..2* C w i no ...3* (.; :)LF !- .1./...t•OliT144I': TOTT1'T ,ulijo '..4+ C Nu DIm“NSIoN StATimt- HTn PrwHIRFLI .viOu -.5* C cAIL U('INTtOmEr•,AlcAlAAY.C.Vx/vY,A1 , A2FL,F1iF2ruINT) wtou ,..64- C ulOu -47* C -_--rxTERNAL cUMMUTINES REOUTRED--- - • 61Liu - As C Oink, ,. , 1+ C AA.A olOu ,0* c 0100 •,1* C - ---AkGUIvFNT 1 IST--- - 0.06 ,24 C 010u -.3t C Mid,. GA A*4 14D INo ULOU :.4* C AK4 R*4 ND TrIu OIOU -)5* C AK•r R+4 viD INp 0100 :,6* C L (Z+4 ND IN,*

Out0u .0* C A2 R*4 ND INo UUtOU • .1* C I.: I*4 NO OUT 00100 02* C Fl R*4 ND OUT tiUtOU .3* C F2 . R*4 0 our uulOU .4* C (4.4T R*4 ND OUT . U0100 • .5* C UUt0u ..6* C NO COmw.oN STORAGE HSFO . uU.LOu "7* C LUiOu .8* C .----INPuTc...... C U0100 "9* U000 ,0* C oMPGA tAme...DLAR'FREOUrNCY IN RADIANS/GEC .)01.00 #1* C AKt =X roMPONENT Or WAVE NUMBER VErTOH TN KM**(-1) u010U /2* C 410 =Y COMOONENT Or 'AVE NUMBER VEC.TOH IN KM**(-1) uulOu #3* C c P.SnUND SPEED Iri KM/SEC t,U10U /4* C VX =X CoMPONENT OP WIND VELOCITY TN KM/SEC . 0010U /5* C VY =Y roMPONENT OF WIND VELOCITY TN KM/SEC UU1OU /6* C Al =CoEFFICIENT Or F1(Z) IN INTEC;PAND U010u /7* C' . A2 =CoEFF1CIENT OF F2(Z) IN INTEGPAND u010u 0* C , UO/Ou .9* C -..--OUTPUTS- -- - U0100 .11* C U0t00 .1* C L =1 OR -1 DEPENDING ON WHETHER UPPER BOUNDARY CONDITION uOtOu .2* C OP F1(Z).F2(Z) DECREASING EXPONENTIALLY WITH INCREASING 00100 h3* C HFIGHT CAN OR CANNOT DE ;ATTSFIEU. IT REPPESENTS THE u0100 ., 11 C STGN OF G**2 wHFRE G TS DEFINED IN THE ABSTRACT. LO1OU .5* C Fl :VALUE OF Fl(Z) AT BOTTOM OF HALFSPACE ► DEFINED AS 0ut0u /,64. C . -cr`RT(G)*A(1./n) FROM EON. (4A). uulOu .7* t F2 =VALUE OF F.2(Z) AT BOTTOM OF HALFSPACE ► DEFINED AS UULOO .S* C SnRT(G)*(G+A(1,1)) FROM EON. (4B) . . UU1Ou #A9* C HImT r_THF INTEGRAL DEFINED HY EONS. (1) AND (5) IN THE 4.)0100 40* C ARSTRACT (101) 41* C tiOtOU 4'* C Uu100 • 43* C --7-PROGRAM FrILLOwS BELOW--- uOtOU 44* C 0u1OU 45* C UU101 •6* SUuROUTImE UPINT(O•EGA.AKX,AKY ► C.VX/VY/AI ► A2eL , F1*F21UINT) OULU 47* HImENSTON A(2+2) 00103 48* C U0103 49* C COmPHT•TIOw OF A m.TRIX AN(t OF X=G**2 u0104 .1.)0* CAIL AAAlt(OMEGA.AKX ► AKY.C,vX.VY.A) 0(1105 11,1* x=A11#1)**4 4- 01,2)*A(2,1) 00105 11,2* C .CHECK ))N STGN uF X uu106 1,,3* 2 IFI X .GT. 0.0 ) GO TO 3 UU106 1.4* C O0106 1 1.5* ' C A TS 1.&GATTVE tiOilU 1.6* L=-1 UUtll 1.7* RE,URN O01111.0* C cONTIN0ING FROM?. wITH X POSITIVE U0114 1.9* . 3 L=1 00113 i0 ,1, 6=t.ORT(X) u0114 1,1* , 6R/=5,01.0 (G) Uu115 112* F1r.-6OrtA(1e2) 0006 1,3* F2=GRT*(G+A( 1 f i ) ) wy.1.4.1

END OF LOmPILATION: NO nIAGAOSTICS. wHoG,P **I.*** SUHC6 ***4** • LIFO'S 1.S,BC61SuBC6 1-08 S11F- 1 ./10/74 - 18:08:h0 (ut)

SUDROoTINE USEAS FN;RY POINT 00n244

SiORAc,E USED: %DOI) 0110314; DATA(n) Ou002A; RiANK COMmON(2) On0000

EXTEPNAI• REFERENCES (HLDCKt NAME)

. oUo3 SORT U0o4 NERR3s

STOkAc,E ASSIG14MENT (HLnCict IYPEt RFLATIVF LOCATION. NAmE) 0001 0,10431 1000t 00n1 000n26 200L nOni 000067 300L nool 000164 700L 0001 000201 800L 00n0 R 000u03 AK 00n0 R 0n0n01 AKSO 0000 R 00000n AKV 0000 R 000002 BOM 0000 000006 INJP$

u0100 1* C uS.7AS (SUDROUTTmF) 7/25/68 LAST CARD IN DECK IS NO. uOinu 2* C UOIDO 3* C ----ABSTRACT- -- - udiOu 4* C 0010u 5* C 1ITLE — USFAS . 1.010i) 6* C THE PURPOSE OF THIS SUB4OUTINE IS TO COMPUTE THE NUMBERS Al, A2 U0100 7* C AND A: «,HICK DEPEND ON ANGU1AR FREQUENCY OmEat ► HORIZONTAL WAVE oulOu 8* ' C • NuNkER COMPw1F.N1S Axx AND AKY, THE SOHND SPEED C ► AND THE WIND u0i0u 9* C • SPE.v0 CoMFONFN% VX AND Vy. THE INTEGEp IT DETERMINES WHICH • UUiOu .0* C FoRmULAs ARr USED FOR At' A2 AND A3 ACCORDING TO THE FOLLOWING GOinu ii* C TAM E U0100 J2s C UOiOu i 3* C (IT) (A1) (A2) (A3) U0 100 14* C ------uulou '5* C 1 1 0 1 • uti100 16* C p 0 . . 1 1 6010u ) 7* C 3 1 0 DOM*(KUOTV)/(C**2*K) LJUIOU i8* C 4 • 1 0 Oo•/C**2 UU100 39* C 5 • 1 0 VX4n0M/C**2 uUlOu .0* C o 1 0 ' VY*140M/C**2 uOJOu .1* C 7 G/C —C K*OmEGA/DOM**3

t,U1Ou (.2* C . 8 C/C —C 1.0/60M**2 b0100 .3* C 90/C —C K**2/00m**3. uUiud ,4* C lo G/C —C VX*K**2/POM**3 uU1Ou ,5* C ' 11 G/C —C VY*K**2/(10M**3 ••••,, me. wi•u Vt-IOCITT. AND K=SORT(AKX**2+AKy**2) IS THE MAGNITUDE UU100 • .10* C Of 110. w.AVE MHNER vFCTnR. THE ACCELERATION OF GRAVITY G IS (40100 .11* C TowN AS yOn90 rsM/SEC**? TN THE COMPUTATION. COMPUTED VALUES u01(1v .12* C . shomn HE It KNFSLC SYSTEM OF UNITS. uU100 .13* C UGIOU .14* C LANGUIV4E — FORTRAN IV (360. REFERENCE MANUAL C28-6515-4) OU1OU .s5* C APTH0R — AyByPTEPCI::* M.I.T. ► JuNE.1968 UUIOu C u0i0U C ----CALLING SvOUENCE - -- - .%8* C LOIOU .19* C SEF SUpRuUTINE rctrmr U0.100 40* C IaO 01PrNSIuN STATEMENTS ARC' REOUIRED b0100 41* C TT: UGIOU 42* C .CAI( UGFAS(OmEnAlAKX.AKY.C,VX,vYrIT.A1rA21A3) ARE Noll AVAILABLE FOR FUTURE COMPUTATTONS 43* C A1 ► 1* ► A3 CUIOU 44* C UU10b 45* C NO EXTwRNAI. LIHRARv SUBROUTINEc ARE REQUIRED UU1OU 46* C UOICU 47* C ----ARGUMOT uOIOU C 0010u C 0MrG4 R*4 ND INP 1,0100 nO* C AK.v R*4 IMP UOIOU n14 C AK. k*4 ND /NP 00106 n2* C C R*4 ND INp LOLOU n3* C VX R*4 ND INN O0.100 n4* C vy R*4 ND INp UUIOU C IT I*4 ND INP u0100 n6* C Al R*4 ND OUT U0100 n7* C 42 R*4 ND OUT u0.100 n8* C 43 R*4 ND OUT UU1Ou n9* C 00100 00* C NO COMpON cTnRAGE uSED UUIOu 01* C UOIOU 02* 0010U 03* C UOIOU n41, omwGA .7.AmcULAR FREOUrNCY IN RAD/SEC OULOU /15* C AKw r- X MAHONE...NT Or wAvF NUMBER VEcTOR IN KM**(—l) 06* C /.Kr .my COMPONENT Ow WAVE NUMBER VEcTOR IN KM**(—I) 00100 07* SPEED Im KM/SEC LU100 08* C VX =K COMPONENT OP WIND VELOCITY TN KM/SEC . U0106 09* VY zY COMPONENT OF OND VELOCITY TN KM/SEC V0100 /.0* C IT =Cw:TROL PARAMETER FOR SELECTION OF FORMULAS (SEE LOIOU il* C ARSTRACT). /2* C v0100 /.3* C ----OUTPUTS---- /4. C UOIOU /5* C Al .7PAnnme..TER DEFINED DY FORMULAS IN ABSTRACT uoino /6* C 42 »PARAMETER DEPTNED DY FORMULAS IN ABSTRACT (JOIOD '7* C A3 =PARAMETER flEFTNFD BY FORMULAS IN ABSTRACT IJUIOu '8* C UUIOu /9* C UOIOU •0* ---PkOGR.M FnLLOwS BELOW---- U0100 01* C vOiOU 02* tnLim EAI1 uului 1,br C V AI lii S . UU103 n7* A1;:1.0 u0104 of.1* A2:11.0 0U10b h9* A3:.1.0 U0105 40* c IF IT lc 1. THESE ARE CORRECTt.HOWEVER. UU106 .1 1* IF(T1 .F.U. 1) PET(iRN 00110 t12* IFill .0. 2) 440 TO 200 00110 43* C uO1lU 44* C IT IS '.... THF CURRFNT VALUFS APE 1,0,1. WE CHANGE THE FIRST TWO. 00112 45* Al-:0.0 UU113 46* A271.0 00114 '.7* RE(UhN 00114 48* C 00114 .9* C IT IS •GT. 2. WE •OMPUTE SOME QUANTITIES FOR FUTURE REFERENCE 00115 1.0* 20n Aka=i1Kx*VXMKY*VY 0011i, 1,1* 0“.0=AKx**2 -1 A1ono u0156 1.)9* C IT=10 1..3* C IT:11 SHONLON-T INPUT IT FOR VALUES OUTSInE RANGE OF 1 TO 11.)• U615/ 1•4r 113011*AS uulou 1.5* 1/0/61 1.,6t

EAJJ OF COmeILATION: NO IITAGNOSTICS• i.11,,C,11 O..*** sUl C/ 4**.** 1.5.6C7rSuuC7► slIF-1,/10/74 - 18:06:b6 tut).

SUh ► ONTINE AINCN FN1RY POINT ► On240

SIORAI,E USEu: CuOL(1) 04.n.;, 74i; OATA(n) OuOnE/I SIANK COMmON(2) On0621

EXTER,A, REeERLNCLS NAML) 01,3 wDFN 00114 Nrin(31

G)oRA(.L AcSlOWA:N1 (1iL"(K, IYPL , RcLATIVE 10CATION, NAmE) 1100 000,,J7 1111' onnl 010nH5 117G 0001 00013n 133G n001 000144 1406 0001 000203 150L noo 00(10/ 131L 0On.: 0n0nn1 CI nOn0 R nollonn nELUM 0000 K 000015 FPP 0002 000455 HI ORO I OnOul? I wOno I OnOn 0 6 1J 0000 I 000007 TJOLD 0002 000000 NAY 0000 00.0017 INJP$ u0N0 I 000005 III onnQ I OnOnn3 J 0000 I 0ouon4 JOLD noon 1000016 K 0000 I 000014 L ,10n0 1 On0,,10 NEW) 00110 I Woo? nnn0 I 000001 NSTAkT n000 R Onn011 OMEGA 0000 R 000013 VPHSE 00014t) VX1 0011.2 000111 VYI

un i Ou it C wI.EN IsiIHROUTT)•E) 6/19/68 LAST CARD IN DECK IS NO. uilLOu 2* . C • U 0 ,0 0 uu,Ou tir c ----AOSTRACT---- untOu 5* U010v 6r C TI1LF - wInEN 01.11011 71, C wInFt: mAtRIlf TMmODE RY ADDING Kw COLUMNS BETWEEN COLUMNS Ni AND 0,10u 3r H1+1 u010u 1,01013 ';A.UEN MDS KW ELEmENTS TO THE: VECTOR OF ANGULAR FREQUENCIES LAWN .1* rr Om I r*IVIUING THE INTERVAL RI:TWFEN OM(N1) AND OM(Nl+I) INTO uuluu ,2* C Ki.+1 1= - UAL PARTS. FOR EACH ,4•W ANGULAR FREQUENCY, A NEW u0100 13+ C CNLUMN IS ADOFn TN THE INmoDE MATr.IX (nEFINFD IN St./HMI- uPLOu T1HE mrOUT). INMnrE IS STORi.:0 IN VECTOR FORMr• COLUMN AFTER 0.1100 15* C CuLUMN. 1JUinu uo/Ou ,7* C LANGOA.,F FoRTRAN IV (3611, wEEERENCE MANUAL C20-6515-4) ulIt00 08t UlUU .2* C ...... •'WSACA- OLOU /3* C Lii ■ it) eq* C AMI.V,IfJ :00r: mUcT nE nImFNSIONFn IN CALLING PROGRAM Oiuu .5* C ilk uNiY SoJR0,1rIN CAI LED I' NMuFN , DEciCRTOCo ELSEWHERE IN THIS ULOU 'hi c -,F,TI-S 010u .7* C u100 ,A* C fORTHA.., USnur UiOu ,9* C LAI L :1,(,3FN(DMrupTtUOnE.NOM,NWIP , NVP,NI/KW , THETK) Ot00 ,0 ► C OiOu 01 1 C IN0HTS 0LOu .,2* C utOo A. C CM Vr_CTOo WHOSE 1-- LF•i-liTS ARE TILE VALUES OF ANGULAR FREWEHCY utOu ,4* C R44(D) CoRREcPOWING TO THE COLUmNS OF MATRIX INMOnE. (RAD/SEC) ()IOU ,,54 C Ulrmi ..34 C V 1:17.CTOw •HuSE I•LEMENTS ARE THE VALHES OF PHASE VELOCITY 310u . ■ 7* C RAIi(D) CuRREc:r'ONDIN• TO rHE ROWS OF MATRTX INMOUL. (KM/SEC) u4Ou .:i*. 1: ulOu ,q.. C INM,iDc. L,,CH rLEMLHT oF TNIS nATRIX CORRESPONDS TO A POINT IN THE 0400 .0. C I , 4(D) 1:1-%EOUENCY (0M) - PHASE VELOCITY (v) PLANE. IF THE NORmAL ;Jou -.1. C Hunr. nTsPLuSIon FHNCTION (FPO, FOHND BY CALLING SUOROUTINE U,0,1 .-24- L tIOEN) T. POSITIVv AT THAT POINT, TILL ELEMENT IS +1 ► IF JI0o .3* C FP-41 lc NFuATIvEr 11LE ELEMENT IS -it IF FPP nOES NOT EXIST, U100 ..4* C THE ElEmENT IS ri. iNmODE IS STORED IN VECTOR FORM' COLUMN Dili; •S.. L AI-TFR roLuMN. Ji0o -.6. C ulOu -7i C Nom m0Brp OF EIL,,ENTc TN OM (AND NO. OF COLUMNS IN INMonE) ul0J •R* L 104 . :.:N wIDEN IS cALI(I:n. °IOU ..9Y C' NvP H0MBEw OF ELFN,FNTc IN V (AND NO. OF ROWS IN INMODE). ULUu •,0* C /. 0t JI0o !,1* C otOu •2* C H1 1P•FI3p OF IN,•nnF rOLUmN IMMEDIATEIY LEFT OF SPACE IN WHICH 310,1 •,3* C I.4 tir.::! CoLUMNS A,iF TB BE ADDEO. .1,0o :,4.. C J.,Ou .,5• C Kw '11 CU •,J* C I..4 f.U•;1EP OF COLLIMNS TO BE ADDED TO TNHODE. )ir)U •,7* C ILO ', ,6* C THF1K PHASE vFLuCITy nIwECTION MEASURED COUNTER-CLOCKWISE FROM i k 0 u !,9* L • R..4 X-AYIc (R/OIANs). IL ou .0 C II0o ,I* L. uUTPNT., 110J ,,2* L: lin0 "3* C THE OuTPUTc APE -)MIJ (= NOM + KW = THE NEW NUMBER OF FLEmENTs TN OM '10J ,,4* 3 AND THE :•1-- NU•w - r2 JF CoLUM.S IN INMODE) ANn REVISED VERSIONS OF OM , I0i; "5* C /114D II:Mil , i: . AU0 "iii C 10u "71, C LOU ,3* C ----LXAMLE ---- , 0 u ,0* C LOJ ,o ► C S0P., 05E Jai z. t.nr2. ► 3.11 ANn ihInEN IS CALLEn WITH KW : 3r AND ?'I = IOU ,1* c IHE.N uHON RETURN TO rALLTNS PROGaAMr On = 1.0.2.0.2.25 ► 2.5.2.75 ► i 0 U t2P C 3.0. 1.0mP = bi Ain INIOnE WILL HAVE THRLE NFW ROWS CORRESPONDING TO 100 ,3* C THE nrW LLLI.;ENT(.: OF OM. LOu /4* C 100 /5* C U01.01 i9* SOMOUTINE WIDFNCOM ► V,INMODE.NOM,NOMP,NVP/Nl ► KW,THETK) 06101 h0* C 00101 .1* C VARIAHIF DTMFNSIONTNG 0)103. m2* DIAENSTON uM(11 ► V(1) ► ImMODF11) 00104 m3* co...moll ImAx,C1(100) ► VX11100.VYI(100)1HIC100) 00104 .4* C 00104 .5* C INTEkVaL AT wH1CH mrw VALUFS OF OM ARE RE PLACFD UETWEEN OM(N1) AND 00104 m6*. C OM(N1+1) Ic DETERMTNED 00105 m7* nElOm:(0)++(N1+1)-0MiN1))/(Kw+1). 60i0U .8* C 0010!) .9* C NOmP Ic NUmUFR OF FLEMLNT TN EyPANDED OM 0010o 40* 140•.,P=NnM+Kw U0106 41* C 00100 42* C NSTART IS TfiF NUMBrP. OF THF FLFHFNT IN THE NFW OM WHICH CORRESPONDS TO 00100 43* C ELFvFNT •1+1 IN T}l OLU OM VECTOR 0010/ 44* NSTART:N1+1 -1- KW 00101 45* C 00107 46* C MOvE At L ELEMENTS nF OM REYOND ELENIENT Ni TO THEIR NEW POSITIONS, BEGIN- 60107 0* C N1mG WtTH THF LAST FLEMENT 0011.0 48* DO 90 nJtN:;TART,NOI P 1 U0114 49* J=u0mP-(NJ-NSTART) H' '00i14 1,0* o0,D=J-Aw vOil 4 1,1* C 00114 1.2+ C MOVE CuL11MD JOLD OF INMODE INTn POSITION FOR COLUMN J uUllu 1,.3, OM,J)=nNI(JJL(1) UUtlo 1..4* !)0 9UTPzIpNvP U0121 105* lJz(o-i)*NvP+(mVP - IP) +•1 --- 00122 106* 1J,ILDF(JL1 - 1)*NVP+(NVP-IP1 + 1 0)124 1,7* IN.101)17 (1J)PINMnnE(IJOLI) ) AW124 1..n* 9n COoTIME• 00124 1 0 n* C UU124 1,0* C 01ANT IS NuNAIO OF FIRST NEW rOLUmN UU127 1,1* NSrAkT:N1+1 00127 112* C U0121 1,3* C NEND Ic NUWIFR OF IAST NEW COLHMN 0013U 1,4* NEmD=1,11+K11 u013u 1,5* C uU130 1,6* C NEW VA,HFS OF u' 1 AnE ESTABLISHED v0131 1,7* oM,-6/1=nm1N1) U0132 118* DO 190 J=NSTARTINEND U0135 1,9* DM(J)=nmEGA + nFLOM 0013o 1,0* OMGA = OM(J) v01.37 1,1* DO 190 1=1,NVP (,U137 1,2* C U6137 1,3* C la IS mUXOFR 01 ELruENT IN VECTOR REPRESENTATION OF INMODE WHICH IS

U0137 • 1,4 * ELFMI-Nr J TN ROW I OF INMOUE UU142 1,5* 1Jz(o-1)*NVP+I 00443 1,6* vPHSF=V(I) 00144 1,7* C 00143 1,J1* - C CAiL N-DFN To EVALUATE THE NORMAl MODE DISPERSION FUNCTION (FPP) 00144 1,9* ' CAAL NMOhMOMEnA,UPHSEITHETKILFFPP,K) u0i44 100* U0144 1,11* C IF FPP noEc NOT FXTST L 7: -1 UP1.405 1.:,24 IF/ L .Eo. - 1 1 GO TO 150 .w a ■ rHr l ) 1:1u* IF (1-PP.LE.0.0) INMOnE(TJ) uu152 1.17* o0 10 ton 00153 108+ 1 5 n IN..01JECIJ)=5 U0154 1.)9* I8n CO.JTINHE v0155 1..0 ► 1 9 R CONTINHE u0160 1...1* kErURN u0161 142* ENn

END OF LOI.O'ILATTON: NO nIAONOSTTCS. koHuGrP •**+*** ****** f.FulitS I .SalultSupc8 FOk SliF- 1 4/10/74 - 18:07:o3 (n1) MAIN PR,0GPAM

STORA6E USED: CODE()) On000p; DATA(n) 000001: B1ANK COMMON(2) On0000

EXTFRNAI REFEkENCES (1LnCK, NAME)

04 3 - NsT ot) 1

STOkAGE ASSIGNMENT (hLhCK ► TYPE' RpLATIVE LOCATION, NAME)

LA4101 twImGt)OsTIC* /* Ic A RAD LAREI LU101 *DI.GHOSTIC* THIS pROGRAM HAS m0 END CARD. u0101 1* /*

END or comPILATTow: 2 nTAGOSTTCS. suiic9 • ****,** 1.17 ,S 1.S,I1C9,SULiC9 F01( Sl1F- 1 4/)0/74 - 18:07:05 (O ► )• MAIN PRi)G'1API

STORAhE . USEU: CODE(1) Onn002; DATA(n) 000001; MANK COMMON(2) 0(10000

ExTERNAL REFEkENCES (fiLfICK. NAME) 001;3 NsToP1

SIORA0E . AcSIGNMENT (FILCK• TYPE' RrLATIVE LOCATION, NAME)

•

END OF LOWIL ► TTON: 2 TIIAGNOSTiCS.' laiu4sP 4.**** sUnKu *44,244 6FuRtS . 1,54,1WOrSU3K0 FOit SIIF-1/10/7 14 - 18:07:n9 (0.) ND SoURCE DECK---CoNTRoL CARD IGNORED. NAiNG ► F 4 4 +4** hUR00 *44+4* wFuR,S 1.5.0 ,10,Su3o0 FOIL SltF-1./10/74 - 18:07:10 (14/)

SNDRONTiNE RTml • ENTRY POINT 00n600

STORAGE USER: C)DL(1) Oy06041 DATA(111 0000741 Iii ANK COMmON(2) On0000

EXTFRNAL REt;ERENCil_S (HLNCK, NAME)

0003 N4Du$ ouy4 NT02$ • 0005 NER1(31;

STORAGE A!-,S1ONMENT (tiLnCK, TYPE ► RFLAT1VE LOCATIONS NAmE) 0001 000173 IlL ciOnI 000207 13L nOni 000211 130L 0001 0041063 1366 0001 000272 1 4 L u 0 o1 000311 16L o0n1 000314 17L 0001 000363 pOL nool onn373-22L 0001 000410 24L 0001 0004212_5L non' 000054 4L 0001 0001n 7L 0000 0000P0 723r 0000 000031 724F 0000 000440 72bF TiOnt 0001 56 9L 0000 R 000011 A n000 R 000003 CONV 0000 R 000015 DX 0010 U 000004 FL nOnn H 0n0n17 FM 0000 R 00o005 FR 0000 1000006 I • 0000 000056 INJP$ 00(01 I 000010 K nOno H 000n13 o 0000 R 000014 nA 0000 R 0(10012 QUEST 0000 R 000002 TOL 0000 R 000007 IOLF 0O00 R 00000s XL 0000 R 000016 xM 0000 R 000001 XR

u0iOu 1 * C RTMI 1 yu100 2 RTMI 2 utiOu 3$ RTmI 3 0010y. 4* RTM1.44 - LAWN 5* UIMI 5 yuluu 6* PuRpuSL RTMI 6 UULOO 71. C TO SaVE• lErTLV . INEAR EMI/0 ION50E-- FORM FCT(X)=0 RIMI 7 uolOu 0$ rty mtANS np muELtER-c, 1 8 Loily 9* RTMI 9 uOinu USA(;L RIO In uoinu rA X ■ F PFCT XI /EPS, LEW) , TER RIM 11 nu .2* NyiAmETEw FCT HERUIRFS AN EXTERNAL STATLMENT. RTMI 12 1 0l1 13* R.ZtT 13 001011 DEScRIPTION 0F PARAmETEoS RTMI 00100 .5, - RESULTANT RnOT OF EoUATION FCT(X)=0. RTMI 15

1 ,, i ri.ui AHV21IT82lv 30 OdAZ . , N3S1 ,0/11I8)? I N '9 73S #30Nj2J3338 NOJ D 1"5 n9 THIN I NOI1dT2J0S70 1V3IIV1'DHIvW du ( 17 4 ) 3V1n6,1803 215 A0v000v • 0 ng 6h 1141HAOIOV3SIIVS NO 1S31 2i0: 1. '(A)10J JO SNOI1VnlVA: Oml 53611102d 0 4h 0f7 •Pb TWIN c13IS NOTIVM11I 3NO . 0147 01 1Vnn3 ION SI X 100N 1V (X)103 0 01 TW121 •.:I0 2AIIVAIN1n 1HI JI 011Vd0Vnd SI 30N30211ANO0 I I6X ONV IlX 0 Or 9i, iftp..1 c.nN(109 1VIIINT 3HI IV 5121V4S HJIHm INOliV1OdU3INI 011000d 3 01L1 9+7 INIu , 3S2i1ANI OUV.SN0I1.03Sid 3AIS530011S 31 00H12W NOIIVM3II D cf, ilh TWIN c-213113nW JO SNV1N Ad 3N00 SI V 1: (x)103 NOTIV1102 A) NorinlOs 0 4, ft cli• TWIN . OM-112w 0 ,t zi/ IND; 0 . ,t Ttl IN1/1 '83sn DH1 AU 3 Tb Of, INIH 02HSINN(13 29 ISniii (X)103 HybotirionS N0II0NI"H 1vN1131X3 3141 0 nt 6C TWIN 0.Inu38 SWV11008ddilS NOI101,1111 ONV S3NI1n02iONS .0 4C PC TWIN . 0 02 LT !WI N I F=N2I 30VSS7W modd2 3Hi 53A10 UNV 03SSVAAG SI 321(10300Md 0 /2 9C INI2I2H1 dINX ONV I1X SlnivA Indla Au 071 13STIVS ION SI NolidvinSsv 0 n2 qc /w12' OISVO SIHI 3I IN0IS 3WV .JH1. 10N 3AVH TUX ONV ilx soNnou 0 (2 liC INIM 1VIIINT IV S3N1VA N0110N(13 IVH1 53vinssv 32indj3Udd 3H1 0 4, CC TWIN SNMVw3N 0 r2 Fc I NIN 0 22 IC TWIN '02T3SIIVS ION SI 02132 01 1V(103 dO NVIII 0 T2 n2 nc TWIN SS31 (in)10:4*(T1x)i33 NuIldoossv oisvn - er.113i 3 6F TWIN #NOI101SI9 D 4e PF TWIN 30 5M315 3AISS303nS Clial Ad 0Jm01103 D oe L7 Top! Sci3IS NOIIVM21T ON'jI 8313V 3ONJ9d -MN00 ON - I=N3I 3 re al IWI21 'MOUd3 ON - 0=N3I D clz GF TWIN SM01103 SV 03000 8.113WvuVd MOMUD LNVI1nS16 - 2i3I 0 e7e -he hi.: THIN °O3TAT013S Sd31S NOI1Vd311 AO 87UKIN imilixv,i - 001 0 IX linS32J 30 U001 0 r-z ce THIN 7z ' TWIN 7411 30 OW.09 212dcfn 3H1 S3IJI33dS H3IHM DrilvA indNi - SJ3 . 3 TF TWIN 'X 10nd 2111 30 D rz n? iwimme IHOIN 1VIIINT 3111 S3IJI03uS. H0IHM 3n7vA indNI - In 0 ne 61 TWIN 'X lOnd 3H1 30 0 41 0T PT TWIN nNnOU 1331 1VTIINT 3HI S3IJIO2dS H0IHM 2n1vA IndNi - I1X D LT IWIN . 03Sn WVMOOMPCIS N0110NnJ "'VW :110 3111 JO 34r'vN - 10J 3 IT QT TWIN I X 1008 IV Jn -IvA NnIIONn3 INVIins3d - - d 0 nT D 9T TWIN In=(X)10J Noi1vn03 30 100u INviinsU - X .QT 4.7T TWIN SU3114V8Vd 30 NOTIJI2i0S3 1 0 +, -1 VT TWIN 3 (71 FT TWIN 'INJIVISN 104u21X3 NV 520n038 103 212I3WV8Vd 0 PT • TT TWIN (031 , ONJI , Sd3iIdX , I1X , 10J , J , X) 'Him 11v0 0 TT OT TWI21 3ovSn 3 OT 6 TWIN 0 4 a P TWIN °00H13n NOIIV2F3II s - w1113nw AO SNV3W 0 3 L Illiii 0=(X)103 1003 3141 JO SNO11Vno3 NV3NI1NON 1V213N39 3A1OS 01 0 i. 9 THIN 3sodand 0 n !WI N 41 11,118 3Nilnou0Nel 3 +1 INLH F. c' TwIN D F !WIN D I TWIN D 7 00UHVT*ONWWKiNI. • __ any— ,, RTMI oU 61 XL=ALI RTMI bl 65 XP=ARI RTMI o2 • 64 XITX6 RTMI o3 64 TOL=X RTMI o4 bo F=FCT(TOL) 65 IF(F)1,1641 =I e: rt.,) 61 t CONIINUE 65 t0101=AQS(F) 6I F=Ft.T(TOL)/CONV 7v• FL=r: 71 X=Xr( RTYI t 74. TOL=X • RTMI u9 74 F=Ft..T(TOL)/CONV 74 IF(r)2,16,2 . RTMI 71 74 2 FP=1-: 75 IF(LIIGN(l..tFL)+SISN(I.,FR)) 25F3,25 71 C ERTMI 7u C BASiC ASSUMPTION FL*FR LESS THAN 0 IS 5ATISFIED. RINI 75 7' C. trNuRATE TOLERANCE FOR FUNCTION VALUES. RTMI 70 Hu 7; 1=0 RTMI 71 81 TOLI-r-EPS*(AUS(F)+1.0)*100.0 85 C RTMI 79 04 C RTMI oil 84 C STAriT ITERATION LOOP RTMI b1 .84 4 1=1+1 . R1M1 62 Flo C RTMI 03 8/ C START 9ISLCTION LOOP RTMI 84 8c, 00 13 K=1,IEND . RTMI t, 5 89 X=.5*(XL+XR) RTMI bu 9u TOL=X RTMI b7 91 F=FCT(TOL)/CONV 9.: IF(F)5,16,5 RTMI b9 94 5 IF(OIGN(1.pF)+SIGN(1.orR)) 706,7 RTMI 90 9* C RTMI 91 94 C INILRCHANCIE XL AND .XR IN ORDER TO GET THE SAME SIGN IN F AND FR RTMI 9i 9u 6 TOL=XL RTMI 9J 91 XL=A R RTMI 94 95 XO7.10L RTMI 95 9v ToLzri_ RTMI 96 10u FL=FR 10.A FP=IOL c,-- 105 7 TOL=F*FL 104 A=FATOL. Efflil 104 A=A+A RTMIlul 104 IF(A9S(A)-1.0E3q) 71 ► 710130 105 71. CONIINUE 10/ 01 1EtaT=FR*(FR—FL) 105 IF(AIISNUEST)-1.0E35) 72 ► 72,130 109 r coolvvE 11u lEln—ER*(FR—FL))8,9,9 RTMI1U2 111 A !EU—IF:NE/117,1709 RIMIlUi 11. Q XR=^ RTMI1U4 114 FR= I: RTM1105 -... fi-AWZ7k ,..Iti R1MI1u9 Ilu IF(A-1.)(1,11,10 RTMII1U 11', 10 TOLIITOLtA RTM1111 12v 11 IFinlISIXR-XL)-TOL)12,12r13 RIMIII4 121 12 IP(mRS(FR-FL)-T(LF)14,14•13 RTMIlla 12s 13 CONiINNE RTMI114 12o C END OF BISECTION LOOP RTMI116 124 C RIMIllil 124., C NO CONVERGENCE AFTER TEND ITERATIOM STEPS FOLLOWED bY IEND RTM1117 12u C SHCoESSIVE STEPS OF BISLCTION rii STEADTLY INCREASING, FUNCTION RTMI116 121 C VALuES AT RIGHT HOUNDS. ERROR R6:TtInN. RTMI119 12o 130 CON1INUE 129 Ir.-R=1 RTMI120 134, F=CuHV*F 131 :FR=CONV*FR 134. FL=LONV*FR 13v• (1=20, (XL-XR)./00.+XR) 134 laA=mns(o) 13Li IF (DA .LT. 1.0E-4) GO TO 16 13u IER = 1 13t WPIIE(O.723) IEn.X 13o 101.1E(6.724) FtYLI ► XRI 139 WPI1E(6.725) FR,PLO(R,XL,OA 1444 GO 10 to 141 14 IFImAS(FR)-AOS(FL))16,16,15 RTM1121 14c 1K X=XL. RTMI122 14o F=Fo RTM112.) 144 FL=40NV*FR 145 FP=,;ONV*FR 14u F=CuNV*F 14, 16 RETURN RTMI124 144 C RTMI125 14'1 C COMPUTATION OF ITERATED X-VALUE BY INVrRSE PARABOLIC INIERPOLATIVNRIM116 15v 17 A=FR-F RTMI127 191 DX=(X-XL)*FL*(1.+F*(A-TOL)/(A*(FR-rt.)))/TOL RTM1126 154. X0.:-.A ; RTMI129 15a FN1- RTMII3d. 154 X=XL-DX RTMI131 15:) loLzX RTMI132 15o F=FuT(T)L)/CONV 15r IFIF)111.16,1a RTMI1.54 15o C RTMI135 15'.; C TESL ON SATISFACTORY ACCURACY IN ITERATION LOOP RTMI166 16v 1n TOL=CPS RTMI137 16s AAuS(X) RTMI138 16s IF(A-1.)20/20t19 RTMI 1V19 16o 1n TOLzTOL*A RTM1140 16'T 20 IF(A9S(DX)-TOL)21,21,22 RTMI141' 16:, 21 IFtm5S(FI-TULF)16,16,22 RTMI144 16u C RTMI143 1O/ C PREHARATION OF NEXT BISLCTION LOOP RTMI144 lOo 22 IFIGNi1.rF)+SIGN(1.,FL)) 24.23.24 RTMI 145 169 23 XR=A RTM1146 17v FP-F RTMI147 1 /LI FR.7&M RTMI1b2 17u GO JO 4 MMUS-5 17' C ENO OF ITERAT/04 LOOP RTMI1h4 17u C RTMI155 17'0 C RTMI156 180 C ERRUR RETURN IN CASE OF WRONG INPUT DATA RTMI1b7 181 25 IER=2 RTM115d 184 FL=LONV*FL 14u FR=CONV.kFR 184 F=CONV*F 18b IER = 2 I8u WPIrE(6 ► 723) IER,X 18"/ WRITE(6,7241 F,XLI,XRI 18u WRIFE(6,725) PRpFL,XR,XL,OA 18i 703 FORMAT(1H ► 4X,I5HTROUSU IN RTMI,3XP4HTER=e16,3X,2HX=FG15.8) 19u 7P4 FORmAT(IH !3)02HF=,G15,d,3Xo8HXLIrYRI=,2G15.8) I91 7P5 FORMAT(1H ► 3)061GFR,FL=r2015.8,3X,614XR,YL.7,2G15.8,3X,3H0A= ► G15.8) 19c RETuRN RTMI1b9 19.) ENO RTMI16O QPRItt) 1.50850

..1(11.4.m.e1-1F=.61b.A.3X.6HYLI.XRI=.2 FO.MAT(111#6X1(14FR.FL=.2G15.3.3x16HXRIXL=.2r,15.b- G 1•2* RErtAiN 13' T.14 gq 0/t" R1MIIA0

END OF LOWILATION: NO nIAGNOSTICS. 4, HOGIP **4*** SUBOI ***4** 4EuR.S 4 SIIF-1 / /10/74 - j6;07:p0 (n t )

SOROHT1NF AAAA ENTRY POINT 00n052

STO..AoE USED: CODE(1) 0i10056; DATA(n1 00001.1 B1ANK COMMON(2) On0000 •

EXTERNA1 REFERENCES (HL - CA. NAME) • 043 NERR31..

STORA0E ASSIGNMENT (HLoCK. TYPE. RFLATIVE LOCATION. NAME) u0D0 R 000000 nOmS0 0010 R OnOnnl CSO 0000 000007 TNJP$ 0000 R 000002 T

uOIOU I* C AAAA (GURRoUTINcl 7/2b168 LAST CARD IN DECK IS NO. u0100 2* C u0 i00 3* C - ---ALISTRACT - --- UulOu 4* C uOiOu 5* C TITLE - AAAA uUlOu 6* C THIS SUnROUTTHE COMPUTEc THE 2-11Y-2 MATRIX A OF COEFFICIENTS u010u 7* C IN THE RESInHAL EQUATIONS - u0i0U S* C. u010u 9* C n(PHII)/DZ = (Ail)*oHII - + (AI2)*PHI2 • U0100 10* . C 00100 II* C D(1 I2)/07. = (A?1)*oHTI + (822)*PHIP oulOu 12* C u010o 13* C OEPTVED OY A. PIERCE, J. COMP. PHYS., VOL. 1, NO. at 343 ► -366 ► U010u •14* C. 1967. (SEE EON. (11".) OF: THE PAPER.) THE EXPLICIT EXPRESSIONS uOlOu .5*, C FUR THE A(I.J) ARE 60inu 16* C U010u 17* C A(1.1) 7.: G*(N/Bom)**2 - GAMmA*3/(2*(-:**2) uui00 18* C 1((1.2) = 1 - (C*KlOoM)**2 u010u 19* c A(2,1) :....((G*K)/(ROM*C))**2 - (HOWC)**2 uulCo .-0* C A(2.2) 7.1.- - A(1,1) U0100 el* C UOIOU ,-2* C WitEPF GAmmA=1.4 IS THE cPFCIFIC HEAT RATIO' 0=-0098 KM/SEC**2 UuiOu r3 4 C TS THF ACCEIERATION OF a.RAVITY , C IS THF SOUND SPEED, K IS THE O010o '44 C HORIZOHIAL wAVE NUMRFR ANn HOM IS THE DOPPLER SHIFTED ANGULAR

..„. . . %JUL I 1%1 00

UUiOU ' Os C 40400 trns C ---,CALLINGG SrOHENcE -..7. vuiOu 01* C bulOu .12+ C SEE SU..ROUTINES ELTNT , MMMm. NaMPDE ► NMDFN vOlOu .N3* • C DImENSTON At2 ► al U0100 04* C. CAvL AAAA(UME0A/AKX ► AKT/CtuX/VY/A) LJUI.OU 05* C uO100 06* C NO ExTRNAT SUIROUTINES ARF REpUTREO VOIOU 07* C uOIOU .N0* C ----ARGUMENT 1 IST---- vOLO0 09* C •vOiOu .0* C ui.GA R*4 ND Trio u0i0u. •1* C AKt li *4 ND INp vt.1100 .2* C AK., R*4 ND INu UU1OU .3* C C K 4 4 N[) UOIOu ..Lot C VX n*4 ND 1111: vU1Ou .5* C . VY ' R*4 UOIOU .6 * C A R *4 2 124- .) jgl; U010u 47* C utii0v ..6* C NO COM-ON sTORAGE TS USED UUiUU .9* C v010v •,0* C ----/NPUTs-- -, u0.Ou hl* C vOiOu •24 C OMEGA .7.Arw;OLAR FAEOUENCY IN RAO/SEC UOIOU h3* C AKv ::X roMPONENT Op HORIZONTAL WAVE NUMnER VECTOR IN 1/14M UULOu h4* C AKY =Y (70MPONENT Op HORIZONTAL WAVE NUMBER VECTOR IN 1/KM UOIOU - n5.4. .O C =SnUND SPEFo Ild KM/SEC LIO•,OU -,6* •C VX =X COMPONENT OF wIND VELOCITY TN KM/SEC uUlOu n7* C VY :fie COMPONENT OF wIND VELOCITY TN KM/SEC 40i00 •,Lis C UOIOU h9* C -_--OUTPUTS- --- vOTOU /.0* C vUlOu ,.1* C ACT....1) =(T,j)-TH 0..Em•NT of MATRIX A GE COEFFICTESTS IN THE uU100 f,2* C Rrr,TDUAL FnUATIONS AS DEFINED IN THE ABSTRACT. 0010u (,3* C u0i0o ,. 11* C ----PROGRAM FnLI.OwS BELOW . uOtOu ' , 5* C U0101 u6* SUo.RUUTINE AAAA(OMEGA ► AKXTAKY/C/VX.VY , A) v0101 . 117* C vu103 nt► s NI•ENSTON A(P1,) v0194 1.9* Ho•,w=tomE6A-AgX*VX -AKT*VY)**2 u010n 1 0* CS4,r-Ctr. OUtni. 114 1=IANX44, 2+AKY*49)/110MSN 00107 ,2* 1.(1 ► 1)=.n098*T-.00606/CSO vu107 /3* C 6AmMA*Ia? T':. *0 0 68A, UOilu it.* 1.(1/2)=1.0-CSOT 601.11 /5* A(211)=((9u.04E-6)*T-BomSO)/cSo vUsll 16* C G**2 I<: 96.04E-6 Km**2/SEC**4 . 40112 17* A(:) ► ?)=..4(1 , 1) uult.s sn* NETum U0i19 /9* F.N. SuUdOuTiNr AKI i, NTRY poIN( 00n134

STORA6E USELA CODE(1) On0153) DATA(n) 00003h; 81ANX CoMmON(2) 00 .0000

EATCRNAE REFERENCES (tiLnCK. NAME)

U003 COS u 0 n4 SIN u 0 (,5 NERR3a,

SIORAGE ASSIGNMENT (HLnCK, TYPE' Rg(:ATIVE . LOCATION, NAME)

0001 000067 20L 0001 Wilt 30L n000 R 000002 AI n000 R 000000 CTRIGT 0000 R 000004 CTRX 0000 R 000001 1)CLAA 00n0 R 000nOn iJELom 0000 000023 TNJP$ 0000 R 000007 STRICT 0000 R 000005 STRX 0000 N OnOoin Si ()Ono R 000n1I S2 0000 R 000003 Y

u0100 1* C MKT (SuHROOTINr) ' 8/15/68 LAST CARD IN DECK IS NO. 00100 2* C 00100 3* C . 0UkOo 4* C ----ABSTRACT_--- uOinu 5* - C u010u 6* C TITLE - AKT u010,) 7* C FvALUATION nF INTEGdAL nF A(OMEGA)*COS(pHI(OREGA)) FROM 0M1 TO 001Uu 8* C 0M2 u0i0U 9* ---- 00100 10* C A(OMEaA) ANn Na(nmFoA) ARE ASSUM•D TO BE LINEAR BETWEEN U0100 i1* L Oml Amn 0012, FOLLnWING THE METHOD OF AKI ( J. GEOPHYS. 00100 12* C r(Es., VOL. 65 (1o60). PP. 729-74n ). THE INTEGRAL IS u010u . 13* C READI; Y EVALUATED AS , uOlOu ,4* C uulOu iti* C (PHT 1 )**(-1) * (AT + A 1 *(0M2-0M1)) * SIN(PHII+X) u0i0o (•). 4, C u0i0u 17*. C + PHI'**(-2) * Al * COS(PHII + X) uU,00 t8* C 0()100 19* C : - pH1 , **(-1) * (AI - A' * (0m2 - 0M1)) * SIN(PHII-X) uOinu .0* C .- 001.0U el* C - PHI'**(-2) * Ai * COS(PHT - X) UO1OU '2* C U010U r.3* C WHERE 00100 '4* C 00.100 .5* c A T = AvEdAGE VALUE OF A IN INTERVAL u0i00 .6* C WATT : AvFRAr4E VALUE OF PHI TN INTERVAL ujiOu .7* C A i = D(A) / n(nMEGA) 00100 .8* C PUT' = D(PHI) / D(OMEGA) UOLOU e9* C y = PHI' * (nM2 - OM1) / 2

- • , UUlOu :14* C AKITHT = 2 * PHT'**(-1) * AI * SIN(X) * COS(PHII) UU100 :6* C u0i0u .16* C + 2 * pHI/**(-2) * Al * (X * COS(X) — SIN(X)) - uUiOU .17* C u0100 :X.* C * SIN(PHII) . U010U .19* C UU10j 40* C WHENEuFR X IS SMALL. SIN(X)/X AND COS(X) ARE EVALUATED BY U0100 »I* C USING THEIR PoWFR SFRIES REPRESENTATIONS. u0100 42* C u010U 43* C LANG11.11•r **FORTRAN IV (360, REFERENCE MANUAL C28-6515...4) uO10u .s4* C • u0100 45* C AUTHORS ". A.D.PTERCE ANn J.POSEY, M.I.T., AtiGUST,1968 uUIOu 46* C UUl00 47t C . u0i0u 48* C •----USAGF---.— UOLOU »9* C uOIOU nO* C ' NO SUBROHTINES ARE CALLED uUJO6 •,li. C uUlt1J :.2* C 1:ORTRAm USAG). u(400 :,3), C u010u n4* C CAIL AKI((.41.0412.A1 , A2.CTRTGI#STRIGIICTRIG2 , STR/G2t uUlOu n5* C 1 .ELPHYAKIINT) uU100 :16* C 0010u :,7* C INPUTS uOIOU •,8* C UolOu n9* C OM1 LWITR LIMIT OF INTEGRATION OVER ANGULAR FREQUENCY uUlt.Jo .0* C R44 (.010IvIS) u0i0u 1.1.- C . utilOU .2* C ' 0442 UPPER LIMIT OF INTEGRATION (RADIANS) UO.tOu .3* C R44 UOIOU .4* C r LWOW uJi C Al VALUE OF A AT OMEGA = OM). • U0i00 .6* C R44 oui00 .7* C UO100 ,•84: •C A2 VALUE OF A AT OMEGA = 0M2. uUtOU .9*- C R•El . uOlOu i t)* C uOtOu /1* C CTRTGI COS(P61I) wHERif OMEGA = OMI. uU1OU (2t C R44 u0i0U /3* C UO;Ou /4* C S1R1G1 SIN(PHI) IvHERF OMFGA = OMI 1.1010u /5* C R.4 UO10u /6* C uOlOu 0* C DE•PH CF1ANGr IN PHI OVER THE INTERVAL ( PHI(0M2) - PHI(OM1) ) u0i0i) /8* C R44 (it6DI4NS) uUlOu ,9* C 0010u s,0* C vUTPUTs 9U1.00 ..l*$.1* C u010J .%2*. C CTRIG2 CuS(PHI) WHERF OMFGA = 0M2 L.0i00 .3* C R44 • UULOU ..iii . C QUIOU r151, C 5111rG2 SIN(PHI) WHLRF OMPGA-= 012 UUiOU A6*. C R-44 ----PROGRAM FOLLOWS FIELOW----. u010u 43* u0i0u 44* 00101 . Su -.ROUTINE AKI(00 , 0m2,A1,A2 ► CTRIGI , STRIGI.CTRIG2 , U0101 46* 1 SIRIG2rhaN'MKIINT) 00103 47A flE10m=nti.2 - oM1 - 0040 00)04 48* OEI 11,4=A2-A1 u0204 49 4, C 00105 1,04 AI-:(A2+Al) /2.0 0090 00106 1,1* x=.1ELPH/2. U U0107 1,12* CTi,X=CnS(X uOilo 1,3* ST,X=sm(x 00111 1,4* CT..IoI=cTR 161*rTRX-STR TR1*cTRX 00112 1,5* SrwIGI=STR IG1*(:T(2XFCTR 1(11*ciPX 60213 1,6* IC, I*rT9X-STR 1Ri*cTRX UU114 1,7* ST.JIo2=STP. 151*rTRX -I. CIT IGI*CTRX u0115 1,3* 1.0E-2) 20,2 nulo 00120 11.94 SI=STRx/X u0i21 110* ,-,2=CS1-CTR XWA*2 u0122 111* (;0 70 30 u0,2S 112* 2n s1=1.0-(t. oi6.n)*X**2+(1.0/120.0)*X**4 00124 113* S2zil.n/3. 0)-(1.0/30.0)4x* ► 2+(1.0/B40.0)*X**4 0180. 00.25 1144 3n 11KI/N7=CAI *51*CTRI6I-(7FLAA*DELPH*0.25*S2*STRIGI)*UELOM u0i26 115* UE uU)1N 00127 116* •

END OF CON,PILATTON: N O n I AONOS T ICS. k,HoGrP A***** sUii06 *** A ** 4,YDR,S 1.SL,8O3rSuBo3 FOx S1)F- 1 ,110/74 - 18:07:30 (6r)

SD&ROuTiNE ALL 0D ENTRY POINt 000310

STORAoE USED: CODE ( 1) On036u; DATA(n) 000125$ RiANK COMMON(2) 0n0624

EAT E RNA L REFER E N C ES (11L,ICK.

00 113 NYMoDF 00114 MOULTR 0005 WDDS, 00u6 N102$ 1) 0 117 NERk3i

STokAGE AcSIGNMFNT (HLoCK, TYPE, RyLATIVE LOCATION. NAME) u0t)1 On0u73 ilr unn0 OnOn4p nonl 000123 151.; nOol 000023 2L 0000 '000057 21F U 0 11 1 000025 3L nOn1 000163 301. nOol 000203 35L 0001 000243 4111. 0001 000250 43L 0000 I 000006 JENO uuliu 1 UOU014 J4 noon 1000003 J50 0000 I 000011 J52 WWUYi KO( 000u I 000010 hRUD 0000 1000u00 NMODE 0002 000621 OMEGAC 0002 R 000623 THETKP 0002 000u2Ae VPHSFC 0002 000)45 0(I 0002 000311 uYI

00100 1* C ALsmOD (SUaROUTTNE) 6/25/68 LAST CARD IN DECK IS NO. .g u010u 2* C 00100 3* C uOtOu ► * C TITLE ALIMOD 00400 5* C . PROGRAM TO TABULATE niSpLRSrom CURVES OF UP TO MAxMOD GUInED 1O100 6* C MODE S. ONLY PORTIONS OP CURVES Wm' OMEGA DETWEEN OM(1) AND bOIOU 7* C 0m(mCoL) Aft WITH PHASE VELOCITY BETWEEN VP(mCOL) AND VP(1) 60100 8* C ARE TAhoLATrD, THE ANOHLAR DEVIATION OF GROUP VELOCITY DIREC- uU100 9* C TACK 1-"Pom PHASE vanciTy nIREcTioN plETK IS NEGLECTED. o0100 10* C SUCcIsSIVE monES NU•BFRED FROM 1 To MnFmD ARE EACH TABULATED BY uUlOu Al* C CALI Imo SUBROUTINE moDErR. STARTING POINTS FOR EACH MODE ARE uOIOU $2* C FoUmo mY CAI LINT, SUHROHTINE NXMonE. THE NORMAL MODE DISPERSION u0100 ,3t C fuNcITON (NHnEY sHomn NE NEARLY ZERO FnR EVERY TABULATED POINT u0100 ,4* C ON FACH PISuERSION cURVE. THE COMPUTATTONAL METHOD IS BASED u0100 7 15* C O; TIIF PREVTIOSLY COMPUTE() VALUES OP THE •MDE SIGN u010u I 6* C INmoJF((J-1)*NRow+I) AT POINTS (I ,J) IN A RECTANGULAR ARRAY OF 0010U A7* C ' NkOw po,vS Amn MCCL COLUMNS• DIFFERENT COLUMNS (J) CORRESPONO 00100 ,8* C To n11.4- OrMy AN3ULA4 FREONENCIEs OW(J) wHILE DIFFERENT ROwS (I) 00100 19* C " CORRLSPoNn To DIFFERENT PHASE VELOCITIES VP(I). IT IS ASSUMED uULOU *0* C THAT vp(t) .GT. VP(2) .GT. vE(2). ETC. DISPERSION CURVES • uOIOU el* C 01, vAwIoUS mODES APPEAR ON THIS ARRAY Ac LINES OF DEMARCATION uOtuo ..2* C FO..TwEEN ADJACENT RE6ION.; wITH OPPOSITE TNMODES• IT IS ASSUMED LUi00 ,3* C THAT. HISPERcION CURvES sLoPE DO4NwARDS. MODES ARE NUMBERED u0i0u '4* C STARTIWERoM LOWER LEFT OF INMonE ARRAY. 00100 e5* C uU101) .6* C PI-1E6RA.* NOTES UG*00 e7* C , UOIOU '13* C . THE ARRAYS OMmOn AND OMOD ARE NSF() TO STORE DISPERSION L0100 i-.9* C CoRVEc FOR ALI THA-- MORES TO CONSERVE STORAGE. FOR THE o0)..00 00* C . MmOmE-TH ROBE. VP%•0D(KST(NMODE)+K-1) IS THE .PHASE VELOCITY 00i0o 01* C CoRpEqroNuIMG TO ANGULAR hRFAUENCy OF OMMOD(KST(NMODE)+ UOitJu :12* C K-I). THAi. PAIR OP VALUES CORRESPONDS TO THE K-TH TABULATED 11O100 .13 -4 C 'POINT roR THE MODE. THE LAST TABULATED POINT FOR THE 00100 ,,4* c NmORE-TH wORE IS AARELED BY THE PAIR VPMOD(KEIN(NMODE)), uU100 05* C ' ()AMOD(KFIH(NmonE)).. THUS OMMOD(K) , VPMOD(K) FOR ulliOu 06* C K .GE. KST(NVOTIE) AND K •...T. KFIN(NMODE) DESCRIBE THE 0010u ..,7* C •ionE-TH •OnE-S DTSPERSION CURVE. 00100 .18* C VOiOU .19* C THE FI AG KWOP IS NORMALLY RETURNEn AS 1. HOWEVER, IF t$UtOu .0* C No nIc.RERSION CURvES ARE TABULATEn ► KWOP IS RETURNED AS union .1* C 1. uUIOU .2* C 1)010o .3* C LAhiGUAAF - FORTRAN IV (3r,n, REFERENCE MANUAL C22-6515-4) UUIOU ..4* C AUTHOR - A.D.PTERCE# M.I.T., JuNE01968 u0i00 .5* C U0100. ..6* C ----CALLING SpOuENCE ---- u0a0u .7* C ,.h.r..z.,ow Li‘inuirvA11100)•VYI(100) ► H1(100) OU'Ou N24- C TH- SU-RUM- IN*: USE< VARIADIF DiMENSIONING. THE ASSIGNMENTS ABOVE APE u0IOU n3 .4 C 1H0S NIVCN 0 PAIN PROGRAm

U0i0u :,4* C CO — Mutt ImAAPCI,VXI/VVI/HI uUlOw !)5*, C ATmOSPNERIC VAI:IAPIF MUST RE IN COMMON BEFORE ALLMOD IS CALLED. uUtOu :,6* C CAIL AlLmON(HRnYINCOLtmAXMnDeMPFND.OM.VP,KgT/KFINIOMMOD.VPM00. 00100 ni* C I I.iMoDFITHKTKeit , OP _ uU1Ou :,8* C IF(KINOP .NE. 11 GO SOM1.:WHEpE uUiOu n9* C UOLOU 00* C ----EXTERNAL cURPOUTINES REOUTREU---- uu100 01* C U010u N2* C NX.A00E,MoDETRINyTFINTIRTMI,FNmOD1PFNMOD2tNMnFN0AAAArRRRR,MMMMrCAT*SAI uOlOu ',3* C u0100 ,4* C WYpILME AND rq)DLTR ARE PXPLICITLY CALLER. THE REST ARE • UuL0u ,5* C. IMPLICITLY CALLED RY CALLING MODETR. FOR FURTHER INFORMATION 0i).1.0o .,6* C ON 1RM SCIrNTIFIC cURRoUTINE PACKAGE ROUTINE RTMI ► SEE DOCU.. LO1C0 ',7* C MENTAT1ON oF MODETw. LJO1OU 00* C 00100 r,9* C - -ApOuMLNT LIST_--- uuiOu ,0* C uOLOu /1* C 1+4 INp u0Clo /2* C = I*4 1, 010u /31. C mAimup I*4 HD UULUU 14t C MOAN[ 144 140 E 00:00 ,5* C oM R+4 VAR INp u010u io* C VP VAR IN0 OULOu / 7t C KS, T:4- VAR OUT oui00 .6* C KFTN 1+4 VAR OUT belot, 19f c oM.ON R*4 VAR OUT 1)0100. e,(J* C vPNOo R*4: VAR OUT u0i0u i,1* C IN., ouE I*4 VAR INp u0i0u r,2* C IHITK R*4 ND !Pp UOIOU (,3 ,. C 1010P 14, 4 ND OUT

u010u /-,4* C u0i0U t054 C COMMON S10dAGE USE() uGlOu ,6* C CO—MON ImAx ► CI.VXI.VYI ► HI ► nMEGAC,VRHSEC ► THi:TKP •. uU40u ..,7* C 00100 ,,ti* _C 1MAX /*4 NU INp tiOlOu ,.9* C Cl R*4 INp 00100 40* C vX1 R*4 100 INp u0100 41* C vY, 100 iNp ■,0100 42 C HI 1-1;4:i ino IMp uOlOu 43* C 014.GAC R*4 ND OUT (USED INTERNAL, Y) LOIDu 444 C VP..SEC R4, 4 ND OUT (USED INTERNALI Y) LiOlOu ,-,5* C 1H.JisP R*4 ND OUT (USED INTERNALtY) iJUIOu 46* C UO“W 47* C - ---INPUTs-- - - uOlOu 48+ C wilflu ,94, C NRuld zMur:nEt< OF RoWq IN INMODE ARRAY. MAXIMUM INDEX OF u010u 1,0* C Vp(N). 60100- 1"1* C NCf,L =NHrRER OF COLHMNS IN INMODE ARRAY. MAXIMUM INDEX OF 1,2* oucm). u01011 C LUIOu 1"3* C mAmoD =MAXIMUM NHMREw OF MODES TO nE TABULATED u010u: 1,4* C °M(N) =AKIGULAR FREOUNcY OF N—TH COLUMN IN INMODE ARRAY 1..e. t0•7•11rMr204/+ .1-TH ELEMENT CoRRESPONDS TO NMDF WHEN uUlOU 1.9* C OmFGA=OP(d). PHASE VELOCIrYzVP(I). uOtOu 110p C UNTK , =PHSE. VELmCITY nIRECTION IN RADIANS RECKONED COUNTER - U0100 111* - - C CtoCKAISE WITu RESPECT TO X AXIS. 00100 1;2*C IMAX . =Hur ,REtt OF ATVoSPHERIC LAYERS oF FINITE THICKNESS U 0 100 1 1 3* -- C - CI SnHttD SPEFn Iii T-TH LAYER UOIOU 114* C VXiti) - =X COMPONENT OP WIND VELOCITY TN I-TH LAYER UOIOU 1 15* ' C VY1(1) • =Y COMPONENT OF wIND VELOCITY TN I-TH LAYER uolou 116* C HItI) • =TOCKNESS OF I-TH LAYER uOIOU 1i7* C UOIOU 1;0* -. C ----OUTPUTS-- -- UOIOU 1,9* C UU100 1,0* C mbom IrIhrolnER OF MODES F0UND C, 0100 1,1* C ' KSr(N) =INICEX OF EIPST TANULATED POINT IN N-TH MODE 00100 1,2* C: KFEN1N) :INnEX OF IASI. TAPuLATFO POINT IN N-TH MODE. IN uOiOu 1,3* C GF!•RALp KFIN(N)=KST(m 4- 1)-1. • UOIOu 1,4k C OM.ON(N) • r...AQPAY STORING ANGULAR FREQUENrY ORDINATE OF POINTS UOIOU 1,5* C Om DISPEINTON CHPvES. TH NMnDE MODE IS STORED FOR UUIOU 1,6k C N r'ETAEL'N KsT(NmonE) AND KFIN1HMODE). 00100 1,7* C• VPA401;(14) =ApI;AY STORI•n PHASE VELOCITY ORDINATE OF POINTS ON u01011 1,8* C - DTSPENSIOw CUoVES. THE N•ODE-TH MODE IS STORED FOR U0100 1,9* C N rrtAEEN KST;NNIO0E) AND KEIN(NMODF). UUiOU 100* C KW0P r.7i IF NO mOOrq ARE TABULATED. OTHERWISE IT IS I. 00100 1.,1* C. OMFGAC =ImTERHALLY NSF() FREQUENCY TRAmSMITTED AMONG SUBROUTINES uOIOU 1,2* C Tt_IPOUL3H CoHMOhl OOIOU 1..,3* C' VP4EC r.ImTER.4ALLy USED PHASE VELOCITY TRANSMITTED AMONG u0tOu 1.,4* C SmPROU1INFS THROUGH COMMON LIOIOO 1.,5* C 7 'THFIKP =SeT. AS THETK 00100 1.,6* C uOiOu.7* C • - ---EXAMPLE- --- 0)106 108* C U0100 109* C SUPPOSE THc TA;ALE OF ItJMODF VALUES IS AS SHOWN nuow WITH U0100 1.0* C ' uOIOU 141* +++--.1.4-•* NROW=6: mCOL=10 uUlOu 142* (; +++...-++,-.. .1..,+4.:._--** u0i0U 143* ' C IF mAxMOD=lo ► YOU SHOULD FIND MDEND=6. uUIOu 144* C • 5---+++--- uOIOU 145* C . 5++-+4++++ KST(1)=1 KFIN(1)=4 OMMOD(1*36) SHOULD BE 60100 1..6* C 5++...*++.14.1. KST(9)=5 KFIN(2)=10 VPMOD(1-36) TABULATED oU/OU 1..7* ' C KST(3)=11 KFIN(3)=21 u010.0 1..8* C KST(4).7.22 KFIN(4)=29 00100 1.9* C . ,ST(S)=30 KFIN(5)=34 u010u ►1•,0* C KST(h)=3q KFIN(6)=66 uOlOu 1•,I* C .,U100 12* C 6U1Ou 1.3* C ----PROGRAM FULIOwS BELOW---- '040u 1:,4* C uUlOu I5* C 0010) 1..t..* SIA.NuUTINE ALLmoO(NROW,NCO; tmAxmOD.MDFNOrOmtVP.KST , KFINtOMMOD, 001011:17* 1 VpmuDi(i.tNiuDEtTHq'TK PKWP)o 1.0101 Loll* C (4010 3 1:i9* 01,EHTum CI(Inn).'fs'I(100).VYT(100) , HI(100) U01.04 1.0* 0ImENSTON wM(1),VP(1) ► AST(1).KFIN(1)e0MmOD(1) , VPM00(1) , INMODE(1) 0010!)' 1.1* Co.,MoN ImAX•CI.VXI.VYI.HI•nMFGAC/VPHSEC.THFTKP . - u010u 1.6+ C AT ThIc.- POTNT, WE wAVEN-T FOUNn ANY monEs uU 107 1.7 * ND.4,1)=0 . u0107 1,8* C • U0107 1.9* C '.F. SIA,T SFARCH FOR rIlisT MODE IN LOWER LEFT CORNER OF INMODE ARRAY. u0107 1/0* C wE SEE ., A ;JOIN) WITH INMODv .NF. 5 WHERE THE NMUF EXISTS. u0110 1/1* NM,nt'=1 u0111 1,2* KSr(NmoDF)=1 00ilk 1,3* 15r=wRow (.:011:-.! 1,4*. C 00112 1,5* C 1H• sE,,PCH GO[ S TO THE RIGHT. IF WE 00N-T•F/No A POINT IN THE DOTTOM uUil?. 1,6* C kOw. w.: IRV NIL (NidnW - 1)-To ROW' ETC. AT:STATEMENT 2 WE ARE STARTING 00112 1,7" C AT THE LE, ET or A GIVEN ROW. u0113 1,8* 2 JSv=1 00114 1/9* 3 J51-,=(Jci- -1)*NRwq+IST u0115 1,0* 10.-...INM(OF(..J50) 0011E, 1,1+ IFtt(1 .1,F. 5) n0 TO 10 uOilt, 1,2* C u0Llo 1,3* C IF JST IS NOT NCOL WE G0 To THE RIGHT. 00,20 1 „4 * IFiJST .1.0. NCnL) u0 To 5 00128 •105* JST=OST*1 U0123 ' 1,6+ (.0 TO 3 u0i26 1.7* C U0123 1,8* C AT THIs POTNT wE HAVE EXHAHSTEn AN ENTIRE ROW. WE GO TO THE NEXT u0123 ,,,)* C HInHFR R()1'1 PROVInEn 1ST .NF. 1. IF 1ST IS 1, THE ENTIRE SET OF 1,J123 1,0t C 1Nm0NEs Arir•s. uuk24 1.,1* IFfI!-;T .F.O. 1) SO TO 7 00120 1 ,,2i ISi=1ST-1 00121 1.13t GO To 2 u0127 14i+4, OUI3u 1 ,,5* 7 wRiTi... (6,e) Uj132 • 1,6* A i- o,MAT0H0 , 51HTNE NORMAL MnOF DISPERSION FUNCTION DOES NOT EXIST 1 u o 132 147* 1 2...H.:OR ANY POTNT iN THE APRAY / 1H ,22NAILMOD RETURNS KWOP: - I) u0 i33 1‘/T3* LI KW,43 =-1 uOis 1,9* kEllJN v0134 2.0* C

. 00134 4.1* C STATFM.NT 10 IS START OF LoOP. EACH PASSAGE THROUGH LOOP CORRESPONDS uji34 2,,2* C TO A GIVi-.N uoDE. u0135 2,,3* .„ In CA1L NV,IONOTSTrUST , NCOLINPOW/INMOnE,IFNO , JFNU/KEX) 00135 2.4* . uui37 2"5o C IF MU CANNOT FIND THE FIRST MnDF YOU ARE IN TROUBLE uu13o 2,,6* IFIIT,•ont_ .,E. 1) GU TO 15 00140 2;,7* IF(KEX .1-, 0. 11 TO 1 u0i42 2,8* wRTTt.. (,111) LA)1 114 2„9* 11 F0•.MATt1u0,3(01N•MOuE COULD NOT FIND THE FIRST . MODE/ 1H t uU144 210* 1i)2viALL.Wn RETURNS SWOP=- r) 00145 211* co To c 1 u0145 2,2* C 0014o 213* C 1F THE HOOF SCJONT TS 140T THE. FIRST AND YOU CANNOT FIND IT. THEN THE 00.45 2L4* C kETURN IS c(„NSIDERrn SATISFACT,Ry. 001q6 215* 1S IF‘ke.X .FO. -11 GO TO Fin 60L00 • 2,6* C u0L iih 2 ■ 7* C wE Now TAONLATC. THr NMuDE - 114 MnDF L,0150 2,8* CALL MODETRCIFNO,JFNDoNmOOriKSTO

-"," ,r1,4i li 1111.1, vu151 2,3* iF(KRUn .E0. 11 GO To 30 LJUiS3 2,4* INRITF (6.21) NmOOE.IFNn.JFND (J0I6u 2,5* 21 H:).;MAT(Iii0/23H14011E01 RwTURHS KRUD=-1.,2)(.25HCURRENT VALUE OF NMODE U0160 2,6* . 1 Do 14, 3H. . 5HIFNO=. 14t3H, P 51-(JENn=, 14/ IH ,27HSEE OOCUME 00160 2,7* ' PNTATION OF ALLHOD) u0161) 2/8* - C 6016o _2 e 9* C wE KFEa Nmniw THE CAME AND TRANSFER CONTROL TO STATEMENT 35 0J161 200 , . 60 TO .35 (.,U16 "2.11* 3n 1,10..NH=mnFNo 4- 1 u0162 202* C id0162 2.-t3t C 1HTS I4 1HF CURRENT NUMER OF HOOES FOUND. 0ni64 204* C wE NOW CHEck IF THTS IS.MAxmOD. IF IT IS/ THE RETURN IS WITH KWOR=1. u0163 2.'1 5* 1FW0FNU .t:(a. mAXMOD) SO To 50 u0165 2064 ...... Nm.DE=wMODOI U0166 2,7* Ic5i(NWILF).7.KFIN(NMODE-1)+1 00160 208* C 00160 2.,9* C.)NE 51-Ew NEW IST ANn JST LIFFORE CALLING NXNODE. 00167 2..0* 3s J5.., =(JP(111-1)•NPOW+1FND U0170 241* 10:LINtinDk(J5P) L,0171 242* IF(I).:(Jn .Eu. 1) GO TO an 00171 2..3t C twin 2,-4* C ipE CHEi.K INMoDE OF PoI•T AUOVE u0173 2 4 5* 03::(JFN:1-1)*NRnW+11.-ND-1 UUITti ...6* 1UP=INAWJE(J3) UUi74 2..7* C L40174 248* C IF THIS IS -TO. THE POINT AriOVr IS THE-ONE DESTRED u017s 249* 1F(I0P .NE. -In) GO TO 40 u0171 2:,,0* IS-r:IFN5-1 u0e00 2 0 1* JSI=JFuli uU e01 2n2* (,0 To 10 U0'.01 2-,3* - C u01(1) 2.,4* C 6E CHErv. INMoDE OF PoIN T To RIrHT.G THERE IS No PLACE TO GO IF JFND= OUI-.01 2!,5* C NCoL. 7HIc TS INTrPPRETED AS 9UCCESS PROVIDTNG MUFND .NF. O. 00e02 2!,6* 4n 1F(J , Ho .NE:. N„OL) GO TO 4 -1‘ u0a4 2:.7* (30 To sO U0.,:04 2:,8* C uu,04 2!,c)* C 1RT IS INvq)DF oE PnINT TO RIGHT tioe:0 2.0* 43 o4=toFH.)*NROW+IFW uUen6 2•1* i12r=1M(..,unE(J4) o0e111 2,2* IFtIliT .NL. -In) 0U• TO 50 u0e.11 2A* iSr=1Fm, uUe12 2,4+ oSI=JFND+1 u0e16 21,5* 60 TO 10 u0,1.) 2.6* C IJO,:l6 2.7* C 114 SE.WLH HAS TERMINATED. IF MnFuD=0, WE. HAVF BEEN UNSUCCESSFUL. uuel4 2"o+ 5n IF( ,41.FNO .10. n) GO TO 9 cU/10- 2.9* KW.P=1 uUe17 2,0* kETilliN UUe2u 2,1* EN•) END or cOmPILOTow: NO nIAGNOSTICS. ti.HDGDP **44** SUft04 ****** SUBROuTINE AMUNT ENTRY POINT 00n107

S1ORA6E USED: CODE(1) On0126: DATA(n) 000013 RANK COMMON(2) 000621 ,

EXTERNAL REFERENCES (BLOCK. NAE)

0003 EXP 0 0 04 NERR35,

S1OkA6E ASSIGNMENT (BLoCK, TYPE. RELATIVE LOCATION, NAME) 6001 000.010 10L 0001 00002•20L n001 000057 30L 6001 000072 40L 0002 R 000001 CI 0000 R 00000u ENPON 0002 R 000t/F,5 HI ' 0002 1000000 TMAX 0000 000005 INJPS 0002 0001 415 VXI 002 000511 01 0000 R 000001 ZT

u0100 1* C AM-1111 (SUBROUTTNE) 7/27/68 LAST CARD IN DECK IS NO. (JOIN' 2* c 00i00 3* C -_...- AJSTRACT - --- u010u 4* C u010u 5* C TITLE - ANRNT uu106 6* . C "relic CUHROUTINE COMHUTEc THE AMBIENT PRESSURE IN DYNES/CM**2 L0100 7* C AT A GIVEN atTITUDE 7 KM.BY USE OF THE EQUATION 00100 8* C uu1OU 9* C PPESUR = (1.E6)*ExP(-TNTEGRAL.FROM 0 TO 2 OF GAMMA*G/C**2) L0100 10v C L0100 11* C WERE 1.EG DYNEs/CM+1, 2 TS THE AMBIENT PRESSURE AT THE GROUND. 00100 12* C GA•14A=1.4 Ic THE SPECIFIC HEAT RATIO FOR AIR? G=v 0098 KM/SFC**2 U0100 13* • C IS THE ACCEIFRAlION OF GRAVITY , ANU C Ic THE ALTITUDE. DEPENDENT: 00100 • 14* C SOUND Sr, ED IN 101/SE.C. THE ABOVE EGUATTON FOLLOWS FROM THE' U0100 15* C HYDROCTATIC •OUATION D(R0)/D2 = -G*RHOO ANU THE IDEAL GAS LAW UOLOO i6* C C**2 = 6AmMA*PO/RHOn. UUI00 4 7v C uU100 18* C THE SOUND SPEED PROFILE IS THAT OF A MUITILAYER ATMOSPHERE AND u0100 19* C • IS PRESumED TO bE STORED IN COMMON BEFORE EXECUTION. THE 0j100 e0* C PROGRAM ALSm RETURNS THE INDEX I OF THE LAYER IN WHICH Z LIES. L,OlOU el* C 00100 e2* C PRnGkAm NOTES UOIOU ,3t C. 66100 ,4v C IN THE EVENT THAT THE INPUT VALUE OF Z SHOULD DE NEGATIVE. u0100 ,5* C •THE FIRST LAYF'R Ic ASSUMED TO HOLD FOR Z .LT. 0 wITH THE uO100 .6* C AMBTEmT PRESSURE cTILL EQUAL TO 1.E6 AT 2=0. THE PROGRAM UulOo e7* . C RETURmS PRESUR .GT. 1,E6 AND 1=1• 00106 eB* C 00100 e9* • C LANGUAf.F. • - FORTRAN IV (3G0. REFERENCE MANUAL C22-6515-4) u010u .W* C AUTHOR - A.D.PTERCLP M.T.T. , JuLY11968 u0i0u .11* C uOiOu s',' t C -L--CALLING SEQuENCE------.„.... Imi , A , L14vxifvYIFHI (THESE MUST BP STORED IN COMMON) oulau :17* C CA1L AsciNT(Z ► PFTSUr( ► I) (JUIN, ,8* C uulOU o g* C --7-ExTERNAL cUHROUTINES RE0UTRED-7- u0100 .0* C UU.tOu .1* C 140 ExTFkNAL SURrOUTINES ARp REwUIRED. uU10.0 .2* C 10100 •3* C ----ARGUMENT IIST--- - u010u .4* C 11101 .5* C / R*4 NU IIglu utiititi .6* C 1,R;Su R R *4 ND OUT 10101 .7* C 1 I*4 ND OUT uOtOu .8* C uUiOu ,.9* -C LOMMON STORAGE USErl

uolpu •,o* C CO,IMON 1mAx ► CI.VXI ► VYI•HI u010u nl* C U0rUu n2c C 1MhX I*4 . ND 'Up U0100 :13* C CI 1 0 0 INP UOIOU n4 * . C. VXT rri: Lt+i 100 'No (NOT USED BY THIS SUBROUTINE) u010u !,5* C VYr R*4 100 7Np (NOT USED RY THIS SUBROUTINE) . uU10U:,6* C HI R*4 100 INo 00100 ,,7* C OUtOu:18* C ----INPUTc,---- LWOW :)(1t. C UO1U0 1.0 ► C / =HrT6Hr IN KM utitou .1 * C ImI,X .7.Hurnr:It OF ATMUSPHFRIC LAYER; wITH FINITE THICKNESS U0100 ,,2* C cIII) :snlimo Spritn (um/SEC) IN I-TH 1AYER uUtOu.3 ► C yX1(1) =X coMRONEwT Or: wIND VELOCITY (KM/SEC) IN I-TH LAYER u0100 0,4* C vY1(1) =Y COMPONENT OP WIND VELOCITY (KM/SEC) IN I-TH LAYER uUlOu f.5* • C HI(I) =THICKNESS IN xM OF I - TH LAYER 10100 f,(i* C uc,lou „7* C ----OUTPUTS---- uUlOu ..rit C uOlOu .9* . C PRI;cUR 7.-AmnIENT- PRESSURE IN DYNES/CM**2 AT ALTITUDE Z - U0100 f0* C I =114r1FX OF lAYFt.? IN WHICH Z LIES u010u fl* C UOiOu ,24 C ----PROGRbM FnLLOwS BELOw---- uulOu /3* C 1)0101 dii, - SU.R.NTINE AmPmT(7•PRE!,URpl) 60101 ,5* C U0101 06* C DIMENSION ANTI LOMMIV1 SIATEIAENTc uU10.5 ,7* 01-ENSICN CI(Inn)•vXI(in0),VYI(100) ► HI(100) 10104 •8* co.AmN ImAX ► CI.VYI ► VYI0HI UOIO4 04 ‘ 2 OF . 10104 /X* C THE iimAL vAiNE OF FNPUN W1LL. ►4E THE INTEGRAL FROM 0 TO OW04 ..1* C -GAMA.G/C4 .62. THr RUN(JINo VAtUF WILL BE THE cUBT0YAL. UU101.4 f.2+ Ft4wON=n.0 10105 m3* C 10105,t4* C THE U.441NGU VALUE nE I WILL RE THE LAYER BEING CONSIDERED u0106 ^54 1=1 0010h ►,6* C. Z i It..S FN LAYER 1 ,P IMAX=0. 10107 ,.74. /Tz0.0 UOilli mlis IFIIMAX. sC ► .• 0) GO TO 30 U011U . 4194

--.- ...... 6 tiq- num) nE A LoOp, THE cURRFNT ZT DENOTES THE TOP OF THE I-TH LAYER. 0011:5 44* in IFt 1 .GT. ZT ) GO To 20 00113 95* ...... C UU1 16 ' 46* C 2 lIt..S IN T-TH LAY 00i16 ,r7* C ZT-H1(11 Is HEIGHT OF aOTToM Or I-TH LAYER UO114 48* C 2 - 7T+HT(1) IS uISTANCE OF 7 ABOVF HOTTOM OF I -TH LAYER 00115 -,9* F.11 ,20N=FNPOi4 - 1.ust.U09S/CI(T)**2)*(2 -2T+HI(I)) 001161,,0* 12 60 To 40 001161„1*6 C -- 60116 102* ---'t z 1 IES Ah0a TuP OF /-7H LAYER 60117 1.3* 20 FN.4N=F1iP014 - 1•u*(.0098/C1(T)**2)*H/(I) 00117 I,,4* C THE CU ,-RENT FN-ON TS THE INITFGpAL OF -1.4*Vc**2 UP. TO THE TOP u0117 1„5* C oF THF I-TH 1 AYER 1.1012u 1,6* 1=r+1 t,u121 1,7* 1Fri .FT. 1NAX1 GO TO 3n Uu1 23 1.8* . ZT=Zi+HI(I) 00123 109* C ZT IS THL TnP uF T-E NEW I-TH !AYER - 00124 110* 60 TO 10 0U124 111+ C iNn oF LOOP uUi24 1 1 2* C U0124 1,3* C i 'IFS IN HPPER 1IA1FSPACE 00125 '114* 3n FN.JON=PNPON-1.u*(.0098/CI(T)**2)*(7. -27) U6.L?1, 1.15* C 00125 116* C CONTINilING fWOM 12 OR 30 ()pia) 117* . 4n pRwS11R=1.E0*EXOENPON) 00127 1 1 8* kETHkN uu13U 119* . Ebill

END OF COHPILATTON:. NO nTAGNOST1CS. 6i1u 6,1) 4*4.*** SUH05 ******

6FuR,S 1.S,005 ► 5uuo5 . f-Ori slIF-1„, /10/74 - 18:07:56 (0,)

SuPROuTINE ATHOS ENTRY POINT 000172

STORAGE USEu: CODL(1) On02201 DATA(n) 00004s1 SIANK COMMON(2) On0621

EXTERNAL. PEI-ERENCE (HLhck, NAVE) 0003 COS u004 SIN 0005 SORT U006 NERii31,

STOkA6E ASSIGNMENT (H1.,1CK. TYPE' RELATIVE LOCATION. NAME) 0001 000046 1150 • oOnl 000105 1256 0001 000147 141G 0001 000067 20L 0002 R 000001 Cl 0000 H 000UO3 finn0 R 000(104 U2 0000 R 000001 n3 0002 R 000455 HI 0000 1 000002 I u0o2 I 000000 IMAX 0000 000n13 INJP% 0000 I 00000n JET 0002 R 000145 VXI 0002 R 000311 WO uuiuu 4* C ---7ABSTRACT--- U010u 5*J C 0010u 6* C OUlOu 7* C TITLE - ATmOS U0i00 8* C TaluLATION nr WIND vELOCITY COMPONENTS AND SPEED OF SOUND FOR U0100 9* C ALL LAYERS nr, MODE). ATMHSPHERES uulOu i0* C u0101.1 11* C • THE MODEL ATMDSPHFRE CONSISTS OF HP TO 100 ISOTHERMAL u010u 12* .0 .LAYER (THE TOP LAYER BEING INFINITE). EACH LAYER MAY .u010u 13* C HAVE a UNIQUE Tr4pERATURE, THICKNESS AND WIND VELOCITY. U(L 14* C SHOR011rINi:. ATh.OS CONVERTS AN INPUT DESCRIPTION OF THE. • U0100 15., C AIMOSpHERc:'S PROPpRTIFS INTO ONE MORE APPROPRIATE FOR THE UUI.Ou 16* C CAI-MATIONS TO PrILLOw (SUCH AS EVALUATION OF THE NORMAL uU l Ou ,7* C MuOC nISPERSION FUNCTION IN NMOFN. DESCRIBED ELSEWHERE IN uuiOu 18+ C THIS FRIES). u010u 19* C u010u ..0* C LANG4)A/4F - FORTRAN IV (3tin. REFERENCE MANUAL C28-6515-4) uOlOu el* C uOIOU e2* C AUTHuRt: ... A.D.PTF.PCL AND J.POSEY , M.I.T.. UHNE,1960 UU1.00 .3* C LOiOu ,4* C 6010U .5* C ---- USAGE---- 6010U ,h+ (.: 6010u ,7 ► C IMAx.MoST HE Sv1 REJ AS THE FIRST VARIABLE TN UNLABELED COMMON WHEN LulOu ,1* C AT.00S TS CALLEn. LOIOu ,9* C UOtOU A* C i.0 FORTRAN SUORc-.IT1NES ARE CALLED. UlliOu ..,1* C u0300 .+2* C FORTRAm OSAuF OulOu ..$3 4 C uu100 • .-,4* C CALL ATmoS(T.VserTX.VKNIYOT.wANGLE.WINDY.LANGLE) uOIOU .15* C U0100 ' .,h* C INPUTS 00100 .17* C u010u oll., C . IMAi NuMpEo OF LAYERS nF FINITE THICKNESS IN THE MODEL ATMOS- 6U100 0 4* C I*L. ()HERE. ( I.L.E.IMAX.LF.99 ) u0i0u .0* C 00100 ..1* C T ' T(I) is TEMPERATURE OF LAYER I IN MODEL ATMOSPHERE. 0000 .2* C R*4(01 (DEGREES KELVIN) 1,0i0u .3* C uOIOU .4* C VKNIX VKNTX(I) IS WINO vELOCITY COMPONENT IN X-DIRECTION (WEST 00100 .5* C R..1.(0) To EAcT) FOR LAYER I. (KNOTS) Ou100 • .6* C u010u .7+ C VKNIY VKNTY(T) IS WIND vELOCITY COMPONENT IN Y-DIRECTION (SOUTH u010u :8* C R.4(0) To NORTH) FOR LAYER I. (KNOTS) v0x0u 9* C uQiOu :.)0* C 21 21(I) IS THE HEIGHT ABOVE THE GROuND OF THE TOP OF LAYER 06100 • ■ 1* • C R*4(0) I.. (wm) uOiOu :12 4 C uulOu n3* C •AN:.LE WANGLr(I) IS KIND VFLOCITY.DIRrCTTON FOR LAYER I. RECKONED

uUlOU '4. 14 C R*4101 CoUNTrn COCKwISE Ft OM THE X-AXIS. (DEGREES ? UUtOU :, 5* C uu,00 • :,6* C WINIly. WiNDY(I) IS MAGNITUDE OF WIND VELOCITY IN LAYER I.

.4.,10. 4, tvirui. 144 IF LAmAILE.LE.0 r vKNTX AND TKNTY ARE INPUT. 0U1OU .1* C IF LANGLE.GT.0 ► wANGLE AND WINDY ARE INPUT. 00inu r,2* C u0100 03* C UUTPUTc u0100 .4* C THE OUTPUTc ARE SToRCO IN UNLABELED COMMON IN THE FOLLOWING tiOiOu .5* - C uRnER' HEGTNNG IN POSITION 2. 0010u CI(100) ► VXT(100).VYT(10).HI(100) (.10100 .7* u0i0u (.8* C CI C1(I) IS rHE SPEE0 OF SOUND IN LAYER I OF THE MODEL ATMOS*► UUIOu .9* C R.4 (D) PHERE. ( Km/SEC 00100 /0* 00101) /1* C VXI VAI(I) TS WIND VELOCITY COMPONENT IN X-DIRECTION (WEST TO 00100 /2t C R*4(0) EAST) FOR LAYER I. ( KM/SEC ) U010u /3* (10.0u /4* VYI VYI(I) IS WIND VELOCITY COMPONENT IN Y-DIRECTION (SOUTH uJ100 /5* R*4(0) TO NORTH) FOR LAYER I. ( KM/SEC ) 00100 /6* 1;0100' /7* HI HI(I) IS THE THICKNESS OF LAYER I. ( KM ) UOIOU s8* C R*4(D) u0.100 / 9* C uulOU .0* C UOiOu .1* _-- -pROGkAM x:01 LOWS 60.100 .2* C u0100 ,3* C 00.01 ,4* SUkRuUTINE ATMnS(T.VKNTX.VKNTY ► ZIFwANGLEIWTNOT ► LANGLF) u0101 .5* u0103 .6* 0/,'ENSTUN C1(1n0),4XI(100).VYI(100).HI(100) ni•.ENSTON f(10n1 ► VKNTX(1001/VKNTY(100),ZI(100) uUlOb 08* I ,I ,'0ENST ON ONG, F(100)1 w INDY(100). 00100 mg* CO..MuN 1mAX0CI.VXIIVyi.Hi uU106 40* C u0106 -,1* C JET Is ToTAL •OMBEn OF LAYERS. u0107 • 42* JET = (MAX + 1 uOilu 43« 1MAX = JFT - 1 00111 .4* IF (LANG1E .LE. n) GO TO 2n U011., 451c 113 =. 3,1415927 / 160.0 0u113 46* C 03 is /HE N,;mrtk,: R OF RAjIANs IN A DEGREE u0113 .,7* C UU113 vat C IF VKNTX ANt ) VKHTY WERE •NOT INPUT, THEY ARE NOw DETERMINED FROM WINDY u0113 49* C ANTI WANGLE. 00114 1.0* NO S T=1 ,JET • 00117 1.1* 1/KmTX(1) = dINnY(I) * cOS(03*WANGLE(I)) u0120 1.2* 5 VK•TY(I1 = WINnY(I) * SIN(03*WANOLE(I)) 00I2 1.3 ► 20 Ul = 1.4 * 8.3144 * 0.001 / 29.0 0012d 1„ 14* C.02 IS THE mUmntR DP KM/SEC PER KNOT. 0, 0i23 1.5* 1)2 = 0.0005148 uu123 1.6* C 00124 1.7* " NO 31) 1 = 1P JFT 00124 1.8*. C 00124 1.9* -- C THE SPrE0 OF SOUND = ( GAMMA * P / RHO ) FOR PFRFECT GAS. AND'( P/RHO ) 00124 liO* C = ( it . T ) • 00124 1.1* C k TS T.E (0NIVERSAI GAS CONSTANT)/(MOLECULAR WFIGHT) 00127 1.2* CItT) = sORT(01*T(I)) • • u0127 1,3* C

1,1,1) = Zitat u012n 119* IF41MAy 11 RETURN 00140 1,0* DO 411 Iz2p1mAy UU14i 1,1* 40 ► I/1) 71(I) 7_I(I-1) U0145 1,2* RETHRN U0146 1,3* ENH

Ei.tp OF COMPILATTOW: NO DIAGNOSTICS. •HuGoP •*4*** Stkillb ***•** 6FuR,S I.Sa3n6,SuBn6 S11F—le/10/74-18;0A:04 (0.)

SUBROoTiNr oBriB ENTRY POIN1 00n1n1

STORA6E USED: CoOL(i) 0A116/ DATA(n) 000025; BLANK COMMON(2) On0000

ExTONAL pEFERENCES (iiLoCK, (4AmE) 0003 SAI 000 CAI 0005 NFRR31

Sl0(tA(X. AsSIONMEN1 (6LI%CK. TYPE+ Rcl.ATIVE IOCATInN. NA,iE)

0 0 01 On0051 3L OnOn6A 4L nOn4 R 00000n cAI 0000 000014 INJP$ 0000 R 000000 S 0003 R On0000 SA1

U0100 1* C MB.01 (cURROUTImF) 7/25/60 LAST CARD IN DECK IS NO. uulnu , 2* C uulOu 3* C ----A6STRACT - --- U0100 4* C 00100 5* C TITLE - — Himo 00100 6* C ThIc SUHROUTTNE COMRUTEc THREE FUNCTIONS RleR2pR3 OF A VARIMILE - 00100 7* C X. THEhE AuF pEFINEn Felfi X ,GE. 0 HY THE FORMULAS 00100 04 C GulOU q*. C rt ic 1.n 4SINH(2Y)/(2y) b0100 0* C u010(, II* C 142= (SINm(PY)/2Y — 1.0)/Y**2 001n0 12* C U0100 t3* . C R3= (COSW2Y/ - 1.n)/Y**2 1.10i00 i 4.1, C 00100 15* C wHERE Yz SORTM. FORMULAS FOR NEGATIVF X MAY BE OBTAINED BY 00100 16* C ANMYTIC CCmTINuATI0N. FOR SMALL VALUES OF )0 THE FUNCTIONS u0100 ,7* C ARE CoMIJUTARLF UY THE PnWFR SERIES 00100 ,8* C - --- ,-... 6 0040U /3* C P3= 4/(2PhCT) + 4*(4y)/(4FACT) + 44, (4X)**2/(6FACT) +,•. LOEOu /4* C UlitOU /5* C THE MANNER TN WHICH THP:GE PARTICULAR FUNCTIONS ARISE IN THE U0100 /6* C THEntiY COMEc FROM INTEGRATIONS OVER VARIOUS PRODUCTS OF CAI (X) uOlOu /7* C AND SAW(/' IN PARTICUIAR, FOR X POSITIVE' 00100 /8* C 00100 49* C ril= (2/YI(IN1EGRAL Om Y FROM 0 TO Y OF (COSH(Y))**2) LUIOU .)0* C U0100 .11* C p,!= (2/Y4*3)(INTFGRAI ON Y Ftlom 0 TO Y OF (SINH(Y))**2)' 0u100 .-.2* C 6Ui.OU .13* C R3= (4/Y**2)(INTFRRAI ON Y FROM U TO Y OF SINH(Y)*COSH(Y)) U0100 ..,Lts C UOIOU ,,,5+ C ulTH Y=sORT(Y). THE CORRESPONDING FORMULAS FOR X NEGATIVE CAN OUIOU . -,64 , C nt,-- OBTAINED NY REPLACINr, SIKH AO COSH RY SIN AND COS' RESPECw 6010U o7* C •I1VELY; AND RY REINTERPRETING Y AS SORT(-X). u0400 , , Sr C 60100 ,19* C LAmGoAi,E - FuRTRaN IV (3•0' REFERENCE MANUAL C22-6515 - 4) 01100 40* C AUTHOR ." A.D.PTERCE , M./.T., JOLY ► 1968 iJOIOU 41* C 00100 42* C ----CALLING SFOHENCE - .- - - U0100 43* C 'ClUiOu 44* C SEE SUROUTINE ELINT 00100 45* C • x= 60106 46* C CA41. FI0ORCX , RI,R2tR3) 66106 47* C - 60100 48* C ----EXTERNAL cURROUTINES REOUTRED-.... 60106 49* 60100 nO* C CAT' SA1 u01(10 n1* C L.,010u H2* C ----ARGUMENT IIST.--- uU106 n3* C 6010G H4* C . X Ht4 ND INS 60100 n5* C Fri R*4 ND OUT 6010U n6* C R2 ti*4 ' ND OUT UOlOu n7* C H3 R*4 .NO OUT 63100 nEir C L,Oi0u H9* C NO CUM..ON STORAGE TS USED 00100 8,0* C G0106 nl* C ----PROGRAM FoLIOwS BELOW---- 6010U 1,2* C 60101n3* SU.JLJOUTINE 9RBACX , R I , R2310 00103 04* S=c.A1(4.P*x) ocin4 n5* IF(AhS(X) .GT. I.E-2) GO Tn 3 00104 ,,64, C 60104 07* C 00mPNTATIOm FOR SMAIL X 6010o '.8* R2.1:2.0/3.04-(2.(1/15.0)*x+(4.0/315.0)*X**2+(2.0/9.0)*X**3/315.0 UU107 4,9* R3=2.0+2.0*X/3.0+4.0*X**2/45.0+2.04X**3/31c40 1.1011U ,0* . oO TO 4 60110 11* C J0110 /2* C COMPUTATION FOR x mOT NEAR ZERn :,v111 '13* 3 R27.(S-1.0)/X i0112 /4* . R3:(CAT(4.0*X)..1.0)/X J0112 /5*

END OF COmPILATTON: u° nIAGwOSTICs. HuG ► P **4*** SUB07 ****** FLA'S 1,5.130715On7 OH s11F-1e/10/74 - 18:0A:10 (01) .

FuNcTIO., CA1 FArRY pOTN1 00_046

SlOkAGE USED: CODE ( 1) 060n59: DATA(n) nuOn12; BIANK COMMON(2) 0(10000

ExTFAIvAl nEFERENCES (NLACKr NAME)

0003 Son( ofJn4 COS 03oS ExP 0006 NERR3I

510RAIZ ASSIGNMENI (KLIICK ► TyPL ► RFLATIVE LOCATION. NAME) bOnl 000u21 11L htfri0 R OnOono CAI n000 R 000001 F 0000 000004 INJP$

010u 1* C CAf! (Fu CTION) 7/25/68 LAST CARD IN DECK IS NO. U100 2* - C JiOu 3* C - -,--ABSTRACT-- - - 0100 44 C Jiou 5* C 1ITLF -. CAT lInU 6* C PROGBAm To EVALUATE FOUrTION CAI (x) FOR GIVEN VARIABLE x. )10u 7* C . IF x IS NFOATTVE , C.,/(X1= COS(SORT(-X)). IF X - IS POSITIVE. 1100 8* C CAI(X)= COS)4(SORT(+x)). THE FUNCTION lc ALSO REPRESENTABLE 1 10u 9* C . Mr TN1-• mOWER SERIES qOu 10* C !nu ,1* C CAI(Xl= 1 + X/(?FACT) + X**2./(4FACT) + X**3/(6FACT) + ... 100 42 ► C . 100 13* C LANGOME - FORTRAN IV (3ho ► DEFERENCE MANUAL C22-6515-4) Inu 14 1, C

10o iSt C AUTHOR • - A.D.PtrRCE ► M.I.T., JULY•1988 10u 16* C -!---CALLING SFOBENCE - --- 10U 0* C • tOu 10* C ' CAr(ANY R*4 ARA:MF:NT) MAY RE USED IN ARITHMETIC EXPRESSIONS ■ flo 19* C tilt., .0* C -_--EXTERNAL cUnRnUTIwES REOUTRED---- 00 .1* C Ou .2* C . NO EXTERNAL SUnnoUlINES ARF RERUIRED nu .3* C DU ,4* • C ----ARGUMFNT 1 IST---- U0100 c9* k; NO COM1.,ON STORAGE 7S USED 00100 00* C UOtOu .-,1* C ".....PROGRAM FnLLOWS BELOW... UOIOU 02* C 00101. ,N5* FOtACTIoN CAI(X) 00101 .14* C 00103 05* 1F(y .oE. 003) CO TO 11 .u0i0;) 06* C 00103 .17* C X IS LESS THAN 0 • uJI05 A* 10 CAI=COctSukT( - Y); u0106- A* RF1UkN U010b 40* . C U0100 41* C x 15 OwEATFk Ok EOLIAL TO 0 uti107 42* 11 F=i-Xia(coRT(X)) 60107 43* C 711F HYcJEkPoL1C COSITE IS COMPUTED 60110 44* CA1=0.5*(E+1./F) u0111 ...5* NEIU(1N 1,0112 .6* )=Nn

END OP CONP1LATTON: NO nIAGNOSTICS• LIL,GtP **.**4 stjah * 4 4 4 ** kFURPS 1.5 ■ ,(308 , 501308 1-01( S1IF-I./10/74 - 1810k:14 (Up)

500kOUT J NE LLINT ENTRY POINT 00n116

STOkA(‘E USEU; CoDE(1) On015P; DATA(n) 000034; SIAM( COMMON(2) .00 .0000.

EXTFrINAi REFEkENCES (11LrICKI NAMO

6 0 t13 AAAA •00t)4 BDtio 00n5 NERR31

S'fORA6E ASSIGNMENT ((1LoCK, TYPE. R,LATIVE LOCATION, NAME)

0000 R 000000 A 00n0 R On0nn4 FP1 nOnO R 000005 FP2 noon 01)0016 INJPS 0000 R 000012 RI 0000 R 000013 R2 00n0 R 000n14 R3 n000 R 0000nh cl 0000 R 000007 52 0000 R 000010 53 00nn R 000011

uOIOU 1* C FLINT (SUBROUTINE) 7/25/68 LAST CARD IN DECK IS NO. uOIOU 2* C 00100 - 3* ' C ...... A65TRACT U010U 4* C 0010u 5* C TITLE - ELINT .-,-,-., ■ 1.1 t AZ.F2(7))**2 (1)

_.„ 1L. C 1i0100 i2t C • THE FuNCTIONS ra(7) AND F?(7) ARE THE SnLUTIONS OF THE COUPLED uuIoO .31tx C ilkDii,ARr DII:TERENTIAL EnUATIONS UG10u 10$ C uulOu 15* C Dri/u = A11*F1 + Al2*.F2 uGiOu '6* C DF2/DI. = A21.*F1 + A22*F2 I'l; uo oo .7, C l u010u ta* C ' wHErw. THE EtEMENTS oF THE MATRIX A ARE INDEPENDENT OF Z. UG1OU 19* C FOR GIVEN SnUND SPEFT) C. WIND VELOCITY cOMPONENTS VX AND VY, u010,u eOs ' C ANGULAR FREnNENCY Ow.EGA. AND WAVE NDMDER COMPONENTS AKX AND AKY, U0.1.0o et* C THE A(I,J) APE COMPUTED BY CALLING AAAA. THE SOLUTION TO THE uCIOU ,2* C OIFFvRENTIAI EOUATIoNS IS FIXED BY SPECIFICATION OF Fl AND F2 UUIOU ..3* C Al 7=H. AUiOu .44, C uOICO .54- C PROGRAA NOTES 0010u .., 6* C uolOu ,-7* C THE GrNERAL SoLUTTON OF EoNS. (?) IS . 1)(1 .10u ..q* C uoiOu As C Fl(?) = CAI(X)*Fl(H)-.(H-Z)*SAI(X)*(All*Fl(H)+Al2*F2(1)) UtoiOU A -* C F2r7) = CATIX)*FP(H)-(11 -Z)*SAI(X)*(A21*FI(H)+A22*F2(H)) uJ1Ou ol* C 0)10u .12* c. WITH y7(A11**2-1-Al2*A21)*(14-Z)**2 c1NCE A22=-All. WE LET uOLOu ...34, C UOiOu 04t c RI =(INTEGRAL nF (CAT(x))**2)*(2/H) U010d "3* C 82 =(INTEGAAL nE ((H-Z)*SA1(x))**2)*(2/H**3) . Uu/Ou .16* C P3 =(INTEG•AL nF ((H-Z)*5AI(X)*CAI(Y)))*(4/H**2) u010u 0 7* C uOiOu .18* C WHERE IN EACH CASE THE INTEGRATION IS OVER 7 FROM 0 TO H. UOIOu o9* • C THE OmAHTITIES RI.RP,P3 ARE COMPUTED BY CALLING BBUB. t,UiOu AO* C THEN LIGIOu .1* C u010o .2* C AINT=(H/2) ,*(FP1)**2*R1+11-(+*3/2)*(FP2)**2*R2 0 100 •.3 C _ - (H**?/?).*(PPI)*(FP2)*R3 1.101Ou 44* C U010u A5* C WITH uUtPU •6* C FPI= Al*F1(H)+A2*FP(H) (,0100 .7* C FPp= A1*(All*F1(H)+Al2*F2(H))+A2*(A21*E1(H)+A22*F2(H)) UulOu - AO* ' C UOtOu •9* C THE LATTER TWo OUANTITIES REPRESENT THE COEFFICIENTS OF UOiOu,,niv . c CAI(X) AND (H-7.) IN A1*Fl+A2*F2. uulou !,14. C uoiou t „2t c LANGuilo.c - FORTRAN IV (3h0, iEFERENCE MANUAL C22-6515-4) • uttinu f).5* C UOInu 'Mt C AUTHUR - A.D.NERCO M.I.T.' JULY01968 4AiOU n54 . C. 0010o 4.)* C' --.--CALLING 5POHENCE ----. 0010u :,7* C uulOu '41* C SEF SIWOUTINE TOTINT UO10u h9* C WO DIMI.NSION SIATE.AFrITS nolar4D u0100 .,0.. C Ctiii. ELINT(OPEAAIAKX.W.C.VX,VY,H.F111,F2H.A1rA20AINT) UUIOU Id* C UOIOU ..at C ....-EXIERNAL cURROUTINES REOUTRE0-..-- .... -_-_ARuUMENT 1 IcT- --- UulOu "7* UOIOU "8* C DM. GA R*4 ND VIP uulOu "9* C AKy R*4 NO INp uOlOu /0* C Asv R1-4 ND INP uUl06 41* C C R*4 ND INN . 00400 ,2* C VY, R*4 ND r tstp utilOu /3* C VY R*4 ND IN uu,Ou /Li* C H R*4 ND IHP • uOIOU '45* C l'Ip •R*4 NO WilOu 46* C VF2_.. R*4 ND• Illt. .41 U0106 17* C Al R*4 ND UU100 /II* C A2 R*4 NO It t l: • u010u /9* L. AINT R*4 ND OUT uu,Ou "0* C

uulnu 4,1* C NO CuM .. ON STORAGE ,ISrD 00100 ,,24, C 60100 "3+ C - --INPUTS---- u010U /.4* C uulOu • "5* C UM. GA =ANGULAR FREOURNCY IN RADIANS/SEC uOIOU ‘,61, C Akv =X COMPONENT OF wAVE NUNIRER VECTOR IN KM**(-1) . G01.00 4,7* C AKY r.Y. COMPONENT OF wAVE NUMBER VECTOR IN KM**(-1) bOlOu "8* C C =SnND SPEFD Im KM/SEC u0,00 "9* C VX :X COMPONENT OF wIND VELOCITY TN KM/SEC k.,C,Oki .O* C VY • z.Y COMPONENT Or wIND VELOCITY IN KM/SEC u0(00 41* C N :INTEG,

ENO OF cOmPILATTON: NO nIAGNOSTTCS. 1.1inGIP **...*** sUB09 ****** 61FoRIS 1.S,4309,SOLH9 ■ /10/7 FO, SlIF-1 ► -18:DA:21 (0 ► )

FuNCTIOw FNMOul KNTRY POINT 00n024

SIORANE USEu: CODE(1) 0110U3n; DATA(n) 0000111 D1ANK COMMON(2) On0624

. EXTERNA1 REI-ERENCE5 (HL(1CK. NAME) 00113 NMDFN u004 14ER1c31)

STORAGE AsSiGNM•.NT (1-0.()CK, 1TPE. RvLATIVE LOCATION. NAME) 0002 000001 CI n0n0 R 000nOn FNMOhl nonO R 001)003 FPP 0002 Onn455 HI 0002 000000 IMAx '0000 000005 I.NuP% 00n0 I 000n04 K nOnO 1 00u002 I n000 R Onoonl OMEGA 0002 R 000621 OMEGAC U002 R 000..124 rHijk 00(1;! 000A22 vPHSFC n002 000145 vXI 0002 000311 VYI

CALOU 1* C FN...Ot)1 (FUNCTIoM) 6/19/68 LAST CARD IN DECK IS NOU LiOLOu 2* C U0100 3* C 00100 4* C ----OSTRACT....-- °LILCO , 5P C 00100 6* C TITLE .. FilmOhl OOLOU '7* C rvALUATATIOm OF NORMAL mOOr DISPERSION FUNCTION AS FUNCTION OF %ALDO 3* C PHASE. VtLOCrTY V LiU100 9* C uUAOu AO* C THE No0 ►,AL MOnE OTSPERSION FUNCTION DEPENDS ON THREE VARI- 00100 11* C AHLFS..AN6ULAR FR.0UFNCY 0NE3A. PHASE VELOCITY V, AND UULOU 12* C DIRECTION OF PROPAGATION THETK. FU 101 OBTAINS V THROUGH, 00100 A3* irs Apcumr.uT, OME6A AND THETK FROM COMMON. SW-MOUT/NE LOLOJ 14* C N•DFN" IS THEN CAL1 En TO EVALUATE THE FUNCTION. (SEF GULC() 15* C P1r:RCF4 J.CONP.RHySICS, FEn.•1967, P.343 -366 FOR DEMI- 0310a 16* C TiON nr NORNAI MOnE DISPERSION FUNCTION.) 00100 A7* C 01400 in* C LANGUM4F ForaReal IV (360, DEFERENCE MANUAL C28-6515-4) U0A00 19* -- C 0010u i, C AUTHOR. ..• A.O.PrERCE ANn J.pOsEY. M.I.T.• .M4E119E18 IJOAOU - el* C 1.10100 l* C FN•401 CALLS SuoROUTINF NMIIEN WHICH CALLS AAAA AND RRRR. RRRR u01.00 ,09* C CA, LS AAAA AND ?,,mMM. ALL THESE SWIROUTINEs ARE DESCRIBED ELSE- uOlOu .%0k C . ■NH.-RE TH THIS cERIES. 0100 .0.* C 00100 .12* C CAILIN,4 SEnUFNCE u010u .+3* C utliau .14* C CO,MoN cm1(401).0MEGAFCM2 , THETK uOLOu 5* C UM-GA 7.4 XXX 00100 ..,6* C fHwTK = xXA 0.0i00 .17* C V = XXx U0k0u .,5* C FUNCIN :.; FwMOD1(v) uU100 ,11* C uOIOU .0* C INPUTS u0i00 .1* C 60i0U .2* C V PHASE VELOCITY (Km/SEC).. uUiOu .3* C R*4 U0100 ..4* C u0100 .5* C OmE"A ANGULAR FREOUENCY (RADIANS/SEC). uniOu .6* . RA4 uOiOu ,.7* C 00100 • 5* C THErK PHASE VELOCITY DIpECTION MEASURED COUNTER-CLOCKWISE. FROM 0010u ...9s C R•4 X-AXIS. 00100 !,0* C 00100 !Nl* C OUTPUT‘; b0100 :12 4- u010u :134 C THE ONLY OUTPUT IS THE VALUE OF THE NORMAL MOUE DISPERSION FUNCTION 0010u :)4* c FOR THE VALUES nF V. OMEGA' AND THETK WHICH HAVE UEEN INPUT. U0100 :)5* C uclOu C 00100 C ----PROGRAM EOLLOWS BELOW-- -- U01.00 nel* cOlOu 60;01 ,O* FwMOD1(0. 0/101 • ,1* t,2* HI,ENST(JN LI(InO) ► VXI(100).VYI(100) ► HI(1110) t,3* COmMON IMAA , CI.VY.I ► VVI.HIPMFGACOPHSECITHFTK . LioiO4 tAt 00104 p,54, C OM FGA THFTK OnTAINED FROM COMMON JG105 t.6* omGA.7.0MFGAC .7. CAIL MmHFULOMEnA,V,THETK,L.FPP,K) JU107 PN.001=PP J0110 .9* REIURN ,Dill /0*

Et.0 Or COHPILATTON: NO nTAGNOSTICS. J u"0,r, ♦ 4 4.**4, SUHSO 4, 4**4, 4 ,FORPS I.S.ar;0,SuU!10 0, s11F-1,/10/74 - 13:0A:26 (It)

ic:Tro / 01 ov (0 , 0 •1.9' 40 1 41 . (ICTO1 ($ 0•0. = f,i7T00 cd (2**Pxo) - le9 = e9 t'£' p1) trOT e44 1PIOU/fe44.1i1 e4 411N0 ) = reg ?VOA e**D/(e**w ef*vvirq = leg r7. 1 0n '4E Ola*av , - wrilwu = "voll 000n ) 5 . ivOJYV = wk?( Le 7 0 1 'Heron (I')00 1 ) = AO • qaT00 . eNiTN- Ocf • ?arnn (I)ml 30 1e100 (I : Tnt nelon L:rl duv: , W. Oft ',i/00 .1/4! NOiv fitron Dii3)ii)N1S.*(TdI)IAA + 0(12H1)S0)+( = N"dA W1300 . 17 . 011W!!" = 1140 ?Tinn J/ 11 600* *.00 - 'NU Ittrin (TWIT) II 01100 I + xtr.:'1 !7! 4 d1 LOtog 0 raj':0(.) 1 )1S\I 1.101.0 007nn (0^01)Y0W' (Aron Jol)Hovq.01(uut)I1 Nol 7 IJI-10 +/Von - 1(1 (0001IA 34(001)IAAI(uot)I0 •ol5m11 'Von DOI3HlitiowdAJowet 4 NIJN'ib:iantugWrA 0.6(1 D 3ml 1 ngtl -ns I on

IAA IU000 8 3000 IXA 5hI000 - 8 3000 3sHdA eI0000 H 0000 HvJA b0u000 0000 "qt8 4-TO000 0"0 0. . V93v40 Tt0000 H 0000 ONO £00000 kl 000U VAU 600000 H uuuu ?m Luu0U0 I 0000 TN 90000U I 0"0° DI1 020000 I 0000 HNHO1 £20000 I 0000 r uI0A00 I 0000 TdT 00 0 000 I OUT) ScinNI L2"000 OUV XV AI, 000000 I 3000 I 500000 10000 Iii SSt000 • ?uuu EU9 9LUOUD d 0 1( 00 re9 0000 H 0 0 0 0 30 LI0000 H 0000 ID tu0000 H TUODOU 2:1 NO t71U000 a nu00 oNV Te0000 H 0 0 0 0 . lona I)1NV 330000 H 0000 10CI 4ST0U0 1000 Oczi u(TOOU IUQU 1031- i.l'At000 Tuou ouTT egnouo

#471.)14) 1.W.1110075SV 3l))/H015 (3 ► VN 1 N011VD01 JATIVild

etcJ11 400n • Nis 900 SOD 5000 AHOS W/0" InOdD £1100

( 141 ►VN ":)"14) /VNLI41:43

9SbOu0 (2)NONWO3 WNW 1950000 ( 0 )Viv0 fLee0u0 (1)21000 :1135n TWHOS

5Ve 0 OU 1010d IHIN4 v&On 3N' 1' 20Vd tartei DIUJ * •** **. 0S5"S *4, 44*

5 DIMENSION KST(10)0(FIN(10) 6 COMMON IMAxicIrVXIO/YI,HI 7 ipi IMAX + 1 8 C = cI(IP1) 9 DMA 0.7*.0n98/C 10 OmR SoRT(O.4)*.0093/C 11 VPAR = VXI(1R1)*COS(THETK) + VYI(IP1)*SINCTHETK) 12 DO 130 1=1,MoFND 13 Ni = KST(I) 14 N2 = KFIN(I) 15 DO 130 J=N1pN2 lb OMFGA = OMmoo(J) 17 VPHSF = vpmOn(J) 18 RKR OMEGA/VPHSE 19 nom OMEGA — VPAR*RKR

20 321 (oMA**2 — ROm**2)/C**2 21 622 (0m(3**p - - npm**2)/R0m**2 22 G2 = 021 — (RKR**2)*G22 23 CKI(0) 0.0 24 IF(G2 •GT. 0.0) GO TO I30 25 CALL. CROOT(OmEGA.RKRtTHETK,LICrANKR,ANKI/LBRNH) 2b FCITF (6 , 1001) IrOMEGAPRKR.LIC 27 1001 FoRMAT(IH ,3x,2HI xtI3t3W)HOMEGALIPG15.81- 28 13X.4HRKR=rG15.8,3X0441LICI-../3) 29 WRITE (hP1002) ANKR , ANKI,LRRNH.VPHSE 30 10O2. FORMAT(1H r3W3HANKR=PG15.8,3X.5HANKI=sG15.0,3X , 31 16HL3RNH.r..I3r3XPoHVPHSE=f05.8) 32 IF(LRRNH . .NE. 3) GO TO 120 33 CKI(J) = 0.0 34 GO TO 130 35 120 . CONTINUE 36 VPmOn(J) = OMEGA/ANKR 37 CKI(J) ANKI 38 130 CONTINUE -- 39 RETURN 40 END QPRToS 1.5US53 ri 4LJ OF COmPlLATTON: NO n/AGNOSTICS. wouG I P slit151 ****** 1.,FuR.S - 1.5-351.Suphl tUR s11r-1„:/10/74-18:01:30 (1,)

SuBROUTiNE uAuOR ENTRY POINT 000154

STORAGE USED: CODE(I) 001201/ DATA(n) 000050; B1ANK COMMON(2) On0000

• EXTFRwAo RE F ERENCES (hLnCK, NAMLO

u003 NERR3i,

STORAGE ASSIGNMENT(RLnCK, TYPE , RrIATIVE LOCATION, NAME)

uOnt 000064 1170 tiOnl 000065 1220 0000.R 000005 ROM 0000 R 000006 BOmS0 0000 R 000004 CSQ 00110 R 00000u D 0000 R 00ent0 UTOKX 0000 R 000011 nTDKY *n000 R 000007 DTDOM '0000 I 000012 I 0000 000022 INJP% 0000 I 000n13 J 0000 R 000014 yAT

tiOIOU 1* C DAuOR (SUBROUTINE) MODIFTED 7/11/74 LAST CARD IN DECK IS NO. UUlOti 2* C u0I00 3* C uuiOu 4* C -- ABSTRACT ---- u0t0u 6* ' C u0 0u 6* C TITLE .. nAouu UUt00 7*• C THE Fu•CTIOm OF THIS SUARoUTINE IS TO CnMPUTF THE COMPONENTS ti0i00 0* C nF Tht io1TRTCCS DADoM ► DADKX, AND DADKY WHICH REPRESENT THE UULOU 9* C PARTIAL D•RtVAT1VES OF THE mATRIX A WHIcH WOULD BE U0100 tO* C COMPJTEii BY SUBROUTINE AAAA. u6100 11* C • HIU )OM TS THE PARTIAL DERIVATIVE MATRIX OF A WRT OMEGA UOIUU t2* C DAOKX rS TUE P.RTIAL DERIVATIVE MATRIX OF A WRT AKX 00i00 83* C DA./KY IS THE P.- 471AL DERIVATIVE mATRIX•oF A WRT AKY uU100 14* - C LIKE At ALL ARE 2-13Y-2 mATRICES. U0A00 15* C tA/00 16* - C LANGuANE - FORTRAN V (UNIVAC 110ArREFERENCE MANUAL UP - 7536 REV. 1) OUIOU 17* C AUTHOR: - ALIAN De PIERCE, CHRISTOPHER KAPPER• G.I.T., JULY, 1974 UOLOU sil* C 1JUIOU 19* C - --CALLING SrOWENCE---- u010U e0* C ul)100 el* C SEF SU.4t4J1)TINE COMoK L.U.Oti e2* C DIAENST6N 0(2 , 2)*DADOM12,21 , 11ADKX(P•2),DAOKY(2.2) t:0100 e3 ♦ C CAIL DAut4RIOMEnA.AKX,ARY.C.Vx,VY,DADOM/DADKX•DADKY) u0100 ,4* C OU1OU e5* C NO EXTFRNAL SUHROUTINES REouIRFO ut1100 •l* AKY R*4 ND INP uU100 •.12* R*4 NO II.IP uOiOU .13* VX R*4 ND INP u010u o4* VY R*4 NO INP U0100 hAoOm H*4 2-RY-2 OHT 00i00 06* DAmKX R*4 2 -0Y-2 OHT 0)100 .1 7* 0AoKY R*4 2 -8Y-2 OHT 00100 U01.00 o9* C NO CoMmON STORAGE uSFD u0100 40* 00100 - --7INPUTS7--- 00100 »2* u04.00 -3* 0MGA . =AN(MLAR FkPOUENCY RAD/SEC U0100 »4* AKi =X COMPONENT OP HORIZONTAL WAVF NUMDER VECTOR IN 1/KM 00100 »5* AKY =Y COMPONENT OE HORIZONTAL WAVE NU•RER VECTOR IN 1/KM 00100 »6k C C =SoNMO SPEFO Im KM/SEC u0100 .7* VX =X COMPONENT OF wIND VELOCITY TN KM/SEC 00100 »8* vY .my COMPONENT OF WIND VELOCITY TN KM/SEC 00100 u010u nr:14, ---O UT PU T S- - - - u01.0u "I* 00,00 n2* N A HCM,t (Tt,J) =(T,j)-TH ELEMENT OF DOOM MATRIX A • u0100 n3* 0A"KX(TrN) =(irj)-TH ELEMc-IIT OF DAOKX MATRIX 1 u0100 WAIIKYITPJE =(T,J)-TH FEEMENT OF DAOKY MATRIX uUlnn 5* u0100 •15* - ---PROGRAM FnLEOwS RELOw--- - u010u 00101 .)11* DA0w-“OMEGA,AKX.AKY,C,VX , VY,DADoM , DAOKX•DADKY) u0101 :.9+ MA"nM , DAIIKAPPAnKY ARE MATRIX OFRIVATIVES OF A WITH RESPECT TO u0101 "0* uM.r6ArAKX,AKY , WHERE A IS AS COMPUTED BY AAAA. u0103 "I* 1110FWSToN 0(2P2).DADOM(2,21POADKX(2 , 2),DADKY(2 , 2) UO1n4 "2* CS , =C 4 C UOLO5 "3* H(1,1)=.11098 U010t1 "4* 0(1,2)=-CSo u010/ "5k M(..)t1)=(g6.04E-A)/ESO . unilu u01.11 "7* HO-;::0MEGA-AKM/X7.AKY*VY 1,011. "8* 1-10-SR=Ao.4**2 U0112 "9* 1 rS AK6w/o0MSn u0113 /0* NT"Or.=-2.0*(AK .4**2+AKY**2)/(BONSO*HOM) ( 7 U0114 HTKX=-DTDoM4-Vv+2.0*AKx/FiOmS0 00116 /2* NTmK1=-NTUW4 *Vy+2..U*AKY/BONSt4 u0116 /3* no 90 f:-10.! 00121 /4* rip 90 j=1,;! 00,24 /5* 0A00M(T,j)=OTOw'*0(I...)) u0125 /6* 1.1A"KX(T,j).=OTOKX*0(I•j) u012o /7* 9n HA•KY(Tr.)=0y0Ky*n(T , J) 1.1012h ta* TN.. AR"vE ELEX;THTS ARE CORIECT EXCEPT FOR 12'1) ELEMENTS u0i,31 /9* xAr=2.nkHO,v1/CS(1 u01.31 "0* xAT IS THE CTRTVATIVE WITH RESPECT TO OMEGA OF ROMSO/CSO u0L3e "1* 0A"nN(2,1);:DADnm(2 , 1)-AAT 00133 n2* DA"KX(2.1)=DADI4X(2PI)+xAT*vX FAD OF LOmPILATTON: 010 nIAGNOSTTCS. folluG.P ****** sUh52 ***•** 1.,F01.5 I.ShUS2rS0h2 FOR S 1 IF-14/10/74 - 10:0n:37 (up}

SUU,IOuTiNE 4JEDUR ENT.RY POINT 00n161

STORAGE USED: CoDE(I) 000221; DATA(n) OU0052I DIANK COMMON(2) On0000

ExTERNAc REFERENCES' IRLocKr NAME? 00(13 DAD.R 0 0 114 DRO,iR 0005 NERR3$

'SIORAGE ASSIGNMENT TYPE , RriATIVE LOCATION. NAME) o0u0 R 000004 DAhKg u0n0 R OnOnln DADKY n000 R 00000n NUM n000 R 000032 DGDKX 0000 R 000033 DGOKY u0on R On0u31 ilOh0m nOnu R 0 . 10(121) oRDKx non() R 000024 nRDKY 0000 R 000014 DROOM 0000 R 000030 GU UOUO OuOu34 INJP%

00100 1* C NEhoR (SOhROUTINE) MODIFTEN 7/11/74 LAST CARD IN DECK IS NO. U040u 2* C uulOu 3* C UulOu 4* C ----AuSTRACT---- u04ft 5* C UOIOU 6* C TITLE - OFDOR utii00 7* C ThE NORMAL NODE DISPFRSTON FUNCTION COMpUTEU NY SHBROUTINI:T11 uu100 6* C hmDFN It. CON;IDERED A FhNCTiON OF ON IFOA,AKX. AND.AKY. ut)10o 9* C THIS SUMOUTINE COMPUTEs THE PARTIAL DERIVATIVES OF FPP uu100 i0* C RESPECT TO omrGAFAKx. Awl) AO R1:.SPECTrVFLy. u01011 II, C nFuOm TS ThE PARTIAL DERIVATIVE OF FPP wRT OMEGA u010u i2* C DFNKX TS THE PARTIAL DERIVATIVE OF FPP wRT AKX uOIOU 13* C DFUKY TS THE PARTIAL DERIVATIVE OF FPP WRT AKY * 00100 ► C ,U0tOu '15* C LANONAI.F - FORMAN v (UNIVAC 1108,REFERENCE MANUAL UP-7536 REV. 1) L0400 46* C uOtOu 47* C AUTHDRc .- A.D. PIERCE' CHRISTOPHFR KARPER ■ 0.I.T, , JULY' 1974 0010u .0* C WII0u 19* C ----CALLING SpONENCE ---- 00100 eLl* C LOiOU .1* C 5tF SMAWAJTIME COMP.'. r.. 0 0010U ,2* DI...rNSToN RPP(9.2).DADOM(2.2).DADKx(2.2),DAOKY(2.2) 0010U .3* C DI•FNSTuN NROOm(2,2).0nKX(2.2).DRoKY(2 ► p),A(2,2) U4.00 ,4* C GAIL OFWA(OMEnADAKX.AKYrGr ► RPPeAPC , VX/VY,DFDOM.DFUKX.-DFOKY) 0100 o0* C - ---ARGUMFNT IIST..--,- u0100 ol*. C UOLOU .12. C oM,;GA R*4 ND LIP uOIOU 03* C AKy R*4 .NO TO' pOiCu .04* C WO R*4 ND INP, u0i0u A5* C GI R*4 NO 'NIP 00100 n64 C C R*4 ND INP u0i0u A7* C VX . R*4 NT) INF' U0100 ,8* C VY . R44 ND . ImP u010u .0* - C A R*4 2 - 11Y-p InP u0100 .0* C kili, R ✓ 4 2-BY-P. INIP 00i0u 41* C OFisolvi R*4 NO ONT L10100 42. C oFoKX R*4 NO .ONT u0i0o .3* • C oFNKY R*4 NO ONT U0100 -4* C UOiUU 45* C NO CUM-ON STORAGE, NSF0 00100 46* C u0i0u 47* c ----INPUTc.---- UOIOu •8* C uUlOu .9* C oMwGA =ANIGULAR FliEOUP4ICY _ RAD/SEC • UplOu t,0* C AKe =X Co%1POHFNET oF HoRIZoNTAL wAvE NUMBER VECTOR IN 1/KM o0.1.00 .,I* C AKy =Y comPOPENET oF HoRIZWITAL wAvE NUMBER VECTOR IN 1/KM 0U LOU :12* C 01 =PARAmiTER FOR PETERMNING Go TN UPPER HALFSPACE 00100 t)3* C C =SnUND SPE,M 1,4 KM/SEC 001.0u A4* C VX =X COMPONENT Or WIND VELOCITY TN KM/SEC 0000 :,5* C VY =Y COMPONENT 01- wIND VELOCITY TN KM/SEC klUithi 5i.' C Api.:(I,j) =(T,j)-TH 1.. 7 LFMI:NT OF MATRIX RPp CONNECTING 00106 •47* L. Sni UTIONS OF THE RESIDUAL EQUATIONS AT THE BOTTOM OF 0010U A* C THr7 UPPER HALp- SPACE TO SOLUTIoNS AT THE GROUND. 00100 :0* C A(T ► J) z(i,j)-TH •LEYWNT oF. MATRIA A uuiOu a* •C 00100 .1* C - ....-ouTPUTS- - - - 00100 .2* C u010o ,,3., C NnlOm =PATTAL OFRIVATIVF OF FPP WRT OMEGA UOIOU .,4* C 0F.KX =PARTIAL DFRIVATIVF OF FPP WRT AKX uujOu "5* C UF.IKY =PARTIAL DHlIVATIVF OF FPP WRT AKY u0100 "6* C uO1n0 ,,7* C ----PROGRAM FoLlOwS TIFLO.1--- - u01.0u "6* C 00101 "9+ SU-PuUrINE (W=Tn^(0i.IEGA,AKXJAKY,GI , (APPIA,CPvX , VY,DFDOM/DEDKX , DFOKY) 0(J10,.y /0* IIIENSTON A(2/91,11 ADOM(P,2) , DADKX(2#2) , )ADKY(20) 00104 'It (iI-EIJSTON•01,1 (2rP)PORDoM(2.2)tODKX(2+2),DRDKY(212) ilUi05 /2* GU7G1 uOAOo ./3* CAiL OACAR(ON^EaP,AKXfAKY,C.VYrvYrOADOmpnAnKX , DAnKY)

U0107 • /4* laill0m=(2.0*A(1.1) .*DADOm(1 , 0 -1-A(1,2)*DADOM(P , 1)+A(2 , 1)*DADOM(1,2))/ ... • U0107 - i5* 1(2.040H) 6011u /6* I;GI.K x= Le . 0*A (1. I. )*DAUKK ( 1 P 1 ) + A (1 r 2 )*DADKx (pr 1)+A (2 r 1 )*DADKX (112 ) )/ uuilo 0* 1(2.04, 0H) 00111 ,0 nG.KYzP.O*A(1.1)*oAnKr(1.1)+A(1,2)*DADKy(2,1)+A(2,1)*OADKy(1,2))/ 00111 19* 1(2.0*Gu) . u011',..! ,,O.* LAiL Diii;oR(OmEnAlAKX,AKYOlpPtA,DRD0MpORDKX,DRUKY) 00112 Al* C • F TS R(1,1) 4 A(1,2)-R(1.2)*tGU+A(1 , 1)) • ....—.,4.44,=Nrrkill)*JADKX(1 ► 2) 1-11.,Ptlein*(DoOKY+DADKX(1,1)) uOilh nF10=a0bKr(1 , 0*A(112)-DRnKY(142)*(GUtA(141))+RPP(1,1)*DADKY(1,2) 00Iih 1-RoP(1.2)*(DONY+DADKY(1,1)) t.#611i) wErUkil 00117 !'NH

END OF LOMPILATTON: NO nIAG,JOSTiCS. wHoGpP ♦ .*** skii4S *41, 4** ofuR,S 1.Si.f1c3eSuilh3 FOK S11F-1./10/74 - 18:04:36 (24)

SJDROuTINF CMCOR EN(RY POTNr 00n306

SINA6E USED: ConE(T) 0,0354; DATA (n) 000103; B1 ANK COMMON (2) 0110000

FxTFRNA,. REFERENCES (hLorK, NAME) 1003 • DADJR 0 0 04 CAI 0005 GAI 6006 Nr1,0$

STOoAciE ASSIGNMFNT (BLACK, TYPE, RrLATIVE LOCATION. NAME) 1,001 000102 I24G 0001 000103 1276 0001 00012n 135(, 0001 000202 I466 0001 000203 151G 0001 00051 I62G hOnl 0n0nr,3 4L n001 000070 sL n000 R 000022 CA 0004 R 000000 CAI 0 0 00 R 000010 nA,,Kx 00no R 000014 uAnKY 0000 R 000004 nADOM 0000 R 000024 DCAIX 0000 R 000000 DMDX 00.00 H 000020 RSAIY 0000 It 000n33 uXOKY 0000 R 000034 nXDKY 0000 R 000032 DXDOM 0000 R 000027 GEM 0000 R 000020 HSO 0006 I OnCn30 I 0000 000051 iNJP$ 0000 I 000031 J 0000 R 000023 SA 0 0 05 R 000000 SA1 0000 R 000n35 T 0000 R 000021 y 0000 R 000025 Y

0(4.100 1* C OM.koR (sUHRGUTINE) MoDITTEn 7/11/74 LAST CARD IN DECK IS NO. 2*. C 00tOu 3+ UU1OU * C UOIOU 5* C 0040u 6* C LE - nMnoir 06100 7* C HF VW m or THIS SUHRONTINE IS TO CnMPUTF THE COMPONENTS u6100 R + C ..4_i4ATRTCFS DMD0m.01:.DKX,•AND DmDKY WHICH REPRFSENT THE ioui00 9* C PARTIAC-f7 VAT1vES OF THE FM MATRIX WHICH WOULD RE uULOu ,0* CoMPuTF0 RY ' TNF mMMM. 60tOu ► l* OM00m TS THE P4R PRIVATIVE MATRIX OF EM WRT OMEGA 001.00 OHLAV is ME PARTIAL ()FR E MATRIX oF EM WRT AKX 400100 43* C pHoKy Ts THE PARTIAL DFRIVATIvi: N F EM WRT AKY 0.11.0 14* C MATRIX EM Ic ALb0 COMPUTEn IN THIS SUnRoUTINr. 45*

5 C 6 C TITLE - IIMDoR 7 C THE FUNCTION OF THIS SUBROUTINE IS TO COMPUTE THE COMPONENTS 8 C OF THE MATRICES DMDOM,DMDKX, AND DMDKY WHICH REPRESENT THE 9 C PARTIAL DFRIVATIVES OF THE EM MATRIX WHICH WOULD BE 10 C COMPUTED BY SUBROUTINE MMMM. 11 C OWDOM IS THE PARTIAL DERIVATIVE MATRIX OF EM WRY OMEGA 12 C DMDKX IS THE PARTIAL DERIVATIVE MATRIX OF EM wRT AKX 13 C DmOKy IS THE PARTIAL DERIVATIVE MATRIX OF EM wRT AKY 14 C MATRIX EM IS ALSO COMPUTED IN THIS SUBROUTINE. 15 C 16 C LANGUAGE - FORTRAN V (UNIVAC 1108,REFERENECE MANUAL UP-7536 REV.1) 17 C 13 C AUTHORS - ALLAN 0 PIERCErCHRISTOPHER KAPPER, G.I.T., JULY, 1974 19 C 20 C ----CALLING SEQUENCE-- -- 21 C 22 C SEE SURROUTINE COMPK 23 CDIMENSION A(2.2).Em(2.2).0mDX(2p2)DMDOM(2 , 2),DmOKX(2.2) 24 C DIMENSION DMOKY(2,2),DADOM(2r2),DADKX(2 , 2),DADKY(2#2) 25 C CALL OMDOR(OMEGADAKXIAKYrCrVXrVY,HrArEMPOMDOM,DMDKX,DMOKY) 26 C 27 C ----EXTERNAL SUBROUTINES REoUIRED---- 28 C 29 . C DADORFCAI,SAT 30 C 31 C ----ARGUMENT LIST---- 32 C 33 C OMFGA R*4 ND 34 C AKX R*4 ND 35 C AKY R*4 ND 1I M P 36 C C R*4 NQ N. 37 C VX R*4 ND VI; .38 C VY R*4 ND 39 C H R*4 ND IrP 40 C A R*4 2-DY-2 INP i 41 C EM R*4 2-BY-2 OUT 42 C DMDOm R*4 2-8Y-2 OUT 43 C DMOKx R*4 2-11Y-2 OUT 44 C DMDKY R*4 2-6Y-2 OUT 45 C 46 C NO COMMON STORAGF USED 47 C 4(3 C ---.-INPUTS-- 49 C St) C OMFGA =ANGULAR FRERUFNCY RAD/SEC 51 C AKX =x COMPONENT OF HORIZONTAL WAVE NUMBER VECTOR IN 1/KM 52 C AKY ..:y COMPONENT OF HORIZONTAL WAVE NUMBER VECTOR IN 1/KM 53 C C =SOUND SPEED IN KM/SEC 54 C VX zX COMPONENT OF HIND VFLOCITY IN KM/SEC 55 C VY . -zy COMPONENT OF wIND VELOCITY IN KM/SEC 56 C H =THICKNESS IN KM OF LAYER

o1 C EM :2-1sY-2 TRANSFER MATRIX WHICH RELATES THE SOLUTIONS 62 C OF THE RESIDUAL EQUATIONS AT THE TOP OF A LAYER 03 C TO THOSE AT THE BOTTOM OF THE LAYER. 64 C DMnOm(II,J) =(IrJ)—TH ELEMENT OF MATRIX OWDOM 65 C DIADKx(1.4) r(I.J)•-TH ELEMENT OF MATRIX DMDKX bb C DMDKy(IrJ) =(I,J)-TH ELEMENT OF MATRIX DMDKY 67 60 C ----PROGRAM FOLLOWS BELOW--- - 69 70 SUDROUTINE DMOOR(OMEGArAKX.AKY.C,VX,VYtH ► AtEMIDMDOM,DMDKX/DMDKY) 71 DIMENSION EM(212),DMDX(2r2)rOMDOM(2r2)/DMOKX(2,2)/DMDKY(2t2) 72 DIMENSION A(2,2),DADOM(2.2)IDADKX(2 , 2),DADKY(2,2) 73 CAL(7..:XL DORIOMEGArAKXIAKY.CIVXIVY,DADOMPDADKX.DADKY) 74 75 X=CA(111)**2+A(1.2)*A(2 ► 1))*HSO 76 CA:CAI(X) 77 SA=SAI(x) 78 DCAIX=0.5*SA 79 Y=A8s(X) 80 IF(Y—I.nE-2) 3r3t4 " 81 3 DSAIX=1.0/6.0+X/60.0+X**2/1680.0+X**3/90720.0 82 GO TO 5 83 4. DSAIX=0.5*(CA—SA)/X 04 5 GEM=HSOSAIX 35 Du 2n I=1.2 36 00 2n J=1,2 87 20 DMDX(I1J)=—GFM*A(I,J) 88 DO 3n 1=1.2 89 30 omnx(I,I)=omnx(111)+nCAIx 90 OXDOM=t2.0*A(1.1)*DADOM(1 , 1)+A(112)*DADOM(2,1)+A(2+1)*DADOM(1,2))* 91 11150 92 OXDKX=(2.0*A(It1)*DADKX(1 1 1)+A(Ir2)*DADKX(2,1)+A(2 , 1)*DADKX(1 , 2))* 93 1HSQ 94 DXDKY=(2.0gA11,1)*DADKY(111)+A(lt2)*DADKY(2,1)+A(2 , 1)*DADKY(1 , 2))* 95 IHSQ 96 T=H*SA 97 DO 90 I 7.7112 90 DO 90 .J51,2 99 DMD0m(I,J):Dm0X(I , J)*DXOOM—T*DADOM(I,J) 100 DMDKOI,J)=DMDX(I.J)*DX0KXT*DADKX(IrJ) 101 OMDKY(I.J)=DMDX(I/J)*DXDKY.4*DADKY(I,J) 102 90 EM(I,J)=—T*A(IFJ) 103 DO 190 r=.1•2 104 190 EM(I.I)ZEM(IsI)+CA 105 RETURN 106 END VIAT'S 1•SUB57 ' A ll uPr'W 0000 1010035 J . nuou i 000045 IL 0000 R ()00024 UPP 0000 R 000037 VX v u uo l D V A I 0000 R 00011 ,10 VY 0002 R 000311 vYI

U010U • 1* • C ORooR (SUBROUTINE) moDIPIFD 7/11/74 LAST CARO IN DECK IS NO•. 0010o 2* C uOIOU 3* . C uOiOU 4* C ----ARSTRACT---- 00100 5* C UulOU 6* C TITLE . oRnuR UuiOu 7* C THE PURPOIE OF THISSUBPOUTINE IS TO COMPUTE THE COMPONENTS OUI0o 8*. C . ' Oi- THE-oiATRTCES DRIMMoOpOKX, OW DRUKY wHICH REPRESENT-THE L., U1Ou 9* C PARTIAL DERTVAT1VES OF THE RPP MATRIX .W•ICH WOULD BE v0100 10* C CUMPUTEH nY SUPROHTINE PRRR. 00100 11* C nPuOm TS THE PARTIAL DERIVATIVE MATRIX OF RPP WRT OMEGA u0100 12* C DRuKX TS THE PARTIAL DERIVATIVE MATRIX OF RPP WRT AKX UOIOu '3* C HRUKY TS THE PARTIAL DERIVATIVE MATRIX OF RPP WRT AKY 60100 ,4* C u0100 i5* C LANGNAe.E — FORTRAN V (UNIVAC 1108, REFERENCE MANUAL UP-7536 REV. 1) . 00100 )6.* C AUTHORk; — A.n. PIEtvT, CHRISTO,JHFR KAPPER, G.I.T. , JULY, 1974 00100 17k C 00100- 10* C ----CALLING SrOuEwCE - --- u0100 19* C U0100 ...a* C SEF SUitROUTiNE COMpw uUiOU .1* C NI.rNSTON cI(In 1 ) , VX1(100).VYI(100) , HI(100) 00100 ...")* C DI..;FISToN d'P(2,7)'A(2.2) , nROOw(2'2),DRnKX(2'2)FDRUKY(2,2) ofliOu ;, 3* C D•AENSTON ,_m(2.2),WADOm(2o)oOmDKX(I8I)01MnKY(2o2) uULOO 44* C OI.4ENSTON UPPt,rPhOPP(2,21PAINT(2,2) 00100 .5* C CGJNOW IMAAPCI ■ VXI0VYItHI U0100 - . .6* : - C CAIL DROOR(OMEaApAKXDAKY'RPP,A,ORDOM,DROKX,DRUKY) u0i0u ..7* C U0i00 e8* C ----FATERNAL cURR0UTINES REQUiRED---- v0Luu .q* C uulOu 001, C oMivOR (UMDwR CALLS DAD0R,CAIrSAI) uOIOU .11* C u0100 ' .)2* C ...... -Ai2PUMi:NT 1 IST- --- ;;OiOU ..,31, C JOiOU ..14* C oM..GA ND INP )0100 :45* C AK. 'f12 * ND nP JUI.Ou ;;C.3* C AKy R*4 ND iUjOu .,7* C RPL, R*4 27BY-7 Ill':il iji0i .16* C A R*4 2—RY-2 INP , 01U0 ...9* C NRHOm R*4 2—RY—P OnT

0100 ,-.0* - C DR.1KX R*4 2 -11Y-2 OHT 0100 ,..I* C nil.)KY R*4 2 — BY—a OuT URDU .2* C Oino .3* C COMMON slopAGr U,En uLOu .4* C CO..AmON 1HAAPCI.VXIoVyI,HT JIOU ..5* C . OiOU .6* C 1MAX 1*4 ND ImP

_., - t. VYI R*4 100 IoP U0400 h0* C HI R*4 100 IoP U0t00 4 1* C uuiOU 52* u -_-..INPUTc.---- u0100 53* C uOinu 44* C OMPGA =ANSULAR FwEOUi.NCY RAD/SEC 00100 45* C NKY =X COMPO0ENT Oc HORIZONTAL WAVE NUMBER VECTOR IN 1/KM 0010) 56* C AKr =Y COMPONENT Ow HORIZONTAL WAVE NUMBER VECTOR IN 1/KM uUlOu 57* C =2..nY-2 TRANSFrP MATRIX WHICH CONNECTS SOLUTIONS W0106 50* C Ow THL REIDNAL EIJUATIONS AT THE BOTTOM OF THE u0100 49* C UPPER HALF-SPAcE To SOLUTIONS AT THE GROUND. u0100 4 ,0 i, C A =MATRIX A 01.7 CnEFFICENTS 0010u .1* 00/00 •2* 1 - _--ooTPUIS- --- 00100.3* C • UU100 .41 C 6RHONCT,j1 .7(i ► j)-TH ELEMENT oF MATRIX nRoOM u010u .5* C -0R.KX(y.j) 7c(T.,0-TH ELEMENT CF MATRIX nR(1KX 0)100 46* C 6RoK1(T.j) =(T,0)-TH FLrIENT OF MATRIX nRoKY uU100 .7* C 011100 .8* _C ----PhOGRAm FnLI Ov,S RELLOw---- 00A0u .9* C u0101 .0* SU•poNT1NE DRDoP(0;ZGA.AKX.AKY,RPP.A.D(tnOM.DRUKXFOROKY) u0103 /1* IiI ,.ENSTON CI(1nrOPVXI(100).VYI(100) , HI(100) u0104 ,2* 61 ,.,ENSTON RPP(912) , A(2.2)AnR00m(2 , 2)10ROKX(2.2).DRUKY(2 , 2) u0105 /3* 6I:.ENS1c,0 tM(2.2) , UM00m(2,2).DmOKX(2.2).nMoKY(2,2) 1,0106 .4* 01.,.ENSI.H urP(.2).DPP(P,21,AINT(2.2) 0)107 .54 CO. MON II AA , CI,VXI.VYI..HI 00.10 '6* •00 10 T'_1 ,r uU11.5 .7* NO 10 0l1 , ..A. kJ011u ifl* '61100H(T.J):.0.0 00117 /9* . 0RoKX(Y...10.0 00120 s,0* 10.(ARIIK1(TIJ)=0.0 00123 ..1* 0PLA1s1)=1.0 00424 .,2* UP.J(1,2)=0.0 11') rr u0,2,..) .3* uP.APII)::Q.0 , . uui2u .4* 0130(2f2):;1.0 t 7 ij :- U0i27 05* no 15 T=1Af! 00132.6• . 00 15 j=1,2 u0i3b .7* lF 0P0(1..0=RPP(I.J) 00140 AA* 60 100 J=1 ► IMAy 00446 .-03* 1=TvAX+1-J /if 0).44 ,10.i c=r1(I) I/ / A 2- 7 Z. — At47 - i (.014b 41* VX.:VXI(1) 0 014h 42* VY=VY1(1) ,21, 2 Z. 427/ 11a I uc.,47 43* ii=14/(I) tui50 44* CA1L DmUORiOmEnA.AKX.A.KY.C.Vx.VY.H.A,ENOnMoOM,DMOKX,DMOKY) b0150 45* C foUITIPIY DPP TVAES THE INV;RSE OF FM , U0151 46* •AI•T(1,1)=uPP(1.1)*EM(2)-OPPIlt2)*EA(2.1) 0015 47* AI.kT(1.'2)=-OPP(1.1)*EM(1,21+nPP11.2)*EM(1#0 00154 48* ALAT12.11=OPP(9PI)*EM(7:2)..OPP(2 , 2)*E01(2.1) b0154 40* AI.T11=-OPP(2.1)*EM(1.2)+OPP(2 , 2)*M(1A1) 00iSb 1.0* 00 20 r1=1,2 u0160 1.1* nO 20 JJ=1.2 UO16a 1.2* 2n 61%.. (11.J.J/;;AINT(IIrJJ) uOlup 1i,3* no 3), TI:;1,2 uow.A ■ ripu,1+=URDKXIII , J,J) .4- 11PP(/IPKO*DMRKX(KK , LL )* UPP ( LL , JJ )

: 04 1.9* 3n 0R,KY(TipciLI)=ORDKY(II,J,J)+nPP(II;KK)*uMnia(KKPLL)*UPP(LL.UU) '11 1.0* HO 411 T1=1,2 el4 1.1* HO 40 .41=1;2 el7 1.2* 4n AI,Jcil,J,I)=FM; IIF1)*UPP(1,JJ)+Em(r/t2)*6Pp(2 , JJ) e2? 113* nn 50 T1=1 , 2 e25 114• nO 50 JJ=1;2 1.5k 5n up,(11;HJ)ATNT(TI;JU) 33 1,6* 10n cO.ITIHHL e35 1,7* REIHWA 1,3*

F:4.1) OF C0mPILATTON: NO nIAGI4OSTTCS. ,G;P 4*...*** sub5b ***,..*• ;R'S 1.S,,U55;Sin155 c s11F- 1 ./10/74 - 18:09:49 ()1.)

SUbROHTINF URoUT • FNTRY P01(11 o0n910

Sic;inoE USEu: CODE(1) 000b52) DATA(n) 0(01231 BtANK CON0N(2) 000000

ExTI-MAAL REFEHENCES (HL,ICKt NAME)

0 0 03 PDIFR 50i14 0 0 05 Rr:Hc-.K 0 0 ,1r> SORT 1,007 NFRR3$

S1NA0E ASSIGNMENT (HL,,OK; TYPE; Rp(ATIVE LOCATION; NAME)

of 0'(31 1001_ (.001 MANI 1 5L 0001 000327 18L n001 Onn336 20L 0001 000436 200L 000.47 203L unnl 0i104611 2061: 0001 00047n 210L 0001 0n0347 22L 0001 000354 30L 000J67 40L nr,n1 000“10 91) L 0000 R 000053 A 0000 R 000070 AKR1 0000 R 000073 AKR2

0o0J75 AKhZ1 nnnu 000(017 ALFA. ;limn R 000054 0000 R 000065 C 0000 R 000056 0 00Ou 4 0 1,ELK 0000 000n52 NELKk 0000 R 000077 nELMIN 0000 R 000051 DR 0000 R 00001(1 DUM1 00000:i nU.2 0000 OnOW-, 0 LPS nOn0 nnninn PMIN 0000 It 000035 FRES 0000 R 000071 FRES1 0(!0.174 1-14.5? nnn0 000n7A 0000 R 000(ifiro rZ 0000 U 000046 02 0000 000103 Itsups 000c30!) K (Inn(, Nr1 01143 0000 I 0000 -0 11 000 1 1 000032 LUOS 0000 1 000044 LETS 0001;6. orm0 000ne,7 nono 0 000060 y1I 0000 R Onno57 X1R 0000 ti 000062 X2I 000u61 al< 0000 000n44 A3I nnn0 000063 Y3k

CROHT(ONac,A,AKS.THETAILIC;AKRvAit1fLOPNH) I 1",`," .011:14<(3) LAIL PniFklCkE,';P;AK5;THE:TNTL-I,TTMTiLET:;-iTqMIF-

•

4 21102 FORViAl (1H t'AHL1.007 , 3)03HF7=4G15.8 , 3X ► H(12=yr,15.8,3X.SHALFA=r 1615.At3x04HEPS=r615.A) 2UO3 FuRMAT (1H 01HCNOUTOX.3H0R=r01ti.11 , 3X/6H0ELK71=rG15.013X,2HAr., 7 1Glii.e1 , 3)(12HrizrUI!i.8,3X/2HC=r415.8,3Xt2H0=,015.8) Is 2014 FLJOIAT (111 fiCROOTC3 )( 1 4HX1)2=“11.5.8 , 3)04HX1I=rG15.813X/4HX2R=, 9 1G.03.A) 10 2 ∎ 24 FuRAAI (1H 0.11C000Tr3X , 4HX2I=•G15.11,3X,4HX3P=,G15.8,3X, 11 1411.01=0615.41F3X0HK=II3)

12 2005 Fud•AT (lit O'UiC ,

Ci,A) OF COmPILATTONT NO nIAONOSTICS. l.11 )Grf 4 4, 4*4-4, 44**•* 1,J=LJR,S t.S"rk6rSoilh6 SITE-1./10/74 - 1ATOgit33 (I')

• SOUo0L,INE CUIc ENINY POTUT 00n4n2

STORA(,E USEuT Coati) OriO457) UATA(n) 000053, Ell ANK COMmONt2) On0000.

(hLoCK, NAME) EATETZNAi REFERE:NaS

0001 PICA:,[ 0004 SoRT u0o9 X 7,11,1 u0o6 COS utio7 • NFHR31

SToaAkiE ASSIGNMENT (HI,A,OK , RrIATIVE LOCATION, NAME)

J001 0n0142 20L fool 000,53 6 0 L. 0001 000317 40L 0001 000343 SOL 0001 000361 60L 00H1 000,71 70L u0n0 R 000n03 ACRC 0000 R 00u017 AMP 0000 R 000004 BCRC 0000 R 000011 CAPA .00n R Un0012. cApU (0110 H 000n13 COPHT 0000 R 000007 rPACB 0000 R 000010 CPaCB 0000 000037 INJPS 040 R 00000u P Inuit R 000016 NFU 0000 R 000001 n 0000 R 000002 R 0000 R 000006 RAD H 00000p RAuS0 00n0 H On0n15 RI 0000 R 000014 cIPHT

4101 1* SUmPuOTILIC CIIP.C(Ar0,ConeXtR,Y1I,X2II ► X2I,X3RPX3I ► K) ?* IF (Anc(A) .LL', 1.E -20) GO To 30 • 010 - 3* P=../A • )01.0o 4r w=q/4 '010i 5* o=“/A • fOilu 64 AC,C=(3.*O-Pti, 2)/3. 0 11* CPACh=-HCM4/2.+ 4 0 Uu117 12* CP'Xii=-1,CPC/2.-PAD UUL20 13* CA,A=SV,M(1.0.17PAC0)*(ARStrPACR))**(1./3.) 00121 Cilori=ST(-A(1•0frINIC(1)*(ARS(CPUCB))**11./3.) 00122 )5* • XI;;=CApA+CAPB - p/3. 000.3 i6* X1r:7.0.n 00124 17* X2,1z-tr.ApA+CAPn)/2. -P/A. 00115 ,8*• x21=C(ChPA-CArk)/2.)*1.732n51 QUL2n 19* 00127 x:i1=-X21 U0130 ,01* K=1 00131 .2* NErUNN 00132 e311, 2o (:0pH1=-GICHC/2.)/SRTC-CACI 4**31/27.). 0013J ,4* SI,HI=c,m7(1. -rnPH1**2) 00.34 .5* LAit PHW,C(C0NT*S1PHI.RItoHI) 00135 Am-, =2.n 4 S0.(TC - ACRC/ 3 .) 00i3h .7* X14-7:AmP*COS(PHT/3.) -P/3. 00137 X11=60 00.140 r9+ 2..,=ArP*COS((PuT+6.2831/153)-/3. ) -P/3• U0i41 A21=u.n 00142 ,11* X31//- AMp ► COS(Wwi+12.566370A)/3.)-P/3. 00143 A317.0.0 U0[44 60,4b ,14* 00140 ,45* 3n iF (11W-4(I-1) •LE. 1.E-201 GO To 50 00150 .)6* HAI1SO=C**2-4.*p*D 00151 .s71, IF (hAnSn .GE. n.) GO TO 4n U0153 .18* Plo,.7.SOPT(-NADSra) U0154 09* X1.4=-C/(:.J.443} 00155 40* xl(=RAo/(2..*13) uuk5n 41* A2./=X1R u0i57 42* X2)=-X1I U0160 43* 00161 44* HErUNN uU/62 45* 4n PA.)=SOPTIRADS01 0016J 46* X1,=( - C+11A0)/(a.*(3) 0(1 164 47* X1T=0.n u(),()5 48* X2., =I - C-RAD)/(2.*(3) uui60 4q* A2r=0.0 00167 nO u0 1 7u )1.* DOL71 5n IF (Anc.(c) 1.E-201 GO To 60 uU170 !,3* Al,=-D/C 00174 :)4.0 0111 T;) 00/71, , 41* HF ono) u0177 6,1 IF (ABc(n) .GT. I.E. -20) GO To 70 0.01 ,* K=o, 00.02 7n K=7 Li 0 r.03 00* pEriaiN i0.04 ml* ...... S•• R-:'o ...... DATE 121074 PAGE 2

F. 1VU OF ~ O,.lill L 1\ T f Ol'~ : ,.. IlliG ,V ... + ~...... ~Uu') 7 ...... ,.fuH,S 1.S.. li':)1,SIJu..,7 ~O~ ~11F-l~/l0/7~-lB!Oq:h7 (~,)

FN rfn PuTN T 'OOnO 00

STO~~hE U5EU: COD~I,) O~OOOnj OATAln) 000000: 01 ANK COMMON(21 On0621

uOI).3 orW,/R 1.10114 HPlh, 1)0 11') DAOuR ULlIIE' fI~AJ\

1,0 110 R OOOUOIl A IInliO I~ OnOn.' (, I\K X nono R 00nUn7 AKY 0000 R 00nO<;3 AOr<1 0000 R 000051 C IJOJ)~ H OOOuOl Cl nooo R OnO(111 11 [;/lflK nonO R non034 nADI\X 0000 Il 0000110 OflDKY 0000 R 000030 DI\OQM lIOIiO R Of IOu 54 111\()~ll)K llOu 0 I( I)nOl?~ 0l.LKH nonO f{ 00 til ?,/l nEPlJOM 0000 R 001)123 OEP<;DK 0000 H 000115 DFIDK 00 .,0 n '01'0117 ilF l On:·) 11000 H unOI"'£> "nlD,\ nOOO R oon10r, nFHuO'~ 0001) R 0001n3 OFpnK 0000 R 00010~ DFPDQr-l tlOIiO f\ o (lIH 1,+ I/rt,D~ UfJJH) n 0001 H, uFfW(,M onllo F! 00(1077 nFZL)K 0000 H 000V76 DFznO".l 0000 R nOO061 OG2DK " t'ol 0 H o n~lLJl,;l I )(i,'UI)f\1 (II' fllJ n onOl1n IJ en!!)" 1·1 nooO P flOl,107 nor.1UKP 0000 R 01)0024 DHDK 0000 R 00001£> onOKX 1)11110 H ,)0(1112'1 Ill1uK Y n(lnO I, 00Un1n uilf)Ot~ nOnO R OOOl?? Frs ~oon H on0113 FI 0000 R 000102 FMNUS UOill') H OOOtOl FI'LUc;. 1I0no R Ono,11 FI< nOIlO R 00nu7~ ~Z 0000 R 000100 G 0000 R 000 112 GI (,OliO R (lnOu57 (,2 unnll H OOOno£> u2rJI\K2 nonO R OOOUn~~ {~2DOr.12 0000 R 00n065 G20~'AK 0002 000455 HI I)Ono I OOOU7d I Ilnll2 I OOOnOO rrl,l\x nooo I (l00n71 .1 0001) I 0001lr;0 I0 ",II.KO,., nanD R OO,ll?(, I/0r~ nooO R 000052 VPAR 0000 R 000073 VX I,OJ)~ H OOtl J.4:1 \lXI l'IOnO Il OOUn71f vY nO()2 R 00il311 IIYI 0000 R 000060 XED

.....---....,..,-..

liVID:' U o! (l't L01O~ 4,'- uO!OO 5t lJUl07 6'- uUtll1 7t uOlll fH "ud;:' '.Ii' "U! 1.:i .01

tr/ 12* CI' H7.-LCPJ2.-0A0 au 12o 13* c4,,,Az:sT6N(1•nirPACH)*(AnS(rPACO))**(1./3.) uji21 14* L1toriST6N(1.0PrPtiC0*(AIIStrP(ICH))**(1./3.) o0122 15* x1p=ChpA+CAPO - 2/3. 00123 16* X1r=0.n Al24 17* ;42a=-CcAPA+CAPrO/2. -P/7). J0i2b 8* X217-(CAAPA-CAPIA/2. ) *1.732A51 Al2r1 i9* x3•=A2P )0127 -0* X31 - -X21 J0i3U el* K=1 J0131 /2* PErUNI1 /0i32 2n LOpH1=-(HCHC/2.)/SR7(-(ACwC**3)/27•) )0136 /4* SI.JIII=c6)0(1. - reTHi**2) ;01.4 /5* 1 A11 Pru,sE(CONT.SIPHI,R1.PHI) I0i3b /6* AM,J=2.n*S(MTI - ACRC/3./ Jui31-1 :7* x1,.1AMP*C0S(PHT/3.) - P/3. J0137 JU .06 /9* xp,.:-. Arp*cOs((P.474-6.2s311153)/3.) - p/6. )0141 .10* X21=0.n Ai42 .11* x3:4740ip4c0S((PHI+12 , 5663706)/30-P/3. )U143 X31=0.A J0144 ,u145 uElukn J01 ,40 0 5* 3n IF (Afic(H) .1.E. 1.E - 20) GO To 50 4150 HAw7,(4=C*2-4.*R*0 J°151 .17* IF (mArISO .GE. n.) Go TO 4n )0153 .10* PAI,=SOoTI-RADSM ;01.54 09* i0i5h X1i=PAA/(2.*11) /u1511 41* A214=X1R /0157 X2i=-X1I 10160 .3* K=4 ,0161 .4* wElUkti IU 162 t.5* 4n kA0.7.5011T(RAD;0) 10163 01164 .7* x1T=0.n U165 .8* x2,=(-c-NA0)/(2.*(3)- Ut6r) A2t=o.n 6167 nk h=a 017u PElliktq 01.71 ,2* 5n IF (i0s(C) •LE. 1. E-20)- GO TO 60 017i xlw=-D/C 0174 J4 t x1r=n.n U175 01.7b HE rUHN 01.77 un IF (AHc(n) •GT. 1..L-20) GO To 70 0eAl K=0. J/'02 09* 7n K=7 0,03 •0* RETURN 0004 ml*

moomt2,2110RUKX(2,2),DROKY(2 , 2),DROK(212),DADOM(2,2) 4 DImENSION DADKX(2,2)tDADKY(2 ► 2),DADK(2r2)fRPPP(2 , 2),WAK(2v2) 5 DIMENSION DOm101) , DUP2121) 6 2001 FORMAT (III 0HPUIFROXOHF7.=,815.8 , 3X,6HDFZDK: ► G15e8,3X, - 7 17HOMOM=s015.8r3Xt9•RPP(It2)= ► G15.8r3X , 6HDG2OK=1815.(1) 2002 FORMAT (1ii pSHPOIFRf3XPIOHDRDK(lt2)=t815.80Xt11HDRDOM(112)=r• 9 IGIS.A13X,5HG2: , 615.8) 10 2(103 FORMAT (111 rhHPOIFRr3X,4HEP5.7, 315.8r3X,6HDELK8= ► 315#8t3X, 11 14HVOM=f615.(3) • 12'. 2004 FORMAT (1H ,SHPDIFROXvIOHDRDK(1,1)=PG15.8r3X0OHDROK(1,2):r815.8 13 1,3)( ► IOHDRDK(2,1)=rG15.813X,10HDRDK(2t2)= ► 815,8) 14 2005 FORMAT (1H 0)HPDIFR/3Xr9HRPP(111).= , 315.13,3X,9HRPP(1,2)=:315,8, 15 13As9HRPP(2p1)=r815.813X,9HRPP(2121=IG15./1) 16 . 2006 FORMAT (1H thHPDIFROXOHA(1,1)=,(515.8,3X ► HA(1 ► 2)= ► 815.8,3X, 17 110HDADK(1,j)=1615.80X,10HDADK(1/2)=FG15.8) 113 2007 FORMAT (1H P5HPOIFR/3X,6HDFZDK= ► 815.8) 19 2008 FORMAT (1H t5HPDIFROX,9HWAK(111)= , G15.8,3)09HWAK(1,2)= ► 815.8 , 20 13Xt9HWAK(2,1)=1(;15.6(3X,9HWAK(2,2)=,815.6) 21 COMMDN IMAX,CI,VXI,VYI.H1 22 KUP=IMAX+1 23 C=CI(KUP) 24 VPAR=VX1(KUP)4COS(THETA)+VVI(KUP) ,rS/N(THETA) 25 Bum=nMEGA-VPAR*AK 26 DI (1f)K=-Vi'/,R 27 0,44=.0096 -cSORT(.4)/C 28 OMA=.0096*0,7/0 29 G2=(0MA**2-(3oMs+2)/C**2.-(0m8**2-80M**2)*(AK**2)/00M**2 30 XED:(0MH*AK*C/BOM**2)**2 31 062DK=(-2.*BnM/C**2)t(1.-XE0)*0B0M0K 32 1-2.*AK*(0M8**2-HOM**2)/80M**2 36 D6PDO1=(-2.*ii0M/C*4- 2)*(1. - XED) 34 G2D0M2 7-(*.2./C* 4 2)*(1.+3.*XED) 35 G2DAK2=(-2aC**2)*(10. 3. 40(ED)*DR0MOK**2 36 1+(8.1, 0M1112)+AK+DOOMuKnoMt*3 37 1-2.*(0MR**2-ROM+*2)/UOM 4 42 33 620•AK=-(2,/C*t'en*(1.+3.*XED)*DBOMOK+(4.*AK/BOM**3)*Oma**2 39 AKX=AK*COS(1HETA) 40 ASY=AK*SIH(THETh) 41 CALL RRRR(OMFGAIAMOAKYOPP,KWH) 42 CALL ORDOR(0MEGArAKX$AKY/RPPrA,DRDOM,ORDKX,DROKY) 43 DO 7n 1:1,2 44 00 70 J=1,2 45 70 ORDK(I,J)40RDKX(I,J)*COS(THETA)+DRDKY(1 , J)*SIN(THETA) 46 WRITE (6'2004) OROK(1o1),DRDK(1r2) , DRDK(211),DROK(2 , 2) 47 3LLTA=3.*10E-.4 40 AKXP=AKX+DELTA 49 CALL RRP(OOMFGA,AKXPtAKYPRPPPIKWH) 50 CALL RRRR(OMFGArAKXthi(YrkPP,KWH) 51 WRITF (6,2005) RPP(111),RPP(1 , 2),RPP(2,1) , RPP(2 ► 2) 52 VA=VXI(KUP) 53 VY=VYI(KUP) 54 CALL DADOR(OME0A,AKX(AKY,C,VX , VY,DADOMPDADKX , DADKY) 55 • DO 9n IL71,2. 56 DO 90 J=1,2

1 .1. j 4vAuom(112) 62 1-DROOM(112)*A(10)-RPP(1,2)*UADom(111) 63 OF7DK=OROK(1,1)*A(1•2)+RpP(1,1)4DADK(1,2) 64 1-UROK(1,2)*A(111) - RPP(1 ► 2)*DADK(1•1) 65 D6 c III=1'7 66 DO 9A JJJ=117 07 93 wAK(III,JJJ) = (RPPP(III,JJJ) RPP(III ► JJJ))/OELTA 58 WRITE (61200(1) 4AK(il1)twAK(1,2).WAK(2,1),WAK(2,2) 59 WRITE (6'2007) OFZDK 70 L=1 71 IF (G2 .LT. n.0) GO TO 150 72 L=0 73 G=s0i1T(G2) 74 FPLUS=F7-Rp)1 (1,2)*G 75 FMNUs=F7+8PP(1'2)*G 76- OFFOK=DPZDK-DROK(1,2)*G-RPP(1.2)*DG20K/(2.*G) 77 DFmN=DFZUK+OROK(1,2)*G+RPP(. ► 2)*DG2OK/(2•*G) 78 OFPDoM=DF7_D0m-O.RDOM(112)*G-RPP(1,2)4OG2D0M/(2.*0) 79 OFmDOM=DFZOOm+DROOM(1.2)*0+RPP(1,2)*OG200M/(2.*G) au DOmDKP=-DFpOK/OFFOOM 81 DO:.1001=-DFMDK/OFMOOM 82 GO To 2n0 33 150 FR=F7 84 GI=SoRT(-G2) 85 FI=-RPP(1,2)*G1 86 OFROK=OFZDK 87 TJFIDK= - 6ROK(1 , 2)*GII- RPP(1 ► 2)*DG20K/(2.*00 88 DFROum=DF2DOm 89 DFIDoM=-DROOki(1,2)*GIA•RPP(1,2)*DG2DOM/(2.*GI) 90 200 IF (0G2DK .LT. 0.0) LETS= - 1 91 IF (nG2nK .GT. 0.0) LET$=1 92 ULER=FZ/DFZDK 93 U0m=nF7.00m/OPZOK 94 EPS=HPP(1,2)*SORT(A5S(DG2DK))/DFZDK 95 DEPSDK=DRUK(1 , 2)*SORT(A8S(DG2OK))/OFZOK 96 0LPD0M=DRO0M(1 , 2)*SORT(AB5(Mi2OK))/DFZOK 97 DELKR=-62/DG7DK 98 VOm=nG2DOM/DG2DK 99 VOmOm = 0.5*G2Dum2/0020)( 100 VAKAK=0.5 4 G2OAK2/DG2OK 101 VAKOm=G20mAK/D02DK 1u2 WRITE (6'2001) h'Z ► OFZDK ► DFZDOM,RPP(1,2).0G2OK 1u3 WRIT( (6'2002) DROK(1 , 2),DROOM(1,2)/G2 134 WRITE (612003) EP5 ► DELKU,VOM 105 OUm1(1) = FZ 106 DUm1(2) = FPLUS 107 DUm1(3) = FmNus 11)8 DUm1(4) = DFpoK 109 OUM1(5) = DFMDK 110 Dum2(1) = oF(JDOA 111 DUm2(2) = DFm00:A 112 OUm213) = nomOKP 113 DUm2(4) = 00m0KM Li/ nUm2(8) = DFTOK 118 DUM2(9) = DFRDOA 119 Dum2(10) = OFIOOM 120 nUm2(1I) = GP 121 num2(12) = U7ER 122 1)Um2(13) = UnM 123 DUm2(14) = EPS 124 OUm2(15) = DFPSDK 125 DUm2(16) = DFPD0M '126 DUm2(17) = DFLK3 127 1)Um2(18) = VOM 128 DUM2(19) = VOMOM 129 DUm2(20) = VAKAK 130 OUm2(21) = VAKOA 131 RETURN 132 END QF1N

RUNIC): MEAKD ACCT: 061A1019 PROJECT: INFRASOUND TIME: TOTAL: 00:0n:02.065 CPU: 00:00:00.069 00:00:00.3S4 CC/ER: 00:0n:01.612 OTT: 00:00:00.000 IMAGES fEAD: 3 RAGES: 4 START: 15:33:00 JAN 09.1975 FIN: 15:33:15 JAN 09.1975

OtmLNSION DITA1(!APPU' 4 2(21) 1001 FoRAT (11i fr,1-(2 ■;HLK03X.GHOMLGA=.G15.3 ► 3X.4HAKR= ► G15.8,3Xt 7 14HAKI=.G15.1) b 1012 FOW0, AT (1H .:',HRC•LK.3)04HFRP= ► 015.8r3Xt4HFRM= ► G15.8•3XAMFTP= ► 1G.I.A , 3Xt4HFTM=.(519.6) 10 1022 FOW:4AT (1)3 r!IHR(.IALK•3X.3HFP=iG15.8 ► 3X ► 3HPM=.1115.8,3X. 11 1hNoUPST=.51.h) 12 1003 FUW,:AT (1H tY(RLHCK , 3X.611FPLUS=.G15.8)3X.6HTIMNUS=PG15.8,3X, 13 17HrIOMDKP=.G1! ■ .-3.3X.7HDOMDKM=oG15.8) 14 1004 F0V4AT (1H .tiHR4HL1t.3X , UHVGINP=rG15.8) 15 CALL POTFRComEGAOKR.THEIA.L.DUMI ► LETS.DUM2) lb IF CL .FO. 0) (3U TO 30 17 FHP=nUm2(5) - n(rm,)(a)tAK2. la FRm=OUMP(5)+IIUMP(b)tAKI 19 FIP=OUMP(6)+nUMd(7)4;AI 20 Fim=-DU).12(7)+OUP(7)kAKI el 10 PP=SoRT(FRP**2 4- FIP ♦ t2) 22 Fi•1:SoRT(FRPt42fPIP**2) 23 odFST = AHS(AK11 24 4RIIF (5,101P) FliP ► FaM.FIP,FIM 25 wRITF (6'102?) FP.FM,OUEST 2b IF (OUEST .LT. 1.C-20) GO TO 70 27 , 12 IF (FP .GT. PM) GO TO 20 20 13 LH=1. 29 FINE5=FP 30 RLTURN 31 20 Lwz.2 32 FRFS=FM . 33 RETURN 34 30 OUFST=AT1S(AKT) 35 IF (()WEST .or. I.E-20) GO TO 80 31 FPLUS=DOM1(2) 37 ENNOS=DUM1(3) 33 OomDKP=DUM2(3) 39 OO•DKM=DUMP(4) 4U WRTTF (6.1003) FPLUS.FMNUSOUMITKP ► DOMOKM 41 IF (Domokp ,LT. o.8) GO TO 5U 41 IF (nOmnKm .(1'. 0.0) GO TO 40 FP = AilS(FPLHS) 44 FM r- 11H5,(FMNH5) 45 G) TO IP 5 40 FP = WI(F('LOS) 47 GO To 13 43 50 IF (OOMOKM .(.T. '0.0) GO TO 8U 49 r r. = ABS(KMH05) 5U 00 In 2u 51 70 VuTNP = -flUM2(7)tDUM2(9) DUM2(A)*DUM2(10) wklIF (f,•10n4) v61NP 53 IF (VGINP .GT. 0.0) GO TO 12 54 30 LH=3 55 Ri:JURN 5u ENO uNm:, %o)*Num2(1()) V./1\ 1- t uu1S7 on* 'GT. 0.0) GO TO 1? • lUlril ,9 . bn

tai l 4, • 1114 1 )101 • 1,01o.T. .1v

ENU OF COMPILATION: NO nIAGNOSTICS. 1..1116.P 44#4 ,4* sUlimu *44.4** 1.=PJR , S I.S"300.SuNnO . slIF-1./10/74-18:1n:118 (0 ► )

FuNcTIO- FNmON2 FNIRY POINT 00no24

SiORA0E USEU: Co0011 0.1003(14 UATA(n) 000011: MANK COMMON (2) On0624

ExT;;RNA. r,EFERENCLS ((ILHCK. NAME)

u 0 o3 NYD1.N u0o4 NrR,c3$

SiOkAtvE A5SIGNMi.NT (10...HCK. lyPE ► R-(.ATIVE LOCATION. NAME) u0112 00001 CI °Orin R OnOnOn ENMON2 0000 R 000003 rPP 0002 Onn455 HI 0002 000000 IMAX o0110 000u05 IMuP% 0000 I 000n04 K n000 I 00000? 0002 000621 OMEGAC 0002 R 000623 THETK uOun R 0n0u01 v nen2 R 000c,??. vPHSFC nOu2 000145 vXI 0002 000311 VII

u010U 1* C 1:N01,2 (FUNCTInv9 " 6/19/68 LAST CARD IN DECK TS NO. LAU.% 2* C LIUIOU 3:*. c vOlOu 4* C ---0,ADSTRAcT---- QuiOu •5* C 1.1.1i0u 6* C TITLE - FNmo112 tilOu 7* C FVAIQATATIOm OF NORNAL mODE DISPERSION FUNCTION AS FUNCTION OF uOlOn 8* C ANGuLAR.FRENHFNCY Ot4EGA u0k0u 9* C uUlOu 10* C THE NnPmAL mOOF DTSPERSION FUNCTION DEPENDS ON THREF VARI- OolOu ,1* C AIILFS. ANGULAR FRI:00ENCY OMEGA. PHASE VELOCITY V ► AND u0100 .2* C DIRECTION OF 1-90PAGAT1ON THETK. FNMUD2 OBTAINS OMEGA • 6040u 13* C TH::0UnH ITS AI.GOMFNI.v AND THETK PROM COMMON. SUBROUTINE LOIOu 14# C NADFN IS THEM CALL En TO EVALUITF THE FUNCTION. (SEE u010b '54 C PIERCE. J.CoMd.PHySICsr FED..19(.7, P.343-366 FOR DEFINI- Uulnu _16* C TION nF MoRMin mO(.E DISPERSION FUNCTION•) LOI.Ou .7* C u1i).00 04 C tAmONA-F FukTPpN IV (3nn. .iEFENENCE MANUAL CU-6515-4) LOLUti 19* C (JOAO() el). C AUTHWN - A.D.('TFRCL ANN dipOSEY ► M.I.T.• JHNE•1968 0.1104) .11 C vviou .... et* UOIthi e51 FM..01)2 CALLS SH9ROUTIW NNnEN WHICH CALLS AAAA AND RRRR. RRRR 00100 ., 9* C CAtLS AAAA AND MMMm. ALL THESE SURROUTINEc ARE DESCRIBED ELSE- 0010u .10* C wH“Ir TH THIS c•RIES. U0100 .'11* C 6010u _12* C CAiLIN SEoUFNCE uOiOu '..13 4, C U0100 .-IF"* C CO-MON CM1(4021tV , THETiC U0100 .15* C nM.GA = XXX uOIOU . .16* C V z.. xXX 00106 ...,7+ C T14- TK = XXX . 00106 • ,5 C 1-"U,•CTN 7. FNMOD2(0MLOA) . o0i0o ..19* C 00100 40* C INpUlS 00100 41* C UOIOU 42* C V PHASE VELOCITY (KM/SEC). 0010u 43* C R•4 00i00 44* C UOIOU 45* C OmEl;.A ANGULAR FREOUFNCY (RADIANS/SEC). U0100 46* C R.4 00100 47* C 00100 45+ C• THEO( PHASE VELOCITY DIpECTION MEASURED COUNTER-CLOCKWISE FROM 00100 49* C R•4 X-AXIc. 00100 nO* C U0100 nl* C OUTPUT ,: UOIOU N2* C 00k0U "031, , C TitE ONLY oUFPUT IS THE vALUp OF THE NORmAL MODE DISPERSION FUNCTION uOiRu 1,4* L FuR THE VALUES nF VP OMFGA, AND THETK 41tICH HAVE BEEN INPUT. u0100 :)5* c UOiOU :16* C 00100 )7., C ----PROGRAM FOLLOWS RELOw---- u0100 t)E3* C UOIOu :,9* C 00101 .,0. FUtoClIoN FIT10D(OMEGA) 00101 ra* C 00104 t.2* HImENSTON CI(lnn) , VXI(100).VYI(100) , HI(100) 00104 t,3* C001MoN ImAxPCI.VXI,VYI.HItnMEGACIVPHSEC,THFIK 00104 .,4* C 00104 n5* C V AND THETK ABUAINPD FROM COMMnN 00105 I,6i V z VP•FC 0010n n7* CAiL NmUFN(OMEaA. V ITHETK.L.FPP,K) 0u:07 t,9* PNAD2=FPP 0011U h9* kErURN U0111 /0 I, FN, )

END OF LOmPILATTON: NO nIAGNOSTICS. ' HuG' P **1' .44* SUH81 *****4 oFURPS 1.Su001,SOUril SlIF ..1 ,410/ 7 413: 1 n:12 (OP) m i..triqmuNt21 0n0621

---- EXTONAI REFERENCE:5 (HL“CKF' WOE) 0003 NMOPN 00u4. NEE 13$

STOliAt*E ASSIGNMEUT (BL6CK. TYPE. RFLAT1VE LOCATION. NAME)

0 0 o1 0.00u47 111G uOnt 0000,7 1170 0001 000107 125G 0001 000127 1316 0001 000251 80L 0)61 00040u 91_ oOn?. On0001 CI nOn0 R 00000n nELVP now) R °noon, FPP 0002 000455 HI 0000 I On0u03 I 0000 1 -0001106 1J 0000 I 0 00007.TJOLD 0002 000000 IMAX 0000 000013 INJP$ 0000 I 000u01 IP 80.00 I 000n05 J 0000 I 000004 JP 0000 I 000012 K 0000 I 000010 L U 0 00 1000001 N2 ((002 000145 VXI 0002 000311 vYI

00100 1* C LN...ThN (SUoRQUTINE) 7/19/68 LAST CARD IN DECK IS NO4 3 U0100 2* C 0001 (Attic, 3, C 0003 uUiOu 41, • 0004' uu1.00 5. C 0005 0)100 6* C T ITLF - LNGTHM 0006 uUlOu 7* C • liNGTHEN THE MATRIX INM/IEJF HY ADDINti xL ROWS BETWEEN THE Ni AND 0007 u0i0u 8* ,C N1+1 0008 u0100 9* U °nog uUlOo 10* C LNGTHN At.)uS K1 ELFMENTS TO THE VErTOR OF PHASE VELOCITIES 0010 u0iUu C • V ' DIVIDING THE INTERVAL BETWEEN V(N1) AND V(N1+1) INTO 0011 00100 12* C FruAL PARTS. FOR EACH NEW PHASE VELOCITY' A NEW POW 0(112 (JOIOu 13* C IS ADnED TO THE ImMoDE MATRIX (DEFINED IN SUBROUTINE, 0013 uOiOu 14* C MPOUT). iNUOIIE IC STORED COLUMN flY COLUMN IN VECTOR FORM. 0014 u0100 45* C 0015 UOtOu ,G* C LANG0ANE FoRTRAN IV (36n, REFERENCE MANUAL C28-65I5-4) 0016 u010u J7* C A.UTHuR J.W.PnsEY. M.I.T.. JUNE/1968 0017 uu.i00 L8* C 0018 uU100 *9* C 0019 uC1OU e0* C ----USAGE---- onn U010U el* C 0021 U010u 2* C OM.V.INMOt.:E MUcT DINII7N5yONED IN THE CALI ING PROGRAM 0 0 22 U0100 e3* C NM.iFN T. ONLY cURRJUTINE CALLE0 0023 uOIDu e4* C 0 0 24 u0100 e5* 1.0RikAm USAGE 0025 on26 L,UtOo C EAIL LNGTHN(OM,V,IMIODV,NOMFNVPINPPINIJKL,INETX) U0101) 17* C 0 0 27 U0100 e8* C 1NPUIS Don G0100 e o* C 0029 (JOiOu C Ot. VECTOo wHuSE vLFMFNTS ARE THE VALUES or ANGULAR FRE-. • 0030 uulOu •1* C R.4(0) CoENCy COkRESPONDTNG TO THE COLUMNS OF THE INMODE MATRIX. 0031 1.010t.;. C 0032 u0100 03* . C V VECTOR WHuSE FLEMFNTS ARE THE VALUES OF PHASE VELOCITY 0033 00100 A4* •C R44(D) CoRREsPoND1NG TO THE ROWS OF THE mMODE MATRIX. 0034

0 1 39 mvl p.X1JT , THE FIEMENT IS 5. IDeHAS 0040 N4P RoWS (INCREAGFD To NVPP) AND NOM COLUMNS. ,MATRIX IS 0000 42* ,.. SIOREn IN VECTOR FORM COLUMN AFTER COLUMN. 00100 43* C NON THE NWRER OF ELEmENTS IN OM. U0100 44* C I*4 uOIOU C NVP THE NeRER OF ELEMENTS IN V (WHEN LNGTHN IS CALLED). 0°02 u01.0U •..•..46* 45* C • I*4 00100 47* NI : .. NuMRE* or INMoDE RCN IMMEDIATELY ABOVE SPACE IN WHICH NEW O°0 :n 4 74 U0100 48* C 1*4 ROWS ARE TO BF ADDED . \ polo u0i0u 49* C 0049 00100 t)0* C KL NuMREp oF ROWS TO BE ADDED . 0050 U0100 :21* C • I*4 • 00100 !)2.* ••• C T HETK PHASE VELOCITY DIL4FCTION (RAOIANS) 0010u .,3** C R*4 U0100 ntis, C 2 00100 n5* C OUTPUT ,: 0055 uU100 •,6*, e 0n56 u0100 b7. C THE OUTPUTS ARE ;WM'. (= NVP + KL) AND REVISED VERSIONS OF V AND 0057 U0100 ._• n8*C INWIDE. 0 0 58 u0100 :,9*C 0 0 59 0u.10ii ,0* C 0 0 60 00100 - ,1* C ---- EXAMPLE ---- U0100 02* C uUiOu e,3* C VALUES OF 1,40Dc:- NOT VALID -- FOR ILLUSTRATION PURPOSES ONLY U0100 ,4* C 0010u A5* C ' V.=1.0 • 2.0 HH v010v nt.y* C OM=1.(Ii2.0 On66 00100 A7* C INMODvn1,-1.-1/1 u010u ,f-ii C CALL INGTHN(Om.VITNmOi/E.2,2.NVPP/1/31THETK) On6S uOlOo A9* C 0069 uulOu /0* C UPnN RETURN TO CALIING PROGRAM THE VALUES OF V AND NVPP ARE v0100 ,1* C V=1.0.1.25t1.S.1•75,2.0 0071 00100 /2* - C NvPPLIs 00100 ,3* C 1NmOnE WM' RE OF THE PORM 00100 /4* C . INMOOF=1 , Y , Y•Y•-1. - 1 ► y•Yfy•I u0i0o /5* C wHpRF IHE vIS ARE NEW ELEMENTS. EACH OF WHICH MAY BE -It 1. 00100 /6* C OM 5 uolOo 17* C UOIOU ,3* C oRTGINAL MATRIX EXPANDED MATRIX LULU /9* C vUlOu AO* C + r +_ 00100 nl* C Yy 00400 ,.2* C Yy JUlou A3t C Yy • ..4. Jolou t ■ li 4' C JOlou 054. C iolou et6* C 0075 'U1OU #.7* C 47-PROGRAM ;;01.LOWS HELOW-- -- 0:0 77: 010u A8. G c riOu A9 * C 0077 ' 0101 40* SURROUTINE LNGTNN(OM,V.INMnDF.NOM , NVP•NVPP,N1 , KOTHETK) 0101 41* C

--...,...- vet _-- --t.,... OEtVP = (1/(41+1)-V(N1)/ / (KL+1). w0106 46* C bEtPV oS THE LITEUNAL OF PNAS wELOCITIES FUR THE AOOEU ROWS. uOlno 47* NVUP = NVP + K1 U0100 -48. C NVIJP I4 THE NE./ NUmrER OF kOWS IN THE TOTAL MATRIX, U1/100 79* C OUIU6 1110+ C N2 IS ,..Ed Num001 OF oLu ROw NO. (NI+1). 0082 UUt07 1.1* N2 = NI + AL + I U0/07 1.2+ C u ti4tri 1.3* C SHTFT 0LN VALULS Op V(I) IN LOwER ROWS TO 1+KL SPOT9 ONE. HAS TO LUI07 1.4* - C SHIF1 THE NVP LLFMrvT FIRST, NOTE THAT I RANGES FROM NVPP TO N2 1J0i07 1.5* C DOwNwAPD WHILE I-Kt RANGES FROM NVP TO N1+1. Oillu 1.6* 00 71 TV =N2pNwPP - 60116 1.7+ I -t. NVPP - (IP•m2) • uOilL. 1.8* 7 1 . v(t) = ,t(I-KL) uu4 I. 1.9* C u0114 1,0* C NEw VA, tk.S of 40 Apr INSFRTED INTO V 0084 Uu116 1,1* 00 7? T0=1 , KL PU.21 112* I v. NI f IN UU122 1,3* 7P 11(1) = v(t41) + IP*DELVP UO2?1 1,4* C u0122 1,5+ C tiEt:INNTNG AT THE RIGHT INMc)PE TS LENGTHENED COI UMN SY COLUMN 0090 vul24 1,by 00 9ti JP=1.NoM uU127 1)71- tl 7- NON - tJp-i) 4013,+4013,+I.A* 00 90 tv.7.1•NVPp uu134413.; 1,9* I = NVpv- (IP - 1) uU13i 1,0* C u01.331,1+ C 1H1; IJ ELEmENT IN T0r INMOOE VECTOR IS THE j EIEMENT IN THE I ROW OF 0095 uu13.) 1,2+ • C THF NEB.. .(NtAchr MATRIX 0096 00134 1,3* IJ = (0-1)*HvPn + I oU134 1,4* • C . UU134 1,5* C IF I CNPREcVnN0:; Tn A (JEW ROW INMODE(IJ) MUST RE DETERMINED FROM NMDFN 0098 uU136 1,6. IF (I.GI.NI.AM.T.LT.NP) Gn TO 9 u0131i 1,7. C uo13t, I,n* C IUALU IS (JO. OF- art ,FNI IN OLD INMODE VECTOR WHICH IS TO DE MOVED INTO u0136 1,9* C IJ PoSrTION oF MEW VrC1OR gIN 40137 1,,..0$ IJ.L0 = (J-1)*t.P.11 + I uU137 1.,1* C NOTE T,.AT TOLD IS AI wAYS I IF T .LT. Ni RUT TO1 D IS I-KL IF I .GE. N2. 44137 1,2* C 1JoLN tS Cof,PUlEn nN THE SasIS OF NW' RATHER THAN NVPP ROWS. uOiqu 1•.3* IF (I.RC-:.(J2) I0OLD = I.JOLD - KL tU141 104* IM.AOHF(AJ) = INr000IJOLD) u0 1.43 1.,5* HO TO AO utiiI,) 1,6* C L0.5144 1.-.7* (4 CALL NmOrN(Om(d)0,(I).THETK ► LIFPP , K) uO444 1.-,At C LIU144 1,9. C 1E 1,-FT LAIgN L = i ANu INmODE(IJ) = (FPP/ARS(FPP)) 0110 01( 10). 140+. IW•fl01:(iJ) = 1 u0140 14•* 11 (L.ro.1.AND.FPP.LE.0.0) INMODE(IJ) = -1 wOilIN 1.44 C vOCNI 1.3* C IF FPP Our, NC;' .-'-77 L = -1 0109 iiii15o 144* IF (L.Fo.-1) INMO%(IJ)=5 uui5u 1.5* uUit),:. I.1")* 8n LoTINIve 04i5.5 147* 9n co•JTINIJF u015o 1•D* AEr040 to citi(7 , r 4-4f** suR82 ****** 10:oRtS t.S,Bra.S0382 f•UR SlIF-1,/10/74 - 18:1n:18 (0p)

SuSRO(,TINE MMMM ENTRY POINT 00n076

srokA6E.usEu: C0001) 0601214i•DATA(n) 000027) 8IANK COMMON(2) On0000

ExTKRNAI REFERENCES (141.0ck, NAME)

00.03 AAAA 0904 CAI u005 SAI u0a1 NOK31,

STORAGE AsS1GNMENT (HUICK, lYPE. RA- LATIVE LOCATION. NAME)

041 On0040 i1ZG nOnl 000041 1150 0001 000057 1236 0000 R 000000 A • 0000 R 000005 CA 0004 R 000000 CAI OW I 000nln i 0000 000015 TNJP1 0000 I 000011 J 0000 R 000006 SA 0005 R 000001 sAI non() R 000n07 TA 0000 R 000004 x

LOIOG 1* c mmmm (cunRoJTINF) 7/25/68 LAST CARD IN DECK IS NO. u0100 2* C 00100 3* C - ---ABSTRACT- - - 00100 4• C utl100 5* C TITL .F. - MMMi uUlOU Cyt C THIS SUBROUTINE COMpHTEc THE 2 -BY-2 TRANSFER MATRIX EM WHICH 00100 7* C CONNECTS THE SOLUTIONS nF THE RESIDUAL FOUATIONS AT THE TOP 00100 8* C . ... OF A LAYER TO THOSE AT THE_ HOTToM OF THr LAYER 3Y THE RELATIONS 0010u 9* C 00100 10* C pHI1(7B)= EM(1.1)*RHII(7D+H)+ EM(1.2)*PH72(a+N) 00100 11* C uOIOU a* C THI2(7.11):: Em(2.1)4 7 41(711+H).1. Em(2 , 2)*•HT2(Za+H) u0100 13* C 00100 14+ C wHaw Zii nrmnTES THE HEIGHT OF THE HOTTnM OF AN ISOTHERMAL 00100 /5+ C , lAYF‘i (tHICvNES5 H) WIT'4 CONSTANT WINDS. THE QUANTITIES u0101, 16* C . PHI117) AHD PHI2(7) SATISFY THE RESIMAi EQUATIONS+ u( LOU 17* C ouinu in* C n(PHI1)/n7 = M1.1)*0H1117.) + A(1,2)4PHI2(Z) 00100 ,94, C u010,1 :.0* C n(PHi2)/nz = A(2.1)*;,HI1(Z) + A(2.p)4PHI2(Z) u010o .1* C 00100 ,2* C WHERE THE Aft+J) ARE CONSTANT OVER THE tAYER AND WHERE 0010u' ,3* C A(2,2)=—A(1.1). ON THIS nAsIS• ONE CAN SHOW THAT u0i0u ..4* C 00100 .5* C rm(1,J) =CAI(X)*KOELTA(I.J)—H*SAI(x)*A(I,J) .... c uOlOu AO* C AND WEkE KnrITA(ITJ) lc THE KRONECKER nELTA (1 IF INDICES U010u .11* C. ENUAL, o OTurRW1SE). THE FUNCTIONS CAI AND cAI ARE DEFINED IN. uOlOu A2* C THE.IIFScRIPT/ON$ OF THE CnRRESPONDING FUNCTION SURPROGRAMS. uUtOu A)* C 0010u 04* C THE mATtaY a IS COMPUTEn FOR GIVEN FREOHENCY. WAVE NUMBER, SOUND 00100 AS* C SPEED. AND HIND VELOCITY RY CALLING SUBROUTINE AAAA. u0i0U 06* C oulOu 07* C LANGUA•4E - FORTRAN Iv (3,,n, PEFENENCE MANUAL C22-6515-4) uulOu 08* C 00100 - A9* C AUTHOR -*A.D.PTFRCEI M.I.T.• JuLY ► 19684 u0t00 • *0* C u0100 • 1+ C - ---CALLING SrOuENCE ---- uOlOu 42* C uutOu •3* C SEF SU..ROUTINEti HAmr1)E.RRRk . uOlOu *4-* C D/•ENST6m t:.•(2.2) . uU100 45* C CAiL MmNM(oNEGA.AKX•AKY.CluX.VY , HPEM) u0i00 ..6* C uOlOu . .7* C ----EXTERNAL. cUHRoUTINES REnUTRED--..-- uOtOu *A* C 001O0 *9* C 'AAAA4CAI4SAI 4.10i0o !,0* C uUtOu nl* C ----ARGUMENT t 1 sT--- - u 0 1 0 u ,,?.* C uu 1 00 •■ 3* C oMi.GA 1,44 IN-' uUj00 4. C AyK 10.4 IN n 00100 . !i5* C AKY R *4 flPc r() IN i.10.10D ”6* • C C •10.4 NO INp uLJlUu :N7* C vk R*4 ND 00100 :, 1.1*. C VY R*4 ND Itr,t uUtOu :, 9* C it R*4 NO U0100 #.04. C FM 2-°Y-7 10Plj 00100 '.1* C R*4 00100 .2* C NO CoMmnN sTOPAGE TS USED u0ADJ .3* C U010U . ► * C r---INRUTC----, 0010] .5* C UOIOU .6* C 0M...GA =A.0.1ULAR FliE0O•NCY IN RAD/SEC 00100 .7* C AKi t..x COMPONENT OE HORIZONTAL WAVE NUMnER VECTOR IN 1/KM u0i00 IA+ C AKY .7.Y COMPONENT OF HORIZONTAL WAVE NUMBER VECTOR IN 1/KM • b04.00 .9* C C =SOUND SPEED IN KM/SEC 0010u /0* C 4X :X COMPONENT OF WIND VELOCITY TN KM/SEC U010.) /I* C VY :-.Y roMPONENT OW WIND VELOCITY TN KM/SEC uOtou 12* C H =TwtCKNESS IN 01 OF. LAYER u0100 /3* C t,0 10o /41 . C - ---OUTpuls___" r utliOu i.)4 C utilOu t6+ C FM =2.PY -2 TRANSFER MATRIX WHICH RELATES THE SOLUTIONS OF u0100 .7*.7' C mr RLSIDUAL FONATIONS AT THE TOP OF A LAYER TO THOSE LJOIOu .t3* C AT THL BOTTOM OF THE LAYER OULOu /9* C oulOo ..0* c - ---nROGR4m FnLLOwS DFLOW---- u0i0u ..1* C UULUI /,2* SUHPUUTINE NmMm(OML:GArAKX,AKYtC,VX.VY,H,EM) LrAIL AAAniot'ILL,APAKX•AKYoCrwX.VY•A) 0010:1 X=IA(1.1)**2+A(to2)*A(Pr1))*H**2 00100 CAtCANX) 00107 SA=SAI(X) 00107 00107 42* C COMPUTI: THE TERMS —H*SAIDO*A(T,J) 00110 43* 1A.:.H•SA L10111 44* no 9u T=10. 00114 45* NO 9(1 J_1 r1 u0117 46* 9n EM(I,J)=—TA*A(T.J) 00117 47* C 00.1.17 48* C On IN CAI(X)*NDELTA(I,J) TERMc HY ADDING CA To DIAGONAL ELEMENTS 00122 no 190 1=102 0012!) 100* 19n Em(I.I)=Fm(I.I1+CA u0i2S - 00127 1„2, RETURN 00130 1u3* EN0

rND OF COMPILATTON: NO nIAGNOSTICS• le,HuG,P sUH8s ****** k.F0R,S 1.St,H3ISL1083 s1iF-1,410/74 - 18:1n:22 (120 .

suaRnTiNF mODETR ENTRY POINT 00n977

5I0kAGE USEu: CODE(1) 0n0331; DATA(n) 0000514: DIANK COMMON(2) 0n0624

ExTRNAi PEFENENCES (hLrICKI NAME)

00(13 FNMuD1 00n4 Er.!MuD2 uOu5 RTM1 u0h6 NYTPNT 0007 NFRk3D

STORAhE AGSIGNMENT (BUICK ► TYPE. RrLATIV• LOCATION. NAmE)

UQO1 101_ u0n1 000221 100L 0001 00011n I5L 0001 000112 20L 0001 000254 200L 0 0 01 000116 251. 6001 On0061 !)L n001 000161 r;LIL 0001 000100 81. 0002 000001 CI U0„0 R 000014 O'S 0006 R 0000.16 F 0003 R 000000 p- NMOD1 n004 R Onnon0 FNM002 0002 000455 HI 000 I 000u1/ TER 0002 000100 IMAX noon 000031 INJP1 n000 I 000006 ISM 0000 1000007 ITYP1 00n0 I 00002';› ITYPP . 0000 I OnOn04 1UP 0000 I 000002 10 0000 I 000010 Ii 0000 1000023 12 0000 I 000011 J1 0000 I OnOn24 J2 nOnO I 000001 j5 0000 1000003 J6 0000 I 000005 ..17 000 I 000u0n < 00n0 I 00002r, KUDOS n000 R 000022 nMA n002 R 000621 OMEGAC 0000 R 000020 OgLEF 040 R 000021 omRIT 00n2 000A23 THETK n0n0 R 000015 vA 0000 R 000012 VDOWN 0002 R 000622 VPHSEC u0(;0 R On0J13 VUP 00n2 000145 VXI 0002 000311 vYI uuiOu 4* C ---,ABSTRACT - - - - uOtOu 5* C 0U100 6* C Ting - monErrt U0100 7* C PROGRAM TO TABULATE A TABLE OP PHASE VEiOCITY.VERSUS FREQUENCY u010.) 8* C FuR A CIVEN GUILE[) monE, THE NORMAL MOnE DISPERSION FUNCTION uU10v. 9* C TS 711./0 FOR rACH LIsTINr, oF THE TABLE. THE COMPUTATIONAL 00100 .0* C mETHoD IS RASED ON THE pREVTOUSLY COMPUTED VALUES OF THE NMDF 00100 11t C SIGN THmODE((J- 1)tBROW+T) PT POTNTS (I'd) IN A RECTANGULAR 00100 t2* C ARRO OF NR(' ROWS AND mCOL COLUMNS. DTEFLRENT COLUMNS (j) . 00100 13* C CoR10.7.PoNn Tn DIFFERENT FREQUENCIES WHI1E DIFFERENT ROwS (I) u0100 ill* C CutlitESPUND TO DIFFERENT PHASE VFLOC1TTEc. DISPERSION CURVES uuLOo ,5* C OF vARToUS mnDES APPEAR ON THIS ARRAY Ac LINES OF DEMARCATION 00100 .6* C BETwEEN ADJACENT REGIONs wITH DIFFERENT INMODES. TwO ADJACENT 0010u t7* C POINTS WITH INMUDES OF nPROSITE SIGN BRACKET A POINT-ON THE uOiDJ .8* C ACTUAL oISPrnSIUH CURVE. IF THE POINTS CORRESPOND TO THE SAME U0100 19* C FkEOuEm0. THEN THE PHASE VELOCITY CORRESPONDING TO THAT OMEGA uUlOu itl* C ON THr OISPIrpSION CURVE IS FOUND BY CAL1ING RTMI, A 360 PACKAGE u0100 ,1* C RoUT1NE FOR SOLVING NONLINEAR EQUATIONS. AND CONSIDERING THE bUtOu .2t C f\kC)F AS A F,INCTION OF VPWIE WITH OMEGA FIXED. SIMILARLY' IF 00100 .3* C THE volwTS rORRESPOND Tn THE. SAME PHASE VELOCITY/ THE APPROPRIATE i,C,Oo „4* C Or CoRREsPONOING TO THIS PHASE VELOCITY IS FOUND BY CALLING 00100 .5* C IlImt wI(H THE NMDF CONSTDFRFD AS A FUNCTION OF OMEGA WITH 00i0o .6* C VPHst: F1XED. 0010d „.7* C uu101. :414 C THE PRO6RAM cuCcEsSIvELy CONSIDERS EACH PAIR OF ADJACENT POINTS 00i0t) ayt C WITH olposiTc INm001-7, BRACKETING A LINE OF DEMARCATION AND uoicA 00* C PlUCI:FC:, IN THE DIRECTImN OF INCREASING FREQUENCY UNDER THE uU100 0It C . ASSugPTi ,)N THAT THE PHASE VELOCITY CURVE SLOPES DOWNWARDS. uOlOu 024. C 00100 034, C PROGRAM (VOTES oUi00 n4* C UUiOu .15* C THE Mn1175 ARE Nt)HHERED. THE INPUT INTEGER Nm0nE DESIGNATES 0010006* C '.. .I/CH mODL IS REImG TABULATED. THE PAIRS OF FREQUENCY 0010u 07* C AND PHASE VELOCITY VA(UES ARE STORED AS OMMOD(KST(NMODE)), u0100 .:18* C 0,40D(KST(NMOuE) 41).0mMOD(KST(NmOnE ) +2) t / uUiOu n9* C 0mMODO

00100 s2* • C FREnUr•CY TO FOUR SIGNIFICANT FIGURES. IF THE SEARCH IS uOiOu N3* C UNSUCrESSFUL A mE(;SAGE IS PRINTED AND THE POINT IS L0100N4* C FAIPPFn OVER. ('100 s5* C .... 10u ' s6* C THE ImPUT PARAMETERS IST'JST ARE cOORDINATES OF A POINT IN ...— .,. ,,nu ► otFERENCE MANUAL C22-6515-4)

11 mo* C AUTHOR - A.D.PTERCE , M.T.T. ► JUNE:•196U 00100 m4* C 00100 m5* C ----CALLING SFONENCE---- 00100 m6 ► . C 00100 ...7* • C SEF SUBROUTINE ALLMCO 00100 m3* C DIAENSION KST(i),KFIN(1),OmM00(1) ► vPMOD(1),INMODE(l),OM(1),VP(1) 0010u m9* C (SUPRoHTINE USES %ORIOLE DIMENSIoNING) 00100 /0*. C CAiL MODETR(ISTIJSTINMODErxST/KFINOMMODIVPM00,NROW , NCOL , INMODEr 00100 /1* C 1 06ADVPIKRUO) . 00100 /2* C IF( KRIM .LO. 1 ) GO SOMEWHERE 6010o /3* C U010u . /4* C - --,EXTERNAL SUBRoUTINES REOUTRED--- , . u01.00 /5* C 00100 /6* C NXTPNT, RTmI ► PNM0011 FNmOn2. NMDFN, AAAA. RRRR. MMMm•CA ► SA/ u0100 17* ' C (FNMoD1 ANn rNvODE CALL NMDFN• WHICH IN TURN CALLS AAAA AND RRRR. 00100 /3* C RHPA CALLS AAAA AND MMMm. DESCRIPTIONS OF THESE PROGRAMS ARE 00100 /9* C GIVEN ELSEWHERE IN 1HIS SERIES.) • 40100 mO* C 00100 m1* ' C RTN'T IS A :-DRRoUTINE CoDED AY IBM TO DETERMINE A ROOT OF A GENERAL 00100 m2* C NON,L1NEAR ECUATION F(X)=0 nY MEANS OF MUELIER -S ITERATION SCHEME 00100 m3* C OF SUCCESSIVE nISECTION ANn INvERSE PARAROIIC INTERPOLATION. A 00100 m4* C CO+..,PLETE DESCRIPTION AND DECK LISTING IS GIVEN ON PAGES 198 - 199 OF 00100 "5* C DOCUMENT H20-0905 -2• SYSTEM/3611 SCIENTIFIC SUBROUTINE PACKAGE 00100 m6* C (3,.nA-CM-03X) v7RSION TI. PROGRAMMER-5 MANUAL ► IBM' TECHNICAL 00100 ,...,7* . C PuLALICATTO,4 DEPARTMENT, 112 EAST POST RoAn• WHITE PLAINS' N.Y. 00100 mEi* C 10k.01 ► PUBLISHED 1966 ► 1967. '0100 m9* C 00100 40* c --- •ApOUMENT LIST _--- UU100 41* U0100 42* C IST I*4 ND INo 00100 43* - C JSr I*4 ND INp 00100 44* C NMelnE I*4 ND INp 00100 45* C . KSr I*4 VAR INp (ONLY KST(NMODE) NEEDED) 00100 46* C KFTN I*4 - VAR OUT (ONLY KFIN(NMOnE) COMPUTED) u0100 47* C OM;AOD(N) R*4 •VAR OUT (COMPUTED FOR N .GE. KST(NMODE)) U0100 48* C VP;,100(N) R*4 VAR OUT (COMPUTED FOR N .GE. KST(NMODE)) 00100 49* C NRmW I*4 . ND INL 00100 11,0* NC/IL I*4 ND INp 00100 1.1* C 4m0DE I*4 VAR INP 00)00 1m2* C R*4 VAR INp U0100 1.34 C VP R*4 . VAR 'No U0100 1.4* C KRHD I*4 ND OUT 0010( 1"5* C uOlou 1"G* C COMMON STORAGE USEn u0100 1,174 c ' COMMON ImAX ► CI,VX1 ► /YIIHI ► nMFGAC ► VPHSEC.THFTK 00100 IA* C 00100 1.9* C JMAX I*4 ND INo 00100 1,0* C CI R*4 100 INp 00100 111* C VXT R*4 100 INp UU1Ou 1,2* C VY7 R*4 100 INp 00100 113* C HI R*4 100 INo UOIOU 1,8* C. ---INFUTs-- - - UULOU 1,9* 00100 1,0* 1ST =RnW INDEX OF cTART POINT' WHIrII MUST LIF BELOW LINE 0010U 1.1* Or DEMARCATIOm UOIOU 1.2* JST =CnLUMN INDEX nF START POINT t0010o 1.3* Nmone = NHmREp LAnFLLING MODE TO HE TABULATED U010u 1,4* Ks/(Nmoof) =ImPFX OF oMMOn AND VPMOD CORRESPONDING TO FIRST UULOt 1,5* ProNT TADDLAT4-70. UU1OU 1,6* NRoW =NDHRER OF POWs IN INMODE ARRAY U0100 1.7* NCHL =NHvRER OF COL41MNS IN INMODE ARRAY 00100 1.8* INmODE =ARRAY WHOSE (.4- 1)*NROW+I - TH EI EMENT IS THE SIGN OF UOIOU 1,9* THE NORMA' MOnE DISPERSION FUNCTION WHEN OMEGA=Om(J). 100* VDHSE=VP(I). UOIOU 1.,1* OM =VrCTOR OF FREnUFNCIES AT v,HICH INMODE IS TABULATED. u0,000 102* VP =VrcTOR OF PHAcE VELOCITIES AT WHICH INMODE IS u0100 1,,3* TAnULATED. 00100 1,4* 1MAX =mirnER OF ATA)SPNFRIO LAYERS OF. FINITE THICKNESS* U01.00 1.15* CI(I) =SmiND SPEED I-1H LAYER IN KMSEC. U0100 C VXv(1) =X roMPONENT Or wIND VELOCITY TN I-TH LAYER IN KM/SEC 00a00 107* C VYt(I) r7oMPONENT OP WIND VELOCITY TN I-TH LAYER IN KM/SEC U0100 108* C HI(I) =THICKNESS IN KM OF I - TH LAYER 0U100 1.,9* C TH.:TK =PHASE VELOCITY DIRECTION IN RADIANS W.R.T. X AXIS U0100 1,0* C QU1OU 1..1* C LOiOU 1,2* C U0i0u 1-3* C KFTNtNmOnE) =INDEX OF OmmnD AND VPMOD CORRESPONDING TO LAST 0010U 1,4* C pOINT TAHULATED. uU100 1,5* C OMmOD(N) =ANGULAR FREOIENCY OF POINTS nN DISPERSION CURVE. UU100 1.,6* C N7KST(NMunE) UP TO KFIN(NMODr) CORRESPONDS TO NMODE - TH u0100 1,7* C mr0E. uULOU 1,8* C =pHASE VEiOCITY OF POINTS ON nISPERSION CURVE. u0i0U 1,9* C N=KSTINMonE) UP TO KFIN(NmODF) CORRESPONDS TO NMODE - TH u0100 1•0* C MORE. 10100 1.d* C KR14D =PLAG INDICATING IF ANY POINTS ON DISPERION CURVE uOIOU C HAVE BEEN FOHNO. I IF YES. -1 IF NO. 0010U 1,3* C 4014GAC =INTERNALLY UczED FREQUENCY TRANSMITTED AMONG SUB.* 00100 1:,4* C PoHTINES THRnUGH COMMON 1,5* C VPHSEC =INTERNALLY UcED PHASE VELOCITY TRANSMITTED AMONG Uui00 1:,6* C cURR0UTINES THROUGH COMMON. '1,0100 1A7* C 00100 1:3 8* C UOIOU 1•9* C 00100 1,0* C SUJPOSF TABLE OF INMODr VALUES IS AS SHOWN RELOW WITH U010.0 1,1* C 1'01(1(1 1,2* C ++4.0.014++++4. NROW=7. NCOL=14 00100 1,3* C UOiOu 1.4* C U010U 1,5* C "-**- ft+++++++++ 1.4 00100 1.,6* C 0400 1,7* C ..... VP=.5..45..40'.35..30 , .25..20 u0100 1,8* C ..... 00100 1,9* C NMonEt7 a. IST=3. JST=1• KST(L)=7 u0100 1/0* C VPM00(19)=.28 VPMOD(11)=.35 OMMOD(2n)=1.1 VPMo0(20)=.27 it)* OmMoD(1)=.01 VPMOD(12)=.34 0mM00(21)=1.2 VPMOD(21)=.265 u0100 1,9* C 0mMn0(13)=.R0 W100(13)=.33 OMmOD(22)=1.3 VPmo0(22)=.26 00100 1m0* C OmMoD(14)=.F,0 VPM00(14)=.32 0mmoD(23)=1.4 VPM0O(23)=.255 00100 1,.1* C OmMoD(15)=.70 VPMOD(15)=•31 00100 102 ► C 00100 103* C 00100 104* C - ---PROGRAM FnLLOWS BELOW---- uUiOu 1,,5* C u0100 1-6* C 00101 1„.7* SU,4RWITINE MODg'TR(ISTooST , NMODE#KST#KFIN.OmMOD , vPMOD,NROW ► NCOL. 00101 1.0* 1 16,):ODFroM,VP.KRUD) • 00101 1 09* 60103 140* 1imEN5TON C1(1r10)iVXI(100).VYI(100) , HI(100) ■ uU104 141* DIN,,ENSTUN KST(I)FKFIN(1)•OmM00(1) ► W3 M00(1),INMODE(1)10M(1) VP(1) 00105 142* CO•MON ImAA , CI.VXI.VYI.HItnMEGACIVPHSEC.THFTK U0105 143* C uui05 1 4 4* C FUNCTInNS FNMOU1 AND FNMOD2 ARF USED As ARGUMENTS OF RTMI 00101, 145* FXrERNAL FNMOD1.FNMOD2 00106 1•)6* C u0101:1 147* C INnEX /IF FIRST POINT ON DIRPERcIoN CURVE IS LARELLED AS K 00107 1.41* K=KsT(NmoDe) 00107 149* C u0i10 2.0* J5=(JST-1 ) *NRO w4 IST 00111 2.1* 10=INMn0F(J5) u0111 2"2* C L0111 • 2,13* C WE CHErK To SEE IF POINT ABOVE (IST•JST) HAS A DIFFERENT INMODE 6Uil2 2.4* 2 1F(IST .r0. 1) GO TO 5 00114 2.5* J6(uST-1)*NROw+TST - 1 00115 2.6* iuo-mINmonE(Jh) uOjlo 2"7* IF(IUP .EQ. -In) GO TO 10 uul1b 21,8* C .J0110 2"9* C 1F IT i3OESNT. all CHECK THE POINT ON THE. RIGHT. WE CAN ALSO ARRIVE AT J0116 210* C S FROM 2 IF ISI=1• ;0120 241* 5 1FtUST .E0. NCO1_1 GO TO A '0122 2,2* J7=(J5T)*NROW+TST 0123 213* 151D=INNOCIE(J7) 0124 2i4* IFrISIn .E0, - TO) GO To 15 0124 215* C 0124 2,6* C fa-tRIvE AT St WE CANNOT FTNn A POINT EITHER ABOVE OR TO THE RIGHT )124 2t7* C OF (ISTtJST) WHICH HAS A INMOOF OF OPPOSITE SIGN. 218* 8 KRII0= - 1 )127 2,9* RETURN ;127 2,0* C 127 2,1* C WE ASStON A TYPE INnEX TO THE POINT (IST/JST)4 SEE DESCRIPTION OF 127 2,2* C NXTPNT FUR OFFINITTON OF TYPE INDEX. 130 2;3* 1r1 1T4P1=1 I3u 2,4* C 130 2,5* C OPPOSITE SIGN ABOVr C3.1 2,6* o0 TO 20 .31 2,7* a.. w,.. NUW %.AN 1OLNIIFY OUR FIRST RACKETING 00133 2.,2* 2n 11.74IST 00134 2.13* J1=JST u0134 2.14* C 00134 2,15* C STATEM.;NT OS IS START OF LoOP TERMINATING AT 100. EACH PASSAGE THROUGH 00134 2.16* C LOnP GI;NERATES A Nrw POINT ON THE DISPERSION CURVE. uU131) 2.17* 2S TfrITYp1 'EQ. o) GO TO 50 0013b 2.1a* C uu13o 2..9* C CAlCuLATION IF ITYp1=1. STORE FREQUENCY IN COMMON. FIND PHASE VELO- 00135' 2.0* C CITY WITHIN BRACKETED INTERVAL. 00137 2.1* om.GAC=Om(JI) 00140 2..2* vn.ww=up(11) u0141 2.3* vUL•=vP(1)-1) 00142 2,4* FP...=1.E-. . 00143 2.5* GAIL RTMI(VA.F.FNMODIlvDOWN.VUP.EPS.6.IER) u0144 2.6* om,40u(K)=0...EGAr u0146 2.7* v114.0D(K)=VA (.014u 2.8* (,O To ion uOL40 2.9* C UUI4t, 2n0* C CAlCuLATION IF ITYp1=2. STORE PHASE VELOCITY TN COMMON. FIND FREQUENCY 00146 2:.1* C IN RRArKETrO IwTERvAL. 00147 . '2712* -- -- 5n vPw.SFC=vp(11) 00150 2N3* omiEF=n1(J1) 00151 2 :04* oMwIT=md(J1+1) uu152 2•.5* • 14.=(1.(.-6)*OmoIT uUlSj. 2.6* cAlL RT,:,TtuMA.r.FNm002.0mL(.F.ONRIT.EPS.6.IFR) 00154 2..7* om400tK)=OmA 00156 2:28* . vPm0D(K)=VPHSEr u01.55 2!,9* C u015. 2..0* IOn CO3TINUE U015b 2.1* C wE HAVg.- NOw FOUND THE K -TH POINT. WE DO NOT YFT KNOW IF THIS IS THE 00150 2.2* C FINAL .41INT 1:001 THr NMuDE-TH MnDF. HOwEVER, WE SET KFIN(NMODE)=K 0015/ 2.3* KFIN(NmonE)=K 00157 2.4* " - C wHFN TwF SHfIROuTINF RETURNS. TwE CURRENT STOREn KFIN(NMODE) WILL RE utJ157 2.5* C THE COPPECT ONE. 00i57 2.6* ' C •00157 .2.7* C WE Now PREPARE. THE SEARCH FOR THE NEXT POINT. UOICIU 2/,0* --- K=v+1 u0161 2.9* 174 CAlL NYTPNI(I1.JIPITYP1.I2,J2IITYPP.NROW.NrOLFINMODErKUDOS) 00162 2/0* 18n IF( KUnoS .EQ. -t) GO TO 2n0 00164 2.1* I1-I2 t,0 1 65 2,2* o1T.1.12 UU16o 2/3* ITYP1=ITYP2 00167 2,4* 19n GO TO PS - 1)0167 2,5* C UU171, 216' On coh, TINM 1.1900 2,74, C v.E CoNIINUF HERE APTFR AN HNSUrCFSSFUL ATTEMPT TO FIND THE NEXT POINT. 00170 2.0* C PROVIDsNG wF. HAVE POUND AT LEAT ONE POINT, WE CAN EXIT WITH KRU0=1. 0001 2.9. IF( K .LE. KST(NMOUE) ) GO TO R 00 1 73 2.0* K.R10=1 00174 2.1* RETURN 00174 ' 2,,2* 00175 2.3* ENn

SODRONTINE mOuLST ENTRY POINT 000121

STORAGE USED: CODE(1) 0.1014p1 DATA(n) 00007h1 DIANK COMMON(2) 000000

EXTERNAl REEERENCES (HLnCK. NAME) 0003 NWDU$ G004 Nr0e$ u 0 05 NERri3$

STORAGE ASSIGNMENT (11LnCK. IYPE• RELATIVE LOCATION. NAME) 0 0 o0 0(10007 ilE 0001 000n26 111G nOni 000056 122G n000 000017 21F 0000 000040 31F 0 0 00 R 000u06 CKI 0On0 I 000nOn II nOnO 00005n TNJP$ 0000 1000003 J 0000 1000001 K1 0000 I 000(102 K2 nOnu R 000n04 OMEGA n000 R 000005 VPHSE

00100 1* c MO -iLST (sUHROUTINE) 6/19/68 LAST CARD IN DECK 15 NO. 00100 2* C uOIOU 3* c u010J 4* C ----ABSTRAcT.--- u0i00 5* C 00100 6* C 00100 7* C TITLE - MOflLST 00100 8* C TABULATION nF SELECTED POINTS ON THE PHASE VELOCITY (VPHSE) VS. 00100 9* C ANGULAR FREQUENCY (nmEGA) CURVES OF SELECTED MODES U0100 10* C uOi0u II* C NJ COMPUTATION OR CHANGING OF UNITS IS PERFORMED BY SUB - uUlOu 42* C RoUTImE MODLST. IT MERELY PRINTS OUT THE INPUT IN LABELED uOiOu 13* C AND ORDERED FASHInN. 00100 14* C u0100 15* C LAmGuAr4F - FORTRAN Iv (3An. DEFERENCE MANUAL C28-65154) u0i00 .5* C ,7* UCIO0 C AUTHORc - A.U.PTrRCE AND J.LIOSEY• M.I.T. ► JDNE•1968 00100 18* C J0100 19* C )0100 .0* C ------USAGF-- )9100 el* C 10100 .2* C NO SURRWT1NE5 ARE CALIED. JoIno e3* C 10100 .4* C KFTN, CIMOO, VPF•°D. KST wItL ASSUME THE DIMENSIONS SPECIFIED IN 10110 .-5* C THE CALLIN6 PRn•RAM• (DIMENSION OF KST ANn KFIN MUST BE .GE. NMFND) 1010u .5* C

, 0100 e7* C FOPTHAm USAGE Oinu .8* c 010U :9* C ' CA1L MODLST (Mor7ND,OMMoD.VpMoD.KST.KFIN) A.)* C. MDFmn NUMBER OF MODES To nE PRINTED OUT. uOL00 ..4* C I*4 u0100 H5* C -u010u .16* c • 0mMnD VcCTOu STuRIN(; ANaULAR FREOUENCY COORDINATE OF POINTS ON •001.0u .17* C R04(0) DISPERSION CUwVES. MODE M IS STORED FROM ELEMENT KST(M) U010u A3* C THROUaH ELEMENT KFIN(m). ( RAD/SFC ) • UOIOU o9* C u010u 404 C VPMnD VECTOP STORING PHaSF VELOCITY COORDINATE OF POINTS ON u0100 41* C R*4(0) DISPERSION CURVES. MODE M IS STORED FROM ELEMENT KST(m) u0- 10u 42t. C THROU6,H ELEMENT 1

Lu0100 ',43* i, u0i00 (,4* c ENn 0F FXAmnLE 00/0u ,,5* C u0100 416* C • u0100 ,-,7* C ----PROGRAM FOLLOWS FIELOW--- VU•100 v3* C U010u v9s luU101 1.0* SUIIROUTINE MODIST(MDFNne0MmOntVPMODoKST/KFTN , AKI) Q0101 -. ' 1,,1* -- 00101 1"2,. C VARIABLE DTMEN!,IONTWI u0103 1.3* nI,FNSTON KFTN(1)•OMMOn(1).VIIMoD(1),KST(1).AKI(1) u0104 1.4* . . wRITE(hell) mnptID 00107 1,5* 11 1-0 ,, MAT(IIII,25X.19HTABULATInN OF FIRST, I6,AH MOnES) U0110 1h6* NO 100 IT=IPm0P1 10 . U011.:1 1,,7* wR1TE (6.20 IT uUtlf, 11,f1* 21 FO,:MAT(1H /// ► in 05X, SHMnDE ,I3// , 1H #12; ► 15HOMEGA (RAD/SEC), 00116 1.9* 110..14)ivPHsE (;W/SEC),10X , I0HAKI (1/KM) /) .., 0117 110* _••• K1=KST(II) UOL20 . 1'1* K2 =KFTN(Ii) 012L 112* nO 100 jr.KI•K2 0124 1,3 4 0m.:T,A=m4mOu(J) Oat) 1,4* vp"SE=vomOD(J) Oah 1,5* CKr=AKT(j) • 0127 116* 31 FOwMAT (1H .12ytF15.6.10X,P14.6.10XtE12.5) 0130 1,7* wiliTE (6.30 OmFGA,VPHsE,CgI 0i36 118.* 10n C0i4TINNE1 O140 119* RETURN 0141 1.0A, FNIf

E1.10 Or COmPILATTON: NO nIAGNOSTICS. **.*** bUHS5 ****** FURS 1.ShBA5pS0B65 OK slIF-1,/10/74 - 18:1n:37 (0•.

SUBROLITINE MPOUT • ENTRY POINT 00n175

SVORAoE USED: COD•(1) ("WWII; nATA(n) 000051; Ri ANK COMMON(2) 0n0621

ExTERNA, REFERENCES (HLoCK, NAME)

0003 NMDEt4 0004 NERk3$

SIORAoE ASSIGNMNT TYPL, Rt=LAT1VE LOCATION, NAME)

0 0 01 00005u 1115 0001 000n60 116E nOnl 000077 122G 0001 000140 50L 0001 000144 SOL

uvuc VUUJII VYI

u0100 1* C MPflUT (SUBKOUTTNE) 7/19/68 LAST CARD IN DECK IS UO100 2* C 00.0U . 3* C u0100 41 C ----AOSTHACT- -.- u0i0O 5* C 00100 6. C TITLE .. ,(POUT (Joie() 7t C TABULATION nF NORMAL mOnE DISPERSION FUNCTION SIGN AT POINTS 00100. 8* C IN A RECTAMAULAR REC,TON OF FREQUENCY - PHASE VELOCITY PLANE LIOLOU 9* C 00100 10* L TuE Vi:CTOR V oF PHASE .VELOCITIES TS CONSTRUCTED tiY TAKING uui00 II* C VALUE, AT INTPRVAtS OF ((V2-V1)/(NVP - 11) FROM V2 DOWN. TO u010u 12* C V. ,IMILARLY, VECTOR OM OF ANGUIAR FREQUENCIES IS CON- U0100 13* C SIRUCITD iiY TAKINA VALUES AT INTERVALS OF ((0M2-0M1)/ u0100 14* C (DOM-1)) FROM 0MI UP TO 0M2. NEXT' MATRIX INMODE IS CON- uUlOU 15* C S/RUCTrD WITH MVP ROWS AND NOM COIUMNS, SINCE INMOOF IS u(110o 16* C STOREn IN vEcrop EORMs COLUMN AFTER COLUMN, ELEMENT J IN U0100 17* C RuW I TS REPRFSENTED AS INMOOE((J-1)*NvP + T). THE VALUE 00100 .8* C OF THTS ELEMENT Ic DETERMINEO ny CALLING SUBROUTINE NMOFN u010u i9* C Tu EVALUATE ThE NnRmAL MODE OISPERSION FUNCTION. FPP, FOR uUlOu ,0* C FkEOUrMCY 10!1(J) AND PHASE VELOCITY V(I).- IF FPP DOES NOT 00100 11* C EXIST, THE ELEMENT IS SET EQUAL Tn 5. OTHERWISE THE ELE - 0010u /2* C MENT 61TLL BE 1 TIMES INC SIGN OF FPP. 0010u /3* -,C u0100 / 4* C LANGIJA,..F - FoRTRAu IV (3);,n, REFERENCE MANUAL C28-6515-4) u010L1 /5* C AUTHOR. - A.D.PT!,'RCE AND J.u0SEY, M./.T., JuNE,1968 uniOU /6* C • U0100 /7* C u010u e8* C ----UGAGF---- uU100 /9* C VAuIAD1ES oM.V.INMODE MNST BF DIMENSIONED TN CALLING PROGRAM 0010u 00* C FOuTRAN. SULROUTINE NMDFN (nESCRIBED ELSEWHERE IN THIS SERIES) IS UULUU 01* C CA, LED 00100 .N2* C L'UlOU .13* C FORTRAN USA3F 00100 A4* C GAIL MPOUT(091.0M2oV1•V2,NoM.NVP.INMOOE , DM.V•THETK) uO400 05* C

• u0100 .16* C INPUTS u0100 07* C 00100 08* C OM1 MINTMIIM ANGULAR FuEOUENCY TO BE CONSIDERED (RADIANS / SEC) u0i00 ..9* C R.:4 vOLOu 40* L uuIOU 41* C 0M2 MAXIMuM ANGULAR FuEOUFNCY TO 13E CONSIDERED (RADIANS / SEC) LJOLOo ..2* C R*4 u0i0u 43* C 00100 44* C VI MINIMuM PHASE VELOCITY TO DE CONSIDERED (KM / SEC) 0010u 45* C • 1144 uOIOU 46. C u01061 .7* C V.J MAX/Mum PHASE VELOCITY TO RE CONSIDERED (KM / SEC) 0010u 48* C R*4 u010u .9* C NOM NuMoEu OF FRCoUENrIES TO BE CONSIDERED '(NC). OF ELEMENTS uOiOU -,0* C I*4 IN OM AND NO. OF COLUMNS IN INMODE)

. _. . vtLuciTY MEASURED COUNTER CLOCKWISE uJ10o ts6* C R.4 FROM Y-AXIS (PADIANS) 00100 :17* C vOlOu !18* C OUTPUTs ()two :19* C 00100 .0* C INW)DE MATRIy OF NORMAL MODE DISPERSION FUNCTION SIGNS (SEE (jOiRu ”llt C • I.444(0) ABSTRACT ABOVE FOP EXPLANATION OF ELEMENT VALUES) 1/0100 .2* C u0100 .3* . C OM VtiCTOn OF NOM VALVES OF ANGULAR FREQUENCY AT EQUAL INTER" (10100 .4* C R*4(01 VALS FROM 0M1 TO WO INCLUSIVE (RADIANS / SEC) 00100 .5* C 00100 .6* C V VLCTOp OF NVP VALUES OF PHASE VELOCITY AT EQUAL INTERVALS uOlOu m7* C R*4(D) FROM v2 TO Vi INCLUSIVE (KM / SEC) IJUIOU oa* C u010u .9* C u0t0u ,0* C _-...EXAMPLE-- -- 6010o /1* C 0010u 12* C CAiL1N,:, PROGRAI 0010u /3* C DIMwNSIoN 014(3),V(3)+INmOD(Q) 00100 /4*. C OM1 = 1.0 00100 /5* C 0M2 = 3.0 00100 /6* C VI = 1.n unino 17* C v2 = 6.n u0100 /8* C NOM = 3 0010) /9* C Nvp = 3' 00100 .0* C TNETK = 0.0 u 0 100 .1* C CALI MPoUT (0m1.0N2011,v2,NoM/NVP,INMODEtoM.VITHETK) ullIOU .2* C END 00100 .3* C 00100 .4+ C uPon R,-TORN FROM MpOUT, OM AND V WILL HAVE THE FOLLOWING VALUES U0100 ts5,4, C Om = 1.n t 2 .0 t 3.J u01.0J .6* C V= 3.0 t 2.0 t 1.0 uOin0 .7* C t..AcH CA: THE NINE EIEMENTS OF ImMODE WILL RE -1. 1 OR 5 AS DETERMINED u0t00 . 5* C 0 THE NORMAL MODE DISPERSION FUNCTION (SEE ABSTRACT ABOVE/. i 00100 .9* , c 1 00100 v0* C 00100 ,41* C ----PROGRAM FOLLOWS BELOW---- 01.1i0J. 42* C. 00100 43. C L0101 44* S1J.ROUTINE.MPONT(00,0M2,V1,V2,NOMoNVP ► INMODE , OMtVoTHETK) 00101 45* C 00101 46. C VAPIARIE DTMENAONWG 00103 47* HIAENSTuN UM(110(1),INMODE(1) v0104 40* COAMO• ImbX ► CI(100) , VX1(100)1VYI(100) ► HI(1o0) 00104 49* C U0l04 11,0* C INTERVAL BETWEEN SUCCESSIVE ELEMENTS OF OM IS oETERmINED (, v) 01S 1..1* NEI0m=f0m2-01r11/(NOM-1) 0010.5 1,2* C 00105 1"3* C INTERVm PrIvEtN SucCESSIVI: ELEMENTS OF V IS DETERMINED u010u 1,4* nEl V =(V2 - V1)/(NVP-1) 00106 11,5* C uulot, 1„6* C VErToR V Is CONSTRuCTEO WIrH VII) DROPPING FROM V2TO VI AS I GOES FROM (Oi0 11,7* C 1 TO NiP

UU113 la* C UM(J) p.OF5 fROm OM' TO 0M2 AS N GOFS FROM 1 TO NOM 0011h 1i3t "0 90 4=1,140m u012u 104* nm/J) = ot.11 +(0-1)*DELOM O0120 115« C uo120 116* C FON u I..TxEn VALUE AF J. ALi VA1UES OF I FROM 1 THROUGH NVP ARE CONSID - o0120 117* C 1..RFOr rouS evALUTI-A COLUMN J nF INMODE U01.21 118*. HO qu 1=1INVP U01.21 li9* C UU121 1,0* C 1J IS -Cf. nF ELEvEmT IN VECTOR REPuESENTATION nF INMODE WHICH CORRES- uU121 1,14 C PONDS 10 F1EmEl*T J OG ROW I IN MATRIX FORM OF INMODE U0124 1..2* IJ=(J-1)*NVP + I (A/125 1,3* VPusE=v()) uUL2b 1,4* C u012t) 1,5« C NNIFN TS CALLED TO FVAUATE THE NORMAL MODE DISPERSION FUNCTION FOR uUl2b 1,6* C FRFONEwCY nm(J) ANn PHASE vFLOrITY V(I) u0120 1,7+ CA1L NmDFN(Om(j).VPHSEiTHETKIL.FPP.K) u012o 1,8* C 00120 1,9* C WHFN N.,RMM MOVE OTSPERSION FUNCTION DOES NOT FXISTAL,E0*-1), INMODE U012„ 1.,0* C (Li) = 5 u0J27 1.,1* 1Ful. .F1). -1) nO Tu 50 uU1 27 102* C U0127 1:13* C Vili•N 1%.,17. FuNCTION nOES EXIST AND Is FPPr INM0DF(IJ) = 1*FPP/AFJS(FPP) u0131 1.*4* INAMNE(IJ) = 1 u0132 1.,,5* IF (IPP.I.E.0.0) INMODE(IJ) = - 1 u0134 1.,64 GO Ti; nil u0135 1.a*. 5n IN.noE(IJ):, 5 u0136 1,18* Ein CO.IYINHL U0137 log* - - 9n CO•ITINHE UO142 1..0* kETURN • UOIIN) 1.1* EN,'..

END OF COmP1LATTOw: • NO nIAGNOSTTCS• .huG,P 4,*.t** soyA„ *t*# ** i, FoRpS 1.S4ip,6,Sobri6 1•OR :-,11F-1,410/74-18:1n:43 (3')

SUDROuTINE NAWDE• FN1RY PQM!' 001225

STO ►tA(*E USED: CODE(1) 00127n1 DATA(n) 001332) DANK COMNON(2) On0621

EXTVRNA1 REI:Er*ENCe:S filLoCKr NAME) 0 0 03 AAAA 004 MvMm 000q TOTINT 0 0 08 COS. u007 SIN uOIO SORI . u 0 11 NWOu$ S l O g AGE ASSON•FNT (H L0CK. TyPt, RI=LATIVF LOCATION, NAME)

0001 0(10036 10L nOnl On0A5n 1201. nOnl 000372 130L 0001 0n0126 1476 0001 000400 1501. 0 0 01 000150 1576 nOnl '000204 176G 0001 000043 20L 0001 001202 200L 0001 000230 2060 00nl• 000831 2110 nOn1 Or0p64 224G 0001 000274 730G 0001 000276 2336 0001 000314 2416 0 0 (11 000315 244G 0001 000132 31L A001 000152 .32L n001 000576 3316 0001 000414 331L 0.001 000460 3 3,5L 00n1 000.,76 340L n001 000571 341L 0001 000175 35L 0001 000642 350G 0011.1 000.23 400L 00111 000A 6n nn01 1013 7.t 4o5L 0000 001260 409E 0001 000754 413G 0001 001041 4201. «001 001n96 43% 0001 001107 450L (1 000o 00000n A 0000 R 001170 AKX 0000 001171 AKY 00110 001237 AP1mx 0000 001243 AP1P 0000 001240 AP2MX 0000 001244 AP2P 0000 001452 ROM 0000 0 0111176 nOn2 000001 rI 0000 011002 CIJZ 0000 R 000026 DELT 00nn 0001)54 DUMMY 0000 001230 Dl 0000 001231 n2 000n 000004 EM 0000 R 000034 EMP . 0000 001851 FON 0000 001204 El n000 N 00121n FlP noon 001205 F2 0000 001223 F2RoT 001)0 001211 F2P nOn0 001202. 0000 001203 r,RT nand 001216 I-I 0002 000455 HI ( (ann I 00121s 0000 I 00l)35 IArlmx 00110 I 001216 TAP2MX 0000 I 0(11207 IC 0000 I 001250 TOA 00c12 I On0uOil VAAX 0000 0013112 INJPI; 0000 001217 TP noon 1 001254 IT 0000 001174 v,!mAx 0000 I 001171, 0000 I 001221 JASA 0000 I 001232 JET noon I 001220 JP 0000 I 001213 JZ •0000 I 001414 J8 0000 I 001225 K 0000 I 001224 KTOUP 0000 I 001255 L 0000 I 000024 LAYJZ 1)000 1 On1248 LN 00110 I 001.141 LNM1 0000 I 001173 NOBS n000 I 001172 NSCRCE 0000 I 001233 NZC1 0000 I 001234 NZC2 0000 000A50 PHI1 0000 001024 pHI2 noon Un1222 OUOT 0000 000030 RPP 0000 rt On1e45 .11 0000 0 001246 R2 nOn0 001247 p3 n000 R 000012 S1 0000 000014 52 u0n0 0 001226 11 oOnn 0 001227 72 nOn0 R 001177 vX 0002 000145 VXI 0000 R 000016 vX1J2 00,10 0 00120n vY. 0002 rt 000311 VYI 0000 000020 vYIJZ 0000 001201 X 0000 001256 X3 0000 001457 X7 00110 001253 ZFN • (1000 R 000010 7I,JZ 0000 001206 ZM 0000 R 001212 ZMP

u o lou 1* C NA,ApuE (SUHROUTI•E) 6/27/68 LAST CARD IN DECK IS NO, u0100 2* C 00100 3& C (.010u 4* C 00100 5* C u010u 6* c TITLE ..‘NAmPnE 00100 7* _ C PRoGRAM TO nFTERMINF AN AMPLITUDE rAcTop AMPLTD OF A GUIDED 00i0u 3* C moDr FXcITEn BY A POINT ENERGY SONCC IN THE ATMOSPHERE. THE 00)0u 9* C SoURCF IS AT ALTITUDE ZcCOCE KM AND THE OBSERVER IS AT ALTITUDE 00100 10* 71tns IN KM. TtIL PARTICHLAR AMPLTD COMPUTED CORRESPONDS TO AN u0i10 11* C ANGULO FREnUENCY 0,4,EGA (RAO/SEC). A PHASE VELOCITY VP(ISE 1.,i0t, ,2* C (KM/sFC), - AND A PHASE VELOCITY DIRECTION THETK (RADIANS) RFC- 00100 13* C KONFO CuUnTr0—CLOCKATSE F1WM THE X AXIS. PARAMETERS DEFINING 00(00 14* C TflE WkIENT ATMOSPHFPF ARF PRES1(iE0 TO RE STORED IN COMMON. 0(1100 15* C TIE NOPAL mnnE DISPrRSTON FUNCTION NMOP7 IS PRESUMED TO VANISH 00.10u 164, C F00 A000mENTS OMEGA.VPHsEITHET.K. 1, 010u 17* C uv100 18* C THE ACTUAL nrFINITIoN Or AMPLTD IS AS FOLLOWS. LET S1(Z) AND uOi00 19* C S2(7) BE THw SOLUTIONS r1F THE RESIDUAL FOUATIONS u0100 .0* C 0010u .1* C 0(50/0/ = (1111)*s1 + (Al2)*52 (1-A) 0010u .2* C D(S8)/D7 = (A21)*s1 + (A22)452 (1—R) 00100 .3* C u0100 .4* C wHEpE THE MATRIA A TS As COMPUTED BY AAAA AND AS DEFINED By 010u #.9* • C an(Z):- (G/C)*S1 - r*R2 (2) OiOu .,0* C oinu .,1 ,1, C WHERI-: 6 IS ACCELERATION OP GRAVITY AND c IS SOUND SPEED. THEN ULOU 02* C 0100 03* C UlOu A4*. C S2(7ScRCE)*ZEN(ZOBS) OLOu .15* C AMPLTD = (1/2)* ------_-_-_-_------_ (3) oinu .16* C 11007SrRCE)*INTEGRAL 010U ..7* C 010u A8* C UHERF 0100 .19* C °Lou ..0* C 1tom(Z)=n0.1FGA -Kx*VX/Z) -KY*vY(Z) (4) UIUU ml* C U1Ou - 1` C IS THE,OOPPIER SHIFIED ANGULAR FREOUEmCy. THE INTEGRAL IS 1/2 U100 •,3* C n' THE I -SUnl Di-:FINED .fly A.n.pIERcc, J. ACOUST. SOC. AMER.. uiUU ..4* C VOL. 57. NO. 2, FER..19A5, PP. 218-227• E0. (51). SPECIFICALLY, Oinu ..b* C OiOu .6* • C INTPGRAL = INTEGRA' OVOR 7 FROi O TO INFINITY) OF 01OU ..7* C 010u .8* C (OOM*((#0:*Vy+KY*VY)/K)*YEN(7)**2. 0100 .9* C +(K*OM1;GA/kOm**3)*7FN(7)**2 ) (5) U1OU rl* C OLOU nl* C WHERE K IS THE MAGN(TUDF OF THE WAVE-NUMBER VECTOR (KX.KY) AND UiOu n2* C 000 s3* C YFN(Z) = (I/C)*11(7.1 (6) 01011 .s4t C . ulOu •5* C'PR ►GRAm NOTES U1Ou n6* C . ULOu n7* C THE INTEGRAL IS CnMPUTED RY SURROHTINE TOTINT IN TWO PARTS UkUu n11* • C As X34.x7. THE FIRc:T IS OPTAINLD OT CALLING TOTINT WITH 0100 •9* C •I(=1. .RILE THE SI=CoN0 IS ODTATNEn BY CALLING TOTINT WITH 010u HO* C Ir=7. THL IT PAROVTFR GOVERNS THE CHOICE oF COEFFICIENTS UtOu -1* C Alf A2, AS REIHRNRD TO TOTINT fly cUBRouTINE USEAS. FOR Jloo 02* C FORTHrP INFOPmAT/nN, SEE THE DOCUmEN1ATION ON TOTINT AND J10u .3* C USEAS. 1100 ,.4* C )tnu ,.5* C THE NnRMALIZAT/ON OF Si AND 52 CANNOT AFFECT AMPLTD, 110u #'6* C HoWFVFR, TOTINT AnONPIS NORMALI7ATION WHERE )10u H7* C • . RI =-SURT(G.J*Al2 1100 1.8* C e2 zSopi(GG)*(GG+All) 100 •9* C Al THP- 110110M OF THE UPPER HALF SPACE. THE NUMERATOR OF 10 ∎ 1 i() C C.(3) IS ACCoRDIHOIY COMPUTED WITH SAME NORMALIZATION. 10u /1* C hLRE AC=SoRT(411*.:2012*A21). 10u /2* C 100 /3* THE ONLY NOUNDARY CONDITIoN EXPLICITLY USED IS THE UPPER 1.00 #4* C IAJUHDAPY CONDITIOm wHFREBY RUtH S1 (Z) AND S2(Z) DECREASE 10U /5* C EAPOIWNTIALLY WITH INCREASING HEIGHT IN THE UPPER 100 ,/6* C ► ALFSpACF. IV THrS CANNOT nE SATISFIED. THF PROGRAM 100 /7* C RETURN' A•LT1,170. THTS •oULD IMPLY THAT THE POINT IOU /8* C CuNSII ►rRED IS PRArTTCALLY IDENTICAL TO ONE WHERE OMEGA i00 /9* C IS THr CUTOFF EREnUFNCY FOR THE GUIDED MODE UNDER AU /4* C CuNSInFRATIuN. .0u ..1* C . . , r---CALLING SrOHENCE -7-* u0100 ,7* C uOlOu “8*. C SEF S(b.ROUTINE PAMpLE L0100 *,9* C NI,,ENSTON CI(1n0),%0CI(100).VY1(100) ► H1(100) U0X00 ',0* C COMMON IMAX ► CIeuXTO/Y1tHI . (THESE MUST OE IN COMMON) u010o 41* C CAIL NAMPDOZSrrCE.20Bsp0MrGA•vPHSE , THETKrAMPLTD,NPRNT) u010u• 42* C u0i0u 43* C ----EXTERNAL cURROUTINES REOUTRED---- 0010U v4* C uUlOU 45* C TOTINT0.MmtAAAA.USEAS.UPINT.ELINT.OBB(OCATFSAI uuiOu ,A* C uClou ,v7* C (THE FIRST T)1RrE ARE ExPLII - IILY CALLED. THE REMAINING SUBROUTINES OUIOu 40* C A0E ImRICITLY CALLED wHEN TnTINT IS CALLED.) uilloo 49* C uOlOtt 1i3O* C ----AoGUMENT . 1.IST---7* uOiOu 1i,1* C liulOU 11.2* C 7SCRCE R* 4 ND INP u0100 1,,3* C /0,1S R*4 ND IMP U0100 11,4* C OMPtlA R*4 ND INp u0100 1,,5* C VP•;5E R44 NO INn (JOIOu 1.6* C 11167K R*4 ND INo uUlOu 1.7. C AMPLTD R*4 ND OUT u0i0U 1,0* C NPwNT 'I*4 ND INn IJOIOu 1.9* C UO1OU 110* C COmMON SIC/PAGE USED LUIOu 1,1* C COMMON IMAX ► CI.VXI ► LYIsHI u0A0u 1,2* C (110u 1,3* C IM,IX I*4 ND TNIJ u0knu 114* C LI R*4 100 INn u010u 1 1 5* C VXt R*4 100 INu u0100 1.16* - C VYT R*4 100 IN0 u010° 1 ( 7* C HI R*4 100 INn uulOo 11g* • C U010i, 1,9* C ----INFUTS---- u010u 1,0* C U010u 1,1* C 7SrRLE 7.1NFIGHT OF SOUnCF TN KM u0i0u 1,2* C. i0(4S • =HrIGH1 OF .ORSwRVER U0.100 1,3* C . OMEGA =AmrULAR FliFOUI-NCY IN RADIANS/SEC u0100 1,4* C VPHSE =FN•SE VELoCITy TN KM/SEC • 6010u 1,5* C IHrTK =PHrSE VELUCITy OTUECTION (RADIANS) RECKONED oj100 1,6* C HP0N1 z.RQTNT OPTION TNOICATOR ISLE NAM1 IN MAIN PROGRAM). oniOu 1 e 7* C ChHNTER-CIOCKNISE FROM X AXIS. oOlgo 1,g* C Im;0( =NwHER OF ATM0SPHER1C LAY(RS'wITH FINITE THICKNESS igilOu .1,9* C CI (T) =5,01ND SrEpo (14. M/SEC) IN 1-TN (AYER • 0010u 1,,21* C VXT (1) =X coMPONENT Or wIND VELOCITY (KN/SEC) IN 1—TH LAYER ouJou 1.,1 C vYr(I) =Y COMPONENT OP WIND VELOCITY (KM/SEC) IN I-TH LAYER OUlOu 1;,2* C H/ (I) =THICKNESS IN kM OF I-TH LAYER uUlOu 1,3* C u0100 1,4* - C - ---OUTPUTS---- u010o 1•5* C u0100 1•6* C AMplID =AMPLITUDE FACTOR FOR GUIDED WAVE EXCITER BY POINT u0100 117* C E,EHGY SOURCE. UNITS ARE KM**(-I). uOiOu 1.13 4 C .- .....“..“•it ,,[1) ItiLl‘r. Attu NO WINDS. THEN WVAUU 444* C T11. RE TS ONLY 'INF wonE, F0.4 WHICH vPHsEr.c. FURTHERMORE, YEN(Z) u0100 . 1 ,3* . C AN.' S1(Z) hRE pOTH ZERo. THE /FN(2) uLCREASES EXPONENTIALLY 0010u 1,4* C WITH HFI611 AS ExP( - (0.3*GiC**2)). THE RESULTING AAPLTD 00100 1,5* .- C " -. SH1.uLD isf . uOIOU 1,6* C AMPLTL=-1.3*A/C**2)*vXM-.3*(G/C**2)*(7.0RS+ZSCRCE)) 00100 1..7* , "c ' NE.AHOtENS OF vtLUES OF oMpGA AND THETK. TF C=113 KM/SEC, 00100 1,8* C 1,=.01 KM/SEC**2, zons=n, zcRcr=o , THEN AMpLTD= .027 KM**(-1). uU10.4 1,9r - c u0100 1•,0*C uU1Ou 1nls — C -,--NOGRAM FnLI.OwS BELOW---- u0i0U 1:32* C

00100 1:,34 C . 00101 1:,44, SU.ROUTINE•NAMPr)EtiSCRCF.7.n0StOMEGA.VPHSF/THETK.AMPLTD,NPRNT) 1.10101 -. 1n5, UU10.5 1,6* 1a•‘ENSTUN CI(1nn),VXT(1n0).VYI(100),HI(1r10) • . 00104 - 104 - — n/,ENST1N A(2,),EM(2,2). 0005 1:18*• HL.ENSTvN /I.g.0.)/51(2).S212).VXIJ7(2)tVYId2(2),CIJZ(2) u0t05 1,-,9* C ONENSION cIAlt•ENTc AuDED IN THE OEBUG PROCESS 0010u 1,0* NI,..Et6Ti-asi LAYJ7(2),DELTC2).RPP(2.2).EMP(10n , 2 , 2) , DUMMY(2.2) 1,010/ 1,1* • lar.ENSTON PHI1(100) , PH12(I(10) 00116 1.2* _ CO.MoN ImAX.CI.vYI , VVI.H1 0011u 1.3* C L,ullu 1,4* C CO(.IPuT.= WAVE NUMUEr VECTOR COMpONENTS 00111 1.5* AK%=(0mr.GA/VPHcE)*COS(IHETu) uU412 1,6* AKC:(0MV6A/VPHcEl*SIN(THETve) uu112 1,7. C U0112 1.0* --...... C THE SOURCE AND OWJFRVER LOcAT/nNS ARE NUMBERED. ACCORDING TO HEIGHT 00113 1.9* IF(ZSCucF .GT. Z013S) Go TO In (JOil 1/0* 7I.17.(1)=7SCRCE ' U0116 1,1* /1.17(2)=700S ' 00117 1,2* N5rFILE=1 00120 1,3* NO:.:=2 u0121 1,4* 60 Tu 211 • UOLPe 1,5* 1n 7/J;(1)L!70iiS U012 I,8* /IAZ(21.74SCRtE 00124 1/7* NcLAS=1 00125 1/0* NS(RcE=2 00125 1/9* C 60125 1..0* C WE DEN.3T• ci AND 52 AT BOTTOM nF UPPER HALFSPACE BY Fl AND F2. THEIR 0012n 1.1* C COmPuT.T1ON TS AS FOLLOWS. UUi26 •1,■2 , 2n lz.dlx=2 uuk27 1.3, J=IMAX+I 3JUOU 1.4* C=rI(J) 00131. 1.5* vx=VAI(J) 0002 1.6* vY-VII(J) 00 1 ,4.; 1.7* CA1L AAAA(wAEGArAKPAKYrCrvX.VYrA) 00 4 ,54 1.4r • XL- A(1,1)**, 1 A(1.2)+A(211) 0(1134) 1.9t 1 • 0( Or. 0,0) cn fO 2110 UULSY 140* (,= ,,ORT(X) 0014u 141* 1Rr=S611(0) 00141 142* F1z-6RT*A(1#2) tW1N2 1,!* F2=OkT*(A(1 , 1)+0) v.J.1.4.2 1•* C 0014 1•,5* C wE COMoUTE 7' REPRESENTING THE BOTTOM OF THE UpPER HALFSPACE uu11 2.0* C 00151 2,1* C wE sro.1E EI(I,F2R,ZmP U053 2"2+ 31 P10=1-1 u0154. 2,13* f:2L.,=1-2 U.0155 2"4* 7 410=IM 0)155 2"5 + C U015 2"6* C COmPuTATION OF LAYH7.(J2) ADD DwLT(a) uj15b 2"7* C LAYJ7(.1Z) TS THE INDEX OF THE iAYER IN WHICH ZTJZ(J7) LIES, uU15::) 2,8* C OTLF. „ELT(J7) IS THE DISTANCE OF /IJZ(JZ) AROVE THE BOTTOM EDGE OF 00155 2,9* C TH LATER U0150 2,0* DO :',.) JZ=1.2 ODifil 2,1* .LA ,rJ1(J7.)=IMAX+1 ' 00ib2 212* 32 nErT(J7)=Z1J7..(07)7ZM . Lu163 2,3* IF(DFLT(JZ) .GT. 0.0) 60 Tn 35 U0i6t) 2,4* IF(LNYJI(JI) •F0. I) GO TO 35 u0167 2,5* 1.Ay,i7(Jz)=LAYJ7(a) - 1 6017C 2,6* J817:1AYJ7(J4) u0171 2,7* 'i_Mz7m-Hi(J(t) u0171 2 , 3.4. c AT THIc: 1-dOT•T 4M DFMOTLS THE OOTTOM OF THE LAYJZ(J2) LAYER u0172 2,9* GO TO 32 L0 173 2,0* 3s IM=ZNP 00176 2,1* C 0017.1 2,2• C COMPUTATION OF EM WITRICES FOR ALL IMAX LAYERS OF FINITE THICKNESS 00i73 2,-3, C Em(IH,JP) FOR I - TH LAYER IS STMED AS EMP(I*IP.JP) L017:1 2,4* DO 3h T=1•IMAX oU,OU 2.5i L=rI1I1 u0 ,-0 1 2,6* VX=VXI(I) u0,!fle 2,7* vY=VYI(I) 00e06 2,3* H=i4(1) 110e04 2,9* CAi L •WN(OMEGAIAKX ► AKY ► C ► vX.VY ► H ► EM) UUe0b 2.10* HO 3h TP=I,2 u0e10 2.14 HO 3o ,IP.71.2. 00.13 2,2. 36 FM:A1oTP,J1-)=Ey(IP ► JP) u1.1,11;) 2,3* C 1;0/.1,', 2,44* C CONPUTATION OF RPP MATRIX. THTS ACCOMPLISHES THE SAME AS CALLING u1)1..) 2,5. C SUkNothINE RRRR uUe.17 2,6* uP:41#11=1.0 u01.2u 2,7* RP,(1,2) -z0.0 OCe21 2,8 * RP0(P.))=0.0 uUe.2'e 2,9* RP.J(P.2)1.1.0 uUe23 2,0* Do 38 T=1,(MAX uei20 2,1* JA ,..A=ImAA41 - I 1,04'27 2,2* 00 37 TP;..1.2 uUe32 2,3* 1 10 37 .11 , 1.2 u0?3;) 2-4* 3 7 IAL14Y(TPIJP)=FuP(JASArTPrli*RPP(1 , JP)+EmP(jASA , IP , 2)*RPP(2,JP) u0,00 2,5* DO 3H TP=1,2 u0,.1 2,6* HO 3h jp=1.2 uOrtio 24.7* 3A RPI:(1P,...-)zDUMmY(I) PJP) u0e:to 2•3t C uUe52 9k (.0(.T = AH;1R(>P(1,1))/(ARS(PPP(1,1))+ABS(00(1 , 2))+ABS(RPP(2 ► 1)) 00c52 2:,,0* 1 4.AltS(RPP(2,2))) u0e53 2:41* IF ( O(JOT .LT. n.1 ) Go TO 120 u0e5!) 22* v2.0I=1:2P/RPP(1.1) UO263 2:,8* •GO TO vin u0 4, 64 2:,9* 7 . 1 3n F2.401=u1P(.!•I)r1P+RpP(2,2)*F2P . U0165 4;0* I5n 1-2•40T=F2ROT 60466 2"1-% PI01(1)=0.0 u0e67 2n2* PHiP(1)=F2nOT 00e7U 2..3* KT6UP=1 • 0071 2.,4* it=1 •oAX+1. , 00e72 2.5* PHII(K)=FIe u0473 2.6* P1-02(K)=i;2P (.10174 2 67 * - 33 1 II:-.F11I1(K) ot),?75 2.,8* . I2=PHI2(K). CoUe7b 2‘,9* K=.:-1 00+:77 2'0* ••• 'FOC .F0 1) Gn TO -46o b0...,01 - 2,1* C=rI(K) U0002 2/2+ vx=1/xI( A ) L10.)03 2,3* VY=VTI(K) UU004 2/4* CAIL AAAA(uHEGA.AM•AKY.C.vX.VY•A) 0.305 2/5* A=A(1.1)**2+t(1.2)*A(2.1) 0U006 2,6* IFiX .Gr. ii.0) 60 TO Sun . 0011u 2,7* — 333 PHr1(K)=Fro(K•1,1)*Ti+Fm(tit ► i,2)*T2 u0.)11 218* 144 ► 20(.1=F•O(K.1)*T1+FmP(v.2•2)*T2 60:;12 2,9* 60 TO AA 00013 2-0* 34n 01:.:A(1,1)*(1+A(1.2)*T2 00314 2.,1..• 02=A(ni1)+11+A(2.2)*T2 u0a15 2..2* IF ( DI .Lf. 0.n .i,WD. T1 .LT. 0.0) GO TO 341 60317 20,3* IF ( 01 .GI. 0." .AND. Ti .GT. 0.0 ) (4 TO 341 Gu621 2.4* IF U DP .Lf. 0.0 *ANT). T2 .LT. 0.0 ) GO TO 341 U0J23 2+,5* IF ( DP ..Gl• 0,n .ANn. T2 .GT. 0.0 ) Go TO 341 U0.52b 2+,5* GO T0.333 - 00320 2.,7+ 3111 LO,ATIVI0E 00O211 2”3+ C AT TnIc. PO1WT VHF r0RRENT vALUr OF K IS NOT 7En0 OR ONE U0a27 2,,9* KT.i0P=V.. 0103u 2-.0* nO 360 K=2,KTOun 00..)33 2y1* JEt=K-1 U0034 2,2+ T1=P0IIIJEI) 00035 2.3* T2=PHI9(JEI) UO336 2,4.. PHi1(K)=FMP(JET.2.2)*T1-EM0(JET•1 , 2)*T2 00037 2..,5* 36 n plii2(K)=-Ewr(JrT.2.1)*T1*.E.AP(JET.1.1)*T2 00.34t 2,8. 40n wZr) = 0 01.:.42 2-,7* 1.12(.2 = U 0U.:,43 2-.8* )ApImX = 1 u0o44 2,9* IA-?.1X = 1 UU.45 3.04 AP%mx = A9,;(PH11(1)) o0.146 3.1+ AP-.''1A = Al. ,(PHY2(1)) 00041 3.2* 00 407 LNN1t=1.T71AX 0005+! 3.3* LW = LK:1 + I 00353 3..4. Aptp = AnS(PHI1(0)) v0054 3.5k IF (APIP.LE..AP16X) GO TO 40 00056 3.6+ IA.J1MX = Lw 00.157 3.7* APIMX = AFL(' v0.)6u 3.8* 403 AP.: 1 = AtiS(Ficl(LN)) 00061 3.9* IF (AP2P.L.AF2vX) GO TO 4,15

.2,6 3,2' 40s IF (Ww11(041)*PHII(LN)).41.0.0) NZC1 = N7CI + I :,67 343* IF (IPRI2(LNM1)*PH12(LN)).41.0.0) NZC2 = N7C2 + 1 .)71 344* 407 LO•TINW :J73 3,5* RI = Fnf1(IAP1uX)/AP2mx 374 3,6* k2 = Pni1(IA 02mX)/ ► P2MX _,75 3,7* k3 = PHI2(1)/AD2mX .)70 ' 348* wRiTE (v.4(49) nmEGAIVPHSE•TAP1mX4R1•NZCI41AP2MX.R24N7C2 ► R3 •11 3,9* 4044.F0wmAT (IH F2F12.5.9)013,F12.549X , I3 , 9X , I34F12.5• 9 X ► 13/F12.5) 41 3,0* 41C nO 450 o1=1 , 2 415 3,1* . IDe=LAY,17(JZ). --ID - 3e2* c=cI(InA) •,17 3,3* vx:VxI(inA) •42.0 3,4* vY:;VYI(IOA) 42i 3.5* CI.47(J7)=CI(IDA) -422 3,6* VX4J[(oZ)=uXI(TDA). 423 3,7* VYIJZAJZ)=VYIITr, A1 .24 — 3,8.47-- 1F(IOA .F.O. IMtX+1) GO TO 420 1 , 420 3,9*. IF(IDA .LE. KTnUP) GO TO 4 -0 43u 3A* 7 JEi=10A+1 ,31 3-41* 41=4.I(In.1) -0ELT(JZ) ,32 3:42* CA4L M4...im(0AEGA4AKX , AKY4C•vX4VY.H•EM) ..35 3..3* .S1(J1)=Fm(1/1)*PHIOJET)+Em(1.2)*PHI2(JET) ,34 - 3..44* S24.11)=im(2 , 1)*PHIOUET)+Em(242)*PH12(JET) 43t, 3,6.0 GO TO n5n -,30 3.16* 42n FON=EXp(-G.DELT(JZ)) -437 3A7*. 5,14JZ)=FIP+EoN H.4u 3.18* 52(J7)=FPR , EON J441 309* - GO To 450 J44z 3,0* ---- 43n h--;FLT(Ji) 4 4,5 3,1* cAIL 11w..;4(omEGA.AKxPAKyrcrvX,vrpH , Em) 444 3.2* = C1iJ7)=Em(2,2)*PHII(IDA)-E4.(1.2)*PHI2(IDA) 0..411 3.3* S2t,),_)=-Fto(41)*PH11(InA)+PM(1.1)*pHI2(IDA) 0440 - 3,4* ' 45n cc:6,41'1,1HE . )4 46 3,5* C. o440 3,6*'-- C AT THIc POINT SI(J)) ► 52(JZ)4CIoZ(J7) , ETC. ARE STORED FOR J.Z=1 AND 2. J.4° 3..7* C tNE CumpUTE TUE DOPoi.ER SHIFTED ANGULAR FREG/UFNrY AT SOURCE ALTITUDE. J450 3,8* 10n u0.-zoMF6A -AKX 4 vXTJZ(N5cRCE) -AKy*VYIJZ(NSCR(:E) .:.)45( 3,9* C . D4511 3:10* C wE CoMwulE Zi=f1 AT nPSE14/ER ALTTTME 4513:41* ;fTN=A.n008/C1J7(NO6S))*S1(mORS) -CIJZ(NOBS)*S2(NOBS) 3451 - 3:42* C -- . -0451 3:43* C itERE W.: TAKE THE ArCELERATION nF GRAVITY TO RE .0098 KM/SEC**2. ,0,51' 3:44* C v0451 345* 'C COgPUT4.4TION OF INTFGRALS 40452 3 : ,6*. IT-3 • :1456 37* CAIL ToTINI(0M- GAFAKY.AKYFTT4L,X3WHI1,PHI,) 0454 ' 3:15* - 1FIL .F(.. -1) an TO 200 J045o 3:0*. • IT=7 4)457 34•0* CA, 1. ToTINT(OM-oAtax.AKY•TT.L.X7PPHII.PHI2) 0.60 3.'1. IFiL .F0. -1) a0 TO 20n 0460 3.2* ' .0 J0460 3,43* C FINAL ANSWFR A462 3,4* -7-- AMoLTD= 0.5*S2(NSCRCE)*ZFN/((X3+)(7)*80m) , 0466 3.5* RETURN 0463 • 3..6t END OF COMPILATION: NO nIAGNOSTICS. wHoGOI **4*** 50687 *4**** &FuR,S 1.5.6117,St.087 FOR slIF-1../10/74 - 18:1o:57 (6.)

SuBRONT.NE NMOFN ENTRY POINT 00n112

STORAGE USU.,: CODE(;) 000137; DATA(n) 000333 BIANK COMMON(2) On0621

EXTERNA. REFERENCES (FLACK, NAME) 00o3 AAA, 0004 ROI( u0o5 COS u0o6 SIN 00o7 SORT 0 0 10 NFRR31.

STORA.,E ASSIGNMENT (HLork, TYPE, RrLAT1VE LOCATION, NAME)

00u1 O00054 111.. 0000,R 000nOn A 0000 R 000010 AKX 0000 R 000011 AKY 0000 R 000013 C 00(12 R 00001 CI • 0000 R 000nI7 GU 0000 R 006016 nUSO n002 000455 HI 0002 I 000000 IMAX 000 O0002o INJP$ nano I 000n12 J n000 R 000004 RPP 0000 R 000014 VX 0002 R 000145 VXI 0000 R 000015 VY 0002 R 000111 vY/

00100 1* C NMnFN (SHBROUTTHE) 7/25/68 LAST CARO IN DECK IS NO. 0010u 2* - C U0100 3* . C .. .. . -_--ABSTRACT ---- 00100 4* C u0100 5* C TITLF - NmnEN U010u 6* C SUBRuuTME: TO COMPUTE THE NORMAL MODE DISPERSION FUNCTION FPP u0100 7* C FOR GIVel ANoMLAR FRFoUrNcy OMEGA, PHASF VELOCITY MAGNITUDE 001.00 8* C VPHSL A,40 PwA,SE VELOCITY DIRECTION THFT<. FOP SHOULD VANISH UO ► OU 9* C IF RoTH'UPPrn AND LOWER PouNDARY CONDITTONS ARE SATISFIED FOR UUlOo 40* C - --- TOE SOLJTIODS OF THI: RESIDUAL EaUATIONS u0400 AI* C 0)100 ,2* C OPHI1)/r2 = A(1,1)*PHI1(2) + A(1,2)*PHI2(Z) uU ► OU 13* C tiOLOU 14* C D(PHI2)/n7. = A(2 ► I)*PHI1(Z) + A(2,2)*PHI2(Z) OO ► OU .5* C 00100 16* C WHERE THE EIFFIENTS nF THE MATRIX A VARY WITH HEIGHT Zr BUT ARE 00..00 .7* C CoNsTANT IN EACH LAYFR nF A MULTILAYER ATMOSPHERE. THE ELEMENTS ,10100 .8* C OF A ARE FUNr:TIONS OF OHEGA, AKX ANO AKY AS DESCRIBED IN ...aNI&O - LI u010u .3* C AtOzOMEr.A*SIN(TsiETK)/VPIISE uClOu ,4* C UOIOu .5* C ThE FUNCTION FPP IS DEFTNED AS THE VALUE OF PHI1 AT THE GROUND uu10U e6* C (2=n) ';',IEH (1) THE uPPEP BOUNDARY CONDITION OV PHI1 AND PHI2 uOiOu ,7* C DECRFASING •YPONLNTIALLy WITH HEIGHT IN TFIL UPPER HALFSPACE PUIOU eli* C IS. SCTISFIEn( AND (.2) P,411 AND PHI2 AT THE BOTTOM OF THE UPPER vUtOu .9. C HALFSPACE Apr GIVEN RY A(1,7) AND -(G+A(1 , 1)) WHERE u0100 AO* C ..... (.“...Sn,ITim(1,1)**2+A(1,2)*A(2.1)). THE EIE•ENTS OF A HERE ARE u0100 Al* C THOSE. APPROpPIATE To THE CIPHER HALFSPACF. CONDITIONS (1) AND tJU100 A2* C (P) ARE NOT INDEPENNFNT. tCONDITION (1) IMPLIES THAT G**2 .GT. 0 u010u A3* C AND CoNolTInN () WITH t**2 POSITIVE IMPLIES (1). IF G**2 IS uUiOU .-,4* L NEGATIVE , FPP DUES NOT rXIST AND L=-1 Is RETURNED. OTHERWISE u010u .15* C L=1 IS RETUWA7D. . . u0100 A6* C u0t0u A7* L PRoGRA* NOTS uUiOU A8* C u4I00 ,-,9* C THE PaPAMETERs DEFINING THE MULTI1AYER MODEL ATMOSPHERE uU100 .0* C ARE PpESU•ED TO ri;. STORED IN COMMON. UUIOU •.1* C uOtOU ,.2* C TOE SHRROUTINV RRuR IS USFD TO GENERATE THE MATRIX RPP u0100 43* C WHIrN CONpiECT'N SOIUTIONS OF IHE RESIDUAL EQUATIONS AT uUtOu •4+ C ME DrITTOM OF THE UPPER HALFSPACF TO SOLUTIONS AT THE u0100 •.5* C (,ROUNn. IN TFRNS OF THIS MATRIX' THE NMDF IS GIVEN BY uUtOU -6* C UU100 47i C 1•PP= PpP(1,1)*A(1 , 2)-RPP(1.2)*(G+A(1 , 1)) uOlOu 41* C o0i0o 49* C LANGUAa.F - FORTRAN IV (360' :4EFERENCF MANUAL C22-6515-4) u0100 •10* C uUt00 nl* C AU7HOR - A.D.PTERCE• M.I.T. , AUGUST , 1960 u0k0u !,2* C uU10d n3* C ----CALLING SEOHENCE ---- u0i00 n4* C UOtOu n5* C SE• !,U..RoUTINE'i LNaTHNoWIDI-N.MoOUT uOIOU :16* C DI•ENSIoN CI(Inn),VXI(100).VYI(100) , HI(IRO) U0100 •1 • * C CO-MON INAA 0 CI.uXI'VYI(H/ (THESE MUST BF STORED IN COMMON) UutOu :,8* C CAll. NmOFN(OmE.VPHSE(THETK.L(FPP•K) UOtOu n9i. C u0100 .0* C ----EXTERNAL sUtIROUTINES RE0UTRED--- - ulii0o .1* C u010u u2* C - aRwRiMmMmIAAAA.CAIFSAI u0100 .5* C u0i0o.44 C -4o..-ARGUMFNT 1 IST--- - tiOtOu .5* C u01011 a)* C ii•BA R*4 ND IN:, tiO1O0 4,74 C , VP,(SE R*4 ND IN' IJOLOO .,6* C 1H-TN R*4 NO UU100 4.9* C L I*4 ND g'1 )* u010u (0* C 1444 R*4 NO OUT uU100 41* C K . 1*4 NO OUT (ALWAYS RETURNED AS K=0) u0t00 (2* C U01011 '(3* C COMMON STO4A6L USEn uOtOd 141 C co..oN lmAX•CI.VXI•VII.HI U0100 (5* -C v ,, it14i Mir L HI R04 in() INo u010o ,1* C uU4 Cu4,2* C - ---INPUT ,,---- U0100 03* C uOtOu 04* C um:GA =AlonOLAR FuroU6. NcY IN RAD/SEC u010u 05* C vP,4Sf =PHASE VFLuCITy mAGNIToDE IN Km/SEC uU100 06* C TH.TK =Pw1sE VELOCITY DIRECTION RECKONED COUNTER CLOCKWISE uU100 07* C. FROM THE x AXIS IN RACIANS U0100 03* C I•AX E-NtimoEtt OF LAYrkc of FINIIE THTCKIJESS uuiOu 09* C CI(I) =SnuND SPEH-) im KM/SEC IN I-TH LAYER u0100 40* C VXT(I) .:.X- comPONENT OF w1w0 VELOCITY TN I-TH LAYER (KM/SEC) T.,0100 41* C VYT(I) =V COMPONENT DT:- WIND VELOCITY TN I-TH LAYER (KM/SEC) u010u 42* HItI) ..1- 714TCKNESS IN KM OF I -TH LAYER OF I-TH THICKNESS UULOu 43* C 00100 44* -_-_auTruIS- --- uulou 45* C uuloo 46* C L =1 IF NORMAL MnDF DISPERSION FHNCTIoN EXISTS, -1 IF 00100 -a* C IT DOES NnT. u01.0u ,FT* C FPP . =NnrwAl. MOoE DTSPERSION FuuCTInN u0100 49* C K =DmmmY PARAmET ► R ALWAYS RETURNFD AS K=0 0010t, 1.0* C u010u 1,1* C ----PROGRAM FoLLOwS BELOW--- - uU106 1,2* C 00101 1,3* SUi.iROUTINE NMDwN(ONIEGA.VPHcEITHETK ■ L,FPP,K) DO1O1 l u ll* C 00101 1,,5* C DImEwSfON ANn LOMMnN STATEI,FNTc LOCATING PARAMFTERS DEFINING MODEL uult11 1.6* - C WI TILaYFR ATMoSPHrPF uU106 1.7* nI„,;ENSTuN 0(1nn)IvXI(Ino).VyI(100)pHI(100) u0104 1,,f1* coAum ImAxICI.vxIO/Y1,11I u0104 1.9* u0105 140* 0Ii.ENSTow A(2 , 2),RPP(242) U0105 1t1* C uOLOS 1,2* c COmPuTTION OF AKX AND AKY uOLOo 143* AKy=0•r6a*CO5(THETK)/VPHSE uulU7 1,4* 1+Ky=UMF6A*SIN(THETK)/VPHSE u0107 1,5* C uJ107 116* C cOmpui.TioN OF MA7PIX A AND G**2 FOR UPPER HALFSPACE b011o 1i7* J=IMAY.-1 u0111 1111* C=rI(J) u011 1,9* vX=VXI(J) u0113 1,0* vY7VYI(J) u0114 1.1. CA.1. AAAA(0MrOa.AKX , AKY.C.vX.VY.A) u0i11)1.2* GUk.0=A(1.1)**2+A(1 , 2)*A(2.0 uOilb 1,3* C. uUilo 1.4* 1F(GUS(a .Gf. 0.n) i.70 Ti 11 u0116 1.5* C u0110 1,6* C 6Us0 lc LESS THAN 7FRO UU12u 1.7* Lz-1 00120 1,8* C 2ECOIIu ORDFR APPROxIMATION FOR FPP u0120 1,-9* C 0=.0.n ANDGuSw=n.n uU121 1.10* GUL:O=0.0 UU122 1.1* (0 TO 11 U0122 1.12*

L/0124 1,7* C cOmptadTlOw oF );PP r.q1TRIX Uu125 1,*(1* CA1L TOMP(uMEGA.AKXFAKY/RPp/K) u0i2ti 1,4* u0125 1.0* C COMPLJT.tTIOW oF FPP u012h 1.1* rP6,7.1WP(I/1)*A(1.2) -RPP(1 ► 9)*(GU+A(1.1)) 0012° 1,2* 1.3* UE;ORM u0130 1.4* END

END OF COmFILATTON: 140 nIAGNOSTics. 40HoG/P 4*.*** sUlom ***.** ..F.JR/S I.S.pcs,sod6 (3,)

SUBoOUTINE NXmODE ENTRY POTNr 00n266

STORAoE U:Eu: CoDE(1) Ono 0c) DATA(.) 01)0021; BIANK COMMON(2) 0(10000

FATI-RoAi IC EuLINCLS (hLoCK. NAME) 0003 NERR3$

SiO;ONGL ASSIGNMFNI (kLoCK, TYPE , RcLATIVF LOCATION/ NAME) (01)1 000u51 10L 0001 000953 100L 0001 00013n 20L n001 On0141 30L 0001 000172 40L 0 0 111 45L 0001 000947 !AL nOni 000250 60L 0001 000251 70L 0001 000252 BOL DOhn I 0n0u02 I 00n0 I 000n05 ICHK nOn0 000006 TNJP$ 0000 I Onno01 10 0000 I 000003 J 0000 1 000004 011) 0000 I 000n00 09

u0lOn 1* C NXmOOE (SU)tROUTINE) 6/24/68 LAST CARD IN DECK IS NO. uniOu 2* C Uu100 3* C uOlOu 4* C uulOu 5* C W0i00 6* C famoDE u0Juo 7* C PROGRAM TO EIND A PotmT WITH CO(,NDXNATEc I=IFNO,J=JFM0 IN AN vOtOo 6* C ARRAY WITH unm LOLUmuS ANN NVP ROWS. FnUNU POINT COPRESPOMDS oUiO4 9* C To sTARrING p0SITI0w FOP CALCULATION oF PHASE VELOCITY VCRCUS 00101.1 ,0* C FhEONFHLY Oc A pAoTiCULAR ()HIDE O MOUE. A TAPIA: OF VALUES OF oUlOu sip C THE SIGN OF THE HORaiAL MOOE DISPERSION FUNCTION IS PRESUMED uUrOu 12* C To tar' StOPEn AS IHNOriE(W-1)*NVP+I) FOR EACH POINT (I/J) IN THE uoino 13* C Ar:RAY. DIFFERENT COLUMNS (J) CORRESPoNn TO DIFFERENT FREOUEM- u0t00 14* • C CIES wHiLE nIFFERFNT ROwS (I) CORRESPONn TO DIFFERENT PHASE LoJIOU .5* C. VFLoCTTIES. - THE SEARCH PROCEEDS FROM AN INITIAL POINT (IST/0ST) uO*OU 16* C TO soCCESSWE AUJACFNT OOINTS HAVING THE SAME INMODE AS THE uF A POINT WITH

.-...,.. el. L OPPOSITE INMOnE UOI.Ou .2* C u0100 r3* C 2. Ip MUST OF THE HIt,HEST POINT (LOWEST I) IN THE REGION uUiOu i4* C SATISFYING COmDTTION I u0100 ...5* C Li(1100en. C 3. It:It: m THAN i POINT SATISFY 1 AND 2 ► THEN THE POINT LOIOU ..7* C DrTERAINEH IS THAT FURTHEST Tn THE LEFT. u0100 ,8* C U0100 19* C 4. ONLY POINTS IN THi; RECTAMAS ARE CONSIDERED U010o 00* C • u010u J1* C THE COMPUTATION ASSIiMFS REGION OF SUCCESSIVELY. ADJACENT POINTS u0100 A2* C HAVING :.,AME TNMODE IS STNPLY CONNECTED AND THAT PHASE VELOCITY ' u0100 .13* C CHRvLs .END DoWNWARiA, T.F., O(VP)/J(OM) •LT, O. (THIS CAN RE • uulOu ,)4* C THE CASE. PRnVIDIN6 vP lc GREATER THAN THE MAXIMUM WIND u010o 05* C VELoCIP(i) TF THE POINT IS FOUND, K=1 ► IF NOT FOUND' K="1. uUtOu A6* C uuiOu A7* C LANGHAi..E — FuRTRAN IV (3H0. PEFERENCE MANUAL C20-6515-4) • uULOU 08* C AUTHOR — A.D.PTERCE ► M.I.T.. JUNE ► 1968 uOIOU ;19* C utJiOO ,.0* C ----CALLING SrOUENCE ---- u0100 4 1* C 1.20.100 42* C SEE cl1.4RoUTINE ALLmOD U0(00 ,..3* C HI,ENSIC:N1NMOnE(1) (VARIAHLE DIMENSIONING) uUtOu •.4* C CAIL NxmO0E(IST,JST , NON,NVP,INmODE,IFODIJEND , K) u040u ,.5* C uU100 ...6. C NO EXTERN/II SUAROUTTNES ARFREnUIRED uOtOo ,.7* C uLI10U ..8* C ----AaGUMENT IIST---- u010u ..9-. C uU100nO* C IST I*4 ND (JOIOU .1* • • C o SrI*4 HO I tts'ilr-) uOlOu ..2* C (40,1 I*4 NO INP u010u 03* C NVw 144 ND 00100 !w4* C IN—ODE 1*4 VAR JIUI u0100 .5k C IFNI) I*4 ND OUT UOIOU :,6* C uFfoN I*4 , NI) OUT (iOlOu 07. C h I*4 OUT ()LIM) h(1* C ' uUIOu !,9* C NO CUM.ON STORAGE itsrn uulou "0* c :AIN, .1* C ----InPuTs---- u010u .2* C 6010u .3* C IS,. :.-Rn,:! INDEX OF START POINT =Cn1,101N INHEX nE START POINT .. VU IOU ,,4* C OUIOu .5* C ;J,IL =Nn. OF COIUMNc oF ARRAY uUIOu . 6* C NVw :No. OF ROwS Ow ARRAY - 00i0U “7* C Ift,!ONE(L) =SIGN OF NoRMAI MOOE DISPERSION FUNCTION, 1 IF POS., 00100 .8* C — 1 IF NEG., 5 IF IT DOESN'T EXIST. LET I=L MOD NVP ► 60100 .9* C J=(L—I)/NvP+1. THMODE(L) IS SIGN OF NmnF FOR uUjOU JO* C OorGA=Om(J), hIIASF VEL. =O(T) , WHERE OM(J) .GE. 014(4-.4) uU.LOO . 11* C AND VP(I) .LE. VP(I-1). UULOu .2* C U0A00 ,3* C ---- ouTpurs---- -cfro IF POINT (IFNn,JFNO) IS FOUND, 1 IF u010o /8* Yre., -I IF NO. U0100 ,9* C UOIOU n0* - ---EAAMPLF---- U0100 nl* C u0100 n2* C SUpPoSw THE ARRAY ne I;JMODF VA1 UFS IS AS SHOwN BELOW V0100 n3* C OU300 , $4t C 4.44++++++-- NVP=St NOM=11 UU1OU C +1- 4 ++++++t~ UUIOu .6* IF IST=8,,NT=:.i THFN IFND-7.3,JFND=2,K=1 1)(1100 ,J1* IF IST=2,JST=S THEN IFND=1,,IFND=9,K=1 U0100 ,03% C 55 ------+ IF IcT=3,JsT=7' TUrN IFND=3/JFNO=2,K=7.1 L10100 IF IcT=8,JsT=2 THFN K=-1 00100 IF IST=2,JsT=11 THFN K=-I uulOa C5D------+ QOIOU -12 u01..0n 43* 44* Fr%LLOWS SFLOW----• oUlOo 45* C u010U C u001 .47J, SU.4RUUTINE NXMnnE(iSTedST,NOm.NVP , INMUOE,IFNO , JFND,K) 00101 UU104 ni.ENSTON INMOnr(1) unL04 1,0* 1 IF( 1ST .Gf. Nur, ,1ST ,GT. NOM) GO TO i00 uU100 1.1* o9a(oST-1)*NVP+IST u010/ 1.2* 10::InhoL*(„(9) ,411U 1.3* IFI 10 .NE. I .AND. TO .NE. -1) GO TO ion uUlIo 1.4* C 00410 1.5* C TH• POINT (isTsJsT) LIES /1.1 THr ARRAY AND THE NORMAL MODE DISPERSION 1,0110 1.6* C FUNCTI.01 EXISTS AT THIS POINT WITH A SIGN 10. WE FIRST GO UP UNTIL uUtlo 1.7* C A fID,TFRENT I•0ODE IS EHCOHNTEREO OR UNTIL WF REACH 1=1 uo112 La* IziSf 1„9s • J=.61 1;0114 1,04- ln fFi I .E(J. 1) an Tu 30 1 , 1=1-1 1/2* Oln=1J-1)*NVP+T 0)120 113* ICNK=IHMODc.(J1n) u0121 1,4* IF( ICHK Tr!) 00 To 10 u0123 115* I=(71-1 1,6* C uUI26 1,7* THE CU4NeNT 1 IS NnT I. IF TH• ICHK OF THE POINT ABOVE IS NOT 5, WE 00123 118k C mOvE T.. fir u1)124 119* 15 1F1 ICHK .E0. 5) GU TO 50 U0i2o le0* IFf J 1) r:0 TO 20 LAJI3U 1,1* 00131 1..2, a1n=1J-11s-NVP+T 1,0134 1,3* ICK=INA00(:(,11n) 0013;: 1,4* C vai31. 1,54 C IF THE ILHK oF THE CON'ADEmEn ON POINT IS Int WE TRY TO 00 HIGHER 0013.J. 1,6t C AGAIN. u0t36 1,7* IF(1E.HK .(:0. In) Gu TO in 1/8i 0.1mi . 0U13;) 0.113:) IA* C >tiL HAVc In AIIUVF tHF CURRENT POINI AND ARE FITHER ON THE FAR LEFT OF u014u- 1,,,5* ,,E: )=J LUt4I loo* REIURN u0141 1,7/ C ullig1 1 • 8* C 1HF LO•SiDFRED NFW POINT Is ON THE FIRST ROW. WE GO TO THE RIGHT. u014,-: 1.1'4* 3n iFt J .Eo. ' ,JONI GO TO nn (10144 1..04 - 0=.t+1 u0145 1,1* 0111=W-1/*AVP+r Uu140 142* ICHK=INAW0J1n) uUI 117 1...3 y - IFt ICHK •F.O. yr, 60 Tn 30 uUlBi 144* J=A - I U0t51 145* C •

U0151 1,0* • C IF THE PoIN• Al TH• RIGHT OF CHRRENT (I•J) IS - 10F wE HAVE SUCCESS UuL52 147* IFi 1CHK .E0. -in) GO TO 2n 00152 14S* C uU152 149r C IF IT tS NnT -10, wF ALLOW FOR POSSIBILITY OF TNMOGES:5 IN UPPER RIGHT U0152 1n0* C HAND CHRNFR nF THE TABLE ANN) TRY TO SKIRT THESE FIVES BY MOVING EITHER uu152 1.1s C pOwNwAw0S oR To THr RIGHT. 00154 1•2* 4n IF( I .Era. NVP1 GO TO 70 u015t1 1,3r 1=i+1 .00157 1,4. J1n=(J-1)*HVP+7 Wilso 1u5* iCHK=TNi4OUF(J1n) uUlG tt 1,6* C U0160 1,7* C IF THIc ICNK IS +In WE ARE IN A POSITION TO MAKE A TRY OF MOVING TO 00160 1 : ,at• C THE RI,HT. uOi61 1,9*- 4u IF( ICHN .NE. TO) 00 To AO 00161 1 0 0* C u0161 1.1* C IF 'vn Aftt- no THE RT(HT HAW SIRE OF THE TABLE THE oEsIRER POINT CANNOT u0t61 1“2* C HE FOU•n. wt-. RETURN will K=-1 u016,, 1,3* . 41.5 tFt J .Co. NOVI GO TO inn OUifIS 1,4* J=J+I ii016B 1,.5* C U016B 1,6* - C IT IS rAKEN FOR GRANTED THAT THE INMODE OF POINT ABOVE CURRENT (I,J) u016b 1 1 ,7* C 1G 5 _GIME IT wAF, ROONo TO RE s TO THE LEFT ANn ABOVE. THE INMODE OF UU16b 1"84. C THE POtNI TO Tri LIFT IS In. TF THE NEW INMoDw IS +IC, wE HAVE TO TRY u016, 1,9* . C TO Mt0b. FURTHER TO THC RIGHT. U0166 1,0r J1.1 7:(J-1)*NVP+T UU167 1,1* IC.,K=INnODE(J1n), 00170 1,2* TFt 1CHk .E0. TO ) Go TO lis UUI72 1,3* Jr.. r- I uU172 1,4* C u01.7/,! 1,5* C IF ThE CURRENT ICll IS 5, ...E TPY TO GO DOWN AGAIN. THE OTHER POSS.* U0172 1,6. C 1BILTTv, ICHK.T-I0 tmnICATES SUCCESS u0t73 1,7* IF( 1CHK .E.Q. ...I0) Go TO 2n , • u0,7!1 1,3* GO TO 40 U017!) 1,91' C U0175 1,0*. C wE CoNlIollr HERE ERnm IS. THE POINT AROVE THE CURRENT (•J) HAS u0175 1,1* C ICNK 0:0. S. THE sITUATION IS 50CH THAT WE CA.,.! RESUME CALCULATION 0017b 1,-,2* C AT 4s. ANti TRY 10 MnVE FURTHER TO THE. RIGHT. u0170 1,3* 5n (;(;) To 41i UU170 1,4* • C , . U0170 1,,S* C, WE CONrINUF HERE WYTH 1=1/0= ,100 FROM STATEMENT 30. SINCE WE HAVE-NO 0017(1. 1,-,6* C PLACE r0 Gn THE SEAOCH IS UNSUrCFSSFUL. wE RETURN WITH K=-1. 00177 . 10* 6n 60 To NO

-...- .11 U) lu luo u0r.Ou 1M2$ .0 u0rOu 143* C 6E CONrINUE HERE ForiM STATEMENT 44 WITH THE POINT BELOW HAVING UUtOU 1444, c ICHK *HE, To. THE POINT AT THw RIGHT HAS ICHK .E0. 5. wE CANNOT uUenu 145* C SKIRT iI U FIVES ANn HENCE wE RpTuRN WITH K=-I. LUe.01 146* 8n GO TO 100 0.)?,01 147* C 00.efl1 1•4B* C hE CuNrINUF HERE FcrOM 1 1 3.45,6n , 70.0R 80. THE SEARCH WAS UNSUCCESSFUL. U0.?0.! 1.9* lOn K.T.- -1 uUe0.5 2u0* kETURN 00e04 21,1* EMI '.

END OF LOmPILATTON: NO nIAGNOSTTCS. *Thit,G,P **.**** ‘0.11;84 ***•** tdFuR r5 1.55/19.50;io9 hOR SIIP-1./10/74 - 18:11:10 (3.)

SuBROpTINE NXIPNT ENrRY POW n0n237

STORAGE USELI ■ CODE(1) 0J1030,; OATA(n1 000032; DIANK COMMON(2) On0000

EXTFRwA, r__E-EFErStr_r_RN NAME) 0003 NERR31

SlORAGL ASSiGNMENT (riLnCK, TYPE , RFLATIVE LOCATION. NAME)

u0n1 00010b 15L 00n1 On013n 2 5L nOnl 000202 PCL 0001 000222 30L 0000 I 000007 ID 000 I 000011 1Dm 00n0 . 000n12 1NJP$ 0000 100u003 TR 0000 I On0005 IRU 0000 I 000001 10 0000 I 00000n jll 0000 I OriOnOP JI2 n000 I 000004 j13 0000 1000006 J14 0000 I 000010 J15

u0101) 1* C NXTPNT (OMOUTINE) 6/24/68 LAST CARD TN DECK IS NO. uOino 2* C 0010u 3* C

UOLOu 4* C - ---ANSTRACT-- -- ovitio 5* c. LJOIno • 6t C TITLE' - NX•PwT t./40u 7* C PHONRAr TO pIND THE MrXr POINT (I2•02) nF AN ARRAY or NROw ROWS uulou 84 C AND HcOE COtHMNS GIVEN THE PRECEDING POINT (II•J1). POINT WILL u01 0 0 9* C 1W DsrD IN cHHSLCIIUENT CALCULATION OF A PARTICULAR POINT ON THE vOin0 10* C PHAs5; V► 0CyTY VERSUS FwEDUENCY CURVE OF A GIVEN GUIDED MOOE. U0100 114, C A TAAL OF uALUiS OE- THF' SIoN OF THE NOwMAL MODE DISPERSION 0010o 12* - c Furlc:TON IS PREt,UMFH TO nF STORED AS IHHI(J-1)*NVr+I) FOR EACH b0100 ' Ji* C. PoINT (l , J) TN ME ARRAY. DIFFERENT COIUMNS (J) CORRESPOND TO 6.10100 14* C 016,YRENT FpEnULNCIFS WHILE DIFFERENT RoWS (T) CORRESPOND TO J 1 1 i 0 u 15* C DIFFrRENT PHASE VrLDCITTEs. SUCCESSIVE POINTS ARE CHARACTERIZED

....„,,,,_ ,/uN. siNLL UOTH ..... u posSIAILITIFS CAN OCCUR. THE prsi&JATED TYPE INDEX ITYP1 DENOTES tiu100 il* C THE PREvIOlk OSE OF THE POINT (II/JI) IN COMPUTATION. THE VALUE u010u .2* C ITYP2 WILL rN GENERAL fl PEN0 ON THE DRE)/10US VALUE ITY('1. • U0i0ii r3* C THE DERIVED VALUES OF I ► H2 ► ITYP2 AitE CALCULATED AS FOLLOWS. uUiOu ,4* C UOiOU .5* C I. IF 'TIP'. IS t AND INM OF POINT TO RIGHT IS OPPOSITE UOLOU e6* C OF /0=INMUJ-1)*NVP+I)t THEN T2=11,J2=J1rITYP2=2. 00100 r7* __. C u010u - fa* C TwE POINT (I2,JP) MUST EITHER HE THE DIRECTLY ADJACENT 0010u e9* C PnINT TO 1HF TIGHT (I1•J1+1)* THE POINT DIRECTLY BELOW u0i0u At c (11f1rJ1). OR THE ADJACENT POINT To THE LOWER RIGHT ulliOu .)1* C (T1+11,11+1) Iy CONDITION I DOES NOT HOLD • uUiOU .12* C uulOu .13* C _. 3. Twr CHOSEN POTNT N.UST HAVE THE SAME INN AS (11fJ1) 00i00. 0 4* C - AND HAVE A POINT EITHER DIRECTLY ADOVE OR DIRECTLY TO 1)00)0 ;,9* C TwE RIGHT WITN OPPOSITE IOM. 00100 .6* C 00100 17* C L. IN THE rv•n- HOWE THAN ONE POTNT SATISFY CONDITIONS uoloo .,O* c 2 ANO 3. PRIMTY OF SELECTION IS (1) THE .POINT TO OulOu ...9* C THE RIGHT, (2) THE POINT DIRECTLY DELOW. (3) THE POINT u610j ,.0* C Til THE LOWER pIGHI. IF THE SFLECTED POINT SATISFIES uUTOu ,.1* C CRITERIA POP uOTH ITYP2=I OR Pt ITYP2=1 IS RETURNED. u010u ,.P* C OTHEMTISE. TH• APPROPRIATE ITYP2 Is RETURNED DEPENDING ' 00100 ..3* C Om WHICH CRITERION IS SATISFIFO. uOtOu •.4* C 00)00 ..5* C THE CoM,IJTATION ASSHmrS REGION OF SUCCEcSIVELY ADJACENT POINTS UUIOA -6* C HAVING sAmr iNN TO sir SIMPLY CONNECTED AND THAT PHASE VELOCITY u010» 47* C CURVES ()END POWNWAPHS/ 1.F./ D(VP)/0(OM) .LT. O. IF NEW POINT U0i0U ..8* C IS FM/NO/ Kzfl. IF IT TS NoT FOUND. K7...- -1. 00100 49* C U0i0u •O* C LAmGUArZ - FoRTRAN IV (3.0/ PEFERENCE MANUAL C28-6515-4) uUtOU nl* C AUTHOR - A.O.PTERCL , M.I.T. , JUNE' 1968 uuiOU n2* C uU100 n3* C - ---CALLING SFOHENCE ---- 00100 n4* C 0010u n5* C SEF ,..th.ROUTINE MODFTP 00100 n6* C HImENSTUN INMOnr(1) (INMODE IS SAMF AS INM) u010u :0* C CALL NxIPNT(I1..)1FITYPI,I2,J2 ► ITYP2 ► NROW,NcOL.INMODE.K) • u010u n8* C IF( h .L•. '- 1) GO SOME4:HFRP 00100 r,9* C US.. 12,J2/1TYF2 • yU100 IX* C UOI.OU .,1* C NO ExTi42NAL LIBRARY SUJROUTINEs ARE REWIRED 0010u .,2* C uUlOu (.3* C ----AqnUMENT LIST---- U0100 ,,4* C 00100 .5* C 11 I*4 NO Ilia-. 00100 . 6* C J1 I*4 N D I tip ' U0100 .7* C ITrP1 I*4 ND I No bUlOu .8* C 12 I*4 NO OUT uu10a .9* C J2 I*4 ND OUT 00100 /0* C ITyP2' I*4 NO OUT U0100 /1* C . NRnW I*4 NO INo uU100 /2* C NCoL I*4 Id0 INP 0610u a* 00i00 C • u010u C 0010u C. 11 INDEX OF cTART. POINT 6U1.00 rd.* C J2 INNEX riF START POINT 0010u .c ITYP1 =TxrE 1NO.Ex OF START POINT, I mEANS POINT ABOVE HAS u0i0u 10* DrrrE“EmT INM. 2 MEANS POINT TO RIGHT HAS DIFFERENT. uulOu C 1(4'4. .00100 -r•S$ F.NirliEft OF ROdc IN ARRAY 00100 ,6* NCmL =Nn"RER OF COLIONS IN ARRAY 0010!) 4.7* Lida =Sr7,H uF NORMA, MONE DISPERSION FUNCT/ONr 1 IF POS., uulOu ,J1* C -1 IF Nrs., 5 IF IT DOESN'T EXIST. LET I=L MOO NVPr• C Jz(L-i)/NvP+1. INMOOF(L) IS SIGN OF NMnF FOR U0i0u ,0* C OmrGAzOm(J), DHASF VEL. =vrtiy ► WHERE Om(J) .GE. OM(J-1) 0010u ANC VP (I) VP(I-1) uU1Ou 42* 00.10U 43* C 00100 44* C UOIOU 45* 12 =R,' INDEX OF rONNO POINT 001.00 ,6* C J2 =ColUMN IMoFX (tF FOUND POINT 60100 -r7* C ITrP2 :Tyr INnEg OF FOUND POINT 001n0 ,46* C K =Ft AG INOICATImG IF POINT (12'J2) IS FOUND, 1 IF YES' 0u1.00 C -1 IF NO 1,0* 0010u 101* u010u 1,2.k 00iDu lo3s C SUPPOS*: THE ARRAY nr INM VALUEF. IS AS SHOWN NE101.4 uUlOu 1"14* 0610u 1,5* C +++4 f NROW=8, NCOL=I1 u0i0u 1,6* C ++f 00100 1o7* C. 5------++++ IF /1=3 ► j1=4,ITYP17.1 THEN IP=3rJ2=5. 00100 1.8* C • ITYP2=1,K=1 0010u 1.9* C UU10t1 1,0* IF II:j(J1:79tITYP1=2 THEN I2=21J2:10. 0000 1,1* Ir(P2=1,K=1 00100 1,2* 60100 113 IF 11=31J1=7,ITYPI=1 THEN I2=3,J2=7, 1/010u 1,4* ITYP2=2rK=1 u0iflu 115* -C 0010u 1,6* IF II=3 ► J1=11,ITYP1=1 THEN K=-1 tJUJAU 1i7* u01.0%) 1,8* 0010u 119* ----Pi'OGRAm FrILLOwS nKLOW---- uUlOu 1..0* UOLDu 1,1* C uut01' S(1.nuUTINE NXTPNT(I1.J1eITYPIPI2 ► J2 ITYP2 NHOW(NCOL,INM,K) U() 101 le3r. ► ► 00106 Jell* D•0ENSTON INm(1) 00iO4 1,5* 01):(j1-I)LNROw*I1 U011)1) 1,60- 10zIwM(J114 uU100 le7t 1 IF( 10 .F.O. 5 .AR. I1 .GT. NROW .01-t. J1 .GF. NCOL) Go TO 30 uoint, 1,8* C poion 1..9* C In IS Inm nO ruINT TO THE NIGHT. 10 IS INN OF POINT (I1,J1). uu114 l.,4* C 00414 1,5* C IR HAS THE f-,Ame.: SIG•; AS IO. WE CHECK TRU REPRESENTING INM OF UPPER 00114 1,6k C 017,H1 001VT. IF TIATS IS -10. THE RIGHT POINT TS THE DESIRED POINT. 00114 1.17* C IF IT IS NnT -10t wE CANNOT FIND (I2 ► J2). 00416 IA* If) J1i=(,J1)*NitOw+.1-1 00117 '1 0 9* 1RH=INAJ16) 00120 1.0* 11 IF( 1RH .NE'• - TO ) GO TO 3n 00122 1.,1* 1TO2=1 00123 '1.2* I2=I1 00124 1.3* ,J2=J1+1 0312a 1.4* K=I 00i2626 1.,5* REIUM 00126 1.6* C UU426 1,7* C wE ARRTVE , HERE FROM STATEMENT As THE POINT TO THE RIGHT HAS A 00126 1,3* C OIFEt:RN1 T,„:M. IF THIS IS -I0 AND ITYP1=1 ► WE HAVE (I2 ► .12)=(II , J1) 00126 1.9* C WITH ITYP2=2. . IF THIS IS St WE CANNOT FIND (I2 ► J2). u007 1;,,r* 15 IF( IR .E0. 5 1 GO To 30 001.27 1, C U0127 1!,2* C IR=-I0 Al TITS POINT 00131 1n3* ir( 1TYP1 .NE. 1 ) GO TO 2s 00133 1•,4* 12=11 00134 1:,5* J2-J1 00135 1')6* IT111 2=2 U013o 1,7* K=1 u61.37 1:-.;i4, REIthiN 004.37 1.19* C 00137 1.,0* C 1R=-I0. ITYP1 IS P. WE CONTINUE FROM STATEMENT 15. IF WE ARE ON THE U0137 1“1 t C HOTTWI ROW; wE CANNOT-FIND NFW POINT 00140 1,2 ► 25 fF (I1.1:0.NROW1 GO TO 31) 00140 1„3* C

. 011140 1”4* C WE CoNcIHE14 POINTS PrLOW AND Tn LOwER RIGHT . 00142 1"5* ul,,=(J1-114NROw+T1+1 Liu143 1,6* ID=INM(J(4) 00144 1,7* J15=1J110w+r14-1 0014:1 1..4* 1014=INO,J1:1) U0145 1r194, C 4J0I4S 1,0* C IF IHR IS s ► Nr: CANNOT Elfin THE NEW POINT u0i46 1,1* 2A TFI. 1nR .Ew• - ri ) GO To 30 U-01 116 1,2* C utfilin 1,3* C IF IhR IS TO ► IHE mFXT POINT lc THE DR POINT 00150 1,4* 27 1Fr Ina .NE. In ) GO To 28 OUtS, 1,5* 12=11+1 00153 1,6* J27...J1+1 UU454 1,7* /TiP2=1 00155 1,3* K=t 00156 1,9* liEfUmN i/Oi5o 1,,0* C UUlbo 1,1* C IR=-10, 1TYP1 IS 2. ION IS -TO. WE CONTINUE FROM STATEMENT 27. 00157 1,,2* 2M 1Ft ID .NE. TO ) GU TO 30 00157 1A3* C 00157 1,4* C THE NE%T PnINT'IS THE UOWN POINT U0161 1,5* ' 12=11+1 00162 1,,G* •J2=JI

****** S,,R$19 ****** DATE 121074 PAGE

1.7* 0)164 1.84 u0165 1m9* 14ETUNN 00165 1,0* C u016b C *.E ARRIVE HERE FMOM 1.7 , 11.15.25.26. THE NEXT POINT CANNOT BE FOUND UUlou 1•02. K=..1 u0161 "143 4 RETURN u0170 1•4* Ehh)

Ft U OF COAWILT.TION: NO MAGNOSTTCS. 6HuG ► P 41-..**** 4UH90 4. ***** 6E0,5 1.S,,B90$S090 • Fuji slIF—le/10/74718:1t:17 (1')

SuligiOuTiNE PAmROE ENTRY POINt 00n3n6

STORA6E USEU: CODE()) 01iO354; DATA(n) 00027P; BIANK COMMON(2) On0621

EATI;RNA, REFERENCeS (hLnCK, NAM;

0)03 NAMPUE 11004 A ,, H;JT 0005 u0o6 NIOeb 0007 SQRI 0 0 10. 0011 NER,i31

STOaAhE ASSIGNMENT (BUICK. TYPE. R•LATIVE LOCATION. NAME)

o001 000031 117G ► On1 000053 132G nOnl 000221 I646 0001 000251 1756 0000 000020 19F o0o1 000.1J22 20L o0n0 000122 22F nOnl 000044 23L 0000 000150 31F 0000 000214 41F 00,0 000d2o S11.: nnn2 R OnOnn] CI 0000 R 000015 cONST n002 000455 HI 0000 1000000 II 0002 .0n0u00 IMuX onoo 000,g11 1NJP$ nno0 I 000013 TORS n000 1 000011 ISCP 0000 I 000003 J 00110 I 000001 J1 0000 I 000nna J2 0000 I 000004 K 000n 1 000016 K1 0000 I 000017 K2 0000 R 000005 oMFGA 0000 fl 000.-12 IJonS 0000 R 000010 oSCRCE noon U On0014 UEU 0000 R 000006 VPHSE 0002 000145 vX1 0002 000411 VYI 0000 R 000007 x

00100 1* C RA•AROE (SUNROUTINE) 7/30/6A LAST CARD IN DECK IS NO. OU ) 00 2* • C U0 100 C GOTO0 C 00100 J* C TITLE . PA ► WOF

_._ ,-• L wHERE CoNST=4.0/SoRT(P*PI). CI (t) IS THE SOUND SPEED AT THE UUtOu 17* C nkOmin ► (P5M7CE/1.E6) IL: THE AMHIENr PRESSURE AT 7SCRCE DIVIDED uULOu IS• C' BY TIIF AHnIwNT PREScHPE AT THE GROUND. THE (QUANTITY UED IS 1)0100 19* C THE soUARE pOoT OF (AMRTENT DENSITY AT 70(35)/(AMBIENT DENSITY AT ' uuloo .ps C 2SCRcF ). THE SCALING FACTOR ALAM IS GIVEN nY uU1Ou el* C u0100 ' .2* C • AtAM w. (I.E6/PSCRCE)**(o.333333)*(CI(1)/CI(ISCR) uUjOu ,3* C u0.10k1 ,4* C WHERF CIIISrR)'IS THE SnUND SPEED AT THE SOURCE ALTITUDE. THE . u0100 .5* C SIG•IFICANCF OF THESE_ OHANTITIES IS EXPIAINEn IN SUBROUTINE ' uUiOu ,6* C PPAMP. uOluu e,7* C uU100.B4 C HRoGRA.A NOTES UUkOu .9* C _ uOlOu .-.0* C THE PARAMt:TERk;JM.X.CI.VX1oVYI.HI DEFINING THE MULTILAYER uUkOU A* C Afm0S0HFIL ARF 11 RrSDMED STORED IN COMMON. THE AMBIENT uU100 .N2* C PRESSHPES APE COMpUTED BY CALLING SUBROUTINE AMONT WHICH k,U1OU .+3* C ALSO rOMDUTFS THE INDICES JOBS AND ISCR OF THE LAYERS 0010u ,,4* C IN WITCH JBSERVFR AND SOURCE, PFSpECTIvELY, LIE. UU.kOd .15* C UOIOU .‘6* C 1w comouTING AMDIFNT DENSITIEStTHE IDEAL GAS LAW u010u .,7* C RHO= aAMMA*P/C**2 IS.uSED. THUS UFD =(CI(ISCR)/CI(10BS))* u0100 ,18* C GoRT(ponS/PSCRCE). uulOu ,,g* t., uOlOu 40* C LAwGuA,,r - FuRTRaD IV (3i3O, PEFERENCE MANUAL C28-6515 -4) uulOu .1* C uU10u 424, C AUTI IURc - A.D.PTERCE AND ..1.pOSEYr . M.I.T.t JDLY ► I968 uUlOu .3* C uUkOU .4* C • - ---CALLING SwODENCE - --- u0100 .5* C UOIOU .6* C SEF TH,. MATH PROGRw UU,k0u .7* C - DI-ENSTON CI(Ino),VAI(100).VYI(100).H1(1O0) ' uU10u .3i C DI,:ENSIIA NST(1) ► 0•IN(1).0mMD011)tVPMOD(1),AMP(1) utaiOu .9* C THE PR.)GRAm tiSr.S VapIAALE DIMENSIONING FOR ODAmTITIES IN ITS uUkUu nO* C ARGUmE4IT LIST. ;JUIOU •,I* C • CO„,,MuN Im4X.CI.VXItVYI,HI THESE MUST OE STORED IN COMMON) U0100 :121, C GAIL PAmPD1::(7SrIT.EoZOnS,MIL:ND.KSTrKFIN,OMMoD , VPM00.AMP,ALAM , o0100 ;)3* C I EhCT,THETKPNPPNT) uuiOU :,4* C uukOU !,5* C -.7-EXTERNAL cUHROUTINES REDUIREU--- - kJULOU .;i5* C LOIOu .,7* C AM..4NTeNAMROEIToTINT , MMM.AAAAtUSEAS.UPINT , FLINT.8BBS , CAI.SAI 00100 nfi* C u0 0u n9* C ----ARGUMENT IIST---- U0kOd .,0* C CulOu ,-.1* C /SeRCE R*4 NO IN uGiUu .$2* C /0..5 R*4 ND INp uOi00 A3* C iiIDND I*4 NO INP 00100.4* C KSr I*4 vAR INp u0100 .5* C KFsN 144 On flBi ()Qin .61F C DMIAD R*4 VAR INp • UOIOU 1,.7* C VID,.00 R*4 VAR VID U0100 ' An* C ' .AMP R*4 VAR OUT u0100 ► 2* C NP.,NT I*4 NO INP 0010u /3s C uU100 ,n* C COmMoN STORAGE USEn 00100 r!is C cO.AuN ImAx ► CI,VxI ► /YI.HI 00100 /6* C 00100 /7s C IMI.X Ito 0 INI) 0310. 0 /8* C . CI R*4 100 INp 00100 /9s C vX1 , R *4 100 IN ► 0010u r.04 C VYJ R*4 100 IND

VOIOu i-, 1* - C HI R*4 100 INp uUtOu 02* C u0100 r,3* C - ---INPUTs-- -- u0100 04* C uOlOu 05* C ?SrHCF =HEIGHI IN KM nF BuRST ABOVE GROUND tJUIOu 06* C iCiAS r-NrIGHI IN KM ,,F OBSERVER ABOVE GROUND 0010u 07* .0 mD.-NH =thr0Ek OF GUIDED MODE'; FOUND ue1Ou AB* C KS1(N) .1.I14-FX- OF 1;TRST TABULATED POINT IN N-IN MODE uUJOO 09* C KFlw(N) =ImnEX OF tAST TAnDLATED POINT IN N-TH MOUE. IN 00100 90* C GrNE•AL, KFIN(N)=KST(Nfl)-1, 00100 41* C oMmow(N) =Apr.AY STORING ANGHLAR ERLuNEHrY ORDINATE (RAU/SEC) OF 00100 92* C Pn+NTS ON DISPERSION cNRvES, THE NMOUE MODE IS STORED' ui) 00 93* C Fnn N PET...FEN KsT(NMODEI AND kFIN(NMODE) , INCLUSIVE. uO106 ,i4 r.. C VP•,On(N) r.Arr.AY STORING PHASE VELOCITY nRDINATE (KM/SEC) OF oUiOu 95s C PnIHTS ON DISPERSION CURVES. THE NMOUE MODE IS STORED 00100 96* C Fn(). N HETv,FFN KST(NMODE) AND kEIN(NMODE). 0010u 97s C THkIK =DTRECTION IN RADIANS TO OHSFRVER FROM SOURCE, RECKONED .00100 48* C CoHNT•R CiOCKw1cE FROA X AXIS. u010u ,,9* C NP”NI =P4TNT OPTION INDICATOR (SEE NAM' IN MAIN PROGRAM). 00100 1.0* C IMO( =NwBER OF LAYERS OF FINITE THICKNESS. 00100 1..1* C (ALI) r.SWIND SPEEr) L.' KM/SEC IN I-TH LAYER u0100 1.2* C VXr(1) ....:X rOMPONEI,T OP WIND VELOCITY TN I-TH LAYER (KM/SEC) L,6i0u 1,3* C vYt()) =Y COMPONENT OP wIND VELOCITY TN I-TH LAYER (KM/SEC) uUlOu 1,4* C HI(I) =THICKNESS IN 04 OF I -TH LAYER OF FINITE THICKNESS uO LOu 1,5* C 0010U 1.6* C - _--OUTPU1S- --- uUlOu 1,7* C uUi00 1.11r C AMP(J) 7.7A•.PNIMPE FACTOR FOR GUIDEo WAVE EXCITED BY POINT uUlOu 1., C Erx- ROY SONRCE, UNITS ARE Km*s(-1). THE J-TH ELEMENT 00100 110* C CroRESPONNS To ANGULAR FREOUENCY OmMOD(J) AND PHASE uUIOu 1 1 1* C vizinciTY IrPNOn(J). THE AAPLITUDE FACTOR IS APPROPRIATE utilOU 112* C To 114: NODE-TH HOOF IF J .GE. KST(NMODF) AND J .L.F. 001011 1,3* C KIN(NMOD(:).. THE AMP(J) IS THE SAME AS AMPLTD COMPUTED 001011 1,4* C BY •ff1ROUTiNE IdAmPnE. uUIOU 1.5* C AL.M =A r,CALING FACTOR NEPENnENT ON HEIGHT OF DURST, MAL uulOu 1.6s C In' CU0E RonT nF (PRESSURE AT W;OUNn)/(PRLSSURE AT uULOU 1,7s C Eli p5T HEIhHT) TtrifS (sOU,4D sPFED AT CiROUND)/(SOUND uUlOo litis c SprEn AT BURST HEIGHT). uUlOu 1,9 ► C FAST =A ex.NLRAL AmPlITUDE FACTOR DEPENDENT ON DURST HEIGHT 00100 1.0c. C ANID 00SFRvpi HEIGHT. A PRECISE DEFINITION IS GIVEN u())0u 1,1 ► C 1.1• THE AUsTRArT. 01)100 1.2* C (Ain(' 1,, 3 ► C - ---PROGRAM FnLLOWS BELOW-- I/L/10U 1:4* C U0i01 115 SUI4RUUT1NC.PAMPOE(ZSCRCErZaStmDFNDO

)0114 1.16* • lq 1-0.,MAT (1H1/41Y,26HPHII ANn PHI?. PROFILE DATA ///63HOIAPIMX = NO. uut14 1.194 loF LAYER PO WHICH AR(PHIOIAP1MX)) IS A mAXIMHM/63H IAP2MX = NO Uuiltl 2 O.. LAYER h'OR wPTCH ADS(1112(IAP2MX)I IS A MAXIMUM/42H P1 = PH U0114 311tI)1P1•x) / AWITPHI2(IAP2mX)) /42H R2 = PHT2(IAp2MX) / ARS(PH 00114 142. 412(TAP2Mx), /371) RJ = 1)1II2(1) / AOS(PHI2(IAP2MX)) /40H NZC1 40114 1,3* 5= oF TiMES CHANGES SIGN /40H N7C2 = NO. Or TIMES PHI2 C LQ11.4 144+ 6HA4,61-3 MG11) 00115 145* 2n COTINnt. uU111) 1,.8* C DO L60., 10 CTNPUTE AMP(J) 00110 1,7+ 1,0 25 II=1 , NDI:ND 00121 1,8* IF (NPRNT.LT,01 GO To p3 40123 1,9* wrifTE tu,110 IT 00.1.2n 1•0* - 2p 1“).,mAT (ill ////// IH osIX0sHmODE '12 /// 1H OX,5HOMFGAOX,5HVPHSE 0O i2„ 1,1* 1,6.,1,11IAP10,1n)(.2HRI,AY,4N7C116X,6HIAP2My110x,2(R2,8X,4HNZC2,1°X U 0 126 1:)2* 2,2•R6 /) Uu.1.27 1L.3* 23 J1:AST(11) UUL30 1:.4* J2:7:KFIAHTI) UU131 1n5* nO 25 J=JI , J2 00134 1,h* K = J 00135 1•7* 0Mw.GA = OMMOD(w) . 00136 1:,8* VP.4SE = vPm0D(k) UU137 1,9• LAiL NAPDE(ZSrPCE,ZO(3S,OMp. GA,vPHSF , THETK*x , NPRNT) 004.40 1,0* AmL..(h) = X • • uUt41. 1.1* 25 CO,TIVu 40141 1.2* C hNn 61: ii)OP 1,3* 001.41 1"4* C COmPuThT1014 OF AmHyrra PREscUR*S 00144 1,5f CAI L AmONTW;CoCF , PSCR(:E,IcCR) (.1014b 1,6* t.A, L Am"NT(ZOOs , POJS,InS) 00(4b 1,7* 06145 1,81 C COmPoniTiOu oF SORr(DrNSITY RATIO) u(114u 1"9+ DE" = (LI(ISCR)/CICIOB(0) * SORT(POS/PScRcE) u(l)+)n 1,0* u014D 1,1+ C LOmPuTnTIOH DF ALA. AND FAcT - 00147 1,2* AL.04=(1.*"(JPSCL,CF)**(0.333;33)*(C1(1)/C1(IcCR)) UUI.47 1,3* C NOTE T-AT cl(1) IS SOUND •SprKo AT THE GROUND 00150 1,4* COST 4.o/SOpT(2.0*3.141s93) 00151 1,5* FAIT = CoHST*Iri(1)*UEn*(PcCRCF/I.E6)**(n.1333333)) U0152 1,6* IF,NPRm1 .14E. RETUR, u01'234 1,7*• .R,TE (",31) ZqCRCE , 70",,FACT,ALAM 0016[ 1,8* 31 i=o,mAr(lHi. 76Y, 361ITMiULATIoN OF SOURCE FuEE AMPL1TuDES, U01.62 1,9* 1 20-I FPOM sUHRnUTINE PAmPD,.: //P1X , 19HHEIGHT OF BURST =,

- 0°162 1 •4.3, 3H KM / 21X, 19HHEIG. HT OF OBSERVER=, F6.3 , 3H KM/ 00162 1.0.* 1 21X, unFACT, 14X, 1H=, F8.3, 7H KM/SEC/ 21X,4HALAmr14X, 1H=, 00162 142* 1 F4.3) KI=KSTiII) u0173 1A8* K2A1-ID(II) u004 1,0* DO FA1 J=k14K2 U1.1177 140* 5n WRITE (6.51) Omvo0(..1 ) ► vPMOn(J) ► AMP(J) (Wail " 1v1* 51 FO.d .4X , p20.5fF2o.50P20.8) o02(17 142* REluRN • uoe10 143* END

END or COmPILATTON:' " NO nIAGNOSTICS. 6HuCIP ******

I.JE0R ► S 1.SuRal.SuL1Y1 FOR SI1F-1,/10/74 - 18:11:28 (2,1

SUBRODT,NE PHASF EN1RY POINT 00n105

STORAGE USELA CODE(1) 80013a; DATA(n) 0000131 8,ANK COMMON(2) On0000

EXTFODAk RE)-ERENCE5 (DLnCK, NAME) u0u3 SORI 004 ASIN u0n5 NCRIt31,.

STORAGE ASSIGNMENT (HLDCK. TYPE' RpLATIVE.LOCATIoN. NAME)

u001 Mali) 200L none OnOn14 30L n001 000064 300L 0001 000074 4001 0001 000052 60L 0000 R 000004 A 0000 R 000nn2 AI 0000 R 000001 AR 0000 020006 INJP$ 0000 R 000000 0

jOIOU I* C - PHASE (SONROOT.E) 8/15/A8 LAST CARD IN DECK IS NO. HilOU 2.* C ,(1 100 3* C ,ulou Li* C _---ABSTRACT---- iOlou 5+ c ulou 64 C TITLE - PHASF 0100 7* . C CONI/LkSION nr A COMPLEX NIIMUER FROM PFCTANGULAR FORM TO POLAR 0100 8* C FDRm °Jou os c ulou. io* C GivEri TWO REAI N ►m0FRS RR AND PIP A MAGNITUDE R AND AN ulOR I I* C ADGLE PHI ARE COM&UTED SUCH THAT UL00 s2* C 0100 13* c on + I*R1 = 0 * FXP( I*PHI ) U101) ,14* C 0100 15* c. wflERE I = (-1) **0.5 .. 1401! 16* C J100 -17* C LAmWAGE - FuRTRAN IV (3110e REFERENCE MANUAL C25-6515-4)

c .."-. - USAGF"'..

U0100 e3 ► C LJUiCIU '4* C NO cUBROHTINES ARE EMBED u0100 e5* C u0100 /6* C VORTRA. USAGE ut)10u /7* C uulOu /8* C CALL PHASE(RR , py*R,PHI) u010o ..9* C u0i00 .10*. C INPUTS uUlOu .,1s C ut.)100 ,12* C RR REAL PART OF THE COMPLEX NUMBER (WING CONVERTED vOlOu .13* C RA4 uUiOu •4-4. .0 u0i00 .15*C. .... RI /1.1AGINARY-: PART OF COMPLEX NUMBER BEING CONVERTED . U010u B6* C RA4 u0100 .17$ C 0U i00 .18* C OUTPUTc utii0U 09* C UU10U 40+ C R MAGNITUDE OF THE COMPLEX NUMBER uu100 .1. C RA4 00100 •.2* C

u0Inu 43* C PHI PHASE OF THE COMPiEX NUMBER (RADIANS) ( - PI.LT.PHI.LE.PI) Oii0u -.4* C R4.4 uulOo .5* C 001011 46* C u0100 47* C ----EXAMpLES---- U0100 4.84 Uu100 .9* C CALL PHAsE(0.0.1.0.01,PHT) u010u :)0* L U0.00 n1P C n = 1.n AND PHI = 1.570796 ARE RETURNED (.10110 n2* C U0100 n3* C CA1L RHASEt1.0.-1.0 , RIpHO u010u :;44, C

UU100 !)5 -4 C R = 1.414214 AND PHI = -0.7b539R2 ARE RETURNED Ou*Ou n6* C 0010U •17* C 00100 -1E3* c ----PROGRAM cOTLOWS BELOW-- -- O0100 n9* 00100 “0* OU101 .1* SU..RNUTINE PHAcF(RR , RIFR,PHI) uu101 r.2* UU103 nay N=hR5(RR) 4 ARS(14r) U0104 "4* IF(0-1.E-2n) 1.1,3U U0107 n5i 1 R=".o u0110 PHI=u.n u0111 .7* PErURH uOile .8* 3n AR=nk/0 U0116 .9* AI=R1/(4 JO1I4 (.0* A=.)RTlAU**2+At**2) 10115 /1* )47.4 ■ *A iUlln /2* PIH=ASTN(ABS(AT)/A) 10/17 /3* IFIRk) ho•1)0,6a 0122 /4* 5n TE4R1) 3n0.300.2n0 RE.IvRN 400 pli1=P111-3.1 4 15Q27 u0135 RE4URN 0013(,. 40n PH1=',43,11 . u0137 RETURN L,U140 F.Nn

CND CF COMP1LATTON: .NO nIAGNOSTTCS. 10.41.) 00) **4.*** Wit92 *** e l.* 1.JF,JR#$ I.S,a(12.Su3if2 roR S11F-1e/10/74 - 18111t42 ()•I

SUBROuTINF PPAMP ENTRY POINT 00n152

SloRAoE USED: CoDE(t) 0n02024 UATA(n) 000057; BiANK COMMON(2) On0000

ExTERNA, REFERENCES (10_0CK ► Ni E) u0o3 SOURCE 00 1 14 xPRR o0,15 S ,.7;111 0006 NrRR31.

STORAGE ASSIGNMENT (HLACK ► •TYPE , RpLATIVE LOCATION. NAME)

Out 0n0U ► 11 1116 00n1 0000 7 1 116G ()On° R 000007 AKAY' 0000 R OnnOol ALAMP 0000 R 000012 DMAG 000 R 000v1:S oPmSF nOnu R OnOnlo FTMAG 0000 R 000011 FTPHSF 0000 I Oon002 II 0000 000017 INJP$ u0,10 I 000005 J 0000 I 0n0n01 K1 • 0000 I 000004 K2 0000 R 000006 OMERAT 0000 R 000000 0

JOI 1* C PPOAP (SUOROUTTME) 7/30/68 LAST CARD IN DECK IS JuLOO 2* C ,uinu wino qt ---- W100 5* ITLF — PPAAP . ....-- g400 6* C PROGRAe 4 To coHPUTE uNn qTORT.AMPUTUDCARRAY AMPLTD ANU PHASE 1,1400 7* C TIRAXPHASO rot: GOINF.D-10AVES EXCITED nY A POINT ENLRGY SOURCE 0100 8* C WITH fimr:--pr„yNOENCi; CORRFSPONDING TO A NUCLrAR EXPLOSION OF 0100 C ENERGY ucPOTill—HX—XJELD IN KT. THE VALHES FOUND ARE TO DE OiOu C SuBsE-Oth:NTLy 10* USED BY IT ACCORDING TO THE RELATION 010u II* ULUU .2* .,---- (PRI7 SsuRE 41 DYNES/CM**2 FOR A i;IVEN MODE)*50RT(R). 010u Ut00 = (NrEGRm OVER OMEGo OF AMFLTO*COS(nMEGA*(T—R/VP1+PNAsq) .5* )100 16* THE o ►AlITITTFS AMPLTD AND PHASO ARE noTH DEPENDENT ON ANGULAR 5 c TITLE - pPAMP 6 .- C - - PRuGRAN TO COMPUTE: ANn STORE AmPLITUDE ARRAY AMPLTD AND PHASE 7 C ARRAY pHASo FOR GUIDED WAVES EXCITED By A WIWI ENERGY SOURCE 8 C . • . wild TINE DEPENDENCE CORRLsPONDING To A NUC(.EAP EXPLOSION OF 9 C ENERGY DENOTED BY YIELD IN KT. THE VALVES FOUND ARE TO BE 10 C spHSEUNENTLy USED BY TNPT ACCORDING TO THE RELATION 11 C ... 12 C - (PRESSURE IN bYNES/cM**2 FOR A GIVEN NonE),,soRT(R) 13 C 14 .-- C = INTEGRAL OVER OMEGA. OF AMPLTD*COS(omEGA*(T-R/VP)+pHAso) 15 C 1G C THE QUANTITIES AmPLTO AND PASO ARE ROTH DEPENDENT ON ANGULAR 17 C FREOUL;NCy AND ARE DIFFERENT FOR nipFERrwr MODES. 1(3 C . . • 19 C PROGRAM NOTES 2u C • 21 C IN THE FORMULATION FOR A POINT ENERGY SOURCE, THE ENERGY 22 C • EnNATIoN Is WRITTEN i3 C 24 -• - C - Dp/nT -(C**2)D(RHo ) /OT = 4wpI*C**2*F(T)*(DELTA FNCTN ) 25 C AN EXPRESS/ON FOR F(T) IS ;:b C • 27 C FIT) =((L0-2)/CS)*PoS*(iNTEsRAL OVER X FROM 0 TO CS*T/L 23 C OF UNIVERSAL FUNCTION FUNTv(x)) 29 C 30 WITH L=(ENERGy/POS)**(1/3) AND Pns,CS REPRESENTING PRESSURE 31 C AND soUNO SPEED AT THE SOURCE. IF FIKT(T) IS THE PRESSURE 32 C AT A DISTANCE OF 1 Km FROM A 1 KT EXPLoSTON AT SEA LEVEL 33 C AND WITH TIME ORIGIN CORRESPONDING TO BLAST WAVE ONSET, 34 C . - THEN 5 C 36 - FUN/V(X)=((L1*P01)**(-1))*F1KT(L1*X/C1) 37 C 3d • C THE FOURIER TRANSPORm OF F(T) IS ACCORDINGLY FOuNn TO BE 39 C 40 c o(omFGA)= (1/(2*PI))*(T**(2/3))*(c)/c5)*(P0s/P01)**(1/3) 41 C 42 -- *(1/(-I*ONEGA)) -$FTmAG(OmEnAT)*EXP(I*FTPHGE(OmERAT)) 43 C 44 - , C • WHERE Y IS YIELD IN KT, IFSORT(-1), AND OmERAT=ALAM* 45 C 0mCOA*Y*4(1/3). THE FUNCTIONS FrlAn AND FTPHSE ARE AS 45 - C ComPuTia) Rye swRouTINE SOURCE. THE o0ANTITy ALAN IS 47 C (Cl/Cs)*(P01/p0S)** 1 /3 AS CONFUTED RY suRROuTINE 4E} ,, FAMPOE. . 49 C A LENGTHY DERIVATION NOT GIVEN HERE'INDICATES THAT 50 • C 51 C AMPLTD*ExP(-I*PHASQ)

'52 - C 53 C = -4*SoRT(K)*6( 0mEGA)*CS*6EO*SORT(2*PI)*AMP 54 C *EXPt-I*PI/4) 55 C 55 •C WHERE AMP IS THE SAME AS THE AMPLTD COMPUTED BY NAMFIDE AND

_ - PHASU =(3/4)*P1 — FTPHSE(OME RAT) 6 C 63 C IWPLTD=FACT*A1PA(Y**(2/ 3 ))*FTMAG(OMERAT)*SORT(K)/OMEGA 64. C 65 C WHERE FACT Is 4/SORT(2*pT)*C1*UED*(PS/P1)**(1/3) AND IS 65 C COMPUTED Py SUBROUTINE PAMPOE, . C7 C 66 C THr ouAnTrTIEs FACT, ALAN, AND Aa ARE IN THE INPUT LIST 69 C OF THE SUBROUTINE. NOTE THAT THF;E APE YIELD INDEPENDENT. 7g C 71 C THE SCHEME OF STORAGE FOR AMFLTD(J) AMD PHASO(J) IS THE 72 C SAME AS FOR OMMOD(J) AND VpM0D(J), SEE SUBROUTINE ALLMOD. 73 C 74 C LANGUAGE ,7 FORTRAN IV (360. REFERENCE MANUAL. C2 -6515-4) ' 75 .0 75 C AUTHORS - Ao.pIERcE AND J.POSEY, m.I.T.. J4(LY.1968 77 C 73 C ----CALLING sEVEncE--- 79 C 80 C SEE THE MAIN PRoGRAM 61 C - DIMENSION KsT(1).KFIN(1).OMMOD(1).VP' 100 (1).A!.• (1) 62 C Ulm•NSION A ,IPLTD(1),P•ASo ( 1) . V3 C THESE OUAUTITIES MUST BE DIMENSIONED. THE PRnnRAM USES VARIABLE R• C OIKENSIONING. rop ACTUAL DImENSIoNS ASSIGNED. St7E THE MAIN PROGRAM. wi C CALL PPAMptyIELD,NIDFNO,KSTOO*IM,Om.m0n,VPM0DpAmPtALA!.1,FACT, t16 C 1 AMpLfp•l'HA90) ti7 C till C ----EXTERNAL SUDROMTIMES REOUIRED---- 89 C 90 C SOURCE. PHASE (PHASE IS CALLED BY SOURCE) 91 C 9;! C ----ARGUMENT LIST-- 9:3 94 C YIELD p*4 No INP ci5 C MUPND - I* 4 r)')) IMP 96 C KST I*4 VAR IPP 1 97 C 0•: ' I*4 VAR Irsq/ 93 C O•MOD P*4 VAR INP 99 C - W.31,10 Rt4 VAR INP 1 0 0 C A 671P n*4 VAR IMP lel C ALAm R*4 Jr) /VP It-2 C FACT P*4 Nn IMP 1 11 3 C AmPLTO R* 11 VAR OUT 304 CPH ► so p*4 VAR OUT 105 C lou C NO OOM'ION STORAGE IS USED 1 1:7 C ' 106 C ----INPUTS-- 109 C 310 C YIELD =FNEROY.RELEASE OF EXPLOSION IN EQUIVALENT KILOTONS OF 11.1 C TNT, 1 KT = 4•21:19 EGGS. 112 C mCFND =NUMBER OF MODES FOUND IN PREVIOUS TABULATION OF 113 C DISPERSION CURVES.

voi clIRVES. THE tHlonE r4OFDE IS STORED FOR 119 H PETwrEN KST(Nn0PE) Apr) KFIrfNflOnE). 120 C • vPMDD IN) ARRAY STGKI ,lC PHASE vELnCITY oRnINATE OF POINTS ON 121 OTSPERsIoN CURVES. THE OoDF. MolE IS STORED FOR 122 O BETWEEN KST(WODE) AND KFIomnD(:). 123 C AMP(N) =AMPLITHOE FACTOR IH0FOEN0ENT nF YIELP COMPUTED 124 SWROUTIHE P.AqPuE CORPESPONDTHG TO ANGULAR FREQUENCY 125 OmMOD(N) AMO PHAsE vcLoCITy vnmoO(M). ' 126 C ALAM =A SOILING FACTOP DEPENDENT OH HEIGHT OF DORSI , EQUAL 127 TO cupE RnOT OF (PRESSURE AT 1RoUN0)/(PRESSURE AT 123 BURST HEIGHT) T1HES (SOHND SPEED AT GROUND)/COUNT) SPEED AT UURST HEIGHT). 130 - C FACT =A GEMERAL AMPLITUDE FACTOR DEPENDENT ON nURST HEIGHT 1 3 1 AND ("TsERVER HEIGHT. A PRECISE' DEFINITION IS GIVEN 132 IN THE LISTING (F SUDROUTINE nAHrIDE. • 133 34 C ----OUTPUTS---- 135 C 13i, • C , AMPLTU(N) =AMPLITUDE FACTOR REpRESF.ITIN TOTAL MAGNITUDE OF 1 3 7 C FOURIER TRANSFOITh OF THE CoNTr'IrHTICH TO THE WAVEFORM 13i C OF 4 SINGLE CHIMED MODE AT FproHENCY OHN:00(N). It 139 . C REPRESENTS THE AMPLITUDE OF TuE NNOPE-TH MODC•IF N IS 1 4 0 C BEr•!FEN KSTMICI (IE) KFINOTIOCE1, INCLUSIVE. THE 1 4 1 C PRECISE DEFINITION IS GIVEN "V! THE AlISTRACT. 142. C PHASO(N) =p11/.;E LAO AT ERLQUENcY OHH0D(H) FOR HODE-TH MODE w1.0", 143 C N IS DETA,FEM KSTINMonE) AND KFIN(NNODE), INCLUSIVE, — 144 C THE INTLORAND IS UNDERSTOOD To HAVE THE FORM 1e'5 C AmPLTD*Cos(oMviOus(TImE-DISTAH(;E/VPMOO)+PHASQ). 146 C 1 4 7 C ----pR1GRAM FOLLOWS RLLOw---- 146 C 149 SUOROUTINE ppAMP(YIELD, MnFMIKST,KFIN,OMMOnIVPMOnt ' 1-'0 - 1AK I i AP, PNANPIALAM, FACT r AMPLTO 1 PNo% SO ) 151 C • 152 C DIMENSION STATEmENTS USING. VARIABLE DIMENSIONING 153 DoI:Ns.ILA„,; KST(1),KFIN(1),OHMOD(1),NPM0D (1),AqP(1) 154 DIMENSION AMPLTD(1),PHAS0 ( )) 155 It NSION AKI(1000)iPHAHP ( 10 0 0) 156 C 1 57 • C=(YIELD)**(0.333331) 1 5 8 ALAmp=0*ALA.1 159 C 160 C START OF DO LoOp, II IS MODE•NUIOER 1 6 1 . DO 20 II=1 MDFNO 1br! L--- ---KI=Ks1'(I1) ► 163 K2=KFIN(II) 164 - C 165 DO 20 J=K1FK2 161, C COMPUTATION OF SCALED FREQUENCY ONFRAT 167 OCERAT=Omio D(J)vALAmP 165 . 0 COMPUTATION OF scnT(K) 16, AKAys.,. = ABs(wimoD(J)/vpmoo(J)) 170 'AKAY =•SoRT(AKAYSO) RETuRu ENO

RuNID: 2A17 ACCT: OGIA1019 PROJECT: INPRAsouND T IME : TOTALS CO:00:06.070 Cpu: 00:00:CO31A1 //0: DC:On:01,218

CC/ER: CC:00:04,668 WAIT: 0 9 :0 0 :00.100 IMAGES READ: 5 PAGEs: 8 START: 10:42:40 DEC 16,1974 FIN: 10:4 3 :10 nEC 16,1974

0003 u0n4 NT0e.$ 0 0 05 NERR31

SIORA6E ASSIGNMENT (HL0C11, TYPE. Rg•LATIVF LOCATION. NAii,E) 00002 000104 153E n000 006165 21F 0001 000036 311. 0 0 00 000155 11F 0001 123G 0001 uOnt 000041 33L U000 000201 41p 0000 00021p '1F 0001 000132 521. 0001 000152 60L 000 Un021t, tilt 0002 R 000001 rI 0002 R 000455 HI 0000 I 000145 I 0000 R 000147 0000 I 000156 IL 0002 1 000000 IMAx 0000 000271 TNJP$ 0000 I 000152 J 0000 I 000144 JET 0000 R 000154 UST En 00on R 000190 vX n002 R 000145 uXI 0000 R 000151 VY 0002 R 000311 VYI 0000 R 00006 XUV 00n0Ac 000n00

00100 1* PRIATMO (SUBROUTINE) 8/1/66 LAST CARD IN DECK IS NO. uU100 2* 00100 3* 0010u 4* C oulOu 5* C TITLE — 1'RATm0 uulOu 6* PRO6RAm TO PRINT OUT PApAMETERS DEFINING THE MODEL MULTILAYER 00i0u 7* ATMnSPHERE' A LISTING 15 . PRINTEO OF LAYER NUMUER, HEIGHT OF . • uU100 (3* LAYFR DUTTON,' HEIGHT OF LAYER TOP. LAYEP THICKNESS' SOUND SPEED. LUIOu 9* AND OF X ANn Y COMPONENTS OF WIND VELOCITY. 00100 )0* 1104.00 11* LANG0A,4F.: — FORTRAN IV (3h0, REFERENCE MANUAL C26-6515-4) OUIOU 12* u0100 13* AUTHURc A.D.PTFRCE ANI) J.p09EY• M.I.T., AtiGUST.1968 uU100 .4* u0100 ,9 x C - ---CALLING SrOWENCE - 7 - -

uU1Ou 15 ► 00100 t7* SEF MAIN PKOGRAN uOlOu 10* 11I0FNSToN CI(1nn) , VXI(160).VY I(100).HI(InO) uu'Ou i9* COMuN ImAX ► CI,vXI•VYI.HI (THESE MUST BE TN COMMON) 00100 e0-* CAll PRATMU uu101) el* 00100 .2* ----EXTERNAL cUOROUTINES REOUTRE0---- uu.tUti e3* 00100 e 4* NO ExTi;RNAL SUI4ROUTME5 ARE REpUIRED. 'u0i0u e5* e6*C. uu100 e7* 4,0100 e0* COmmoN sYORAGE UsEn

uOtnu e9* CO„IMuN IMAX ► CIO/XI.VYItHI uUi.0U I*4 ND 00100 .11* 1MAX INp R*4 100 IMP u0100 ..$2* LI R*4 100 INp u0100 .13* VXt u010u 04* VYr R*4 100 INp 0010u .15* HI R*4 100 INP IN 1-01 LAYER vXt(I) =X roMPONENT Or wIHD VELOCITY tt4 1-TH LAYER (KM/SEC) rith) 42* C vYr(II =Y roMPONENT OP wINO VELOCITY IN I-TH LAYER (KM/SEC) [100 45* C HI(I) =TNITCXWESS IN uM OF I - TH LAYER OF FINITE THICKNESS 1100 .4* C 1100 45* C ----OUTPUTS---- LOU1 .6* C / Oil 47* C THE ONO' OUTPUT IS A PRINTOUT Linu 4434 C Ilau ..q* C -EXAMPLE- - -- 110u .,04. C ILOu ,I..4, C MODEL ATMOSPHERE nF 10 LAYERS (TOP OF NEw PAGE) 11011 :42* C (IMAX = 9) t i nu r.3* C 110u :144, C 1.,iYtR Zit . Zr H C VX 11.00 .,54 C 10 22.50 INFINITE INFINITE 0.2 9 72 0.0062 liou •,5* C 9 26.00 22.50 2.50 0.295m 0.0093 46 u 474 C 8 17.50 2a.n0 2.50 0.2 9 36 0.0116 .40u -,01- C 7 16.00 17.50 2.50 0.2 9 31 0.0144 I Ou a9t C 6 12.50 15.00 2.50 0.2931 0.0165 100 "0* ' C 5 10.00 12.50 2.50 0.2951 0.0160 lou "I* C 4 /.50 10.00 2.50 0.3012 0.0149 400 .2* C 3 :).00 7.50 2.50 0.3117 0.0116 L00 .3* .0 2 2.50 5.00 2.50 0.3266 0.0038 LO0 .4* C 1 U. :J.50 2.50 0.3394 0 .0057 10u "5* C 00 "6* t 0=HIGH1 oF LAYrR OOTTON IN KM Z,T=HrTGlif nF LAYwil TOP IN KM (THE VY COLUMN IS !101.1 1.8* C =Wv•TH OF lAYEi4 IN KM • NOT SHOWN BECAUSE ilq0 "9* C C =SAMMO SPEF!) IM 10/SEC OF LACK OF SPACE. 'tOO ,0* C VX=X cOMP. OF WIND VFL. 111 KM/SEC. IT DOES APPEAR ON 1(10 41* C VY=Y romP. OF WI•D VEL. IN KM/SEc PRINTOUT.) 10u ,2* C 100 ,3* C . FoLLOwS BELOW---- 100 ,4* C A04. ,5* 3,Uit0OUTINE PRAY ,. 0 inl 1 64. C L01 17* C OlmENSION AHD ► 0.:4MnH SIATEmENTc LOCATING INPUT !Pa iSt w•ENStow CI(lnn) ► vXI000).VYI(100)401(100) ► Z1(100) LO4 $9+ LO,MoN 104 #.0+ C .U4 (4t C LET JE, DEmOYE THE YMOc.X OF TViv UPPER HALFSPACF • M!) ft2* OFJ=IMAXI-1 il 3/ C ('!) /.4 4 C PRINt1mG OF DEADINn Ors “5* wRiTi. JPI 11 "6,, 11 F0,MAT(1H1.14X,19HMODEI ATmO5PHERE OF,I4 ► 7H LAYERS//) 12 la* 1".21 1 ' 14 al* 21 FO...MAT(1H #2X•cHLAYER/7X.2,-antIOX•2HZTr11X,IHH , 11X , 1HCf11X , 2HVX, 14 "9* 110),.2)1),Y) 14 40* I. .41* IFIIMAX .Ew. 0) GO TO 20 15 4244. • J. •- Ho 30 T:-.IIMAX 0012L) y7. 3n 1III):7I(1 - 1)+1.4[(I) u0127 i8* ii CO“,T1NUE IA/127 ./9,* C 00127 1.0* C PRTNTOHT Fn(:UPPER )tALFSPACE 00130 1.1* •xU..7.41(1•AA) U0131 1.2* 33 1FirmAY .Eq. 0) XUV=Oen 03133 1.3* C=rI(JFT) 00134 1.4* VX=VX.ItUFT) 00135 1.5* VY.7..\/Y1tUFT) 00135 106* wRITF 10,41) JET,XU),/fC,vx,L,Y U010b 1.7* 4i 1.01.4MAT(IH ,17.p:12.4X.t1HImPINITE , 4X , UHINFTNITE.3F12.4) uUitt:) 1.8* utligh 1 0 9* IF(IAY .E.J. 0) GO TO GO 00150 110* 1FiImAy .Fo• 1) GO TO 52 00i5u 1,1* C 14015u 142* C TAkULATIoN FoR LAYERS 2 THROUGH IMAX . 1,0152 113* DO 50 J=2,IMAy U0155 1,4* 1=IMAX+p-J 00156 1,5* 1L=I-1

00157 146* 5n WRITE 16f5i) I.7T(1..),/ItI) ► H/(I)/t/(I) ► vXT(I),VYI(I) 0017I 1,7* Si E0.4MAT(IH #17 , 1F12.2,3F12•u) 00171 118* C 00171 119* C TARULATTON FOR LAYER 1 0002 1,0* 52 1=1 00176 1.1* 1STrItr.n.0 00174 1,2* WRITE 1h,51) I.USTED,Z1tT),H7(T).CI(I),VXI1I) , VVI(I) 00174 1,3* C 00174 1.4* C PRTN1OHT OF FYPLAHOTONS U0,:06 1.5* 6n wRITE (6,61) 00..07 1,6* 61 FOuMATc1H0.1,k,11Hlt)=HEIGHT OF LAYER HOTTOm IN KM/ 1H /151.2RHZT=N . u0.07 1.7* 1F.1(.4HT OF LAYFA TOP IN 04/1i4 r15)023HH =WIDTH OF LAYER IN KM/IN r UUe07 1.(i* 215“24HC =SOUNn SPLEO IN Km/SEC/1H P1ti)1/33HVX=X COMP. OF WINO M. 00e7 1.9* 3-It KM/sFC/1H .15XP33HvY=Y COMP. OF WINO VFL. IN KM/SEC) uU407 7_1 0* C 00,,,Au 1A* RETIIRN 00.11 1:,2* FAH

END OF COMPILATION: NO OIAGNOSTTCS. 1&HoG8P **4*** sUri94 ***.** aFoR'S 1.S44094•SUB94 FOR s11F-1./10/74 - 18:11:53 (1.)

SUBROUTINE RRRR . ENTRY PU1NT 00n132

STORAGE UsEu: CODE(1) 0110160; DATA(n) 060046; DiANK COMMON(2) On0621 •

EXTERNAL REFERENCES (hL,", NAM(=) u0n3 MMMA u1101. 000ulh 114G uOnl 000.1F1 125G O0O1 000054 130G 0001 000074 1366 0001 000075 141G Ong R 0 00 0 0 0 AINT 6000 R Onerap C non2 R 000001 cI n000 N 0n0004 EM 0000 R 000015 H (. 0 0? N 000455 HI 00nn 1000n16 1 n0n0 I 000011 TASA 0002 I 000000 IMAX 0000 000023 INJP$ (10(10 I 000017 J tiOnn I 000nin JASA nOnO R 000013 vX 0002 N 000145 VXI 0000 R 000014 vY 110(12 R 000611 VYI

U0100 1* RRwR (coHRoUTINE) 8/1/6A LAST CARD IN DECK IS NO. UUI00 2* uOlOu 3* .- .444A)ISTRACT. UUIOU 4* uUlOu 5* C TITLE! 4 HRRR U010U 6+ THIc SiLiNOUTINE COMRUTEc A p-BY-2 TRANSFER MATRIX WHICH CONNECTS UOJOu 7* C SoLuT10i.iS Or TH• REt,TOUAL EoUATIONS AT THE BOTTOM OF THE UPPER U0.100 0* C HALFSPALE Tn sOLUTIonS AT THE DROUNO AY THE RELATIONS UUJOU 9* C UULOU 10+ C PHII(GPOUNL)= RPP(1,1)•PHI1(ZT(ImAy))+RpP(1.2)*PHI2(7T(IMAX)) UGLOO tl* C UOIOU 12* C PHI2AGROUNO)= HPP(241)*FHI1(ZT(IMAy))+R0P(2,2)*FRI2(ZT(IMAX)) UU100 '34 C UUI00 14* C hhCRE 7CINAY) IS Thr HvIGHT OF THE ToP OF THE IMAX LAYER AND U010U 5* C CONSF000,TLy THE HEIGHT OF THE HOTTOM OF THE UPPER HALFSPACE. 00100 16* C THE FuNt}ONE; PHI1(Z) ANn PHT2(Z) SATISFY THE RESInUAL EQUATIONS. 00100 17+ • C UU1OU D(P)(I1)/nZ = A(1,11*PHII(Z) + A(I ► p)*1'HT2(Z) uULOU .9* u01.00 e ns WHHI2)/nZ = A(2.1)*RH11(2) + A(pf9)*PHT2(Z). uOIOU IJUIOD .2+ WHERE THE Aly.J) ARV FUnICTIONS OF ALTITUDE BUT CONSTANT IN EACH u.OLOU rat UOIOU C U0106 e5+ C It wE L T EM(I) HE THE •M mATRIx (COMPUTED DY SUBROUTINE MMMM) b000 .6+ C FOR TI,E I-TH LAYER. THEm (TN MATRIX NOTATION) u0i0o .7+ JUI0o en* C PPP = E14(1)*LM(2) ,11.....*FM(IMAX-1)*Em(IMAX) JUi00 ,9* • C • AliOU .)0* THE AnOvE FWIMULA IC USED To COMPUTE THE RPP(IPJ). ;0400 014 C A.1100 024 C THE PAP.AmETrrS uEFINING THE MULTILAYER ATMOSPHERE ARE PRESUMED 010u 03* To HE SORE.) IN COMMON. .14*. C 0100 05* C LA, GUAaE FORTRAN IV (300 ► REFERENCE MANUAL C28-6515-4) 010o 0 6+ C 0100 ..1 74 c AUTHOR - A.D.PTERCE , M.I.T. ► AUGUST ► 1968 0100 .134 C- 11100 ;10* C ----CALLING SFONENCE ---- 1100 40* C /IOU 414 'C SEF SU.POUTINE mmDFM )100 42* C • DI.4ENSTO1., cItIn0) , vXI(1nn).VyI(100)•M1(1(10) 1100 43* C CO•MON 1mAA , CI.VXI•VYI.HI • (THESE MUST HE STORED IN COMMON - ---tAIERNAL IWRRoUTINES REOUTRED---- U0100 48* C uUlOo 49* C mM.Am ► AAAA•CAI ► GAT uUlOu HO* C 00100 :II* C -_--ARGUMEMT IIST---- 6U100 :12* C uU100 n3* C (1):1-, 'GA . R*4 ND INo 6040u :,4* C R*4 NO IN:, Vu ,,5* C AK. R*4 WO INo 6010u no* C RP1.. R*4 2-BY-9 OUT 00100 ,7* C K I*+ ND OUT (ALWAYS OUTPUT AS K=0) LULOu ntl* C UuinU n9* C COMMHN STORAGE USEn 00100 t.0* C CO•NON ImAA•CI.VYI.VYI.HI U0i0u o1* C 6010J ,2* C IMAX I*4 ND ' Iii uu10u. ,3* C CI R*4 100 • 1M, u0100 H4* C VX( R*4 100 INN U(14Uu ,..5* VYT R*4 100 UJ.100 /.6* C 'HI R*4 = uUlOu ,7* C 0)100403* C ----INPUTS---- u1100 H94, C u0100 ,0* COwGA =ANOOLAR FkEOUvNCY IN RAD/SEC uOIOU il* C AK: =X COMHOmUJT Ov HORIZONTAL WAVE NUmnER VECTOR IN 1/KM LUIOU ,2f, C JK. =Y CoMiJO•ENT Ov HORIZONTAL WAVE NUMBER VECTOR IN 1/KM 6010U ,3* . C IMAX .-:-.NlwnE OF LAY;.RS uF FINITE THICKNESS, UOIOU 14* CCI(I) =SnliND SPEEn I,.,KM/SEC IN I-TH LAYER 00100 ,5* CvXi(I) • - =X ("oMPONETN OE WIND VELOCITY TN I-TH LAYER (KM/SEC) 00100 /6* C . VYT(I) ..7.-Y COMPONENT OF WINO VELOCITY TN I-TH LAYER (KM/SEC) LOiOu a* C HI,I) =THTCKNESS IN uM OF 'I - TH LAYER OF FINITE-THICKNESS 00100 /8* C 00100 ,9* C - ---OUTPUTS- ---- 60100 HO* C oUi00 nl* C RPo r.2.01Y-2 TRANSFER MATRIX WHICH cONNECTS SOLUTIONS OF 60100 H2* C THE Rc:SIDHAL FOLIATIONS AT THE BOTTOM OF THE UPPER . utliOu n3* C H4IFSPACE TO cO[UTIONS AT THE GROUND.' OuIOU f.4* C K =011!,,mY PARAMETER ALWAYS RETURNED AS 0. 00100 H5* C u01.0j t,6* C ----PROGRAM FOLLOWS DELOW---7 0010U 0,7* 60101 H8* SU4RuUTINE RRROOMOA.AKX•AKY.RPP•K) 00101 4,9* C 60101 40* C DIMENSION AND COMMWI STATENENTc LOCATING PARAMETERS DEFINING THE MODEL 00101 'd* C' MUt TIL4Yi7.R A`fi•oSPHwnr 00106 42. OL,ENSY6N CI(lnn) ► 0(I(100).VYI(100).H1(100) Uu104 43* CO:•MON WAXICI.VXIiVY/tH1 UU104 44* C 001011 .45* HI ,.ENSToN EM(242) , AINT(2.2}eRPP(2 , 2) UJ106 •6* K=.i 00106 47* C U0I0h 48* C,RPP AT TOP uF IMAX !AYER IS THE IDENTITY MATRIX U0107 49* RPo(1 , 1)=1.0 uj110 1,.0* HP.,(1:2)=0.0 _ ...,” .... &Hi- 1NutX nr THE LAYER CURRENTLY UNDER CONSIDERATION ■ vilo 1“6* C ;olio 109. C COmPOT.T1ON oF EM MATRIX FOR IASA LAYER 10111 1)0* trrI(IaiA) .ZAP0 111* VX.-:VXI(IASA) ,v0.1 112* VY.:VYI(LASA) ;0I21 1.3* Hlt.-HT(IiaA) 10123 1.4* CAL MmMM(OMEGAFAKX , AKYrCluXIVY , H+FM) Wi23 115* C ;u123 116* C MUIT1PilLATION OF uPP.-AT TOP Or IASA LAYER BY FM FOR IASA LAYER ;01244 1.7* Ho 80 T=1 , 2. ;0127 1,8* NO 8o J.7.1•e . ;0i32 119* On AI•T(I.J)=EM(1.1)*RPP(I,J)+Em(I•2)*RPP(21J) J0132 1/0* C d0132 1/1* C CURRENT AINT IS RPp AT BOTTOM nF IASA LAYER ;013L) 1/2* 1)0 8S T=IPZ ;014U 1,3* HO 8S J=1 t2 pu14:i 1,4* tit; uPp(1,J)=AINT(T,J) W143 1/5* C kilqu 1,64 100 CO.aT1NUF i0 t46 1,7* C ENn oF OUTFR DO LOnP fultH1 1,8, C 10140 1,9* C CURRENT RPP TS THAT AT BOTTOM nF FIRST LAYER , 0150 IA* RETURN 10151 1.11* 1-:N11

ENO Oc- COmPILATTON: NO nIAGNOSTICS. 11 .X/ , P 4 ***** WHY:, ***4** +oRFS 1.5“BP5rSuBq5 fp.)

FuNcT10.. SAI ENTRY POINT 00n055

SlOkA6E USEol CODE(1) 000063; DATA(n) 00001AI BiANK COMMON(2) 080000

ExTFMA REFERENCES (HLoCK• NAME) 0003 0004 SIN 0005 EXP 0 0 0G NCRR31

SlORA(.E ASSIGNMEHT (HLNCK# (YPE. RrLATIVE LOCATION. NAME) 0001 0n0u56 11L (10(11 OnOnIP 9L 0000 R 00000? F noon Onnonb INJP$ 0000 R 000000 SAI 0 0 00 R 000001 Y uuluo 4* C QUiOu 5* C TITLE - SAT u0i0u 6* C PkOnkAM TO EVALUATE FUNCTION SAI(X) FOR GIVEN VARIABLE X. uU100 .7* C If x TS Nr.GATIVE ► SAT(X)=SIN(T)/Y wITH y=SORT( 4- X). IF X IS u0100 8* C • POSTTIVE ► SAI(X)=SINH(Y)/y wITH Y=SoRT(y)• THE-FUNCTION TS u0100 9* C ALSo PrEPRESrNTAIiLE mY Ti4E POWER SERIES u0100 JO* C uUiOU 11* C SAI(X)= 1 + X/(3FACT) + X**2/(SEACT) + X**3/(7FACT) + .. uti10u 12* C o0100 i3* C LANGWU.E -- FURTRvJ TV . ( 3441 ► wEFERENCE MANUAL C28-6515-4) uU.i.Ou li* C u010U 15* C AUTHOR - A.D.PTrRCE , M.I.T., JULY,19.68 Uu1Ou 16* C uUiOu 17* C ----:CALLING Sc:OHENCE---- . 00i0o L8* C u0100 19* • C SAT(ANy R*4 ARr;umENT) iiAY utE USED IN ARITIWETIC EXPRESSIONS u0i0u e0* C u0100 el* c !-_--EXTERNAL c.UHROUTTNES REOUTRED--- - u0JOU ea* C uOIOU r3* C NO EXTFRNALSURP.OUTINES ARr REGUIRED, u010u /4* C u0100 •5* C ----ARGUMENT IIST- ---, u0i0u .6* C u0i0U e7* C •X R*4 ND INS) 00100 e3* C SAT R*4 - ND OUT uUlOu 49* C UolOu 00* C NO CuMmON STORAGE TS USED uOLOu 01* C UU1Ou .,2* C ----PROGRAM FnLLOwS BELOW---- IJOI0u ..s3* c UU101 04* FUN.CTInN SAI(X) u0101 05* ' C u0106 06* 1 IF( ANS(X) .GT. 1.E -15 ) Go TO 9 U0103 07* C o0103 08* C ADS(x) IS sU SMALL THAT SAT IS VIRTUALLY 1.0 UULOb 09* i .0* u01U6 IS4ril110:16 W.1106 41* C Od106 .2* C'CONTIN.IINO FROm 1 0007 .,3 .c) T=cORT(ARS(X)) UOilU ..4* 1F(X) 111.10 , 11 u011u .4.5* - C u011u -6* C x IS Li:SS THAN O. uOi13 .7* lo sAr=sIN(y)/Y uu114 .a* HErUkN U0114 4.9* C u0114 nO* C X TS P0SiTTVE. SAT= S1NH(Y)/Y. UUil.....) ail* 11 1,..=.:XP(y) U0116 :)2* SAI=U.c1 0 (E-1 ■ /P)/Y UU117 n3*. REWHN u0i2U LA* FICA

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ST 01tA0E ASSIGNMEN1' TYPC.RELATIVE LOCATION , NAME) (1000 R 000u0.5 uENOm 0000 000013 !NUN. nOnO R 000002 nmo 0000 R.000000 PAS 0000 R 000005 PHI u000 H 000u01 TAS 0000 .R 000004 X

0010u 1* C SOuRCE (SUoRCPTINE) 8/15/68 u0) 0u 2* C uU)00 3. C UU4OU 4* C ----AUSTRACT---- uotOd 5* C 1 U.1100 6* C TITLE --SOuRcE ! LJULOu 7* C EVA)UATION nF FOURIFR TRANSFORM OF NEAR FIELD ACOUSTIC RESPONSE uulOu 8* C To ExPL0SIV1: SOURCE 00100 9* C 00100 .0* C • SoUPCr COMPUTES T-E FoURIER TRANSFORM OF THE NEAR FIELD uut00 .1* C PriESSNRF AT 1 KM pRom A 1 KT EXPLOSION AT SEA LEVEL. THE 00..00 .2. C AmDIEmT PRESSNRE TS ASSUMED TO fiE 1.•6 DYNES/CM**2 AND 6040u '3* C THE TIME LAPSE FRnM TIME 7.ERO IS NEGLECTED. AN EMPIRICAL uulOu 0* C FoRmUlA FOR THIS PRESSURE IS 00100 .5* C 0010016* C F(T) = PAS * Ii - (T/TAS)) * EwP( -T/TAS ) . . T .GT. 0 00100 --.7* — c, 60100 18* C = 0 1 1 .LT. 0 UOIOu 19t . ' C 0000 .0* C WiTH pAS = (34.45r+3) * (1,61) DYNES/Cm**2 uUlOu .1* C AND TAS = 0.33 SwC 4 00400 ..2* C OUICii - ' '3*. C THEREFORE. ITS FOuRTER TRANSFORM TS uU100 .4* C- u0100 .5* . C FT(OMEGA): -/ * nMEGA * PAS / (1/TA, - I * OMEUA)**2 UtilOu ,6* C uU/00 ..7* C WHEPE I = (-1)**0.5 . u0i0u .8* C • 001.0U .9* " C LANGNAA.E - FoRTRAN IV (360, REFERENCE MANUAL C28-6515-4) 001.0o AU.C. U0100 Al* C AUTHOR ,: - A.D.PIERCE AND J.0oskY ► M.I.T.. AHGUST.1968 • uU100 ..6* C suknouTimr PHAcF IS CAI LED tJUIed :11* c UU1Ou of)* C FOu";:.:A. OSAGE Uu10U .,9* C UOIOU ..0* C CAIL SO1,RCE(OMF(;A0ETMAG1FTpHSE,DMAG0DPHSE) uOt.OU 41* C u010t1 42* C INPUTS u0100 43* C W.I.LOu ,.4* C OmEo.A •ANGULAR FREOUKNCT (RADIANS/SEC) ' 00100 45* C R44 u010u 46* C u010u .47* C OUTPUTS u010u 48* C uolo ► 49* C FTMA MAGNITHOE OF FT(OmEGA) DEFINED ABOVE IN SULTtROUTINE ABSTRACT u010u nO* . 0 R+4 ( (DYmrS/CM**p) / (RAD/SEC) ) UU100 n1* C U0i0u n2* C FTPL4SE PmASE OF FT(OmFGA nEFINEi) ABOVE TN SUr1ROUTINE ABSTRACT 00100 n3* C R44 (t(AOIANS) ► u010u 64* C UOtOtt n5* : C DMA,. DERIVATIVE OF FTMAG WITH RESPECT TO OMEGA ( (DYNES/CM**2) uOiOU n6* C R4 / (RAn/SEC)**p ) ut)10U n7* C uotou A* C DPHsE OER/VATIVe. OF FTN4SF wITH RESPECT TO OMEGA (BAD / (RAD/ u010u t194: C R44 SEC) ► uUIOu "0* C u0 10u 01* C uOiUJ „2* C ----PROGRAM gOILOWS BELUW---- uU100 03* C uOIOU not c

00101 "5* SUnROUTINE SOU4CE(OMEGAtFTioAG ► FTPHSEtOMAG ► oPHSE) U0101 06* c wE ASSHMF INVEDSF N DEPENDwNCE Mg 00103 "7* PAs=(34.45Ef3/1.0)*(1.01) u0/0..1 . n8* C PAS IS IN DYNES/CM**2 u0103 "9* C THts Is THE PEAK OVERPRESSHRE AT 1 KM 0120 u3104 ,0* TA,.=0.3o 00104 11* C TA' IS THE LENnTH OF THE PnSITIVE. PHASE u0105 r2* umn=10/TAS U1)106 i3* HE..0(.i=nkFGA**20M0**2 * U0101 it ► FTAA(,=pAS*JmFGA/PENOm ffg • u0110 /5* HM;X=i'AS/DE•OM-2.0+PAS*OMFAA**2/DENOM**2 019n 00111 /6* LAIL PHASE(011.0,0mF6AIX,PHI1 - • u0111 17k C PITT IS THE ARCTAN DE c)IEGAiomo. Ng u0114 /3* FT:JHSE=-3.1415nP7/240+2.O*PHI u01.1.5 /9* OPHSL=2.0*uMOinFNOM aril 00113 00* C THi.: DERIVATIVE OF THE ARMIN IS 1•/(1•*Y**2) 0270 . 00114 01* kETURN oPao 00115 h2* ENo 0290

F7No OF -COmPTLATTON: NO nIAGNOSTICS.. 6:HuG,P **4*** 5UI197 *** 4 ** 1.S•an7ISuB97 FOtt 811F-1./10/74 - 18:1p:09 (o t ) NV SuV.CE NEcK-",-CnNIR.4, CART} IGNOorD. **..** SUHOy *** • *4 JEuRiS 1.5,4399,50399 :OR slIF-1/10/74- 18:12:19 (h0

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STORA6E ASSIGNMENT (11L•ICK, TYPE' RrIATIVE LOCATION. NAME) 00,11 000133 134G until 000p10 152G 0000 I 000024 T 0000 I 000012 ICEN 0000 000027 1NJP$ u000 I n0000 1PH 0000 1000025 ISUM 0000 I 000011 J18 0000 1000013 JI7 0000 I 000014 J18 0060 1000015 J19 0000 I 000n16 J20 n000 I 000017 J21 0000 I Oon020 J22 '0000 I 000021 J23 000 1 000022 J24 00n0 I 000n23 NX

,0100 it C SU'PCT (SUhROUTINE) 7/19/68 LAST CARD IN DECK IS NO. ,0100 2* C )0100 ' 3* •C )010u 4* C .. --ABSTRACT.--- wiOu 5* C wi00 5* C TITLE - SticPCT 0I00 7* C EVALUATION ni: SUSPICION INDEX OF ELEMENT ((Om) OF MATRIX INMODE 010o 8* C OlOu 9* C SUSPCT EVALUATES ThF SUSPICION INnLX. ISUS, OF THE ELEMENT utOu 10* C IA ROw H. COLNMN m OF THE MATRIX (NMObE ( (u, M) MUST BE 0100 11* C AN INTERIOR EIEMErJ). THE NEIGH'Ink5 or ((NM) ARE DEFINED uloo 12*. C 'To OE THE EIGHT CiEmENTS wHICH FORM THE - THREE BY THREE 010u 13* C ELEMENT SOARE WINCH HAS (N.M) AT ITS cEN1ER• THEY ARE Ut00 14* C NUmNEnr0 FROM ONE TO NINE DEGINNING IN THL UPPER LEFT AND jt00 IS* C PROCEEPING CLOCKWTSE (NO. I Al4n Nn. 9 ARE SAME ELEMENT). )100 16k C EACH FLEMENT oF MA TRIX INDEW) MUST HAVE ONE OF THREE )106 17* C VALMEr -IP I. OR S. (N , M) IS NOT SUSPICIOUS AND ISUS > ,i;i0 • 18. C 0 IF ANY OF ,IF Lit. FOLLOWING CONDITIONS HOLDS. +10o 19* C 1 .100 ,0* C I. LLEM1•NT (HIM) > S lint; ,1* C P. ANY oF ITS NEIGHBORS > 5

liOu .2 1 C 1. NOWHERE TN THE 3x3 ARRAY OF (N ► M) AND ITS NEIGH.. AOU ,3* C UORS O0Ec THF.RE APPEAR Tn DE A DISPERSION CURVE Int) f it. C WITH POSITIVE SLOPE 100 ,5* C J00 ,ti* C OMEN/ISE ISUS IS SET EQUAL TO THE NUMnER OF THE QUADRANT

LWIOU ml* C LANGUAI.F — FORTRAN IV (3mOs REFERENCE MANUAL C28-6515-4) LAUD() A2*• C uOIOU .13* C MUTHORc - A.D.PTERCE ANN .1.u0SEY• M.I.T. ► JuNE , 1968 0108 A4* C 1./U1OU .15* C ----USAGE---- u010U .16* C uU100 .17* C NO FORTRAN SHOprATINES ARE CALLED u010o .18* C U0i0u A* C FORTRAm USAGE . . 601 .0U 40* C CALL SNSPCT(Nsm/NROWrINMODp•ISUS) uU1Ou 41* C u0.1.0u 42* C INPU1S u010u 43* C u0100 44* C N RuW NtrnER OF ELEMENT UNDER CONSInERATION NAY NOT RE uO10u 45* C 1-4 • FIRST OR LAST ROW)- OU1Ou 46* C U0i0u 47* C M COLUMN NUMBER OF ELEMENT UNDER CONSIDERAION (MAY NOT BE uUlOu 48* C ' I*4 FIRST OR LAST COLmMN) UUJOU 49t C u01.08 nO* C NR0-.1 TuTAL NUMuCR or RnWS IN INMODE uolOunt* C 144 uOiOu n2* C uu100n3* C INMmDE MATRIy UNDER CONSTDFRATIOt4 STORED IN VECTOR FORM, COLUMN uOlOo n4* C I...4(01 AFTER COLUMN. EArH ELEMENT MUST RE —1, 1s OR 5. L1010u n5* C u010(.1,56* C OUTPUTc U0100 n7* C uU100 nO* C ISOc SUSPIrION INDEX OP ELEMENT "(NsM). SEE ABSTRACT ABOVE FOR u010un9* C I*4- DEFINTTION• Otii.Ou mO* C li0i0o mI* C 0.110U 02* C _ ----EXAMPLES---- U00.) n3* C uOIOU nii* C CAIL1W. PRoGRAm u0100 m5* C uUiOU ,n6f C 1)I—El STON INMOmE(9) UulOu0* C IN.DEA : —Is —Ti Is 1, —I' 1. Is 1, ti(JiOu ,,8t C CAil. SUSPCI(2 , 2,3 , INMOUEtIRUc) u010,1 09* C 011K (m000) TSUS uU100 /0* C 20A FOL,MAT (10H EXAOILL ],hx, sHISUS >sI2) uoiau 414 C INA•UE > `IP ... II ll 1, .1• .. 1.• ltlt 1 uoIou at C CAtL SHSPCf(P•.3•INMONE,IUs) u0100 /3* C . WRITE (m.300) TSUS • OUlAu /4* C 30n FO-MAT (11111 EXAvPLE 2thX, AHISUS >,I2) L0100 ,5* ENS, ,6*' C ut,IOU ,7* C TARLES OF 1NMOUE uUtOtt ,8* C u010u (9* C EXAWPLE.1 EXAmPLC 2 uulOu nO* C i1010u nl* C• +++' —++ • uulou m2* • C --+ --+ )010u m3* C ++— +—+

uu.t00 #.9* C 00101) -JO* C 00/0u ,A* C ..---PROGRAM FOLLOWS HELOW---- 0010J .42*- C U010p •43* C 00101 •.,4* • SUwPOUTINE SUSPCT(NPM.NROW.INMODEPISUS) U(J011 ..,..5* C 00101 46* C VARIABIE DT•FNtiIDNING OF INMODE 00106 ,7* • HI,ENSTON IPP(0),INMOPF(1) uu103 '.8+ C • 00103 .49* C LLEWNr (H.;•)) uF IwODE IS TCEN U0104 1,,0* J1k=1M...1)*HROW+ uUt05 1.1* IC‘N=INmOBE(J1h),. u 11106 1.2* ISHS= n uu l ob 1.3* c uulnp 1„4* C IF iccm is 5, IT lc NOT SUSPICTOUS AND ISUS = n u0107 1,5+ JEtILEN .Co. 5) RETURN uUtO7 1.6* C U1) '7 1.7* C IPP(N) IS NETG,WOR No. N (SEE ABSTRACT ABOVE FOR NUMBERING SCHEME) 10111 1.S* • di,=(r-:3 )t.ROW.(1d*1) .JOt12 1.9* 1P.(1)=IN ,L.:DE(o17) 0Uilo 110* u14=(M-1)*.ROW*(H - 1) 00114 1,1. IP..(?)=0.1m.,0L7(.11A) uutth 112* u1.4:.(M-1))*14. 110W+(N - 1) u0116 1,3* IP,J(.5).-...IN•t)0E(mq) U0117 114. J2n=(.1-U)*UROW*(N - U) 0012(1 1,5* I.P1,(4)=11-040E(J20) u0121 1,6* J21=( 41-0)*NROW*(N+1) u0122 117* IP,..(b).-.:INM0DE(JPI) uO12.5 1,A* u.2:;=(M-1)*.IROW+(N 4. 1) 60124 1,9* IP.•4(•Jr.INMoUE(o22) UUL2D 1,0* J21.=(M-?)*NHOW.*(M+1) 01)12o 1,1* 1Po(7)=INVoDE(,3) 0U127 1,2 ► . .12u=(M-.2)*NR01'!+(N+0) u013u 1,3* 1P,..(m):-.1NmoDE(. -1 , 4) ItOt31 1,4* IP.., (9): IPP(1) uti132 lett* 10: = n 4)U13.5 1,6* no Io T=1 ► a u0130 1,7* IFtIPP(I) .EO. q) NX=Nx+1 00140 1,6* In COmT10111.. 001'{0 1,9* C 4-X N THE. NUMBER OF NEIGHBORS WHICH EOUAL +5 . U0&'4U 1.0* C 6014u 1:11* C IF Mott,. 1HAo ONE NFir,H5OR TS FeaUAL TO *S. THEN ISUS=0 U0 1.4e! 1.2. IF 00: .(T. 1) RETURN 60142 1.13. ♦ C 6014 1A4* C IF NEL4HHOR A IS THE OHLY (,NE EQUAL TO +5 ANn FITHEP NEIGHBOR 2 OR u0142 1..6k C NETGHTIta 4 noEs NOT ACH(EE 4.ITH ICFN , THEN ISUS=2 00144 1.,64 ISHM 4 I/01S( IrFN * IPP(2) + IPP(4)) uu14h 1.17* IF (IPP(3).EO.s .ANU. IS(JM.NE.A) ISU5=2 60147 1. ,o+ IF (NX.i.T.u) RETURN U0151 1.•9* • in no So Tz1.9 00154 140* 5n ipu(1)=ITAnS(IpP(I)+ICFN))/2 IPP(1) •E0. 0 $ANO. 1N (2) .EO. 1 .AND. IPP(8) .F0. 1) 00157 146s I fit;TURu tJUJ.61 1..7* IFt IPp(A) •FO. .AND. IPp(2) .En. 0) RFTNRN 00163 1,.3* ISuS 1.9* IF( IPp(2) .FO. 0 .ANO. IPP(3) .E0. 1) RETNRN uOibh 1•0* TEl 1PP(3) .EO. I .AND. IPp(4) .E0. 0) RETNRN u017u 1•1* 1SuS = 3 00171 1h2* fF( IPP(h) .E(4. 0 .AND. /Ppt4) 6E0. 1 .AND. IPP(6) .F0. 1) 0)171 1031, I RwToRN uU17 .5 1.b4* 1F( 1rP(4) .E0. 0 .AND. IPp(h) .E0. 0) RETURN uOLTS 1h51, ISt6 = 4 Utt170 1h5* IF( IPP(6) •EQ. 0 .AND. IPP(7) .E(. 1) RETURN 1-,0 0 11 IF( IPP(7) ,E0, 1 .AND. IPp(A) .EO. 0) RETURN Ih8* 1SuS = O . uudO3 1h9* RETURN UO204 1,.O* Ftin

END Or CONPILATTON: NO nIAGNOSTTCS. 6110O$N 6F1N

RUNIn: 3,18 ACCT: 06IA1019 PROJECT: INgRASoUND NO Rh FOR rORTRAN hLT S11I-th7 TIME: TOTAL: 00:03:45.861 CPU: 00:0o:29.620 I/n: U0:01:14.555 CC/LR: 00: 9 2:01.705 WATT: u0:0t1:02.233 IMAGES hEADI 107 PAGES: 14A START: 18:01:43 OFC 1Rt1974 FIN: 10:12:29 DFC 101974 APPENDIX B The 88th Meeting of the Acoustical Society of America

'ark Plaza Hotel St. Louis, Missouri a 4-8 November 1974

Y, 5 NOVEMBER 1974 CHASE CLUB, 9:30 A.M.

Session A. Physical Acoustics Atmospheric Acoustics

10:45 A5. Asymptotic high-frecueney behavior of guided infrao.r.ic • modes in the atmosphere. Wayne A. Kinney (School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332)

Refinement of previous theoretical formulations and numerical computations of pressure waveforms as applied to atmospheric traveling infrasonic waves could include a description of their asymptotic behavior at high frequencies. In the present paper, calculations based on the W. K. B. J. approxi- • 'nation and similar to those introduced by Haskell Appl. Phys. 22, 157-1G7 (1951)1 are performed to describe the asymptotic behavior of infrasonic guided modes as generated by a nuclear explosion in the atmosphere. The results of these calculations are then matched onto numerical solutions which have been given by Harkrider, Pierce and Posey, and others. It is demonstrated that the use of these asymptotic formulas in conjunction with a computer program which synthesizes infrasonic pressure waveforms has enabled the elimination of problems associated with high-frequency truncation of numerical integration over frecluency. In this Way, small spurious high-frequency oscillations in the computer solutions have been avoided. [Woi:k sponsored by Air Force Cambridge Research Laboratory. 1 Recently, Allan D. Pierce, Ch0.stopher Y. Kapper and Wayne A. Kinney

the Georgia Institute of Technology have been working to refine a computer

•ogram which synthesizes infrasonic pressure waveforms at the ground as

nerated by large explosions in a wind- and temperature- stratified atmos- 1 Rre. Shown in Fig; 1 are three such pressure waveforms along with the

dal waveforms from which each of the three individual total waveforms has

en superposed. Corresponding to each modal waveform is a particular disper-

on curve (i.e., a plot of phase velocity versus angular frequency). Any

ven dispersion curve defines what is referred to as a mode. Fig. 2 shows

spersion curves as they are generated by a portion of the computer program.

labels given to these correspond to the labels given to the modal wave-

Elms in Fig. 1.

Due to temperature stratification, the earths atmosphere possesses

Ind speed channels with associated relative sound speed minima. Fig. 3

)ws a standard reference atmosphere wherein two such sound speed channels

indicated; one with a minimum occurring at approximately 16 km altitude

I the second with a minimum occurring at approximated 86 km altitude.

'en the presence of such a channel, an acoustic ducting phenomenon can

:ur, as is demonstrated in Fig. 4, wherein the energy associated with an

∎ustic disturbance can become trapped in the region of a relative sound ed minimum) It is this mechanism of propagation only that is of interest

In the computer program, the computation of modal waveforms involves

numerical integration over angular frequency of a Fourier transform of

ustic pressure where this integration is truncated at the high-frequency

. 1 It has been speculated that this abrupt truncation leads to the GR GR GR o

So f\JVA,Ao.V ■ANNAN ,

S2 U S3 S ilif4 1 10 10PPRVINVIAANN U LJ S4 S Ct Ss 0 H (I) 75 0 TOTAL TOTAL.

-75 UPWIND DOWNWIND

510 540 570 510 540 570 510 P40 570

TIME AFTER BLAST (min)

Fig. 1 Superposed infrasonic pressure waveforms (with contributing modal waveforms shown) as generated r.

by the computer model for ground locations 10,000 km upwind, crosswind and downwind from a nuclear ex— plosion. 0.45

C) 0.40

S: 0.35 0 C)

0.30

0.25 0 0.01 0.02 0.03 0.04 0.05, 0.06 0.07 0.08 0.09 0.10 Frequency,w (sec-1 )

Fig. 2. Dispersion curves generated by the computer model. The labels given these curves correspond to the labels given the modal waveforms of Fig. 1. 4 — — SUBTROPICAL SUWER U S. STANDARD ARCTIC SUMMER ATMOSPHERE .1962 ARCTIC WINTER

/ 3°

t 10

•OP

i 200 240 _--200 t

11 -r 1 200 400 600 800 1000 -50 0 50 TEMPERATURE (°K) WEST-TO-EAST WIND {m /sec.)

Fig. 3. Standard reference model atmosphere showing two sound speed channels. SOUND CHANNEL DUCTING

TURNING w POINTS

•••••• a■• 00.1.

Vp ACOUSTIC PRESSURE SOUND SPEED PROFILE

Fig. 4. Griphic illustration of acoustic ducting in a sound speed channel. The energy of an acoustic disturbance can concentrate in the region of a relative sound speed minimum. 1 neration of what might be called "numerical noise" in the computer output.

was felt useful, therefore, to extend this integration beyond the heretofore per angular frequency limit by means of some appropriate high-frequency proximation.

The approximations associated with the W.K.B.J. method of solution 2 ply to the analytical model on which the computer program is based at fre- ancies above approximately 0.1 radian/sec. Below that limit, effects due

density stratification in the atmosphere and gravitational forces cannot be

3lected. Such effects therefore are not germaine to the discussion here.

To the best of the authors' present knowledge, the application of the

C.B.J. method of solution to the problem of describing propagation of

)ustic disturbances in an atmosphere that contains two adjacent sound speed

=els has not been approached in the literature to date in the manner to

presented. To be specific, the approach taken here is to seek a W.K.B.J.

for each of the sound speed channels separately, then to combine the

ults rather than to treat the problem with a single model.

The W.K.B.J. model for propagation of acoustic disturbances in a single

nd speed channel consists of solving for the acoustic pressure divided by

square root of the ambient density expressed as

iwt ikx = Ip(z)e e po 1/4

re w is angular frequency, k is the wave number associated with the hori-

tal dimension x, z is altitude, and where g ) satisfies the reduced wave

ation, 2 [ 2 d w 2 k 1) = 0 2 dz c2 (z)

!re c(z) is sound speed as a function of altitude. The W.K.B.J. approxima-

In as applied to this model would appear to be valid provided

re l is some representative wavelength of interest. This approximation

tes that substantial changes in sound speed should not occur within distan-

corresponding to a typical wavelength of interest if the model is to apply.

Particular insight into the high-frequency behavior of guided infra-

ic modes was gained when the following integral was solved numerically by

?uter

j top 1 1 I 2 _ (n+1/4)Tr ez (z) dz [ p zbottom

•e v p is phase velocity, n = 0, 1, 2, 3, ..., and where and z zbottom top tify the lower and upper bounds of the sound speed channel, respectively. 2 integral is a direct result of the W.K.B.J. method of solution , and its

rical solution enabled the plotting of high-frequency dispersion curves.

In the lower portion of Fig. 5 are shown two sets of dispersion

es generated by integrals of the above form; one set (the dashed curves)

ppropriate to the W.K.B.J. model for the lower channel and the other set

solid curves) is appropriate to the W.K.B.J. model for the upper channel. COMPUTER MODEL

DISPERSION CURVES

0 0.1 . ANGULAR FREQUENCY (SEC -1 ) 0.41 W.K.B.J., MODEL

DISPERSION

CURVE

Ii LOWER P) C HANN EL UPPER CHANNEL 0.27-

0 0.1: " 0.2 -1 ANGULAR FREQUENCY (SEC )

Fig. S. Comparative dispersion curves as generated by the computer model and the W.K.B.J. models. the upper portion of the same figure are shown again dispersion curves as aerated by the computer model. It should be mentioned that the computer

1e1 solves a more complex problem in the sense that the simplifications inher- t in the W.K.B.J. model are not present.

As is illustrated in the lower portion of Fig. 5, the two sets of

;persion curves generated by the W.K.B.J. models intersect with one another

various points. A comparison of the dispersion curves shown in both the er and lower portions of Fig. 5 reveals that these points of intersection

k regions of resonant interaction in the phase velocity-angular frequency

ne between adjacent modes of the computer model. To better illustrate this

ervation, in the right hand portion of Fig. 6 is shown one such region of

eraction with its corresponding point of intersection between two dispersion

ves of the W.K.B.J. models shown to the left. It should be mentioned that

dispersion curves of the computer model never intersect with one another.

analytical explanation of this fact is given in reference 1.

The above observation may be stated differently by saying that, for

Itively high angular frequencies, the dispersion curve corresponding to a

tia mode of the computer model is comprised of portions of dispersion curves

I both sets of the curves generated by the W.K.B.J. models. Two important

trences about the asymptotic high-frequency behavior of guided infrasonic

s can be drawn from this statement. First, for some frequency ranges, and

nding on how dispersion curve portions match between curves of the computer

1 and the W.K.B.J. models, it can be inferred that the acoustic energy

dated with a given mode is comprised of energy associated more with propa-

on of acoustic disturbances in one sound speed channel than in the other.

with increasing frequency, this association alternates back and forth

DISPERSION CURVES DO NOT CROSS

APPROXIMATION COMPUTER MODEL 0.34--

UPPER CHANNEL 0.32

•■•■•■•■•.. 0.30

LOWER.' CHANNEL RESONANT I NTERACTION BETWEEN ADJACENT MODES

0.08 0.10 0.12 0.28 0.06 0.08 0.10

JLAR FREQUENCY ANGULAR FREQUENCY (SEC -1 ) (SEC - I)

Blow-up of a section of Fig. 5 showing a region of resonant interaction two adjacent modes of the computer model. To the left are shown the cor- ig intersecting curves of the W.K.B.J. models. ween channels. To illustrate, if for a small range of frequencies a portion a dispersion curve of the computer model matches (in the phase velocity-

Aar frequency plane) a portion of one of the W.K.B.J. model curves for the

.yr channel, then that implies that, for that mode and for that small fre-

Icy range, the acoustic energy density associated with that mode is greater

:he upper channel than in the lower channel. Secondly, in standard reference

spheres the sound speed minimum for the upper channel is shown to be less

agnitude than the sound speed minimum for the lower channel. It can be

rred therefore that those acoustic disturbances for which phase velocities

less in magnitude than the sound speed minimum for the lower channel are

aiated more with acoustic energy trapped in the upper channel than in the

channel, and thus for this reason do not contribute significantly to the..

;tic energy at the ground. This inference implies that care must be taken

which modes are chosen to superpose in the attainment of the final pres-

waveform at the ground, as some may not contribute.

In addition to providing a new analytical tool, the manner in which

.K.B.J. method of solution has been applied to the two-channel problem has

fied the physical interepretation of a mode as defined in the computer

It is hoped that the computer program can now be modified accordingly

in better high-frequency resolution in the pressure waveform output. REFERENCES

1. Pierce, A. D. and Posey, J. W., "Theoretical Predictions of Acoustic Gravity Pressure Waveforms Generated by Large Explosions in the At- mosphere", Report No. AFCRL-70-0134 (1970), see in particular pp. 32, 38, 41-45, 93-99.

2. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, McGraw- Hill, New York, 1953, see in particular pp. 1092-1094, 1098-1099.

3. Pierce, A. D., C. A. Moo and J. W. Posey, "Generation and Propagation of Infrasonic Waves", Report No. AFCRL-TR-73-0135 (1973).

INFRASONIC WAVE PROPAGATION IN THE ATMOSTE

Quarterly Status Report No. 6 January 15, 1974 to April 15, 1975

Contract No. F19628-74-C-0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff

Prepared for

Air Force Cambridge Research Laboratories

Laurence G. Hanscom Field

Bedford, Massachusetts 01730

This report is intended only for the internal the contractor and the Air Force. OF OBJECTIVES

1.n- objectives of the study are to develop new analytical and computa-

1 techniques for the reduction of infrasonic data recorded at moderate

greater distances form atmospheric explosions. Present research is

earned with Line Item 0001 of the contract and in particular with the re-

ion of the multimode synthesis program.

7STIGATIONS IN PROGRESS

In the previous report, discussion centered on the work in progress to iify the existing modal synthesis program to include leaking modes. At at time, it was pointed out that the program modified to include such modes d a number of bugs. Subsequent work during the past quarter has pinpointed e principal cause of the bugs. The cause was a fundamental error in the

: thod of solving for the complex wavenumbers and a modification of the theory

.s required to overcome this. The revised theory, which represents our incipal accomplishment for this quarter, is outlined below.

eigenmode dispersion function which appear in the theory is given by

(1) 7 ") = Al2R11 (All --C) R12

1 and 251 1 are elements of the [A] matrix (coefficients in the residual 'e 4 Al 2

ua ions) for the upper half space, RI, and R12 are elements , of the trans-- s; on matrix [R] going from the bottom of the upper half space to the ground.

uantity G is 2 G = ±[A-.+ A_ 1 1/2 (2) 12A21'

ymbols are the same as used in previous reports and paper and, in particular,

iler's thesis. . All of these various quantities nay be regarded as functions

and k and, implicitly, as functions of the sound speed and Wind velocity profiles. The quantity F also depends on the horizontal direction of propagation, although this is considered fixed in any given waveform synthesis.

The roots of Eq. (1) (i.e., values of k) correspond to poles in the com- p/ex plane which may or not be included in an evaluation of a contour integral by the method of residues. The so called nonleaking modes correspond to roots for which k is real and positive when w is real and positive. The square of G in such instances is real and positive at the root location and the square 2 root G of G is to be taken with a plus sign when Eq. (1) is satisfied. Also the root should be such that dk/dw>0 (positive group velocity). When to is continuously reduced in value, the various roots k n (w) may be considered to continuously change also. For some modes, a point is eventually reached at 2 which, for some value of co, the root k (w) is such that G as well as F(63 -,,k ) n n is zero. Our interest is in finding values k(w) (possibly complex) which correspond to lower values of to for the same mode. Doing this without excessive computational modifications is the principal objective.

Since the earlier versions of the computational program deal with_ phase velocities v (henceforth denoted as v) rather than wavenumbers, for the purpose of discussion, we may regard Eq. (1) as an equation relating w and v. The line G2 =0 has a segment going obliquely downwards in the v versus w plane at low frequencies, as is sketched in Fig. 1. In the region immediately to the right 2 2 of this curve, G >0; while to the left G <0. The portion of a given mode's dispersion curve at higher frequencies has a phase velocity which decreases monotonically with increasing w. Modes of interest here are those referred to as S0, GRo , GR1 , etc. If one follows the curve for any such mode downward in w (upward in v), eventually a point is reached (point A in the figure) where 2 the curve touches the G =0 line. The computational procedure assumes that G is version, the branch cut is taken in such cases as shown

Initially swinging down and then crossing the real axis upwards to the left of kBR , leaving the possibility of a root on the real axis between the cut and km. Since G changes sign at a branch cut, one would have to have G real and negative in the vicinity of such a pole. In either case, the net result with branch cut contour integrations included would have to be the same. Our premise (which probably would warrant some more careful mathematical examination) is that the branch cut contour contribution is smaller when the branch cut is selected as in Fig. 2b.

An interesting consequence of this analysis is that the mode does not i ,riediately become leaking when w is reduced to values below 0 A' The phase velocity and thus k stays real. What does appear to be the most likely possiblity is that v stays real and increases with decreasing w until a point (B in Fig. 1) is reached where the slope is infinite (dv/dw=-- °°). If such a point is reached, then the mode must be leaking for values of w less than Lo w. (This is proven below.)

If one thinks in terms of the complex k plane representation, then there are actually two real roots on the second sheet (see Fig. 3a) for wB 0 and is included in the tallying of poles to be included and represents the logical extension of the mode to frequencies less than w The other root is not included in the residue tally since it has A* dk/dw<0 and would lie below the real axis were w to have a small positive imaginary part. What happnq is that, as w is reduced to wB , these two roots approach each other and momentarily coalesce at w=w8 . When w is decreased to less than w the two roots again separate, one moving above the real axis, the B other moving below the real axis (Fig. 3b). Since the Eq. (1), however intricate, has only real coefficients, it is apparent that these roots must be complex conjugates of each other. The root of interest is that where the imaginary k

zrkra n a, /In W 74.'6 4 r.

011 4-Ae reed ctX1.7 an 4( --tAen

z -/P/ Com, le k pci i's . part of k is positive or, alternately, where the imaginary part of v is negative.

Near wB' vB' it is evident that the dispersion function of Eq. (1) (With G negative) should be of the general form F Z;.a(w-wB) 8(vB-v) 2 (3) where a and 8 should have the same sign. Thus one has 1/2 1/2 1." vc(a/8) (co-wt)

for w slightly larger than vB , while 1/2 1/2 v ,A,.vB-i(n/8) (wB-(0)

for w slightly less than wB . Thus the real part of v should have zero slope (dotted

line in Fig.l) in'the limit as w4wB from below while the imaginary part should

grow abruptly when w decreases below toB , i.e.

+ d(Imv) => -0 as w4wB (6) dw

It is difficult to saya priori whether a given mode's disperson curve will

encounter a point of the type B. Nevertheless, it is obvious that any viable

computational scheme should recognize this possibility.

The revised computation scheme we suggest to take into account the behaviour

outlined above is as follows. Let us write Eq. (1) in the form 1/2 F(w,v) = A.(w,v)-B(w,v)[GSQ(co,v)] (7)

where Al1R12 (8a) B(w,v)=R12 (8b) 2 GRI=G =A 1+Al2A21 (8c)

Let us suppose we have found a point wF , vF on the dispersion curve which is to

the left of w v but that GSQ(w -Aw,v ) is less than zero. Then the estimate A' A' F F of the location is where wA' vA

+ (Ci.)—(.01, ) + 3 I (V—V (9a) 3; v

GSQF 3 GS° 3 G$Q ) F (v-v F F (9b) d v

or where

0 AF /3 v) (GSQ)_2, - ( 9 GSQ F / 3 v) (AF) A— (10a) 0 GsQF/Bv) ci) — (a GSQF/ 63) AF/av)

v GSQ0 w) (AF) — ( 3 AF / 3 w) (GSQ) F A F GSQF/ 3 v) ( 3 A_F/ 3 w) ( GscIF/ 3 w) ( 3 AF/ 3 v) (lob)

If it should turn out that- -w >Aw then the root is estimated as F

w -Aw, vFco (vgv F) F A

and the process is repeated. Eventually, a point is reached at which Aw>w F-wA.

In this case, the tentative root is taken as given by above with the values wF

taken as those currently corresponding to the last found and v F root.

The root just found is regarded as tentative since we must allow for the

possibility of one's overshooting the root point w B . To check this, we compute

the slope of the aspersion curve at the new found point w v (recalling that T! T we are now on the second branch). Thus we set

1/2 F (w v) = A(w,v) B(w,v) [GSQ(w,v)] (12) 2 ' and calculate

(w ) / 3 to and 3 F (w v ) / 3 v 2 T' T T 2 T' T T (13)

If these have the same signs then the dispersion curve is still with negative

slope end we have not yet reached the point B. In this event we regard the

tentative point as an actual point and proceed to look for the next point.

Its tentative position is at (01,-°w , (' to)Aco (14) where a v/ 3 w is calculated from the ratio

F /ay. 3v 2 (15) 3w 3 P2 /a w

In this manner we arrive at a new tentative root and the process is repeated.

If - we eventually arrive at a tentative root which corresponds to the wrong slope of the dispersion curve, two cases must be distinguished; depending on whether the last actual point found was to the right or the left of w A. either case, let the last actual point be denoted w v Then, for the first case, L' L. we estimate the point B as having coordinates determined by the simultaneous solution of the two equations

(GSQ) / a w a (GSQ) / 3 v 6113 wL

3 A/ 3 w 3 A/ a v v B —v L

B2 ( aGSQ/ 3 v) 2 / ( 3 Al 3 v) 2 — GSQ (16) 2 B ( osq/ v) 2 / ( 3 A/ a — A where all of the coefficients are to be evaluated at In the second case, w1.2vL. the point wB , is estimated as that where w/3 v 0 or where

w/ (vT—vL) vB ( a v)L (17a) ( a (a/ av) T — ( a w/ 3v) L

2 (v ) - (v -v )2 LB T T B L = (vL (17b) B 2 2 -vB ) - (v -v ) T B

) 2 The second follows from fitting a parabola of the form co-L B = a (v-v B

[where a is to be determined] which passes through the points w v and where T' L is as determined from the first equation.

v Once coB' B has been determined, the first complex root is taken as

/2 1 w)1/2 B v = v (18) B - B 771.) where w = wcAw is that value of to for which the root is currently being sought.

Successive complex roots for decreasing successive values of co are found

from the following considerations. SupposecoL' vLR -ivLI corresponds to the

last complex root found. Then the next root would be estimated from

T=Lc/Su (19a)

• F:-.;-(1/2)Fww AL Av=v -v = Au T L m ] - Fwv AL (19b) Fv ± (1/2) Fvv[(Av) - 2iv

whore Fw, Fww, etc., refer to first and record derivatives of F 2 at w=6t, v=v LR' Since the right hand side depends upon Av in the latter, the latter should be

regarded as a quadratic equation. The root to be taken is that where Av reduces

to zero as Aw-40. Alternately, an iteration process can be employed. Note that

we avoid the explicit calculation of the function F for complex arguments. 2 Instead we estimate (at least in principle) its values off the real axis by

analytic continuation. •

GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN THE ATMOSPHERE

Quarterly Status Report No. 7 April 16, 1975 to July 15, 1975

Contract No. F19628-74-C-0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff

Prepared for

Air Force Cambridge Research Laboratories (LWW) Laurence G. Hanscom Field Bedford, Massachusetts 01730

This report is intended only for the internal management use of the contractor and the Air Force. SUMMARY OF OBJECTIVES

The objectives of the study are to develop new analytical and computational techniques for the reduction of infrasonic data recorded at moderate to greater distances form atmospheric explosions. Present research is concerned with Line Item 0001 of the contract and in particular with the revision of the multimode synthesis program.

INVESTIGATIONS IN PROGRESS

Progress during the past quarter is summarized in the attached draft

(Appendix A) of a scientific report which is nearly complete.

PUBLICATIONS

The following paper (abstract only) is presently in preparation for oral presentation only at the forthcoming November 1975 meeting of the Acoustical

Society of America. A written version substantially the same as Scientific

Report No. 1 is intended (pending appropriate approvals) to be submitted for publication in the Journal of the Acoustical Society.

Atmospheric acoustic gravity modes at frequencies near and below low frequency cutoff imposed by upper boundary conditions. W. A. Kinney, A. D.

Pierce, and C. Y. Kapper, Box 34775, Georgia Institute of Technology, Atlanta,

Georgia 30332. Perturbation techniques are described for the computation of the imaginary part of the horizontal wave number (k I ) for modes of propagation.

Numerical studies were carried out for a model atmosphere terminated by a constant

0 and GR sound speed (478 m/sec) half space above an altitude of 125 km. The GR 1 modes have lower frequency cutoffs. It was found that for frequencies less than

0.0125 radian/sec, the GR1 mode has complex phase velocity; kI varying from near 4 zero up to a maximum of 3 x 10 with analogous results for the GR mode. There 0 is an extremely small frequency gap for each mode for which no poles in the complex k plane corresponding to that mode exist. These mark the transition from

undamped propagation to damped propagation. In the complete Fourier synthesis,

branch line contributions compensate for the absence of poles in these gaps.

Computational procedures are described which facilitate the inclusion of the

low frequency portions of these modes in the waveform synthesis. [Work supported by Air Force Cambridge Research Laboratory].

PLANS FOR THE NEXT REPORTING PERIOD

The work reported in the attached manuscript will continue. The work on

the higher frequency arrivals mentioned in proceding quarterly progress reports will be continued also.

FISCAL INFORMATION

There is presently a free balance of about $7000. We have requested and anticipate a no cost extension of the contract to December 31, 1975.

Allan D. Pierce Principal Investigator APPENDIX A (Draft of Scientific Report)

The Leaking Mode Problem in Atmospheric Infrasound Propagation

Allan D. Pierce, Wayne A. Kinney, and Christopher Y. Kapper School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia 30332 INTRODUCTION_

One of the standard mathematical problems in acoustic wave propagation

is that of predicting the acoustic field at large horizontal distances

from a localized source in a medium whose properties vary only with height.

This problem, as well as its counterpart in electromagnetic theory, has

received considerable attention in the literature, 1 is reviewed extensively 2-7 in various texts , and, for the most part, may be considered to be well

understood.

A typical formulation of, say, the transient propagation problem 8-9

leads (at sufficiently large horizontal distance r) to an intermediate result

which may be expressed as a double Fourier integration over angular frequency

w and horizontal wave number k; i.e. for, say, the acoustic pressure, one

has an p = S(r) Re 1f(w)e_ [Q/D(w,k)]e ikr dkdw (1)

Here S(r) is a geometrical spreading factor, 1/3/T- for horizontally stratified 1/2 media, lna sin(r/a )] if the earth curvature (a =radius of earth) is to e e e A be approximately taken into account. The quantity f(w) is a Fourier transform

of some function characterizing the time dependence of the source; Q(w,k,z,z 0)

is a function of receiver and source heights z and z as well as of w and k, o possibly also of horizontal direction of propagation if, say, winds are included

in the formulation, but, in any event, should have no poles in the complex

k plane for given real positive w, and given z and z . The denominator D(w,k) o is independent of z and zo , may be zero for certain values kn (w) of k, and is

termed the eigenmode dispersion function.

Typically , in order to uniquely specify both Q and D(w,k) for all complex 2

values of k (given w real and positive), branch points must be identified

and branch cuts must be placed in the complex k plane. The general rule

may be taken to be that no branch cut should cross the real axis, and, if

a branch point should lie on the real axis (when w is positive real), the

branch cut either extends into the upper or lower half plane depending on whether the branch point moves up or down when w is given a small positive

imaginary part. The integration contour for the k integration goes nominally

along the real axis but skirts below or above (see Fig. la) those poles

lying on the real axis which move up or down, respectively, when w is given

a small positive imaginary part. The placing of the branch cuts and the

selection of the contour in this manner is one method of guaranteeing

causality in the solution, or, equivalently, of guaranteeing that the solution

dies out at large distances if a slight amount of damping (Rayleigh's virtual viscosity) is added in the mathematical formulation. The necessity of branch

cuts only occurs if the medium is unbounded either from above or below and

a choice of phases can always be made such that (given, say, that the medium

is unbounded from above) Q dies out exponentially as z 4* c° when w has a small positive imaginary part and when k is real.

The so-called guided mode description of the far field waveform arises when the contour for the k integration is deformed (permissible because of 10 Cauchy's theorem and of Jordan's lemma ) to one such as is sketched in Fig.

lb. The poles above the initial contour are encircled in the counterclockwise manner. There are also contour segments which encircle each branch cut lying above the real axis in the counterclockwise sense. The integrals around each pole are evaluated by Cauchy's residue theorem and one is left with a sum of residue terms plus branch line integrals. Each residue term may be considered as corresponding to a particular guided mode of propagation. The branch line

contributions in some contexts are considered as corresponding to what may be termed lateral waves. 11 (The term may be unappropriate unless there is a

k R

1. Contours in the complex k (wavenumber) plane for evaluation of individual frequency contributions to waveform synthesis. (a) Original contour. (b) Deformed contour. 3

sharply defined interface separating two types of media, such as a water- muddy bottom interface in shallow water propagation.)

In regards to the guided mode description, one type of approximation frequently made is to neglect all poles (i.e. roots k n (w) of D(w,k)which are ikr above the real axis, the argument being that the corresponding e factors in the residues will die out rapidly with increasing r, the bulk of the contribution at large r expected to come from the poles which lie on the real axis. In a similar manner, it is argued that the branch line contour contribution also dies out relatively rapidly (a factor of 1/r3/2 in addition to the geometrical spreading) so it too may be neglected at large r compared to the terms coming from the real roots. The net result for Eq. (1) would then be f

p =2f(r) An (w) cos[wt-kn (w)r+,n (w)] dw (2) rt i LA. where An (w) and (I)n (0 are defined in terms of the magnitude and phase of the residues of the integrand in Eq.(1); the kn (w) being the real roots of

D(w,k)=0, numbered in some order with the index n=1, 2, 3, etc., and it being understood that, for fixed n, kn (w) should be a continuous function of w over some range of w from a lower limit wim up to an upper limit wun . The remaining integral over w can then be approximately evaluated by the method of stationary phase or integrated by suitable numerical methods.

In the present paper, a somewhat subtle set of circumstances intrinsic to low frequency infrasound propagation in the atmosphere is discussed for which the arguments leading to the approximation of Eq.(1) by (2) are not wholly valid, even at distances of the order of more than a quarter of the earth's circumference. We suspect that comparable circumstances may arise in other contexts, but the present discussion is, for simplicity, illustrated only 4

by examples from atmospheric infrasound propagation.

I. INFRASOUND MODES

An atmosphere model frequently adopted for infrasound studies is one in which the sound speed c varies continuously with height z in a more or less realistic manner (Fig.2a) but is constant (=cT ) for all heights above some specified height z T . [If winds are included in the formulation, their velocities are also assumed constant in the upper half space, z>zT .] Conceivably, one has some latitude in the choice of zT and of the upper halfspace sound speed c T , although computations of factors such as Q(w,k,z,z 0) and D(w,k) in Eq. (1) become more lengthy with increasing z T . Also, it would seem that the most logical choice of c would be that which would realistically correspond to height z T T' so the profile c(z) would be continuous with height across z T , as in Fig. 2a.

Another conceivable choice would be one (Fig. 2b) in which c T 4- 00, such that the surface of air nominally at zT would be a free surface or pressure release surface (corresponding to the model generally adopted for the water-air interface in underwater sound studies). A somewhat intuitive premise which may be adopted is that, if the source and receiver are both near the ground and if the energy actually reaching the receiver travels via propagation modes channeled primarily in the lower atmosphere, then the actual value of the integral in Eq. (1) would be somewhat insensitive to the choices of z T and cT. This, however, remains to be justified in any rigorous sense, so we would be somewhat hesitant to take cT = = at the outset. In typical calculations performed in the past, z T is taken as 225 km, cT is taken as the sound speed (z, 800 m/sec) at that altitude.

Since one is often interested in frequencies (typically corresponding to periods greater than, say, 1 to 5 minutes) at which gravitational effects are important, the formulation leading to the infrasound version of Eq. (1) is based on the fluid dynamic equations with gravitational body forces and the associated nearly exponential decrease of ambient density and pressure with height included. I50 • IS 0

100- _100 E

5 0 < S 0 -

260 340 420 SOO 340 420 1 SOUND SPEED (ril/sec SOUND SPEED m /sec)

2. Idealizations of model atmospheres (altitude profiles of sound speed) used in acoustic-gravity wave studies. (a) Atmosphere terminated by an upper half space with constant sound speed. (b) Atmosphere temperature formally going. to infinity at some finite altitute corresponding to a free surface (13:0) at that altitude. The incorporation of gravity leads, among other effects, to a somewhat complicated dispersion relation for plane type waves in the upper half space when c T is finite, i.e. one can have solutions of the linearized fluid dynamics 8 9 equations for z > z T of the form '

ikzz pl(117 = (Constant) e -iwt ikx o e (3) where the vertical wave number k (alternately written as iG for inhomogeneous z plane waves) and the horizontal wave number k are related by the dispersion relation (neglecting winds)

2 2 2 2 2 2 2 2 k = -G = [w - co ] / c - [w - w ] k (4) z B / w2 where w = (y/2)g/c w A , B = (y-1) g/c are two characteristic frequencies [w > w ] for wave propagation in an isothermal atmosphere (g = 9.8 m/s A B is acceleration due to gravity, y=1.4 is specific heat ratio). Here, for brevity, the subscript T on c T has been omitted. For given real positive w, 2 2 positive or negative (G negative or positive). The real k, one can have kz 2 2 values of k at which k or G go to zero turn out, as might well be expected, z to be the branchpoints in the k integration in Eq. (1), i.e., synonymous with the branch points of G. Along the real axis, G is either real and positive (eik GZ zz or e dying out with increasing z) or else G is a positive or negative imaginary quantity. In the latter case, the phase of G may be either 7/2 or -7/2, in accordance with the well known fact that, for acoustic-gravity waves, wavefronts may be moving obliquely downwards (negative k z) when energy is flowing obliquely upwards. In particular, for 0 < w < w one has G real B' and positive for k in between the two branch points on the real axis, the phase of G is 7/2 (k < 0) on the remainder of the real axis; the two branch z points are, from Eq. (4), at

2 2 1/2 w[w - w ] k1ac (w) = ± A 2 2 1/2 (5) c[w - w]

The branch lines extend upwards and downwards from the positive and negative branch points, respectively. [See Fig. 1.]

The dispersion function D(w,k) in the atmospheric infrasound case can be written in the general form

D(w,k) = (6) Al2R11 A11R12 R12G where R and R are elements of a transmission matrix [R], these depend on 11 12 the atmosphere's properties only in the altitude range 0 to ZT , they are independent of what is assumed for the upper half space. In general, their determination requires numerical integration over height of two simultaneous 8 9 12 ordinary differential equations (termed the residual equations " in previous literature). They do depend on w and k (or, alternately, on w and phase velocity v) but are free from branch cuts, they are real when w and k are real and are finite for all finite values of w and k. The other parameters A l2 and A depend only on the properties of the upper half space (in addition to 11 w and k). Specifically, these are given (for the no wind case and with the subscript T omitted on c r)

2 2 2 A = gk /w (7a) 11 - yg/[2c ]

(7b) One may note that, since every quantity in Eq. (6) is necessarily real when w and k are real (with the possible exception of G), the poles lying

on the real k axis (real roots of D) must be in the regions of the (w,k) plane 2 [or (w,v) plane] where G >0. Since the integrand of Eq. (1) divided by fr -Gz should vary with z above z as e T one may call the corresponding modes T fully ducted modes. There is no net leakage of energy for such natural modes

into the upper halfspace. If one considers D as a function of w and phase velocity v (or simply v), where v = w/k, the locus of real roots v versus w

(dispersion curves) has (as has been found by numerical calculation) the general

form sketched in Fig. 3. The nomenclature for labeling the modes (GR for gravity, S for sound) is due to Press and Harkrider. One may note from Eq. (4)

that there are two "forbidden regions" in the v vs. w plane, i.e.

2 2 2 2 v < c[w - w ] 2 / [co - w ] (8a) B A for w < wt and

, 2 2, 11 2 2,1/2 v > cLw - (413 .1 Lw - wAJ (8b)

for w > wA. Within either of these regions G would have to be imaginary and there would accordingly be no real roots for v of D(w,v) = 0. In the high frequency limit, this simply implies that the phase velocities of propagating modes are always less than the sound speed of the upper halfspace, the branch points in the k plane are simply at ± w/c T . The low frequency lower phase velocity "forbidden region" appears to be due to the incorporation of gravity effects into the formulation. However, if cT is allowed to approach 00, this lower left hand corner region disappears. We have done numerical studies on the effects of varying c T on the dispersion curves. Briefly, the result is that the form of the predicted curves for GR 0 and GR1 change very little

-TE,Ivip R 500 V lit 'I,

400 1,.. U : solialkIlKilLivoL im E

0 .J

X 30 C AM h llIllll IIlIllIll'II '" 0. igIIIII* 'illu

rill GR0 20 • • 0 ANGULAR FREQUENCY, rod/sec

10 5 3 2 1.5 PERIOD IN MINUTES

3. Numerically derived plots of phase velocity v versus angular frequency w for infrasonic modes in a model atmosphere corresponding t. Fig. 2. The labeling of modes is with the convention introduced by Freon Land Harkridor (J. Geophy. ROO. 67, 3889-3908 (1962). Thu lino' (1 2,.0 rogimin of

the v versus w plane whura a rota rout of Om olguomodu dimparuJo ► Noel :loll cannot be found. with increasing cT ; the lower forbidden regions shrink insofar as frequency

range is concerned and the curves extend to successively lower frequencies.

Thus we see that the fully ducted modes GR and GR both have a lower frequency o 1 cutoff [WI, in Eq. (2)] which depends on cT . The larger one makes c T , the

smaller is this cutoff frequency.

We thus have the following apparent paradoxes. Given that frequencies below w may be important for the synthesis of the total waveform, an apparently 13 plausible computation scheme based on the reasoning leading to our Eq. (2) will omit much of the information conveyed by such frequencies. Also, in spite

of the plausible premise that energy ducted primarily in the lower atmosphere

should be insensitive to the choice for c one sees that this choice governs the T' cutoff frequencies for certain modes and that certain important frequency

ranges could conceivably be omitted entirely by a seemingly logical and proper

choice for cT . The resolution of these paradoxes would seem to lie in the nature of the approximations made in going from Eq. (1) to Eq. (2). The latter may not be as nearly correct as earlier presumed and it may be necessary to in-

clude contributions from poles off the real axis and from the branch line

integrals. Even if r is undisputably large, it may be that the imaginary parts of the complex wavenumbers are sufficiently small that le ikr i is still not small compared to unity. Also, a branch line integral may be appreciable

in magnitude at large r if there should be a pole relatively close to the branch cut. II. ROOTS OF DISPERSION FUNCTION

In order to understand the manner in which the solution represented by

Eq. (2) should be modified in order to remove the apparent artificial low frequency cutoffs of the GR and GR modes, we first examine the nature of the o 1 dispersion function D at points in the vicinity of a particular mode's dispersion

curve. The curve vn (w) of phase velocity v versus w for a given (n-th) mode is known at points to the right of the lower cutoff frequency w L . Given this, one can find analogous curves v a (w) and vb (w) for values of the phase velocity w/k at which the functions R 11 (w,v) and R12 (w,v) in Eq. (6), respectively, vanish. Since there may be more than one such curve in each case, we pick v a (w) and vb (w) such that these curves are the closest of all such curves to the curve vn (w) for w > wL . One may note, however, that one may apparently define and identify va (w) and vb (w) for frequencies much less than wL , simply from analytical continuation.

A premise which we have checked numerically (see Fig. 4) for a specific

case is that the curves vn (w) , va (w) v (w) defined above with reference to , b a particular given mode all lie substantially closer to each other than to

the corresponding curves for a different mode. In retrospect, this is obvious,

although it took some time for us to realize that it was so. Briefly, the argument goes that, if the mode is predominantly guided in the lower atmosphere, then

there should be a decay of modal height profiles beyond some point substantially

lower than. Ar . Thus, both the 047: and p ov z profiles for a guided mode would have values at zr substantially less than their peak values at lower

altitudes. The same would be true for the profiles of the auxiliary functions

01 and 02 which satisfy the residual equations. Consequently, if guided waves are excited, the inverse transmission matrix connecting 0 1 and 02 at the ground

to those at height 41, would have to have very small [1,2] and [2,2] components. 0.014 0.002 0.106 0110 0. 14 0.002 0.006 0.110

7 FIESIliENCY IrEidiar_ijsec ) A N G_U_L U d sec ANGULAR L 1

4. Curves in phase velocity (v n ,va ,vb ) versus angular frequency (w) plane along which R11=0 (giving v (w), R12=0 giving vb (w), and D(w,k)=0 a ( (giving vn (w). Curves are shown for (a) the GR0 mode and (b) the CR1 mode. N .te the changes in scale and the relatively close spacing of curves corresponding to the same mode. The lines along which G 2=0 are also indicated;

vn(0 is not a real quantity for w values below the indicated lower cutoff frequency. (Recall that (I)1 = 0 at the ground.) Since the transmission matrix has unit determinant, it follows that elements R 12 and R11 of the transmission matrix

proper [from height Z down to the ground and whose elements appear in Eq. (6)] T have to be small.

Given the definitions v a(w) and vb (w)(w), the dispersion relation D=0 for

a single mode may be written

= 0 D '- (Al2 )(a)(v-va ) - [All G] C) (9 ) where a = dR11/dv a = dR12 /dv evaluated at v = v and v respectively. (For , , a b' simplicity, we here consider D as a function of w and v = w/k rather than of

w and k.) The above equation may also equivalently be written in the form

v = va + (va-vb)X/[1-X]

G) X = ("")(All /Al2 which may be considered as a starting point for an iterative solution which in

essence develops v in a power series in v a-vb ; G may be considered as a defined

function of w,v. One starts with v = v a as the zeroth iteration, evaluates the

right_hand side for the value of v to find the starting point for the next

iteration, etc. The considered procedure should converge provided v a or vb

is not near a point at which G vanishes and providing G in the vicinity of v a

or v is not such that the variable X is close to unity. Among other limitations, b the iteration scheme would be inappropriate for values of w in the immediate

vicinity of coL .

In regards to establishing the general trends represented by the iterative

type solutions, two relatively general theorems may be of use. These (whose 13 proof follows along lines previously used by one of the authors in deriving an integral expression for group velocity) are that for real positive w and v,

R DR /av - R aR /av > 0 12 11 11 12

R 8R /Dw - R aR12 /aw > 0 12 1i 11 or, alternately, if one inserts R 11 = (a)(v-v a ), R12 = (0)(v-vb), he finds

cif:(v - v ) > 0 (12a) a b

(v - v )(v - v ) (0a'-0'a) + 0a[v - v ) - v ' (v - v )] > 0 (12b) b a b a a b where the primes represent derivatives with respect to w. The second of these should hold for arbitary v in the vicinity of v a and vb and lead, upon setting v = va , v = vb , or v = (va D v. -va 'vb )(vb ' -va '), along with the use of Eq. (12a), t o

< 0 (13a) vb

< 0 va (13b)

(a/0)' > 0 (13c)

Equation (12a) implies that as long as a or a do not vanish (which would seem unlikely) the two curves va (w) and vb (w) do not intersect. If a and a have the same sign the va curve lies above the vb curve; the converse is true if a and 0 increases with w.

To demonstrate the general utility of the perturbation approach, a brief table of values w, v vb, a 0 v (1) , and v are given in Table I for the GR a' ' ' n o modes for the case of a U.S. Standard Atmosphere without winds terminated and GR1 (1) at a height of 125 km by a halfspace with a sound speed of 478 m/sec. Here v is the result of the first iteration for the phase velocity and v n is the actual numerical result obtained (only if the phase velocity is real) by explicit numerical search for roots of the eigenmode dispersion function. One may note (1) that, for those frequencies where v is computed, the agreement between v n and v is excellent. A more detailed listing of the perturbation calculation n results is given in Figs. 5a and b. The plots there give w/k R or the reciprocal (1) of the real part of 1/v (i.e., w divided by the real part of the horizontal wave number k) and the imaginary part k I of k = w/v versus angular frequency.

is zero above the corresponding cutoff frequencies. The relatively Note that kI small values of the k are commented upon in Sec. IV. I

III. TRANSITION FROM NONLEAKING TO LEAKING

The iteration process described by Eqs. (10) in the preceeding section may fail to converge when G is near zero and in any event gives relatively little insight into what happens to a modal dispersion curve in the immediate vicinity of wL . To explore this transition region, it would appear sufficient to approximate G in Eq. (9) by di 1/2 G = [(p)(w-wL) + (q)(v_v (14) where p and q are readily identifiable [from Eq. (4)] positive numbers taken independent of w and v. ' vL is the phase velocity on the dispersion curve in the limit as w w from above. The bracketed quantity in Eq. (14) may be re- L 2 garded as a double Taylor series expansion (truncated at first order) of G about 2 the point wL , vL at which G vanishes (hence no zeroth order term). The fact that 2 both p and q are positive follows since G is positive to the upper right of the I va " v V'\. v vn --- I b P r I j 0.31202121 + 0.0052 I 0.31203 0.31207 917.4 ' -2783.7 ___ ,,::3. 184 .. x .. 10:~i. __ --'-- ." .. ., .,.--.L ..... -"" _...... ,-- ... --- '--~- .. --.. --.-~-- 1---_._... -..... ------t------·--'-· .. 0.31189059 + 0.0113 0.31190 0.31194 767.9 -3254.2 -1. 6 ,~. ~.,_~ ,_,~. 721 x 10- i "C~._'''''''''_ .r"'.~' a ____"· ___ __ • __ •• _,,_,_ __,_,. '" •.• __ .. ' "". __ " __ .. " __ . ___ . ,._.___ __ ...... T ._.,.~ _L_~'.h ___ ._~_ .. -~-,~.-- ' .-.... ," _._-~.- .,~ 0.0155 0.31176 0.31181 621.9 -3644.3 • 0.31173763 ==-~-l0.31172882 '. I ., GR '''''~' -,,~- ,..~ ~ ' .. "~ ~~.'" 0 --~,.- .. .---.,.."- - _. ----_.-_ ..... ----' ---...... - ...... ,- ...... '" ..... ------j 0.0165 0.31172 0.31177 581.5 -3738.2 0.31167504 0.31167509 ___ T_~ ____

----,~...... -~. "-_._. 1------_._--.. - ---.. __ ._-_ - --'.~--.-,~--.-- .. _--,_.,---- -.--_ ..... __ .. _- 0.0186 0.31162 0.31168 497.5 -3910.1 0.31153369 0.31153394

'''''WV' __ .

u.25267 0.0052 0.24229 0.24816 87.8 -3633.0 + -2.715 x 10-3i . -.-.------0.0103 0.23433 0.23844 94.7 -3990.0 0:i4218 + -1. 337 x 10-3i , ------. GR 0.0144 0.21842 0.22037 150.7 -5307.0 0.21431 0.22178 --~ .. ~--~.~ 1 ------'-- '--~ ! 0.0165 0.20252 0.20345 265.0 -7767.3 0.20016 0.20463 I 0.0175 0.19058 0.19111 418.9 -10~858.0 . 0.19226 0.19212 I I I

1. Frequency dependent parameters corresponding to GR and GR modes; w is , 0 1 angular frequency in rad/sec, va is phase velocity root of R =O, in 11 kro/sec, vb is analogous root of R12=O, a is dR /dv at v=va in sec/km 11 f3 is dR /dv at v=vb in sec, v(l) is first order pel:turbation solution for 12 phase velocity from equations given in the text (units are km/sec), vn is the real root determined by direct numerical solution for zeros of eigenmode dispersion function. Note that v (defined only when phase n . ( velocity is real) agrees exceptionally well with v 1).

Lf.-6-2

CUTOFF GR D.26 • 1

---- - 0;24 ■ GR i • CUTOF 4,

0.002 0.004 0.006 0.008 0. 10. 0.012

ANGU AR FREQUENCY (radian/sec) . . 1 1

• . 3 10

_1 6:4 •

_ GR i . . .

-s , -10--- 1 ---r-----.....-...... ••7 -10- . / . G R 0

. . 4 -10-.9 r 0.002 0.004 0.6 0 6 0.008 0. 10 0.012

ANG_U LAR F RECIU N C Y LradlEn/se c

5. Numerically derived plots of phase velocity w/k R and of the imaginary part k of the complex wavenumber k versus angular frequency for the GR I 0 and GR1 modes. Previous theoretical lower frequency cutoffs for these modes are as indicated. Note that k I is identically zero above the cutoff frequency. line in the w,v plane where G2 = 0 and also since the G 2 = 0 line slopes obliquely downwards. (See Fig. 3).

Let us next note that, in the vicinity of the point w vL , the denominator L' D given by Eq. (9) may be further approximated as

D z (A12a-A110) (tv + piw) + e(Av + vAw ) 2 (15) where we have abbreviated Av = v-v Aw = w -w v = p/q; the quantity p is L' L either -dv /dw or -dvb /dw the two being assumed to be approximately equal. a , (The use of the minus sign here assumes that p be positive.) The remaining quantity E is (q 2) ($) (v-vb ) (16) $A - clA12 11

One should note that e depends on v, although, for purposes of initial analytical investigation, one may set v = vL here. All of the above quantities may be considered to be evaluated at w = w L and v = vL . Note that p and v are both positive quantities. Furthermore, it should also be noted that v > p since 2 the G = 0 curve slopes downwards more rapidly than the lines along which

R or R = 0 in the v vs w plane. (See Fig. 4.) 11 12 The roots of Eq. (15) without regard to the sign of the radical are readily found to be

2 - (17) Av = -thw + We + c(v-11) 2 [Aw 4 where

o = e 2 /C4(v—u] (18)

Alternately, if 'Awl << a, the above may be approximated by the binomial theorem to give 22 2 Av = -vAw + [(v-p) ](Aw) (19a) or

2 2 2 2 Av = +e - (2p -v) Aw - [(v-p) /6 ](Aw) (19b)

for the upper and lower signs, respectively. The first of these (since Av = 0 when Aw = 0) is clearly the description of the disperson curve in the vicinity of w = wL , v =

Equation (19a) shows that, as Aw + 0 from above, the dispersion curve 2 becomes tangential to the line G = O. The two curves do not intersect. The general trend is as indicated in Fig. 6. The solution represented by Eq. (19b) is not a proper root of Eq. (15); it corresponds to the wrong sign of the radical and accordingly lies on the second branch. Furthermore, one can readily show that, for values of Aw slightly less than zero, both roots lie on the second branch. Hence, there must be a gap of finite frequency range in which, for the choice of branch cuts represented by Fig. 1, there are no poles in the k

(or v) plane corresponding to the n-th mode.

To determine the order of magnitude of this frequency gap, it is appropriate to consider the trajectory of the second branch roots in some detail and to determine just where one of them should cross the branch cut, reappearing on the first branch. As long as Av is real and Av + vtw >0 the criterion for a root to be identified with the first branch is Av + uAw > O. According to

Eq. (17), this would automatically place the second root on the second branch for all Aw > -a and would place the first root on the second branch for

-a < Aw < O. Consequently, if either root is to reappear on the first branch, it must be at a value of 1w < -a.

One should note from Eq. (17) that at 1w = -a the two real roots on the second branch coalesce. For values of Aw < -a the two roots separate again,_but teak

T./ 2 -Two ?0Z-Sine Pv9S o42 C4 00 Sin ,% idranclz. cths 1t c p/anean real and positive in Eq. (1) for the determination of the points to the right . of A.

A fundamental analysis of the properties of Eq. (1) shows that the dispersion 2 curve is actually tangent to the G =0 curve at the point A. The two curves, contrary to what we had earlier supposed, do not intersect. A simple proof of this is that the gradient of F(.,7,3) in a cartesian coordinate system where w 2 and v are regarded as coorainPtes has a term proportional to VG /G which 2 dolllintes In the vicinity of the G =0 line. Since a line F=0 is always perpendicular to VF, it follows that such a line (dispersion curve) must be per- 2 2 pendicular to VG when G is srlp 1 l and hence parallel to the G =0 line. Also, 2 since the dispersion curve is Dresumed to touch the G =0, it must of necessity be parallel or tangential to it when it touches it.

The next question which may be considered is that of just what is the logical extension of the mode's dispersion curve to lower frequencies than that value 2 w (not to be confused with Hine's (y/2)g/c) where it touches the G =0 curve. A 2 On. may note that the value of k at which G =0 for a given to is a branch point

(hence call it k (w))ia the cc.,)1.ex w plane. Consequently, when w=w ts the BR corresponding pole has coalesced with a branch point. The logical coninuation of the v versus w curve must consequently be onto a second branch. In other words, the roots for w slightly less than w should be based on that version of Eq. (1) A in which G is taken as real and negative (rather than real and positive). Note 2 that G is still positive. The branch cut previously had been assumed to be as sketched in Fig. 2a with a braac'h line contour included implicitly (but actually neglected in the nodal synthesis). In this old version of the theory, G was positive real for kkio. on the real axis, and one would see a root approaching the branch cut and then disappearing. In the newer Co

I • Gen era./ -Par/7z cyP d sper s c4.4.1- v e ill cle G=O kne. L

6. Sketch illustrating nature of a.singie mode's dispersion curve in the vicinity 2 of the G =0 line. At point A (angular velocity wL , phase velocity vL) 2 the dispersion curve is tangent to the G =0 line; for frequencies below wL down to that corresponding to point B 12n the sketch there are two real roots for v of the eigenmode dispersion function on the second branch. For frequencies lower than that corresponding to point. B, there is a complex root for v on the first branch (which is the complex conjugate of a second root on the second branch). are now complex conjugates. The root in the upper half of the v plane (lower half of k plane) can never cross the branch cut so it remains on the second branch indefinitely. The one in the lower half of the v plane will cross the branch cut at a point which may be approximately estimated as that where

Re(Av) = -vAw or where

2 Aw (1/2.) E = -2C (v-p) with a corresponding value of Av of

= (e 2 /2) [v/(v-u)] - i3

For subsequent frequencies successively lower than w L-2a there is a complex root on the first branch with a negative imaginary part which increases with decreasing frequency.

The discussion up to now has assumed that lAvl << Iv L-vb l and hence that c may be taken as constant. This would seem appropriate for describing

the transition region since all values of Av of interest in this region are of 2 second order of c . However, if an improved numerical estimate is required, we recommend that one regard Eqs. (16) and (17) as a iterative pair. Success-

fully computed values of Av may be used to recalculate c and the new value of

c may then be used in obtaining the next higher estimate for Av. w v p, q, u, v, c, and a are given for In Table II the values of L' L' the GR and GR modes for the model atmosphere corresponding to Fig. 2a. 0 1 The extremely small values of a should be noted. The corresponding plot

of Av versus Aw (i.e., both branches of Eq.(17)) corresponding to their

values for the GR mode is given in Fig. 7. For simplicity, this is plotted 0 in a nondimensional form, i.e.

1/2 V = - {p/[2(v - p)]10 + [1 + 0] (20) TABLE 2

GR GR o

toL (rad/s) 0.0118 0.0125

vL (km/s) 0.31188 0.2323

p(s/km2) 0.14 0.35

q(s/km3) 1.84 x 10-3 1.86 x 10-3 -2 p(km) 2.94 x 10 4.15

v(km) 76 190 1/2 1/2 -6 -3 e(km /s ) 9.6 x 10 1.02 x 10 -13 -9 a(rricb5/s) 3.04 x 10 1.41 x 10

Parameters characterizing the eigenmode dispersion junction near points in the phase velocity versus angular frequency plane at which the GR 0 and GR modes undergo transition from leaking to non-leaking. 1 7. Graph of normalized phase velocity versus normalized frequency in the vicinity of the point (v L, y for the GR0 mode. The imaginary and real parts are both plotted. The dotted line corresponds to real roots on the second Riemann sheet. where v = Av/[2(v-p)a] and Q=Aw/a. Both real and imaginary parts are shown on the same graph. The corresponding plots for the GR 1 mode differ only slightly from those in the Fig. 7 because of a different value of the parameter p/[2(v-p)] in Eq. (20); in both cases this parameter is small compared to unity, i.e. 11<

IV. THE BRANCH LINE INTEGRAL

Since there is a gap in the range of frequencies for whibh a pole corresponding to a mode may exist, it is evident that evaluation of the k integration in Eq. (1) by merely including residues may be insufficient for certain frequencies. Thus it would seem appropriate in such cases to include a contribution from the branch line integral. It may be anticipated that such branch line integrals are significant at larger values of r only when w.is close to some mode's wL (say the n-th mode), in which case the branch point of greatest interest (i.e., that which may have a pole in its immediate vicinity) is at k=w/vL. Consequently, it would appear that an adequate approximation to the branch line integral would be

S;E Branch line [Q/D(w,k)]eikrdk contribution of

ikr Q e dk (21) Al2a -All° fc x+(u-v)Aw+ex1/2 0 where the denominator D(w,k) has been approximated by Eq. (15) with the abbreviation x for Av+vAw. The quantity outside the integral is assumed to be evaluated at w=w1, and k=w/vL . The contour CB runs down the left side of the branch cut, around the branch .point (where x=0), and then up the right side. If one next changes the variable of integration from k to x, nothing that for small x/vL ,

(22)

he finds approximately that 2 -i(w /v)x Branch line 0 q , LL = (Residue) contribution v2 (23) x+(p-v)Am+ex CI; where (Residue) o is that residue which the integrand (Q/D)e ikr would be expected to have at the n-th mode's pole in the k plane were the parameter c identically

equal to zero. The mapped contour ci3 in the x plane may be considered to go

up on the right and then down on the left of a branch cut extending vertically

downwards from the origin in the x plane. If we set x=-iC, then, on the right 1/2 e-in/401/2 side of the cut, x should be while, on the left side, it is -17114 1/2 -e E . Consequently, the total integral combines to 2 -(w /v )r r- Branch lines 2ce e L vE dC 0 (24) contribution = -(Residue) 2 2 [-iC+(p-v)Aw] +ie a This in turn, with an obvious change of integration variable, may be expressed as 06 i 7/4 -n 1/2 Branch line = (Residue) 2K2K e e (111 (25) contribution (n-n 1)(11-n2 ) 0 where

1/2 K=ev / w r) (26a) L L 2 f l , ri = i(K /2)(1+[Aw/2a]) 2 2 1/2 i(K /2)(1+[Aw/a]) (26b)

with a as defined by Eq. (18).

In regards to the ri integration, the integral can be expressed in general

in terms of Fresnel integrals of complex argument after some considerable

mathematical manipulation. One may note, moreover, that In i i and 1n21 are, for

most cases of interest, considerably less than unity. In this case, the

appropriate approximate result (derivation omitted for brevity) is

e iF do in 1/2 1/2 (27) (n-n 1)(n-n 2 ) 0 -n 1 +n2 where the choice of square root should be such that the imaginary part is positive. The net result in this limit then is that the branch line contribution is independent of the parameter K. (The dependence on range r comes only in the residue.) Thus one may write

Branch line 2ui(Residue) B Ow/a) (28) contribution o rh { where_the function B (6w/a) is given by rh

B (0) rh 1/2 1/2 (29) 1/2 1/2 []+(1/2)0+ 1+0) ] +[1+(1/2)0-(1+0) ]

Here any consistent choice may be made for the sign of the inner square roots but the outer, square roots should be taken such that the resulting phases are between -u/4 and 3u/4. The quantities in square brackets turn out to be the 1/2 squares of (1/1/2-.)[(I+0) ±1], respectively. The phase restruction then gives

1/2 B rh (0) = (1+0) if n>o (30a) = 1 if 0>0>-2 (30b) -1/2 = -i(-0-1) if 0<-2 (30c) where here all square roots are understood to be positive;

To completely describe the transition it is appropriate to add to Eq. (28) that contribution (which is zero for 0>6(1)>-2a) from the pole on the first branch in Eq. (21) which lies in the general vicinity of k=co livL .,If the pole is present, its contribution to the,integration over k is 2iri times the residue

(which is not what we have been referring to as (Residue) 0 unless c is identically zero). The evaluation of the residue is moderately straightforward and omitted here for brevity. The net result is that Branch line 3 Pole contribution contribution

= 2ffi(Residue) o B rh (Aw/a)+P ot (Aw/c) (31) where the "pole function" P ot (Aw/a) turns out to be given by

P(Aw/a)=1-B (Aw/6) rh (32)

We accordingly have the remakable (although, in retrospect, not unexpected) result that

Branch line , Pole contribution contribution = 2ffi(Residue) (33)

The above gives one a relatively simple prescription for evaluating a given mode's contribution t'o the k integration in Eq. (1). First, all branch line integrals are formally neglected. If a pole exists on the first branch, the residue which would normally be utilized is replaced by

Seeikri? Qe ikr Res — D d'D/dk (34) k=pole where

D (A R -A R ) dk dk 12 11 11 12

-G — (R ) dk 12 (35)

..e. it differs from the actual derivative of D in that G is formally considered is constant. Doing this when co is somewhat removed from the transition region tear w should make very little difference since R is small at values of k which L 12 ire poles. Near the transition, this neglect should almost exactly compensate

or the neglect of the branch line integral. REFERENCES

1. J. E. Thomas, A. D. Pierce, E. A. Flinn, and L. B. Craine, "Bibliography on Infrasonic Waves", Geophys. J. R. astr. Soc. 26, 399-426 (1971).

2. C. B. Officer, Introduction to the Theory of Sound Transmission with Application to the Ocean (McGraw-Hill, New York, 1958).

3. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon Press, Inc., New York, 1962).

4. L. M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1960).

5. K. G. Budden, The Wave-Guide Mode Theory of Wave Propagation (Prentice Hall, Inc., Englewood Cliffs, N.J., 1961).

6. I. Tolstoy and C. S. Clay, Ocean Acoustics (McGraw-Hill, New Yprk, 1966).

7. M. Ewing, W. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957).

8. A. D. Pierce and J. W. Posey, Theoretical Prediction of Acoustic-Gravity Pressure Waveforms generated by Large Explosions in the Atmosphere, Report AFCRL-70-0134, Air Force Cambridge Research Laboratories, 1970.

9. A. D. Pierce, J. W. Posey, and E. F. Iliff, "Variation of Nuclear Explosion generated Acoustic-Gravity Waveforms with Burst Height and with Energy Yield" J. Geophys. Res. 76, 5025-5042 (1971).

O. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Clarendon Press, Oxford, 1935) p. 137.

1. L. M. Brekhovskikh, loc. cit., pp. 270-280.

2. A. D. Pierce, "The Multilayer Approximation for Infrasonic Wave Propagation in a Temperature and Wind-Stratified Atmosphere", J. Comp. Phys. 1, 343-366 (1967).

3. A. D. Pierce, "Propagation of Acoustic-Gravity Waves in a Temperature and Wind-Stratified Atmosphere", J. Acoust. Soc. Amer. 37, 218-227 (1965) GEORGIA INSTITUTE OF TECHNOLOGY Atlanta, Georgia 30332

INFRASONIC WAVE PROPAGATION IN THE ATMOSPHERE

Quarterly Status Report No. 8 July 16, 1975 to October 15, 1975

Contract No. F19628-74•C-0065

Project No. 7639

Contract Monitor, Elisabeth F. Iliff

Prepared for.

Air Force Cambridge Research Laboratories (LWW)

Laurence G. Hanscom Field

Bedford, Massachusetts 01730

This report is intended only for the internal management use of the contractor and the Air Force.

Appendix

A Proposal of a Doctoral Dissertation

Wayne A. Kinney DISSERTATION PROPOSAL

Mathematical Models of Acoustic and Acoustic Gravity Wave Propagation in Fluids with Height-Dependent Sound Velocities

Wayne A. Kinney School of Mechanical Engineering Georgia Institute of Technology Advisor: Professor Allan D. Pierce

December, 1975 ABSTRACT

Several problems related to acoustic and acoustic gravity wave propagation in a medium whose properties vary only with height will be considered. The contribution from very low frequencies to a modal synthesis of an acoustic gravity waveform will be clarified. Also, an acoustic pressure amplitude predicting scheme based on acoustic ray concepts and for propagation to large distances will be devised. A computer program based on this scheme will be written and used as a tool to test amplitude predicting schemes based on statistical concepts.

This program will also be used to consider the particular case of propagation in a medium with two adjacent sound speed channels. Finally, this particular case will also be considered using the modal approach by means of the W.K.B.J. approximation. PROPOSAL

It will be the intent of this dissertation to present a number of problems and associated new results related to acoustic and acoustic gravity wave propagation in a medium whose properties vary only with height (e.g., the atmosphere and oceans). One of these will involve the clarification of the contribution from very low frequencies to a modal synthesis of waveforms associated with acoustic gravity wave propagation (see Appendix A). The computer program Infrasonic Waveforms )

has previously been devised to synthesize an infrasonic pressure-time trace as might be generated at large horizontal distances by a large explosion in the atmosphere. This program will be modified to include low frequency contributions which have been previously neglected, and which are now understood.

Another problem will involve formulating an acoustic pressure amplitude predicting scheme based on acoustic ray concepts and for propagation to large distances. Although there are computer programs that calculate ray paths (notable among these are those used in underwater acoustics), there appear to be no readily available ray ampiltude predicting programs of sufficient accuracy to predict wave amplitudes after propagation over very large distances 2 . At such distances, one would expect that there be many ray paths connecting source and receiver, implying that there might be some erratic variations in actual wave fields. However, one might also expect that these results could be explained at the outset using models based on statistical notions and involving spatial averages. Such models, for example, have been formulated by P. W. Smith3'1'. A computer program designed to provide large range ray amplitude prediction will be developed as a research tool. Its main purpose will be to test simpler models, as would be based on statistical concepts.

Innovative in this program will be the use of cubic splines to accurately model continuous sound speed profiles 5,6 . Use will be made of ray path periodicity to reduce computation time, and effects taking place near caustics will be explicitely incorporated.

This program will, in addition, be used to consider the particular case of propagation through a medium with two adjacent sound speed channels. By means of the W.K.B.J. approximation, this same problem will be considered using the modal approach as well (see Appendix B). 7

A listing of that portion of the ray amplitude predicting program that has been completed to date is given in Appendix C. Finally, a bibliography of related works is attached following Appendix C. REFERENCES

1. A. D. Pierce and J. W. Posey, "Theoretical Prediction of Acoustic- Gravity Pressure Waveforms Generated by Large Explosions in the Atmosphere", Report AFCRL-70-0134, Air Force Cambridge Research Laboratories, 1970.

2. L. P. Solomon, et. al., "Final Report of the Acoustic Sub-Group of the Technology Group", [Planning Systems, Inc., 7900 Westpark Drive, Suite 507, McLean, Virginia 22101], Section III, Part E.

3. P. W. SMith, "The Average Impulse Responses of a Shallow-Water Channel", J. Acoust. Soc. Amer., 50, 332-336, 1971.

4. P. W. Smith, "Averaged Sound Transmission in Range-Dependent Channels", J . Acoust. Soc. Amer., 55, 1197-1204, 1974.

5. Private communication on Nov. 4, 1975 with L. P. Soloman of Planning Systems Inc., McLean, Virginia.

6. C. B. Moler and L. P. Soloman, "Use of Splines and Numerical Integration in Geometric Acoustics", J. Acoust. Soc. Amer., 48, 739-744, 1970.

7. C. Eckart, "Internal Waves in the Ocean", The Physics of Fluids, 4, 791-799, 1961. Appendix A ATMOSPHERIC ACOUSTIC GRAVITY MODES AT FREQUENCIES NEAR AND BELOW LOW FREQUENCY CUTOFF IMPOSED BY UPPER BOUNDARY CONDITIONS

Allan D. Pierce, Wayne A. Kinney and Christopher Y. Kapper

School of Mechanical Engineering Georgia Institute of Technology

Contract No. F19628-74-C-0065 Project No. 7639

SCIENTIFIC REPORT NO. 1

Contract Monitor: Elisabeth F. Iliff Terrestrial Sciences Laboratory

This document has been approved for public release and sale;'its distribution is unlimited.

Prepared for

AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSPACE RESEARCH UNITED STATES AIR FORCE BEDFORD, MASSACHUSETTS 01730 ABSTRACT

Perturbation techniques are described for the computation of the ima!ginary part of the horizontal wave number (k I ) for modes of propagation.

N=erical studies were carried out for a model atmosphere terminated by a constant sound speed (478 m/sec) half space above an altitude of. 125 km.

The GR and GR modes have lower frequency cutoffs. It was found that for 0 1 frequencies less than 0.0125 radian/sec, the GR, mode has complex phase velocity; k I varying from near zero up to a maximum of 3 x 10 -4 with allalogous results for the GR0 mode. There is an extremely small frequency gap for each mode for which no poles in the complex k plane corresponding to that mode exist. These mark the transition from undamped propagation to damped propagation. In the complete Fourier synthesis, branch line contributions compensate for the absence of poles in these gaps. Computa- tional procedures are described which facilitate the inclusion of the low frequency portions of these modes in the waveform synthesis. INTRODUCTION

One of the standard mathematical problems in acoustic wave propagation is that of predicting the acoustic field at large horizontal distances from a localized source in a medium whose properties vary only with height.

This problem, as well as its counterpart in electromagnetic theory, has received considerable attention in the literature, ) is reviewed extensively 2-7 in various texts , and, for the most part, may be considered to be well understood.

A typical formulation of, say, the transient propagation problem 8-9

leads (at sufficiently large horizontal distance r) to an intermediate result which may be expressed as a double Fourier integration over angular frequency w and horizontal wave number k; i.e. for, say, the acoustic pressure, one has am oo . -iwt p = S(r) Re cf(L.)e [Q/D(w,k)]eikr dkdw (1) o co

Here S(r) is a geometrical spreading factor, 1/3 7i7- for horizontally stratified 1/2 if the earth curvature (a =radius of earth) is to media, 1/[a sin(r/a )] e be approximately taken into account. The quantity f(w) is a Fourier transform of some function characterizing the time dependence of the source; Q(w,k,z,z ) o is a function of receiver and source heights z and z as well as of w and k, o possibly also of horizontal direction of propagation if, say, winds are included in the formulation, but, in any event, should have no poles in the complex k plane for given real positive w, and given z and z o . The denominator D(w,k) is independent of z and z , may be zero for certain values k (0 of k, and is o termed the eigenmode dispersion function.

Typically, in order to uniquely specify both Q and D(w,k) for all complex values of k (given w real and positive), branch points must be identified and branch cuts must be placed in the complex k plane. The general rule may be taken to be that no branch cut should cross the real axis, and, if a branch point should lie on the real axis (when w is positive real), the branch cut either extends into the upper or lower half plane depending on whether the branch point moves up or down when w is given a small positive imaginary part. The integration contour for the k integration goes nominally along the real axis but skirts below or above (see Fig. la) those poles lying on the real axis which move up or down, respectively, when w is given a small positive imaginary part. The placing of the branch cuts and the selection of the contour in this manner is one method of guaranteeing causality in the solution, or, equivalently, of guaranteeing that the solution dies out at large distances if a slight amount of damping (Rayleigh's virtual viscosity) is added in the mathematical formulation. The necessity of branch cuts only occurs if the medium is unbounded either from above or below and a choice of phases can always be made such that (given, say, that the medium is unbounded from above) Q dies out exponentially as z 0. when w has a small positive imaginary part and when k is real.

The so—called guided mode description of the far field waveform arises when the contour for the k integration is deformed (permissible because of 10 Cauchy's theorem and of Jordan's lemma ) to one such as is sketched in Fig. lb. The poles above the initial contour are encircled in the counterclockwise manner. There are also contour segments which encircle each branch cut lying above the real axis in the counterclockwise sense. The integrals around each pole are evaluated by Cauchy's residue theorem and one is left with a sum of residue terms plus branch line integrals. Each residue term may be considered as corresponding to a particular guided mode of propagation. The branch line contributions in some contexts are considered as corresponding to what may be termed lateral waves. 11 (The term may be unappropriate unless there is a kR

(b) k1

1. Contours in the complex k (wavenumber) plane for evaluation of individual frequency contributionS to waveform synthesis, (a) Original contour. (b) Deformed contour. sharply defined interface separating two types of media, such as a water- muddy bottom interface in shallow water propagation.)

In regards to the guided mode description, one type of approximation

(w) of D(wA)), which are frequently made is to neglect all poles (i.e. roots k n ikr above the real axis, the argument being that the corresponding e factors in the residues will die out rapidly with increasing r, the bulk of the contribution at large r expected to come from the poles which lie on the real axis. In a similar manner, it is argued that the branch line contour con- 3/2 tribution also dies out relatively rapidly (a factor of 1/r in addition to the geometrical spreading) so it too may be neglected at large r compared to the terms coming from the real roots. The net result for Eq. (1) would then be tin

p = S (r) AA (w) cos[wt-k (0r44 (01 dw 1 n (2) Y'L 6.3 where An (w) and ¢ n (w) are defined in terms of the magnitude and phase of the residues of the integrand in Eq.(1); the kn (w) being the real roots of

D(w,k)=0, numbered in some order with the index n=1, 2, 3, etc., and it being understood that, for fixed n, kn (w) should be a continuous function of w over some range of w from a lower limit w im up to an upper limit w un. The remaining integral over w can then be approximately evaluated by the method of stationary phase or integrated by suitable numerical methods.

In the present paper, a somewhat subtle set of circumstances intrinsic

to low frequency infrasound propagation in the atmosphere is discussed for which the arguments leading to the approximation of Eq.(1) by (2) are not wholly valid, even at distances of the order of more than a quarter of the

earth's circonference. We suspect that comparable circumstances may arise in

other contexts, but the present discussion is, for simplicity, illustrated only by examples from atmospheric infrasound propagation.

I. INFRASOUND MODES

An atmosphere model frequently adopted for infrasound studies is one in which the sound speed c varies continuously with height z in a more or less

realistic manner (Fig.2a) but is constant (=c T ) for all heights above some

specified height z T . [If winds are included in the formulation, their velocities

are also assumed constant in the upper half space, z>z T .] Conceivably, one has

some latitude in the choice of z and of the upper halfspace sound speed cT, T although computations of factors such as Q(w,k,z,z 0 ) and D(co,k) in Eq. (1)

become more lengthy with increasing z T. Also, it would seem that the most logical

choice of c would be that which would realistically correspond to height z T T' as in Fig. 2a. so the profile c(z) would be continuous with height across z T' Another conceivable choice would be one (Fig. 2b) in which c 03, such that T the surface of air nominally at z would be a free surface or pressure release T surface (corresponding to the model generally adopted for the water-air interface

in underwater sound studies). A somewhat intuitive premise which may be adopted

is that, if the source and receiver are both near the ground and if the energy

actually reaching the receiver travels via propagation modes channeled primarily

in the lower atmosphere, then the actual value of the integral in Eq. (1) would

be somewhat insensitive to the choices of z and c This, however, remains to T T. be justified in any rigorous sense, so we would be somewhat hesitant to take

cT = co at the outset. In typical calculations performed in the past, zT is taken

as 225 km, c T is taken as the sound speed (1", 800 m/sec) at that altitude.

Since one is often interested in frequencies (typically corresponding to

periods greater than, say, 1 to 5 minutes) at which gravitational effects are

important, the formulation leading to the infrasound version of Eq. (1) is based

on the fluid dynamic equations with gravitational body forces and the associated

nearly exponential decrease of ambient density and pressure with height included. 150-

100 E a

'zt 50

260 340 420 500 260 340 420 500 SOUND SPEED (m/sec) SOUND SPEED (m/sec) (a) (b)

2. Idealizations of model atmospheres (altitude profiles of sound, speed) used in acoustic-gravity wave studies. (a) Atmosphere terminated by an upper half space with constant sound speed. (b) Atmosphere temperature formally :going to infinity at some finite altitude corresponding to a free surface (pz0) at that altitude. 5

The incorporation of gravity leads, among other effects, to a somewhat complicated dispersion relation for plane type waves in the upper half space when c is finite, i.e. one can have solutions of the linearized fluid dynamics T equations for z > z T of the form8'9

e ikx e ikzz P/470 = (Constant) e -iwt (3)

where the vertical wave number k (alternately written as iG for inhomogeneous plane waves) and the horizontal wave number k are related by the dispersion relation (neglecting winds)

2 2 2 2 2 2 2 2 k = - = [03 - 0.1j5i] / c - - w ] k / z B (4) where w (y/2)g/c, w = (y-1) , A B arere two characteristic frequencied [w > w ] for wave propagation in an isothermal atmosphere (g = 9.8 m/s 2 A B is acceleration due to gravity, yz1.4 is specific heat ratio). Here, for brevity, the subscript T on cT has been omitted. For given real positive w, 2 2 real k, one can have k positive or negative (G negative or positive). The 2 2 values of k at which k or G go to zero turn out, as might well be expected, z to be the branchpoints in the k integration in Eq. (1), i.e., synonymous with the branch points of G. Along the real axis, G is either real and positive ik z -Gz (e z or e dying out with increasing z) or else G is a positive or negative imaginary quantity. In the latter case, the phase of G may be either 7/2 or -7/2, in accordance with the well known fact that, for acoustic-gravity waves, wavefronts may be moving obliquely downwards (negative k z) when energy is flowing obliquely upwards. In particular, for 0 < w < , one has G real B and positive for k in between the two branch points on the real axis, the phase of G is 7/2 (k < 0) on the remainder of the real axis; the two branch z

points are, from Eq. (4),

2 2 xi w[w A - w ] BR 2 2 (5) c[w - w ]

The branch lines extend upwards and downwards from the positive and negative

branch points, respectively. [See Fig. 1.]

The dispersion function D(w,k) in the atmospheric infrasound case can

be written in the general form

"j' k) = Al2R11 A11R12 - R12 (6)

where R and R are elements of a transmission matrix [R], these depend on 11 12 the atmosphere's properties only in the altitude range 0 to 4r , they are

independent of what is assumed for the upper half space. In general, their

determination requires numerical integration over height of two simultaneous

ordinary differential equations (termed the residual equations8 ' 9 '12 in previous

literature). They do depend on w and k (or, alternately, on w and phase

velocity v) but are free from branch cuts, they are real when w and k are real

and are finite for all finite values of w and k. The other parameters A l2

and A depend only on the properties of the upper half space (in addition to 11 w and k). Specifically, these are given (for the no wind case and with the

subscript T omitted on cr)

2 2 2 = gk /w - yg/[2c ] (7a)

2 k2 /w2 A = 1 - c 12 (7b) 7

One may note that, since every quantity in Eq. (6) is necessarily real

when w and k are real (with the possible exception of G), the poles lying

on the real k axis (real roots of D) must be in the regions of the (w,k) plane 2 [or (w,v) plane] where G >0. Since the integrand of Eq. (1) divided by r e-GzT should vary with z above z as one may call the corresponding modes T fully ducted modes. There is no net leakage of energy for such natural mcdes

into the upper halfspace. If one considers D as a function of w and phase

velocity v (or simply v), where v = w/k, the locus of real roots v versus w

(dispersion curves) has (as has been found by numerical calculation) the general

form sketched in Fig. 3. The nomenclature for labeling the modes (GR for

gravity, S for sound) is due to Press and Harkrider. One may note from Eq. (4)

that there are two "forbidden regions" in the v vs. w plane, i.e.

2 2 2 2 1/2 v < c[w - w ] / [o - w ] (8a) A

for w < w and B

2 2 2 2 ] 1/2 v > c[w - w ] / [w - co (8b) A for w > w Within either of these regions G would have to be imaginary and A' there would accordingly be no real roots for v of D(w,v) = 0. In the high frequency limit, this simply implies that the phase velocities of propagating modes are always less than the sound speed of the upper halfspace, the branch points in the k plane are simply at ± w/cT . The low frequency lower phase velocity "forbidden region" appears to be due to the incorporation of gravity effects into the formulation. However, if c is allowed to approach co, T this lower left hand corner region disappears. We have done numerical studies on the effects of varying c T on the dispersion curves. Briefly, the result is that the form of the predicted curves for GR0 and GR1 change very little 0.01 0.03 0.05

ANGULAR FREQUENCY, (rad/soc)

3. Numerically derived plots of phase velocity v versus angular frequency w for infrasonic modes in a model atmosphere corresponding to Fig. 2. The labeling of modes is with the convention introduced by Press and Harkrider 2 (J. Geophy. Res. 67, 3889-3908 (1962). The lines G =0 delimit regions of the v versus w plane where a real root of the eigenmode dispersion function cannot be found. 8

with increasing cT ; the lower forbidden regions shrink insofar as frequency range is concerned and the curves extend to successively lower frequencies.

Thus we see that the fully ducted modes GR and GR both have a lower frequency o 1 cutoff [wl, in Eq. (2)] which depends on CT . The larger one makes cT , the smaller is this cutoff frequency.

We thus have the following apparent paradoxes. Given that frequencies below w may be important for the synthesis of the total waveform, an apparently B plausible computation scheme based on the reasoning leading to our Eq. (2) will omit much of the information conveyed by such frequencies. Also, in spite of the plausible premise that energy ducted primarily in the lower atmosphere should be insensitive to the choice for c one sees that this choice governs the T' cutoff frequencies for certain modes and that certain important frequency ranges could conceivably be omitted entirely by a seemingly logical and proper choice for cT . The resolution of these paradoxes would seem to lie in the nature of the approximations made in going from Eq. (1) to Eq. (2). The latter may not be as nearly correct as earlier presumed and it may be necessary to include contributions from poles off the real axis and from the branch line integrals. Even if r is undisputably large, it may be that the imaginary leikri parts of the complex wavenumbers are sufficiently small that is still

integral may be not small compared to unity. Also, a branch line appreciable in magnitude at large r if there should be a pole relatively close to the branch cut. II. ROOTS OF DISPERSION FUNCTION

In order to understand the manner in which the solution represented by

Eq. (2) should be modified in order to remove the apparent artificial low

frequency cutoffs of the CR and GR modes, we first exagrine the nature of the 1 dispersion function D at points in the vicinity of a particular mode's dispersion

curve. The curve vn (w) of phase velocity v versus u for a given (n-th) mode is known at points to the right of the lower cutoff frequency wL . Given this,

one can find analogous curves va (w) and vb (w) for values of the phase velocity w/k at which the functions Ril (w,v) and 1112 (w,v) in Eq. (6), respectively, vanish. Since there may be more than one such curve in each case, we pick v a (w)

and vb (w), such that these curves are the closest of all such curves to the curve n(w) for w > wL . One may note, however, that one may apparently define and

identify va (w) and vb (w) for frequencies much less than wL , simply from analytical

continuation.

A premise which we have checked numerically (see Fig. 4) for a specific

case is that the curves v (u) va (w), vb (w) defined above with reference to n ' a particular given mode all lie substantially closer to each other than to

the corresponding curves for a different mode. In retrospect, this is obvious,

although it took some time for us to realize that it was so. Briefly, the argument

goes that, if the mode is predominantly guided in the lower atmosphere, then

there should be a decay of modal height profiles beyond some point substantially

lower than zr . Thus, both the p/ 45: and o ov z profiles for a guided mode would have values at zT substantially less than their peak values at lower

altitudes. The same would be true for the profiles of the auxiliary functions

0 and 0 1 2 which satisfy the residual equations. Consequently, if guided waves are excited, the inverse transmission matrix connecting 0 and 0 at the ground 1 2 to those at height zT would have to have very small [1,2] and [2,2] components. GR o

0.002 0.006 0.010 0.014 ANGULAR FREQUENCY (radial/sec)

0.23 GR,

0.21 0.002 0.003 0.010 0.0 I4 ANGULAR FREQUENCY ircd;cniscc.) 4. Curves in phase velocity (v ,v ,v ) versus angular frequency (w) plane n a b along which R11=0 (giving v (4, R12=0 (giving v (4, and D(w,k)=0 a b (giving vn (wl. Curves are shown for (a) the GR0 mode and (b) the CR 1 mode. N .te the changes in scale and the relatively close spacing of curves corresponding to the same mode. The lines along which G 2=0 are also indicated;

vn (0 is not a real quantity for w values below the indicated lower cutoff frequency. 10.

(Recall that 01 = 0 at the ground.) Since the transmission matrix has unit determinant, it follows that elements R12 and Ril of the transmission matrix proper [from height Z T down to the ground and whose elements appear in Eq. (6)] have to be small.

(w) and v (w) the dispersion relation D=0 for Given the definitions va b ' a single mode may be written

2)(a)(v-va) - [A11 0 ](43)(v-vb) = 0 (9) where a = dR /dv, S = dR12 /dv evaluated at v = v and v respectively. (For 11 , a b' simplicity, we here consider D as a function of w and v = w/k rather than of w and k.) The above equation may also equivalently be written in the form

v = va + (vaTvb )X/[1-X]

= ("a) X (All G)/Al2 which may be considered as a starting point for an iterative solution which in essence develops v in a power series in v a-vb ; G may be considered as a defined

as the zeroth iteration, evaluates the function of w,v. One starts with v = v a right:hand side for the value of v to find the starting point for the next iteration, etc. The considered procedure should converge provided v a or vb is not near a point at which G vanishes and providing G in the vicinity of va or v is not such that the variable X is close to unity. Among other limitations, b the iteration scheme would be inappropriate for values of w in the immediate vicinity of coL .

In regards to establishing the general trends represented by the iterative type solutions, two relatively general theorems may be of use. These (whose 11

13 proof follows along lines previously used by one of the authors in deriving an integral expression for group velocity) are that for real positive w and v,

R DR /Dv - R 3R /3v > 0 12 11 11 12

R aR /aw — R DR /aw > 0 12 11 11 12 or, alternately, if one inserts Ril = (a)(v-va ), R12 = (a)(v-vb ), he finds

aa(va - vb ) > 0 (12a)

(v - v )(v - v ) (aa'-a'a) + aa[v (v - v ) - v ' (v - v )] > 0 (12b) b a b a a b where the primes represent derivatives with respect to w. The second of these should hold for arbitary v in the vicinity of v a and vb and lead, upon setting use of Eq. (12a), v = va , v = vb , or v = (vavb' -va'v.1))(vb -va '), along with the to

< 0 (13a) vb

va' < 0 (13b) (a/a)" > 0 (13c)

Equation (12a) implies that as long as a or a do not vanish (which would seem unlikely) the two curves va (w) and vb (w) do not intersect. If a and a have the same sign the v a curve lies above the vb curve; the converse is true if a and a increases with w.

To demonstrate the general utility of the perturbation approach, a brief 12

(1) table of values w, va, v a, 8, v , and v are given in Table I for the GR b' n and GR modes for the case of a U.S. Standard Atmosphere without winds terminated 1 at a height of 125 km by a halfspace with a sound speed of 478 m/sec. Here v(1) is the result of the first iteration for the phase velocity and v n is the actual numerical result obtained (only if the phase velocity is real) by explicit numerical search for roots of the eigenmode dispersion function. One may note that, for those frequencies where v is computed, the agreement between v (1) n and v is excellent. A more detailed listing of the perturbation calculation n results is given in Figs. 5a and b. The plots there give w/k R or the reciprocal (1) of the real part of 1/v (i.e., w divided by the real part of the horizontal wave number k) and the imaginary part k I of k = w/v versus angular frequency.

Note that k is zero above the corresponding cutoff frequencies. The relatively I small values of the k are cc=nented upon in Sec. IV. I

III. TRANSITION FROM NONLEAKING TO LEAKING

1/2 = [(p ) (w-wd + (q)(v_vdi (14) where p and q are readily identifiable [from Eq. (4)] positive numbers taken

Independent of w and v; v is the phase velocity on the dispersion curve in L the limit as from above. The bracketed quantity in Eq. (14) may be re- L 2 garded as a double Taylor series expansion (truncated at first order) of G about 2 the point w v at which G vanishes (hence no zeroth order term). The fact that L' L 2 both p and q are positive follows since G is positive to the upper right of the D.32

GRo CUTOFF D.26

).24 GR 11 1 0.002 0.004 0.006 0.003 0.010 0.012 ANGULAR FREQUENCY (radian/1s oc)

10 4

6 10

- 10

-o 10

0 0.002 0.004 0.006 0.003 0.010 0.012 .ANGULAR FREQUENCY (radian/sec)

5. Numerically derived plots of phase velocity w/k R and of the imaginary part k of the complex wavenumber k versus angular frequency for the GR I 0 and GR modes. Previous theoretical lower frequency cutoffs for these 1 modes are as indicated. Note that k is identically zero above the I cutoff frequency. V n

0.31202121 + 0.0052 0.31203 0.31207 917.4 -2783.7 -3.184 x 10-6 i 0.0113 0.31190 0.31194 1 767.9 -3254.2 0.31189059 + -1.721 x 10-6i 0.0155 0.31176 0.31181 1 621.9 -3644.3 0.31173763 0.31172882 GR 0 0.0165 0.31172 0.31177 581.5 -3738.2 0.31167504 0.31167509

0.0186 0.31162 0.31168 497.5 -3910.1 0.31153369 0.31153394

0.25267 + 0.0052 0.24229 0.24816 87.8-1 -3633.0 -2.715 x 10 -3 i 0.24218 + 0.0103 0.23433 0.23844 1 94.7 -3990.0 -1.337 x 10 -3 i 0.0144 0.21842 0.22037 150.7 -5307.0 0.21431 0.22178 GR 1 0.0165 0.20252 0.20345 265.0 -7767.3 j 0.20016 0.20463

0.0175 J 0.19058 0.19111 418.9 -10,858.0 I 0.19226

Frequency dependent parameters corresponding to GR and GR modes; to is 1 angular frequency in rad/sec, v a is phase velocity root of R11=0, in km/sec, v.. is analogous root of R a is dR /dv at v=v in sec/km 11 a 8 is dR /dv at v=v in sec, v (1) is first order perturbation solution for 12 b phase velocity from equations given in the text (units are km/sec), vn is the real root determined by direct numerical solution for zeros of eigenmode dispersion function. Note that v (defined only when phase n velocity is real) agrees exceptionally well with v (1) 14

2 line in the w,v plane where G 2 = 0 and also since the G = 0 line slopes obliquely downwards. (See Fig. 3). 12 v , the denominator Let us next note that, in the vicinity of the point 0 L D given by Eq. (9) may be further approximated as

7- (A a-A 6) + pAw) + e(Av + vAw ) 1/4 (15) 12 11 where we have abbreviated Av = v-vL' Aw = w -w , v = pig; the quantity p is L /dw or -dv /dL., the two being assumed to be approximately equal. either -dva b (The use of the minus sign here assumes that p be positive.) The remaining quantity c is

(q 2) (6) (v-vb) (16) 6A - aA12 11

One should note that e depends on v, although, for purposes of initial analytical

v = v here. All of the above quantities may be investigation, one may set L and v = v Note that p and v are both considered to be evaluated at w = w L 1,* positive quantities. Furthermore, it should also be noted that v > p since 2 the G = 0 curve slopes downwards more rapidly than the lines along which

R or R = 0 in the v vs Li plane. (See Fig. 4.) 11 12 The roots of Eq. (15) without regard to the sign of the radical are readily

found to be

2 - (17) = + ()c + e(v-p) 2 [Aw +a] where 2 a = c /G(v-P7 (18)

Alternately, if << c, the above may be approximated by the binomial

theorem to give 15

2 2 2 Av = -vAw + [(v-p) /6 ](Aw) (19a) or

2 2 2 2 Av = +£ - (2p -v) Aw - [(v-p) /6 ](Aw) (19b)

for the upper and lower signs, respectively. The first of these (since Av = 0 when Aw = 0) is clearly the description of the disperson curve in the vicinity of w = wL , v = vL .

Equation (19a) shows that, as Aw 0 from above, the dispersion curve 2 becomes tangential to the line G = O. The two curves do not intersect. The general trend is as indicated in Fig. 6. The solution represented by Eq. (19b) is not a proper root of Eq. (15); it corresponds to the wrong sign of the radical and accordingly lies on the second branch. Furthermore, one can readily show that, for values of Aw slightly less than zero, both roots lie on the second branch. Hence, there must be a gap of finite frequency range in which, for the choice of branch cuts represented by Fig. 1, there are no poles in the k

(or v) plane corresponding to the n-th mode.

To determine the order of magnitude of this frequency gap, it is appropriate to consider the trajectory of the second branch roots in some detail and to determine just where one of them should cross the branch cut, reappearing on the first branch. As long as Av is real and Av + vAw >0 the criterion for a root to be identified with the first branch is Av + pAw > O. According to

Eq. (17), this would automatically place the second root on the second branch for all Aw > -a and would place the first root on the second branch for

-a < Iw < O. Consequently, if either root is to reappear on the first branch, it must be at a value of Lw < -a.

One should note from Eq. (17) that at Aw = -a the two real roots on the second branch coalesce. For values of Aw < -a the two roots separate again„but ImV <0

B G < 0

G>0

DISPERSION CURVE

W L

6. Sketch illustrating nature of a single mode's dispersion curve in the vicinity 2 of the G =0 line. At point A (angular velocity to phase velocity v ) L' L 2 the dispersion curve is tangent to the G =0 line; for frequencies below co L down to that corresponding to point B in the sketch there are two real roots for v of the eigenmode dispersion function on the second branch. For frequencies lower than that corresponding to point. B, there is a complex root for v on the first branch (which is the complex conjugate of a second root on the second branch). 16

are now complex conjugates. The root in the upper half of the v plane (lower half of k plane) can never cross the branch cut so it remains on the second branch indefinitely. The one in the lower half of the v plane will cross the branch cut at a point which may be approximately estimated as that where

Re(4v) = -viw or where

2 Aw = (1/2.) c = (v - p) with a corresponding value of Av of

2 Av = (e /2) li[v/(v-p)] -

For subsequent frequencies successively lower than w -2a there is a complex L root on the first branch with a negative imaginary part which increases with decreasing frequency.

The discussion up to now has assumed that lAvi << Iv L-vb i and hence that e may be taken as constant. This would seem appropriate for describing the transition region since all values of Av of interest in this region are of

2 second order of E However, if an improved numerical estimate is required, we recommend that one regard Eqs. (16) and (17) as a iterative pair. Success- fully computed values of Av may be used to recalculate e and the new value of e may then be used in obtaining the next higher estimate for Av. In Table II the values of w L' vL , p, q, p, v, e, and a are given for the GR and GR modes for the model atmosphere corresponding to Fig. 2a. 0 1 The extremely small values of a should be noted. The corresponding plot of Av versus 4w (i.e., both branches of Eq.(17)) corresponding to their values for the GR mode is given in Fig. 7. For simplicity, this is plotted 0 in a nondimensional form, i.e. 1 N 1/2 V = -{11/[2(v_0]) ; [ (20) 17

TABLETS

GR GR o 1

wL (rad/s) 0.0118 0.0125

v L(km/s) 0.31188 0.2323

p(s/kn2) 0.14 0.35 q(s/km3) 1.84 x 10-3 1.86 x 10-3 -2 1.1 ((m) 2.94 x 10 4.15

v(km) 76 190 c(1=21/2 1/2 -6 -3 ) 9.6 x 10 1.02 x 10 -13 -9 a(rads/s) 3.04 x 10 1.41 x 10

Parameters characterizing the eigenmode dispersion function near points in the phase velocity versus angular frequency plane at which the GR o and GR 1 modes undergo transition from leaking to non-leaking.

••• 2(v-ti)o- *O. .00

.00 Qv 2(v-vt)o-

F -4 -3

7. Graph of normalized phase velocity versus normalized frequency in the vicinity of the point (v L , w ) for the GR mode. The imaginary and L 0 real parts are both plotted. The dashed line corresponds to real roots on the second Riemann sheet. 18

where v = Av/[2(v-p)a] and 2=Aw/a. Both real and imaginary parts are shown on the same graph. The corresponding plots for the GR 1 mode differ only slightly from those in the Fig. 7 because of a different value of the parameter p/[2(v-p)] in Eq. (20); in. both cases this parameter is small compared to unity, i.e. p<

IV. THE BRANCH LINE INTEGRAL

Since there is a gap in the range - of frequencies for which a pole corresponding to a mode may exist, it is evident that evaluation of the k integration in Eq. (1) by merely including residues may be insufficient for certain frequencies. Thus it would seem appropriate in such cases to include a contribution from the branch line integral. It may be anticipated that such branch line integrals are significant at larger values of r only when w. is close to some mode's wL (say the n-th mode), in which case the branch point of greatest interest (i.e., that which may have a pole in its immediate vicinity) is at k=w/v L . Consequently, it would appear that an adequate approximation to the branch line integral would be

03 Branch line [Q/D(w,k)]eikrdk contribution of a° Q eikrdk (21) Alta Allf3 f x+-v) A w+Ex1/2 C i where the denominator D(w,k) has been approximated by Eq. (15) with the abbreviation x for Av+vAw. The quantity outside the integral is assumed to be evaluated at w=w and k=w/v . The contour C Bruns down the left side of the branch cut, L L around the branch point (where x=0), and then up the right side. If one next changes the variable of integration from k to x, nothing that for small x/v , noting

2 k=kB-(w/v )x (22)

he finds approximately that 2 -1(w /vL)x Branch line e L contribution = Resid(ue dx (23) x+(p-v)Aw+cx Ca where (Residue) ikr o is that residue which the integrand (Q/D)e would be expected to have at the n-th mode's pole in the k plane were the parameter c identically

equal to zero. The mapped contour cg in the x plane may be considered to go

up on the right and then down on the left of a branch cut extending vertically

downwards from the origin in the x plane. If we set then, on the right 12 -ir/4 1/2 side of the cut, x should be e C while, on the left side, it is -e-in/4 1/2 C . Consequently, the total integral combines to as 2 2ce+iw/4 -(w /v )r B ranch lines e L L d = -(Besidue) (24) contribution 0 2 2 [-iC+(p-v)Aw] +le C a This in turn, with an obvious change of integration variable, may be expressed

as 0610 1.7/4 -n 1/2 Branch line e e n dn = (Residue) 2K (25) {contribution) o (n-n 1)(n-n2 )

where

1/2 K=cv /(w r) (26a) L L 2 n = i(K /2)(1+[Aw/2a]) 2 2 1/2 ± i(K /2)(1+[Aw/c]) (26b)

with a as defined by Eq. (18).

In regards to the n integration, the integral can be expressed in general

in terms of Fresnel integrals of complex argument after some considerable

mathematical manipulation. One may note, moreover, that ki l l and 111 2 1 are, for

most cases of interest, considerably less than unity. In this case, the

appropriate approximate result (derivation omitted for brevity) is 20

...' e-I V-ri- in do (27) (n-n )(n-n 1/2 1/2 1 2 ni -"2 where the choice of square root should be such that the imaginary part is positive. The net result in this limit then is that the branch line contribution is independent of the parameter K. (The dependence on range r comes only in the residue.) Thus one may write

{ Branch line = 2ni(Residue) B (Lao) (28) contribution o rh where the function B (Aw/a) is given by rh

B rh = 1/2 1/2 (29) 1/2 1/2 [1,÷(1/2)-S-2÷-(l+2) +[1+(1/2)n-u+o) ]

Here any consistent choice may be made for the sign of the inner square roots but the outer square roots should be taken such that the resulting phases are between -n/4 and 3n/4. The quantities in square brackets turn out to be the 1/2 squares of (1/1i)[(1+0) ±1], respectively. The phase restriction then gives

1/2 h (Q) = (I-1-52) if Q>0 (30a) = 1 if 0>Q>-2 (30b) -1/2 = -i(-S2-1) if 52<-2 (30c) where here all square roots are understood to be positive/

To completely describe the transition it is appropriate to add to Eq. (28) that contribution (which is zero for 0>Aw>-2a) from the pole on the first branch in Eq. (21) which lies in the general vicinity of k=115 1, /vc .'If the pole is present, its contribution to the integration over k is 21i times the residue

(which is not what we have been referring to as (Residue) unless c is identically o zero). The evaluation of the residue is moderately straightforward and omitted here for brevity. The net result is that 21

Branch line Pole contribution contribution

27ri(Residue) O B rh (Aw,c)+p oz (Aw,G)} (31) where the "pole function" P oz (Aw/o) turns out to be given by

P ok (Aw/6)=1-B rh (Aw/a) (32)

We accordingly have the remarkable (although, in retrospect, not unexpected) result that

Branch line Pole + = 271.(Residue) contribution contribution o (33)

The above gives one a relatively simple prescription for evaluating a given mode's contribution to the k integration in Eq. (1). First, all branch line integrals are formally neglected. If a pole exists on the first branch, the residue which would normally be utilized is replaced by

LQe ikri Qe ikr Res - D d'D/dk (34) k=pole where

D dk = dk (Al2R11-A11R12 )

-G (R ) dk 12 (35) i.e. it differs from the actual derivative of D in that G is formally considered as constant. Doing this when w is somewhat removed from the transition region near w should make very little difference since R is small at values of k which L 12 are poles. Near the transition, this neglect should almost exactly compensate for the neglect of the branch line integral. 22

REFERENCES

1. J. E. Thomas, A. D. Pierce, E. A. Flinn, and L. B. Craine, "Bibliography on Infrasonic Waves", Geophys. J. R. astr. Soc. 26, 399-426 (1971).

2. C. B. Officer, Introduction to the Theory of Sound Transmission with Application to the Ocean (McGraw-Hill, New York, 1958).

3. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon Press, Inc., New York, 1962).

4. L. M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1960).

5. K. G. Budden, The Wave- Guide Mode Theory of Wave Propagation (Prentice Hall, Inc., Englewood Cliffs, N.J., 1961).

6. I. Tolstoy and C. S. Clay, Ocean Acoustics (McGraw-Hill, New Yprk, 1966).

7. M. Ewing, W. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957).

9. A. D. Pierce, J. W. Posey, and E. F. Iliff, "Variation of Nuclear Explosion generated Acoustic-Gravity Waveforms with Burst Height and with Energy Yield" J. Geophys. Res. 76, 5025-5042 (1971).

10. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Clarendon Press, Oxford, 1935) p. 137.

11. L. M. Brekhovskikh, loc. cit., pp. 270-280.

12. A. D. Pierce, "The Multilayer Approximation for Infrasonic Wave Propagation in a Temperature and Wind-Stratified Atmosphere", J. Comp. Phys. 1, 343-366 (1967).

13. A. D. Pierce, "Propagation of Acoustic-Gravity Waves in a Temperature and Wind-Stratified Atmosphere", J. Acoust. Soc. Amer. 37, 218-227 (1965), Appendix B The 88th Meeting of the Acoustical Society of America

Chase-Park Plaza Hotel o St. Louis, Missouri • 4-8 November 1974

TUESDAY, 5 NOVEMBER 1974 CHASE CLUB, 9:30 A.M.

Session A. Physical Acoustics I: Atmospheric Acoustics

10:45 A5. Asymptotic high-freeuency behavior of Taided infrasonic mociLls in the aLnoz,chere. Wayne A. Kinney - (School of .Nicchanical Engineering, Georgia Institute of 'Technology, Atlanta, Georgia 30332)

Refinement of previous theoretical formulations and numerical computations of pressurc waveforms as applied to atmospheric traveling infrasonic wat -es could include a description of their asymptotic behavior at high frequencies. In the present paper, calculations based on the W. Ic13. J. approximation and similar to those introduced by Haskell (J. Appl. Phys . 22, 157-167 (1951)] are performed to describe the asymptotic behavior of infrasonic guided modes as generated by a nuclear explosion in the atmosphere. The results of these calculations are then matched onto numerical solutions which have teen given by ilarkrider, Pierce and Posey, and others. It is demonstrated that the use of these asymptotic formulas in conjunction with a computer pro r;Tani which synthesizes infrasonic pressure waveforms has enabled the elimination of problems associated with high-frequency truncation of numerical integration over frequency. In this way, small spurious high-frequency oscillations in the computer solutions have iNeeti avoided. [Work sponsored by Air Force Cambridge Research IAboratory.] Recently, Allan D. Pierce, Christopher Y. Kapper and Wayne A. Kinney

at the Georgia Institute of Technology have been working to refine a computer

program which synthesizes infrasonic pressure waveforms at the ground as

generated by large explosions in a wind- and temperature- stratified atmos- 1 phere. Shown in Fig. 1 are three such pressure waveforms along with the

modal waveforms from which each of the three individual total waveforms has

been superposed. Corresponding to each modal waveform is a particular disper-

sion curve (i.e., a plot of phase velocity versus angular frequency). Any

given dispersion curve defines what is referred to as a mode. Fig. 2 shows

dispersion curves as they are generated by a portion of the computer program.

The labels given to these correspond to the labels given to the modal wave-

forms in Fig. 1.

Due to temperature stratification, the earth's atmosphere possesses sound speed channels with associated relative sound speed minima. Fig. 3 shows a standard reference atmosphere wherein two such sound speed channels are indicated; one with a minimum occurring at approximately 16 km altitude and the second with a minimum occurring at approximated 86 km altitude.

Given the presence of such a channel, an acoustic ducting phenomenon can occur, as is demonstrated in, Fig. 4, wherein the energy associated with an acoustic disturbance can become trapped in the region of a relative sound 1 speed minimum. It is this mechanism of propagation only that is of interest here.

In the computer program, the computation of modal waveforms involves the numerical integration over angular frequency of a Fourier transform of acoustic pressure where this integration is truncated at the high-frequency 1 end. It has been speculated that this abrupt truncation leads to the GR o

ANWWV 4 A S A 111 10q 1IF %MANNA V

S ---.40NOMMA ■ 1,PAINArwr omivw 'Zi": 41 S4

(r) 75 75 0 TOTAL AA1 0;i1j),0 OA A

-75 -75 CROSSWIND DOWNWIND

510 540 570 510 540 570 510 540 570

TIME AFTER BLAST (min.)

Fig. 1 Superposed infrasonic pressure waveforms (with contributing modal waveforms shown) as generated by the computer model for ground locations 10,000 km upwind, crosswind and downwind from a nuclear explosion. •• • - ,

0.45

Q1 02 0.40

c" 0.30

0.25 I 1 1 1 I I i 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 Frequency,w (sec 1)

Fig. 2. Dispersion curves generated by the computer model. The labels given these curves correspond to the labels given the modal waveforms of Fig. 1. 4 ALTITUDE 150 100 50 yr Fig. 3.Standardreferencemodelatmosphere showingtwosoundspeedchannels. 200 I 1 1 ( ATMOSPHERE ,1962 U S.STANDARD l --1 I 1 I I

TEMPERATURE ( 400 20 30 10 600 200 ° K) „ 240 \ N 800 ■ '1 , N. 1000 ------ARCTICSUMMER — SUBTROPICALSUMMER — ARCTICWINTER WEST-TO-EAST WIND(m/sec.) -50 4- 0 50 Om. SOUND CHANNEL DUCTING

TURNING POINTS

williall•••••■••••■•■■•••••••

Vp ACOUSTIC PRESSURE SOUND SPEED PROFILE

Fig. 4. Graphic illustration of acoustic ducting in a sound speed channel. The energy of an acoustic disturbance can concentrate in the region of a relative sound speed minimum. 1 generation of what might be called "numerical noise" in the computer output.

It was felt useful, therefore, to extend this integration beyond the heretofore upper angular frequency limit by means of some appropriate high-frequency approximation. 2 The approximations associated with the W.K.B.J. method of solution apply to the analytical model on which the computer program is based at frequencies above approximately 0.1 radian/sec. Below that limit, effects due to density stratification in the atmosphere and gravitational forces cannot be neglected. Such effects therefore are not germaine to the discussion here.

To the best of the authors' present knowledge, the application of the

W.K.B.J. method of solution to the problem of describing propagation of acoustic disturbances in an atmosphere that contains two adjacent sound speed channels has not been approached in the literature to date in the manner to be presented. To he specific, the approach taken here is to seek a W.K.B.J. model for each of the sound speed channels separately, then to combine the results rather than to treat the problem with a single model.

The W.K.B.J. model for propagation of acoustic disturbances in a single sound speed channel consists of solving for the acoustic pressure divided by the square root of the ambient density expressed as

P -iwt ikx - ip(z)e e 2 P o where w is angular frequency, k is the wave number associated with the horizontal dimension x, z is altitude, and where ip(z) satisfies the reduced wave equation,

2 [ 2 d w 2 k IP = 0 2 dz c2 (z)

where c(z) is sound speed as a function of altitude. The W.K.B.J. approxima-

tion as applied to this model would appear to be valid provided

where A is some representative wavelength of interest. This approximation

states that substantial changes in sound speed should not occur within distan-

ces corresponding to a typical wavelength of interest if the model is to apply.

Particular insight into the high-frequency behavior of guided infra-

sonic modes was gained when the following integral was solved numerically by

computer

z top 1 1 1 2 (n+1/2)7r [777 7-2 dz -

z bottom

where v is phase velocity, n = 0, 1, 2, 3, ..., and where and z p zbottom top identify the lower and upper bounds of the sound speed channel, respectively. 2 This integral is a direct result of the W.K.B.J. method of solution , and its

numerical solution enabled the plotting of high-frequency dispersion curves.

In the lower portion of Fig. 5 are shown two sets of dispersion

curves generated by integrals of the above form; one set (the dashed curves)

is appropriate to the W.K.B.J. model for the lower channel and the other set

(the solid curves) is appropriate to the W.K.B.J. model for the upper channel.

0.41

0 COMPUTER MODEL N DISPERSION CURVES

>- F- 0.34 0 0 -J LIJ

CL 0.27 • 0 0.1 .0.2_ ANGULAR FREQUENCY (SEC -I ) 0 .41

0 W.K.B.J. MODEL

DISPERSION

CURVE >- 0.3 0 a _J

LIOWER Li CHANNEL UPPER a. CHANNEL 0.27- • 0 0. 1 0.2 ANGULAR FREQUENCY (SEC )

Fig. 5. Comparative dispersion curves as generated by the computer piodel and the W.K.B.J. models. In the upper portion of the same figure are shown again dispersion curves as

generated by the computer model. It should be mentioned that the computer

model solves a more complex problem in the sense that the simplifications inher-

ent in the W.K.B.J. model are not present.

As is illustrated in the lower portion of Fig. 5, the two sets of

dispersion curves generated by the W.K.B.J. models intersect with one another

at various points. A comparison of the dispersion curves shown in both the

upper and lower portions of Fig. 5 reveals that these points of intersection

mark regions of resonant interaction in the phase velocity-angular frequency

plane between adjacent modes of the computer model. To better illustrate this

observation, in the right hand portion of Fig. 6 is shown one such region of

interaction with its corresponding point of intersection between two dispersion

curves of the W.K.B.J. models shown to the left. It should be mentioned that

the dispersion curves of the computer model never intersect with one another.

An analytical explanation of this fact is given in reference 1.

The above observation may be stated differently by saying that, for

relatively high angular frequencies, the dispersion curve corresponding to a

given mode of the computer model is comprised of portions of dispersion curves

from both sets of the curves generated by the W.K.B.J. models. Two important

inferences about the asymptotic high-frequency behavior of guided infrasonic modes can be drawn from this statement. First, for some frequency ranges, and

depending on how dispersion curve portions match between curves of the computer model and the W.K.B.J. models, it can be inferred that the acoustic energy

associated with a given mode is comprised of energy associated more with propa-

gation of acoustic disturbances in one sound speed channel than in the other.

Also, with increasing frequency, this association alternates back and forth DISPERSION CURVES DC NOT CROSS

W.K.B.J., AP P RO X1 MAT I ON COMPUTER MODEL 34— 0.34

32 UPPER CHANNEL 0.32

30 4' 0.30

RESONANT I NTFRACTION LOWER, CHANNEL BETWEEN ADJACENT MODES

28 0.28 0.06 0.08 0.10 0.12 0.06 0.08 0.10 ANGULAR FRE QU ENCY ANGULAR FREQUENCY (SEC - I) (s E c -1)

Fig. 6. Blow-up of a section of Fig. 5 showing a region of resonant interaction between two adjacent modes of the computer model. To the left are shown the corresponding intersecting curves of the W.K.B.J. models. between channels. To illustrate, if for a small range of frequencies a portion of a dispersion curve of the computer model matches (in the phase velocity- angular frequency plane) a portion of one of the W.K.B.J. model curves for the upper channel, then that implies that, for that mode and for that small frequency range, the acoustic energy density associated with that mode is greater in the upper channel than in the lower channel. Secondly, in standard reference atmospheres the sound speed minimum for the upper channel is shown to be less in magnitude than the sound speed minimum for the lower channel. It can be inferred therefore that those acoustic disturbances for which phase velocities are less in magnitude than the sound speed minimum for the lower channel are associated more with acoustic energy trapped in the upper channel than in the lower channel, and thus for this reason do not contribute significantly to the acoustic energy at the ground. This inference implies that care must be taken as to which modes are chosen to superpose in the attainment of the final pressure waveform at the ground, as some may not contribute.

In addition to providing a new analytical tool, the manner in which the W.K.B.J. method of solution has been applied to the two-channel problem has clarified the physical interepretation of a mode as defined in the computer model. It is hoped that the computer program can now be modified accordingly to gain better high-frequency resolution in the pressure waveform output. REFERENCES

2. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, McGraw- Hill, New York, 1953, see in particular pp. 1092-1094, 1098-1099.

3. Pierce, A. D., C. A. Moo and J. W. Posey, "Generation and Propagation of Infrasonic Waves", Report No. AFCRL-TR-73-0135 (1973).

4. Posey, J. W., "Application of Lamb Edge Mode Theory in the Analysis of Explosively Generated Infrasound", Ph.D. Thesis, Department of Mechani- cal Engineering, Mass. Inst. of Tech. (August, 1971). Appendix C PROGRAM MAIN (INPUT,OUTUTirAPE5=TNPUT,TAP6=0WPUT) 'DIMENSION ZTS(10) COMMON VP,I1,NCS,ZI(100),CI(100),ASOL(100),ZLOW,ZUP READ(5 ,* )NCS,(1I(I),I=1,NCS).(CT(I),I=1,NCS).VP,7OL,Z0U,NSCAN WPITF(6, * )NCS,(ZI(I),I=1,NCS),(CI(I),I=1,NCS),VP.Z9L 1 ZE3U.NSCAN P.EA0*,(7TS(I),I=1,10) dRITE*,(ZIS(I),I=1,10) CALL OASOL PRINT*,"ASOL=",ASOL 00 5 I=1,10 5 PRINT*,"CSP=",CSP(ZIS(I)) CALL INPNT(VPIZOL,ZqU,NSCAN,NRTS,ZLOW,ZUP) PRINT*9"NPTS=" 9 NRTS CALL SHIFT(ZLO4,ZUTI) PRINT*,"ZLOW=".7LCW,"ZUP=".Z1'P CALL RANG (TIME,RLNIHI7LOW $ ZUP) PRINT' ,"RTIME=",RTIME,"RLNTH=",RLNTH = 1 Z = 71(5) CALL DRVTNP(I,Z,VP,DXDVPUOTOVPU,7LON,ZUP) PRINT*,"OXOVPU=",CX0VPU,"CT7VPU=" ‘ OTOVPU I =-1 Z = ZI(3) CALL 0 -):VTNP(I,Z,VP.OXOVFLOTOVPL I ZLOW,ZUR) PRINT*,"0X0VPL=".CXDVPL,"OTOVPL=",OTOVPL Z1 = ZI(3) Z2 = ZI(5) CALL MOLINT(71,Z2.AMXIN,AMTIN) PRINT*,"AMXIN=",AMXIN,"AMTIN=",AmTIN CALL OSDV 1 (I,Z,VP,DXDVP,OTDVPIZLOW,ZUP,A'9XIN,AMTIN, 10X0VPT.OTCVPT) PRINT*,"0x0VPT="911X0VPT I "DTOVPT=" 9 0TDVPI CALL EXIT ENO SUBROUTINE SHIFT(ZLOW 1 ZUP) N = 0 5 CHKL = CMVP(ZLOW) IF(CHKL .LE. 0.0) GO TO 10 ZLOW = ZLOW + 1.E-8 N = N+1 IF(N .GE. 1000) RETURN GO TO 5 10 CHKU = CMVP(ZUP) IF(CHK!J .LE. 0.0) RETURN ZUP = ZUP - 1.E-8 N = N+1 IF(N .GE. 1000) RETURN GO TO. 10 END FUNCTION CMVP(Z) COMMON VP CMVP = CSP(Z) - VP RETURN END SUROUTINE TNPNT(VP,ZBL,Z9U,NSCAN,NRTS t ZA,Z9) EXTERNAL CMVP DIMENSION GUESS(3,1),ANS(1),FANS(1) COMMON VPC VPC = VP DELTA = (ZOU - 7BL)/(NSCAN + 1) F1 = CMVP(ZBL) Zi = ZRL NRTS = 0 10 Z2 = Zi + DELTA

F2 = CMVP(Z2) ,TEST = F1 *F2 IF(TEST .GT. 0.0) GO TO 15 GZ = Z1 - F1*DELTI/(F2 • F1) GUESS(191) = GZ SUE SS(2,1) = Z1 - 1.E-6 GUESS(3,1) = Z2 + 1.E-6 CALL ZAFUP(1,GUESS,10,1.E.•7,1.'T-7,CMVP,-1,ANS.FANS) NRTS = NRTS + 1 IF(NRTS .ED. 1) ZA = ANS(1) IF(NRTS .EQ. 2) = ANS(1) IF(NRTS .E0,. 2) GO TO 20 15 Z1 = Z2 Fl = F2 IF(ZAU .GE. 71) GO TO 10 20 RETURN END SUBROUTINE RANG (RTIME,RLNTH,ZLOW I ZUP) EXTERNAL ROTOZ,RDX07 RTINIE = RAINT(ROTC7,7LOW,7UP) RLNTH = RAINT(RDXCZ I ZLOW,ZUP) RETURN ENO SUBROUTINE DASOL COMMON VP9I1.NCS97I(100),CI(100).ASOL(100) N = 1 DELI = 1.0 DELC = 0.0 AKM2 = 0.0 ALM2 = 0.0 AKM1 = 0.0 ALMi = 1.0 NSTP = NCS - 1 10 DELZP = ZI(N+1) ZI(N) DELCP = CI(N+1) ••• CI(N) ALPHA = OELZ GAMMA = DELZP `SETA = 2.0*(ALPHA f GAMMA) DEE = (OELCP/OELZP) - (OELC/DELZ) IF(N .EQ. 1) GO TO 30 AK = (DEE ALPHA*AKM2 qETA*AKM1)/GAMMA AL = ( - ALPHA*ALM2 - BETA*ALM1)/GAMMA IF(N .EO. NSTP) GC TO 100. AKM2 = AKM1 ALM2 = ALMi AKM1 = AK ALM1 = AL 30N=N+1 OELZ = DELZP OELC = OELCP GO TO 10 .00 ASOL(1) = 0.0 ASOL(2) = -AK/AL OELZ = 1.0 DELC = 0.0 N = 1 10 OELZP = ZI(N+1) ZI(N) OELCP = CI(N+1) CI(N) ALPHA = OELZ GAMMA = DELZP BETA = 2.0*(ALPSA + GAMMA) DEE = (DELCP /DELZP) - (DELC/DELZ) IF(N 'E(). 1) GO TO 130 M = N + 1 ASOL(M) = (DEE - ALPHA*ASOL(N-11 BETA*ASOL(N))/GAMMA 1F(N NSTP) GC TO 200 130 N = N + 1 DELZ = OELZ 0 DELL = DELCP GO TO 110 200 RETURN END FUNCTION CSP(Z) COMMON vp,I1,Ncs,71(100),c1(100),AsoL(100) 7_ = Zr(1) ZP = ZI(NCS) IF (7_ .LT. ZL) GO TO 50 IF (7 .GT. ZP) GO TO 60 I = NCS 10 J = I-1 ZTEST = ZI(J) IF (7 .GT. ZTEST) GO TO 40 T = J GO TO 10 40 CONTINUE Z IS BETWEEN 7_I (I-1) AND 7_I (I) DELZ = 7_I (I) - ZI(J) = (7 - 7I(J))/DELZ WBAR = 1.0 - N TERM1 = wfw?*ci(,J) + W*CI(I) GUT1 = W3AR**3 - WBAR GUT2 = W 4 *3 - W TERM2 = (E?ELZ**2)*(ASOL(J)*GUT1 + ASOL(I)*GUT2) CSP = TERM1 + TERM2 RETURN 50 CSP = CI(1) RETURN 60 CSP = CI(NCS) RETURN END FUNCTION OCOZ(Z) COMMON VP,I1INCS 9 7I(100),OI(100),ASOL(100) = ZI(1) ZP = ZI(NCS) IF(Z .LT. ZL) GO TO 50 IF(Z .GT. ZB) GO TO 50 I = NCS 10 j = I-1 ZTEST = ZI(J) IF(7. .GT. ZTEST) GO TO 40 I = J GO TO 10 40 CONTINUE Z IS BETWEEN ZI(I-1) AND ZI(I) DELZ = 7I(I) — ZI(J) DELCI = (CI(I) CI(J))/DFLZ W = (Z ZI(J1)/DELZ WBAR = 1.0 — W TRM3A = ASOL(I)*((3.0 4 1W* 4 2)) - 1.0) TRM33 = ASOL(J) 4 ((3.0*(W9AR**2)) - 1.0) TRM3 = DELZ*(TRM3A - TRM39) DCDZ = DELCI + TRM3 RETURN 50 OCOZ = 0.0 RETURN END SUBROUTINE ORVTNP(19ZIVP,OXIDVP,OTOVP,ZLOW,ZUP) nnmmnm WPA- T1 _mrC-7T f nn _nT I I 11111 _Acni f 1 nni EXTERNAL FDTDV',FDXDVP,CMVP VPA = V° A = ZLJW 9 = Z IF(I .LT. 0) GO TO 100 A = ZUP 9 = PRINT 4 ,"A=",A .00 VPSD = VP**2 CSPSO = CSP(3)**2 ONTR = (CSP(0) 31. 0C07(9))*(SORT(VPSO CSPSO)) TRMOUT = VP/DNTR D = CALL QUAD(A,171,0,REL,1,AINTX,FDXOTP,NERR,O) IF (I .LT. 0) GO TO 200 1X0V? = -TRMOUT + AINTX !00 DXDVP = TRMOUT - AINTX CALL DUAD(A,8,D,RFL,1,AINTT,FOTOVP,NERR,O) IF (I .LT. 0) GO TO 300 DTDVP = -TRMOUT - AINTT ;00 DTOVP = TRMOUT + AINTT RETURN ENO SUBROUTINE MOLINT(71,729AMXIN,AMTIN) EXTERNAL FAMXIN,FAMTIN A = 71 3 = Z2 0 = 1.E-6 CALL QUAD(A,9,D,REL,O,AMXIN,FAMXIN,NERR,O) CALL QUAO(A,9,D I REL,0,A1TIN,FAMTIN,NEPR,0) RETURN END SUBROUTINE OSOVP(I,7 1 VP,OX0VPIDTDVP I ZLOW I ZUP,AMXIN,AMTIN, 10XDVPT I DTDVPT) COMMON VPA,I1,NCS,7I(100),CI(1001,ASOL(100) EXTERNAL FOTOVP,FOX0VP,FAMXIN,FAMTIN I = 1 Z = 71(5) CALL DRVTNP(I / Z,V 0 ,0X0VPU,OTOVPU I ZLOW 7UP) I = -1 Z = 7I(3) CALL ORVTNP(I,Z,VP,DX0VPL,OTOVPL,7LOR,ZUP) Z1 = 71(3) 72 = 7I(5) CALL MOLINT(71,72,AMXIN,AMTIN) OX0VPT = DXDVPL + AMXIN + OXDVPU OTDVPT = DTDVPL + AMTIN + OTOVPU RETURN END FUNCTION FAMXIN(Z) COMMON VP,K VPSO = VP**2 CSPSO = CSP(7)**2 IF (VPSO .GE. CSPSO) GO TO 20 K = 1 10 TRM1 = 1.E-50 GC TO 30 20 K = 0 TRM1 = (SDRT(VPSO - CSPSO)) 44- 3 IF (TRM1 .LT. 1.E-50) GO TO 10 TPM2 = CSP(Z)*VR 30 FANXIN = -TRM2/TRP1 RETURN END FUNCTION FAMTIN(Z) COMMON VP,K VPSD = V 0 **2 CSPSO = CSP(Z)**2 IF (VPSO .GE. CSPSQ) GO TO 20 K = 1 10 TRMA = 1.F-50 GC TO 30 20 K = 0 TR?IA = SQRT(VPSO -CSPSO) IF (TRMA .LT. 1.F-50) GO TO 10 TRM3 = 1.0/(CSP(7) 4- TRNA) TRM4 = VPSO/CCSP(7)*(TRNA**3)) 30 FAMTIN = TRM3 - TRM4 RETURN ENO FUNCTION CODZS(Z) COMMON VP.I1,NCS•ZI(100),OI(100),ASOL(100) ZL = ZI(1) ZP = ZI(NCS) IF(7. .LT. ZL) GO TO 50 IF(Z .GT. ZP) GO TO 50 I = NCS 10 J = I-1 ZTEST = ZI(J) IF(Z .GT. ZTEST) !O TO 40 I = J GO TO 10 40 CONTINUE IS BETWEEN ZI(I-1) AND 7_I(J) OELZ = ZI(I) - ZI(J) W = (Z - ZI(J))/DELZ WBAR = 1.0 - W OCOZS = 6.0*((WPAR*ASOL(J)) (W*ASOL(I))) RETURN 50 DCDZS = 0.0 RETURN END FUNCTION FDX0VP(Z) COMMON VP,K CSPSQ = CSP(Z)**2 VPSQ = VP**2 OCOZSQ = OCOZ(Z)2 IF(VPSQ .GE. CSPSO) GO TO 50 K = 1 40 ON = 1.E-50 GO TO 60 50 K = 0 ON = OCOZSQ*(SORT(VPSQ CSPSQ)) IF(DN .LT. 1.E-50) GO TO 40 60 FDXDVP = (VP 4 DCDZS(Z))/ON RETURN END FUNCTION FOTOVP(Z) COMMON VP,K REAL NMA,NMO,NMC,NM CSPSO = CSP(Z)* 4 2 VPSQ = VP*2 DCDZSQ = OCDZ(Z)**2 CSPCUB = CSP(Z)**3 IF(VPSQ .GE. CSPSO) GO TO 70 K = 1 60 ON = 1.E-50 GO TO 80 7L, K = 0 ON = SORT(VPS( CSPSC) IF (ON .LT. 1.E-50) GO TO 60 NMA = 1.0/CSP(Z) NMI = (2.0*VPSQ)/CSPCUB NMC = (VPSQ*OCOZS(Z))/(CSFST,'CCUZSO) rlN = NMA - NMC 80 FDTDVP = NM/ON RETURN ENO FUNCTION PDX0Z(7) COMMON VP,K CSPSO = CSP(Z)**2 VPS = V°**2 IF-(CSPSO .LC. VPSQ) GO TO 10 K = 1 5 OSQ = 1.E-50 GO TO 20 10 K = 0 DSOC = 1./CSPS0 OSQV = 1./VPSO DSQ = OSOC OSOV IF (OSQ .LT. 1.E-50) GO TO 5 20 ROXOZ = (1./VP)/SCRT(OSC) RETURN ENO FUNCTION RDTOZ(Z) COMMON VP,K CSPSQ = CSP(Z)**2 VPSO = VP 4 *2 IF (CSPSQ .LE. VPSO) GO TO 30 K = 1 20 OSQ = 1.E-50 GO TO 40 30 K = 0 OSOC = 1./CSPSQ DSQV = 1./VPSO OSQ = OSOC - OSOV IF (OSQ .LT. 1.F-50) GO TO 20 40 ROTDZ = (1./CSPSO)/SQRT(OSC)) RETURN END FUNCTION RAINT(DSCZR,ZLOW,ZUF) EXTERNAL CSOZR ZAVE = (7UP + ZLOW)/2.0 0 = CALL QUAO(ZLOW.7AVE,C,REL.1,ANS1,0S0ZR,NERR,O) CALL QUAD(ZUR,ZAVE,O,REL.1,ANS2.0SOZR,NERR.0) RAINT ANS2) RETURN END Bibliography of Related Works Albers, V. M., Underwater Sound (Dowden, Hutchinson and Ross, Inc., Stroudsburg, Pa., 1972).

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ATMOSPHERIC ACOUSTIC GRAVITY MODES AT FREQUENCIES NEAR AND BELOW LOW FREQUENCY CUTOFF IMPOSED BY. UPPER BOUNDARY CONDITIONS

Allan D. Pierce, Wayne A. Kinney and Christopher Y. Kapper

School of Mechanical Engineering Georgia Institute of Technology

Contract No. F19628-74-C-0065 Project No. 7639

SCIENTIFIC REPORT NO. 1

Contract Monitor: Elisabeth F. Iliff Terrestrial Sciences Laboratory

This document has been approved for public release and sale; its distribution is unlimited.

Prepared for

AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSPACE RESEARCH UNITED STATES AIR FORCE BEDFORD, MASSACHUSETTS 01730 ABSTRACT

Perturbation techniques are described for the computation of the imaginary part of the horizontal wave number (k I ) for modes of propagation.

Numerical studies were carried out for a model atmosphere terminated by a constant sound speed (478 m/sec) half space above an altitude of 125 km.

The GR and GR modes have lower frequency cutoffs. It was found that for 0 1 frequencies less than 0.0125 radian/sec, the GR mode has complex phase 1 velocity; kI varying from near zero up to a maximum of 3 x 10 -4 with analogous results for the GR mode. There is an extremely small frequency 0 gap for each mode for which no poles in the complex k plane corresponding to that mode exist. These mark the transition from undamped propagation to damped propagation. In the complete Fourier synthesis, branch line contributions compensate for the absence of poles in these gaps. Computa- tional procedures are described which facilitate the inclusion of the low frequency portions of these modes in the waveform synthesis. INTRODUCTION_

One of the standard mathematical problems in acoustic wave propagation

is that of predicting the acoustic field at large horizontal distances

from a localized source in a medium whose properties vary only with height.

This problem, as well as its counterpart in electromagnetic theory, has 1 received considerable attention in the literature, is reviewed extensively 2-7 in various texts , and, for the most part, may be considered to be well

understood.

A typical formulation of, say, the transient propagation problem 8-9

leads (at sufficiently large horizontal distance r) to an intermediate result

which may be expressed as a double Fourier integration over angular frequency

w and horizontal wave nir-lber k; i.e. for, say, the acoustic pressure, one has co

p = S(r) Re f(L.:)e-it71` [Q/D(w,k)]e ikr dkdw (1) .

-CO

Here S(r) is a geometrical spreading factor , 1//i7 for horizontally stratified 1/2 media, In a sin(r/a )] if the earth curvature (a =radius of earth) is to e e e be approximately taken into account. The quantity f(w) is a Fourier transform

of some function characterizing the time dependence of the source; Q(w,k,z,z 0)

as well as of w and k, is a function of receiver and source heights z and zo possibly also of horizontal direction of propagation if, say, winds are included

in the formulation, but, in any event, should have no poles in the complex

k plane for given real positive w, and given z and . The denominator D(w,k) - zo , may be zero for certain values k (0 of k, and is is independent of z and z o n termed the eigenmode dispersion function.

Typically, in order to uniquely specify both Q and D(w,k) for all complex 2

values of k (given w real and positive), branch points must be identified

and branch cuts must be placed in the complex k plane. The general rule

may be taken to be that no branch cut should cross the real axis, and, if

a branch point should lie on the real axis (when w is positive real), the

branch cut either extends into the upper or lower half plane depending on

whether the branch point moves up or down when w is given a small positive

imaginary part. The integration contour for the k integration goes nominally

along the real axis but skirts below or above (see Fig. la) those poles

lying on the real axis which move up or down, respectively, when w is given

a small positive imaginary part. The placing of the branch cuts and the

selection of the contour in this manner is one method of guaranteeing

causality in the solution, or, equivalently, of guaranteeing that the solution

dies out at large distances if a slight amount of damping (Rayleigh's virtual

viscosity) is added in the mathematical formulation. The necessity of branch

cuts only occurs if the medium is unbounded either from above or below and

a choice of phases can always be made such that (given, say, that the medium

is unbounded from above) Q dies out exponentially as z co when w has a small positive imaginary part and when k is real.

lb. The poles above the initial contour are encircled in the counterclockwise manner. There are also contour segments which encircle each branch cut lying

above the real axis in the counterclockwise sense. The integrals around each pole are evaluated by Cauchy's residue theorem and one is left with a sum of

residue terms plus branch line integrals. Each residue term may be considered

as corresponding to a particular guided mode of propagation. The branch line

contributions in some contexts are considered as corresponding to what may

be termed lateral waves. 11 (The term may be unappropriate unless there is a kR

(b)

1. Contours in the complex k (wavenumber) plane for evaluation of individual frequency contributions to waveform synthesis. (a) Original contour. (b) Deformed contour.

F 3

sharply defined interface separating two types of media, such as a water- muddy bottom interface in shallow water propagation.)

In regards to the guided mode description, one type of approximation frequently made is to neglect all poles (i.e. roots k n (w) of D(w,k)), which are ikr above the real axis, the argument being that the corresponding e factors in the residues will die out rapidly with increasing r, the bulk of the contribution at large r expected to come from the poles which lie on the real axis. In a similar manner, it is argued that the branch line contour con- 3/2 tribution also dies out relatively rapidly (a factor of 1/r in addition to the geometrical spreading) so it too may be neglected at large r compared to the terms coming from the real roots. The net result for Eq. (1) would then be c")tItt (63) cos[wt-kn p =):2(r) An (w)r+On (w)1 dw (2) YI (4) i Let. where An (w) and ¢ n (w) are defined in terms of the magnitude and phase of the residues of the integrand in Eq.(1); the kn (w) being the real roots of

D(w,k)=0, numbered in some order with the index n=1, 2, 3, etc., and it being understood that, for fixed n, kn (w) should be a continuous function of w over some range of w from a lower limit w im up to an upper limit w un . The remaining integral over w can then be approximately evaluated by the method of stationary phase or integrated by suitable numerical methods.

by examples from atmospheric infrasound propagation.

I. INFRASOUND MODES

are also assumed constant in the upper half space, z>z ] Conceivably, one has T. some latitude in the choice of z Tand of the upper halfspace sound speed cT , although computations of factors such as Q(w,k,z,z 0) and D(w,k) in Eq. (1) become more lengthy with increasing z T . Also, it would seem that the most logical choice of cT would be that which would realistically correspond to height z T , so the profile c(z) would be continuous with height across z as in Fig. 2a. T' Another conceivable choice would be one (Fig. 2b) in which cT co, such that the surface of air nominally at z would be a free surface or pressure release T surface (corresponding to the model generally adopted for the water-air interface in underwater sound studies). A somewhat intuitive premise which may be adopted is that, if the source and receiver are both near the ground and if the energy actually reaching the receiver travels via propagation modes channeled primarily in the lower atmosphere, then the actual value of the integral in Eq. (1) would be somewhat insensitive to the choices of z and c . This, however, remains to T T justified in any rigorous sense, so we would be somewhat hesitant to take c = ... at the outset. In typical calculations performed in the past, z Tis taken T as 225 km, c T is taken as the sound speed (z 800 m/sec) at that altitude.

ikzz = (Constant) e -iwte ikxe ( 3)

Where the vertical wave number k (alternately written as iG for inhomogeneous z plane waves) and the horizontal wave number k are related by the dispersion relation (neglecting winds)

2 2 2 2 2 2 k = -G = 2 - wA] / c - [w - (4) z WB where wA = (y/2)g/c, wB = (y-1) 2 g/c are two characteristic frequencieS [wA > wB ] for wave propagation in an isothermal atmosphere (g = 9.8 m/s 2 is acceleration due to gravity, yz1.4 is specific heat ratio). Here, for brevity, the subscript T on c T has been omitted. For given real positive w, 2 2 real k, one can have k positive or negative (G negative or positive). The 2 2 values of k at which k or G go to zero turn out, as might well be expected, to be the branchpoints in the k integration in Eq. (1), i.e., synonymous with the branch points of G. Along the real axis, G is either real and positive ik z -Gz (e z or e dying out with increasing z) or else G is a positive or negative

imaginary quantity. In the latter case, the phase of G may be either ff/2

or -Tr/2, in accordance with the well known fact that, for acoustic-gravity waves, wavefronts may be moving obliquely downwards (negative k z ) when energy is flowing obliquely upwards. In particular, for 0 < w < w one has G real 13' and positive for k in between the two branch points on the real axis, the phase of G is 7/2 (k < 0) on the remainder of the real axis; the two branch z

points are, from Eq. (4), at

2 2 !

EwA w 3 2 2 2'-2 (5) " = c[w B - w ]

The branch lines extend upwards and downwards from the positive and negative

branch points, respectively. [See Fig. 1.]

The dispersion function D(w,k) in the atmospheric infrasound case can

be written in the general form

(6) D ( J, k) = Al2111 A11R12

where R and R are elements of a transmission matrix [R], these depend on 11 12 the atmosphere's properties only in the altitude range 0 to they are T' independent of what is assumed for the upper half space. In general, their

determination requires numerical integration over height of two simultaneous 8 9 12 . ordinary differential equations (termed the residual equations " Ain previous

literature). They do depend on w and k (or, alternately, on w and phase

velocity v) but are free from branch cuts, they are real when w and k are real

and are finite for all finite values of w and k. The other parameters A 12 and A depend only on the properties of the upper half space (in addition to 11 w and k). Specifically, these are given (for the no wind case and with the

subscript T omitted on qr )

2 2 = gk2 w - yg/[2c ] (7a)

2 2 2 A = 1 - c 12 k /w (7b) 7

One may note that, since every quantity in Eq. (6) is necessarily real when w and k are real (with the possible exception of G), the poles lying on the real k axis (real roots of D) must be in the regions of the (w,k) plane

[or (w,v) plane] where G 2 >0. Since the integrand of Eq. (1) divided by 0 -Gz should vary with z above z as e T one may call the corresponding modes T fully ducted modes. There is no net leakage of energy for such natural modes into the upper halfspace. If one considers D as a function of w and phase velocity v (or simply v), where v = w/k, the locus of real roots v versus w

(dispersion curves) has (as has been found by numerical calculation) the general form sketched in Fig. 3. The nomenclature for labeling the modes (GR for gravity, S for sound) is due to Press and Harkrider. One may note from Eq. (4) that there are two "forbidden regions" in the v vs. w plane, i.e.

2 2 2 2 % v < c[w w ] 2 / [w - w ] 2 B A (8a) for w < w and B

2 2 2 2 1/2 v > c[w - w ] / [w - co ] (8b)

for w > wA . Within either of these regions G would have to be imaginary and there would accordingly be no real roots for v of D(w,v) = 0. In the high frequency limit, this simply implies that the phase velocities of propagating modes are always less than the sound speed of the upper halfspace, the branch points in the k plane are simply at ± w/cT . The low frequency lower phase velocity "forbidden region" appears to be due to the incorporation of gravity effects into the formulation. However, if c is allowed to approach co, T this lower left hand corner region disappears. We have done numerical studies on the effects of varying c T on the dispersion curves. Briefly, the result is that the form of the predicted curves for GR and GR change very little o 1 0.01 0.03 0.05

ANGULAR FREQUENCY, (rad/soc)

3. Numerically derived plots of phase velocity v versus angular frequency w for infrasonic modes in a model atmospherc corresponding to Fig. 2. The labeling of modes is with the convention introduced by Press and Harkrider 2 (J. Geophy. Res. 67, 3889-3908 (1962). The lines G =0 delimit regions of the v versus w plane where a real root of the eigenmode dispersion function cannot be found. 8

with increasing cT ; the lower forbidden regions shrink insofar as frequency range is concerned and the curves extend to successively lower frequencies.

Thus we see that the fully ducted modes GR and GR both have a lower frequency o 1 cutoff [wl, in Eq. (2)] which depends on cT . The larger one makes c T , the smaller is this cutoff frequency.

We thus have the following apparent paradoxes. Given that frequencies below w may be important for the synthesis of the total waveform, an apparently plausible computation scheme based on the reasoning leading to our Eq. (2) will omit much of the information conveyed by such frequencies. Also, in spite of the plausible premise that energy ducted primarily in the lower atmosphere should be insensitive to the choice for c one sees that this choice governs the T' cutoff frequencies for certain modes and that certain important frequency ranges could conceivably be omitted entirely by a seemingly logical and proper choice for cT . The resolution of these paradoxes would seem to lie in the nature of the approximations made in going from Eq. (1) to Eq. (2). The latter may not be as nearly correct as earlier presumed and it may be necessary to include contributions from poles off the real axis and from the branch line integrals. Even if r is undisputably large, it may be that the imaginary ikr parts of the complex wavenumbers are sufficiently small that le is still

not small compared to unity. Also, a branch line integral may be appreciable in magnitude at large r if there should be a pole relatively close to the branch cut. II. ROOTS OF DISPERSION FUNCTION

In order to understand the manner in which the solution represented by

Eq. (2) should be modified in order to remove the apparent artificial low

and GR modes, we first examine the nature of the frequency cutoffs of the GRo 1 dispersion function D at points in the vicinity of a particular mode's dispersion curve. The curve vn (w) of phase velocity v versus w for a given (n-th) mode is known at points to the right of the lower cutoff frequency co L . Given this, one can find analogous curves v (w) and vb (w) for values of the phase velocity a w/k at which the functions R (w,v) and R (w,v) in Eq. (6), respectively, 11 12 vanish. Since there may be more than one such curve in each case, we pick v a (w) and v (w) such that these curves are the closest of all such curves to the curve b vn (w) for w > wL . One may note, however, that one may apparently define and identify va (w) and vb (w) for frequencies much less than wL , simply from analytical continuation.

A premise which we have checked numerically (see Fig. 4) for a specific case is that the curves vn (w) , va (w), vb (w) defined above with reference to a particular given mode all lie substantially closer to each other than to the corresponding curves for a different mode. In retrospect, this is obvious, although it took some time for us to realize that it was so. Briefly, the argument goes that, if the mode is predominantly guided in the lower atmosphere, then there should be a decay of modal height profiles beyond some point substantially lower than 41 . Thus, both the p/ and o profiles for a guided mode sr o v z would have values at z,1 substantially less than their peak values at lower altitudes. The same would be true for the profiles of the auxiliary functions

0 and 2 which satisfy the residual equations. Consequently, if guided waves are excited, the inverse transmission matrix connecting 0 1 and 02 at the ground

to those at height 41, would have to have very small [1,2] and [2,2] components. 0.3119

G R 0

0.3117 I 0.002 0.006 0.010 0.014 ANGULAR FREQUENCY ( - akin/sec) 0.25 1 G 2 -0

' 0.23

N ' 0.21 0.002 0.006 0.010 0.014- ANGULAR FREQUENCY (rcidcn/z-..). 4. Curves in phase velocity (v ,v ,v ) versus angular frequency (w) plane n a b along which R11=0 (giving va (4, R12 =0 (giving vb (4, and D(w,k)=0 (giving va (4. • Curves are shown for (a) the GR 0 mode and (b) the GR, mode. N .te the changes in scale and the relatively close spacing of curves corresponding to the same mode. The lines along whica, G 2=0 are also indicated; v (0 is not a real quantity for w values below the indicated lower cutoff n frequency. 10,

(Recall that 0 1 = 0 at the ground.) Since the transmission matrix has unit determinant, it follows that elements R12 and R11 of the transmission matrix proper [from height ZT down to the ground and whose elements appear in Eq. (6)] have to be small.

Given the definitions v a(w) and vb (w), the dispersion relation D=0 for a single mode may be written

va ) - D (9 ) "'s(Al2)(a) (v- -I- G i (13)(v-vb ) = 0

and v , respectively. (For where a = dR11/dv, = dR12 /dv, evaluated at v = va b simplicity, we here consider D as a function of w and v = w/k rather than of w and k.) The above equation may also equivalently be written in the form

v = va + (va-vb )X/[1-X]

(R/a"A X = ll G)/Al2 which may be considered as a starting point for an iterative solution which in essence develops v in a power series in va-vb ; G may be considered as a defined

as the zeroth iteration, evaluates the function of w,v. One starts with v = v a right_hand side for the value of v to find the starting point for the next iteration, etc. The considered procedure should converge provided v a or vb is not near a point at which G vanishes and providing G in the vicinity of v a or v is not such that the variable X is close to unity. Among other limitations, b the iteration scheme would be inappropriate for values of w in the immediate vicinity of cob .

In regards to establishing the general trends represented by the iterative type solutions, two relatively general theorems may be of use. These (whose 11

13 proof fellows along lines previously used by one of the authors in deriving

an integral expression for group velocity) are that for real positive w and v,

R 3R /3v - R 3R /8v > 0 (11a) 12 11 11 12

R aR/ato — R aR12 /aw > 0 (lib) 12 11

or, alternately, if one inserts Ril = (a)(v-va ), R12 = (s)(v-vb ), he finds

agva - vb ) > 0 (12a)

(v - v )(v - v ) (Ra"-ra) + f3a[v (v - v ) - v " (v - v )] > (12b) b a b a a b where the primes represent derivatives with respect to w. The second of these

should hold for arbitary v in the vicinity of v a and vb and lead, upon setting v = v , v = vb along with the use of Eq. (12a), a , or v = (vavb -va 'vb )(vb -va t o

v ' < 0 b (13a)

va ' < '0 (13b)

(a/0" > 0 (13c)

Equation (12a) implies that as long as a or 13 do not vanish (which would seem

unlikely) the two curves va (w) and vb (w) do not intersect. If a and f3 have the

same sign the v a curve lies above the vb curve; the converse is true if a and $

increases with w.

To demonstrate the general utility of the perturbation approach, a brief 12

(1) a , vb , a, 8, v , and v are given in Table I for the GR table of values w, v n and GR modes for the case of a U.S. Standard Atmosphere without winds terminated 1 (1) at a height of 125 km by a half space with a sound speed of 478 m/sec. Here v is the result of the first iteration for the phase velocity and v is the actual n numerical result obtained (only if the phase velocity is real) by explicit numerical search for roots of the eigenmode dispersion function. One may note that, for those frequencie s where v is computed, the agreement between v (1) n and v is excellent. A mo re detailed listing of the perturbation calculation n results is given in Figs. 5a and b. The plots there give w/k R or the reciprocal (1) of the real part of 1/v (i.e., w divided by the real part of the horizontal wave number k) and the imaginary part k I of k = w/v versus angular frequency.

Note that k is zero above the corresponding cutoff frequencies. The relatively I small values of the k are commented upon in Sec. IV.

III. TRANSITION FROM NONLEAKING TO LEAKING

)] 1/2 G = i(P)(to -Lo la) + (q )(v_via (14) where p and q are readily identifiable [from Eq. (4)] positive numbers taken independent of w and v; v L is the phase velocity on the dispersion curve in the limit as co -3- w from above. The bracketed quantity in Eq. (14) may be re- L 2 garded as a double Taylor series expansion (truncated at first order) of G about 2 the point w v at which G vanishes (hence no zeroth order term). The fact that L' L 2 both p and q are positive follows since G is positive to the upper right of the 0 CUTOFF

0.002 0.004 0.006 0.008 0.010 ANGULAR FREQUENCY (radian/sec)

0.002 0.004 0.006 0.008 0.010 0.012 .ANGULAR FREQUENCY (radian/sec)

5. Numerically derived plots of phase velocity w/k and of the imaginary R part k of the complex wavenumber k versus angular frequency for the GR 0 I and GR modes. Previous theoretical lower frequency cutoffs for these 1 modes are as indicated. Note that k is identically zero above the I cutoff frequency. 0-

GR o

Frequency dependent parameters correspondin[; to GR and GR modes; w is , 0 I angular frequency in rad/sec, va is phase velocity root of Rll=O, in

km/sec, vb is analogous root of R "'O, 0', is dRl1/dv at v=va in sec/km 12 13 is dR /dV at v=vb in sec, vel) is first order pCl·turbation solution for 12 phase velocity from equations given in the text (units are km/sec), v 11 is the real root determined by direct numericnl solution for zeros of

eigenmocie dispersion function. Note that v (defined only ~.;rhen phaGe 11 . velocity is real) a&rees exceptionally well with vel). 2 line in the co,v plane where G = 0 and also since the G 2 = 0 line slopes obliquely downwards. (See Fig. 3).

Let us next note that, in the vicinity of the point w v, the denominator L D given by Eq. (9) may be further approximated as

D (A12a-A11a) (Av + pAw) + c(Av + vAw (15) where we have abbreviated Av = v-v Aw = w -w v = p/q; the quantity u is L L' /du or -dv /du, the two being assumed to be approximately equal. either -dva b (The use of the minus sign here assumes that p be_positive.) The remaining quantity e is

(q 2) (6) (v-vb ) - aA 11 12

One should note that c depends on v, although, for purposes of initial analytical investigation, one may set v = v here. All of the above quantities may be L considered to be evaluated at w = col, and v = v.L . Note that p and v are both positive quantities. Furthermore, it should also be noted that v > p since 2 the G = 0 curve slopes downwards more rapidly than the lines along which

R or R = 0 in the v vs u plane. (See Fig. 4.) 11 12 The roots of Eq. (15) without regard to the sign of the radical are readily found to be

- % (17) Av = -pLu +(%J 2 + c(v-p)2 [Aw +a] where a C4 (18)

Alternately, if 'Awl << c, the above may be approximated by the binomial theorem to give 15

2 2 Av = -vAw + [(v-11) i(Aw) (19a)

2 2 2 2 Av = +e - (21.1 -v) Aw - [(v-p) /e ](Aw) (19b)

for the upper and lower signs, respectively. The first of these (since Av = 0 when Aw = 0) is clearly the description of the disperson curve in the vicinity

of w = 6112 v = vL .

Equation (19a) shows that, as Aw 0 from above, the dispersion curve 2 becomes tangential to the line G = O. The two curves do not intersect. The

general trend is as indicated in Fig. 6. The solution represented by Eq. (19b)

is not a proper root of Eq. (15); it corresponds to the wrong sign of the radical

and accordingly lies on the second branch. Furthermore, one can readily show

that, for values of Aw slightly less than zero, both roots lie on the second branch. Hence, there must be a gap of finite frequency range in which, for

the choice of branch cuts represented by Fig. 1, there are no poles in the k

(or v) plane corresponding to the n-th mode.

To determine the order of magnitude of this frequency gap, it is appropriate to consider the trajectory of the second branch roots in some detail and to determine just where one of them should cross the branch cut, reappearing on the first branch. As long as /Iv is real and Av + vAw >0 the criterion for a root to be identified with the first branch is Av + pAw > O. According to

Eq. (17), this would automatically place the second root on the second branch for all Aw > -a and would place the first root on the second branch for

-a < Aw < O. Consequently, if either root is to reappear on the first branch, it must be at a value of Aw < -a.

One should note from Eq. (17) that at Lw = -a the two real roots on the second branch coalesce. For values of Aw < -a the two roots separate again,_but ImV <0

- B < 0

G>0

DISPERSION CURVE

U.)

6. Sketch illustrating nature of a single mode's dispersion curve in the vicinity 2 of the G =0 line. At point A (angular velocity w phase velocity v ) L' L the dispersion curve is tangent to the G2=0 line; for frequencies below w L down to that corresponding to point B in the sketch there are two real roots for v of the eigenmode dispersion function on the second branch. For frequencies lower than that corresponding to point. B, there is a complex root for v on the first branch (which is the complex conjugate of a second root on the second branch). 16

Re(Av) = -vLw or where

2 E Aw = — (1/2) = (v—p) with a corresponding value of Av of

2 Av = (e /2) ii[v/(v-p)] -

For subsequent frequencies successively lower than w -2u there is a complex L root on the first branch with a negative imaginary part which increases with decreasing frequency.

The discussion up to now has assumed that lAvi << IvL-vb 1 and hence that c may be taken as constant. This would seem appropriate for describing the transition region since all values of Av of interest in this region are of 2 second order of c . However, if an improved numerical estimate is required, we recommend that one regard Eqs. (16) and (17) as a iterative pair. Success- fully computed values of Av may be used to recalculate c and the new value of c may then be used in obtaining the next higher estimate for Av.

In Table II the values of w v p, q, p, v, 6, and a are given for L' L' the GR and GR modes for the model atmosphere corresponding to Fig. 2a. 0 1 The extremely small values of a should be noted. The corresponding plot of Av versus Lw (i.e., both branches of Eq.(17)) corresponding to their values for the GR mode is given in Fig. 7. For simplicity, this is plotted 0 in a nondimensional form, i.e. [1 4. s. 1/2 V = - {p/[2(v - 0]}Q + 2] (20) 17

TABLETS

GR GR 1

w (rads) 0.0118 0.0125 L v (km/s) 0.31188 0.2323 L p(s/km2) 0.14 0.35

q(s/km3) 1.84 x 10-3 1.86 x 10-3

u(km) 2.94 x 10-2 4.15

v(km) 76 190 1/2 1/2 -6 -3 c(km /s ) 9.6 x 10 1.02 x 10 -13 -9 a(rads/s) 3.04 x 10 1.41 x 10

Parameters characterizing the eigenmode dispersion function near points in the phase velocity versus angular frequency plane at which the GR 0 and GR modes undergo transition from leaking to non-leaking. 1 7. Graph of normalized phase velocity versus normalized frequency in the vicinity of the point (v , WL ) for the GR. mode. The imaginary and L 0 real parts are both plotted. The dashed line corresponds to real roots on the second Riemann sheet. 18

where v = Av/[2(1) -11) ] and f2=Aw/o. Both real and imaginary parts are shown on the same graph. The corresponding plots for the GR 1 mode differ only slightly from those in the Fig. 7 because of a different value of the parameter n/[2(v-v)] in Eq. (20); in both cases this parameter is small compared to unity, i.e. 1.1<

IV. THE BRANCH LINE INTEGRAL

Since there is a gap in the range of frequencies for which a pole corresponding to a mode may exist, it is evident that evaluation of the k integration in Eq. (1) by merely including residues may be insufficient for certain frequencies. Thus it would seem appropriate in such cases to include a contribution from the branch line integral. It may be anticipated that such branch line integrals are significant at larger values of r only when Ellis close to some mode's wL (say the n-th mode), in which case the branch point of greatest interest (i.e., that which may have a pole in its immediate vicinity)

. Consequently, it would appear that an adequate approximation to is at k=w/vL the branch line integral would be

Oo Branch line N/D(w,k)]eikrdk contribution of c0 ikr Q e dk 1/2 (21) Al2a-A32 x-1-01-v)Lco+cx JCr 0 where the denominator D(w,k) has been approximated by Eq. (15) with the abbreviation x for Av+vL.w. The quantity outside the integral is assumed to be evaluated at w=w1, and k=w/vL . The contour C runs down the left side of the branch cut, B around the branch point (where x=0), and then up the right side. If one next changes the variable of integration from k to x, nothing that for small x/ , noting

2 kzkB -(wLL /v)x (22) 19

he finds approximately that 2 -i(w v )x Branch line e L ' L = (Residue contribution dx (23) x+(1-v)Aw+cx Ca where (Residue) ikr o is that residue which the integrand (Q/D)e would be expected to have at the n-th mode's pole in the k plane were the parameter c identically

equal to zero. The mapped contour ( 3 in the x plane may be considered to go

up on the right and then down on the left of a branch cut extending vertically

downwards from the origin in the x plane. If we set x=-1_, then, on the right 1/2 e-17/4 1/2 side of the cut, x should be E while, on the left side, it is -in/41/2 -e • Consequently, the total integral combines to 2 AAW/4 -(W /v )rrr Branch lines 2ce e L L d Jr (20) i:contribution = -(Residue) o, 2 2 [-1- 4. (1.1-v)1w] +ic

This in turn, with an obvious change of integration variable, may be expressed

0 as 5

eini4e-nr1 1/2dn Branch line (21) contribution = (Residue) o 2K (n-n 1)(n-n 2 ) 0 where

1/2 K=cv /(w r) (26a) L L 2 n l , n = i(K /2)(1+[Aw/20]) 2 2 1/2 ± i(K /2)(1+[Lw/a]) (26b)

with a as defined by Eq. (18).

In regards to the n integration, the integral can be expressed in general

in terms of Fresnel integrals of complex argument after some considerable

mathematical manipulation. One may note, moreover, that H i ( and In 2 1 are, for

most cases of interest, considerably less than unity. In this case, the

appropriate approximate result (derivation omitted for brevity) is

in e_nrr, an . (27) (n-n )(n-n ) 1/2 1/2 1 n1 +n 2 o where the choice of square root should be such that the imaginary part is

positive. The net result in this limit then is that the branch line contribution

is independent of the parameter K. (The dependence on range r comes only in

the residue.) Thus one may write

{Branch line = 27i(Residue) B (Aco/a) (28) contribution o rh

where the function B (Aw/a) is given by rh

(29) B rh (2) - 1/2 1/2 1/2 1/2 [14-(1/2)s]+-a74-s1) ] +[1+(1/2)0-(1+0) ]

Here any consistent choice may be made for the sign of the inner square roots

but the outer square roots should be taken such that the resulting phases are

between -7/4 and 37/4. The quantities in square brackets turn out to be the 1/2 squares of (1//i)[(1+2) ±1j, respectively. The phase restriction then gives

Brh (2) = (1+2)1/2 if c2>0 (30a) = 1 if 0>2>-2 (30b) -1/2 = -i(-2-1) if fl<-2 (30c)

where here all square roots are understood to be positive;

To completely describe the transition it is appropriate to add to Eq. (28)

that contribution (which is zero for 0>Aw>-2a) from the pole on the first branch

in Eq. (21) which lies in the general vicinity of k=w /v ,Jf the pole is L L. present, its contribution to the integration over k is 271 times the residue

(which is not what we have been referring to as (Residue) unless e is identically o zero). The evaluation of the residue is moderately straightforward and omitted

here for brevity. The net result is that 21

Branch line I Pole + contribution contribution

= 2ui(Residue) o B rh (Aw/a)+P ot (Aw/o) (31) where the "pole function" P ot (Aw/o) turns out to be given by

P(1w/a)=1-B Ow/a) , rh (32)

We accordingly have the remarkable (although, in retrospect, not unexpected) result that

Branch line Pole + = 2ui(Residue) (33) contribution contribution o

Qeikr Qe ikr Res (34) D d'D/dk k=pole where

D _ d dk dk (Al2R11-A11R12 )

-G (R ) (35) dk 12 i.e. it differs from the actual derivative of D in that G is formally considered as constant. Doing this when w is somewhat removed from the transition region near w should make very little difference since R is small at values of k which L 12 are poles. Near the transition, this neglect should almost exactly compensate for the neglect of the branch line integral. 22

REFERENCES

1. J. E. Thomas, A. D. Pierce, E. A. Flinn, and L. B. Craine, "Bibliography on Infrasonic Waves", Geophys. J. R. astr. Soc. 26, 399-426 (1971).

2. C. B. Officer, Introduction to the Theory of Sound Transmission with Application to the Ocean (McGraw-Hill, New York, 1958).

3. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon Press, Inc., New York, 1962).

4. L. M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1960).

5. K. G. Budden, The Wave-Guide Mode Theory of Wave Propagation (Prentice Hall, Inc., Englewood Cliffs, N.J., 1961).

6. I. Tolstoy and C. S. Clay, Ocean Acoustics (McGraw-Hill, New Yprk , 1966).

7. M. Ewing, W. Jardetzky, and F. Press, Elastic Waves in Layered Media (McGraw-Hill, New York, 1957).

9. A. D. Pierce, J. W. Posey, and E. F. Iliff, "Variation of Nuclear Explosion generated Acoustic-Gravity Waveforms with Burst Height and with Energy Yield" J. Geophys. Res. 76, 5025-5042 (1971).

10. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Clarendon Press, Oxford, 1935) p. 137.

11. L. M. Brekhovskikh, loc. cit., pp. 270-280.

12. A. D. Pierce, "The Multilayer Approximation for Infrasonic Wave Propagation in a Temperature and Wind-Stratified Atmosphere", J. Comp. Phys. 1, 343-366 (1967).

13. A. D. Pierce, "Propagation of Acoustic-Gravity Waves in a Temperature and Wind-Stratified Atmosphere", J. Acoust. Soc. Amer. 37, 218-227 (1965). 7- 6 37 AFGL -TR-76 -0055

GEOMETRICAL ACOUSTICS TECHNIQUES

IN FAR FIELD INFRASONIC

WAVEFORM SYNTHESES

Allan D. Pierce and Wayne A. Kinney

School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia 30332

SCIENTIFIC REPORT NO. 2

Approved for public release; distribution unlimited.

March 7, 1976

Prepared for

AIR FORCE GEOPHYSICS LABORATORY AIR FORCE SYSTEMS COMMAND rNITED STATES AIR FORCE HANSCOM AFB, MASSACHUSETTS 01731 Unclassified -1- SECURITY CLASS.FIr:ATION OF THIF = A GE i+'t*rt Dare Entered) ' - RFAD INSTRUCTIONS REPORT DOCWAENTATION PAGE BEFORE COMPLETING FORM . I. REPORT NI,MDER (2. GOVT ACCESSION NO. 3. PECIP'FFT'S CATALOG NUMBER AFGL-TR-76-0055 _...L__ 4. TITLE (and Subtitle) 5. TVFE OF REPORT & PERIOD COVERED GEOMETRICAL ACOUSTICS TECHNIQUES IN FAR FIELD INFRASONIC WAVEFORM Scientific Report No. 2 i SYNTHESES 6. PERFORMING ORG. REPORT NUMBER.

7. AUTHOR(s) B. CONY RACT Ori GRANT NUMBER(,) Allan D. Pierce F19628-74-C-0065 Wayne A. Kinney

9. PERFORMING ORGANIZATION NAME AND ADDRESS IO. . PPO.GRAM ELF-MENT.PROJECT. TASK School' of Mechanical Engineering .4 .z':A Pt WORK UNIT NUMBERS 62101F . Georgia Institute of Technology 76390102 Atlanta, Georgia 30332

11. CONTROLLING OFFICE NAME AND ADD R ESS 12. REPORT DATE Air Force Geophysics Laboratory 7 March 1976 - i Hanscom AFB, Massachusetts 01731 13. NUMBER OF PAGES Monitor: Elisabeth F. Riff, LWW 68 7.- ' 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report.) Unclassified

15e. DECLASSIFICATION/DOWNGRADING• SCHEDULE

16. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited. . _

17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Acoustics Ray Acoustics Geometrical Acoustics Guided waves Infrasound Atmospheric Acoustics Wave Propagation .Waves in inhomogeneous media

20. ABSTRACT !Continue on reverse side ff necessary and identify by block number) A ray acoustic computational model for the prediction of long range infrasound propagation in the atmosphere is described. A cubic spline technique is used to approximate the sound speed versus height profile when values of sound speed are input for discrete height intervals. Techniques for finding ray paths, travel times, ray turning , . - - • - EDITION OF t DO 1 FORMJAN 73 1473 NOV 55 IS OBSOLETE

SECURITY CLASSIFICATIID4 OF THIS PAGE (When Data Entered) -2-

SECURITY CLASSIFICATION OF THIS PAGE(Whon Dal. Entered) points, and rays connecting source and receiver are describa; A parameter characterizing the spreading of adjacent rays (or ray tube area) is defined and methods for its computation are given. A method of determining the number of times a given ray touches a caustic is also described. Formulas are given for the computation of acoustic amplitudes and waveforms which involve a superposition of contributions from individual rays connecting source and receiver and which incorporate phase shifts at caustics. The possibility of a receiver being in the proximity of a caustic is considered in some detail and distinction is made between cases where the receiver is on the illuminated or shadow sides of a caustic. It is shown that a knowledge of parameters characterizing two rays at a point in the vecinity of a caustic provides sufficient information concerning the caustic to allow one to give a relatively accurate description of the acoustic field in its vicinity. The resulting theory involves Airy functions and uses concepts extrapolated from a theory published in 1951 by Haskell. The net result is a detailed computational scheme which should accurately cover the contingency of the receiver being near a caustic in the calculation of amplitudes and waveforms. A number of FORTRAN subroutines illustrating the method are given in an apprendix. Limi- tations of the theory and suggestions for future developments are also given.

SECURITY CLASSIFICATION OF THIS PAGE(WA•n Date Entered) -3-

I. INTRODUCTION

The present report is concerned with the development of a computational model for the prediction of long range infrasound propagation in the atmosphere. The computational model discussed here is one which is partly based on ray acoustic concepts; it should be applicable to wave periods less than three minutes and is intended to complement the guided node model of acoustic gravity wave propagation which has been extensively discussed in previous reports and 1-5 papers. The ray acoustic method has a sizable literature pertaining to it; most of the published work is concerned with applications to underwater sound. (X brief bibliography of relevant papers is given in Appendix A.) Discussions of ray acoustics which are particularly germane to infrasound propagation in the atmosphere are an article , 7 6 , a 1966 AFCRL report by Pierce , published in 1951 by N. Easke11 4 and a 1973 AFCRL report by Pierce, Moo, and Posey. In the present .report, the details of the pertinent theory are assumed to be already known; the emphasis is on the computational implementation of the - theory. Particular innovations discussed here, not generally included in ray acoustic models, are (1) the presence of many rays which connect source and receiver, (2) a method of computing ray amplitudes based on analytical differentiation of ray formulas appropriate to a stratified medium, (3) the inclusion of caustics into the formulation, and (4) the inclusion of Lamb's atmospheric edge mode. The general model used as a starting point may be taken (Fig. 1) as a height stratified atmosphere above a flat rigid ground. The sound speed c(z) and ambient density p o (z)are assumed to be continuous functions of height z above the ground. For simplicity, winds are not included in the present formulation, although we believe that this limitation can easily be overcome with only a modest degree of effort. The pertinent governing equations are taken as the linearized equations of atmospheric compressible fluid dynamics (gravity included). 3 Nonlinear effects are neglected other than in the selection of a source term. Now such a source term appropriate to nuclear explosions may be selected has previously been discussed in some detail by Pierce, Posey,

SOURCE T z SC ATMOSPHERE / / / / / I I / SOUND SPEED GROUND . X

Figure 1.

Sketches illustrating general model used in the analysis. (a) Typical sound speed versus height profile. (b) Sketch of point source above a flat rigid ground, with a height stratified atmosphere. -5-

8 and Iliff. It suffices here to only state that the source is assumed localized at a point whose coordinates may he taken as x = 0, y = 0, z= ZSC A modest analysis of the governing equations suggests that the wave portion with periods less than approximately three minutes may be described at moderate distances from the source (greater than, say, 50 kilometers) by an acoustic pressure which is separable as follows

p(r,t) = {Lamb mode portion} + {ordinary acoustic portion}

(I .1) where the Lamb mode portion may be computed by techniques such as 9 10 discussed by Pierce and Posey and by Posey. The ordinary acoustic portion (which is the only portion considered here) may be taken as the ray acoustic (excluding the edge mode) solution of the wave equation

v2 (01470) (11c 2)a 2 (04-0)lat 2 4711(06a - 1. I - SC' (1.2) where the function f(t) is characteristic of the source. In addition, p/6-osatisfies approximately the boundary condition 2 3p/3z + (g/c )p = 0 at the ground (z=0). The justification for separating out the Lamb mode portion at the outset follows from a 1963 11 paper by Pierce which nay be construed as showing, for the special case of an isothermal atmosphere, that such a separation is possible at the frequencies of interest here. The rays proceeding from the source are lines, each of which lies in a vertical plane including the source (Fig. 2). Since.the geometry is circularly symmetric, we may limit our consideration to rays which lie in the Y,Z plane. A typical ray path passes through the source, bends downwards when the ray is proceeding up and the sound speed is increasing with height, bends upwards when the sound speed is decreasing, etc. This phenomenon of ray bending is known as refraction and makes it possible for more than one ray to pass RAYS

SOURCE \

zsc

GROUND

Figure 2.

Sketch of acoustic ray paths emanating from a source in an atmosphere in which the sound speed varies with height. -7--

through a given far field point. For distances and receiver locations of interest, one may regard this possibility of multi-ray arrivals as typical rather than the exception. The equations for computing such ray paths are well known and are discussed in particular in the 1966 7 report by Pierce. Computer programs which compute such paths are also in widespread use, especially in underwater sound studies. However, most such programs do not compute ray amplitudes. A somewhat lower order (or, strictly speaking, nonuniform) ray acoustic approximation to the solution of Eq. (1.2) is that

Z pray rays where the sum extends over all rays which connect the source and receiver. Here individual terms have signatures and amplitudes which 12,13 may be computed from tae eikonal approximation and from the condition that p reduces to

P//P 0 f(t- R/c)/R (1.4)

in the immediate vicinity of the source. However, the straightforward application of this procedure leads into difficulties if ray tube area, along any ray connecting source and listener, should vanish at any intermediate point along the ray. This difficulty, however, may be 14,15 largely overcome (although this seems to be rarely done) by simply adding a phase shift of 7/2; ie.

f(t) = Re fr(r) dw (1 .5) 0

is replaced by = f Shift (t) = Re e 1-7/2 f(w) e-1-63t dw • (1.6) 0 this shift being applied each time the ray tube area goes to zero along the ray. This is in addition to the normal shift due to travel time along the ray from source to listener. The successive shifting of phase by intervals of 7/2 is a relatively simple matter; the principal challenge in the application is that of determining the number of such phase shifts to be applied. There are two further modifications to Eq. (1.3) which, if incorporated into a computational model, should guarantee that results be good approximations down to relatively low frequencies and for large propagation distances of the order of 1000-18,000 km. These modifications include the explicit taking into account of caustics and lacunae (voids, skip zones, shadow zones, etc.) in the vicinity of the receiver. A caustic is a surface formed by a locus of points at which ray tube areas vanish or, alternately, at which adjacent rays intersect. The eikonal approximation breaks down at any point on a caustic and should be suspect near a caustic. The manner in which the computational method may be revised to incorporate an accurate theoretical model valid near caustics is one of the central topics in the present report. Examples of lacunae (see Fig. 3) occur whenever two adjacent rays split. The splitting leaves a shadow zone or a region in which there is one less ray than in adjacent regions. Lacunae occur in particular if there is a maximum in the profile of sound speed versus height. They also occur near the ground when the sound speed near the ground decreases with height. (The consideration of an image source and an image medium indicates the latter may also be regarded as a case where adjacent rays split.) The present report does not consider the lacuna problem. This is a limitation we hope to overcome in subsequent studies. The inclusion of caustics is regarded as a higher priority and it seems appropriate to thoroughly check out the techniques for including caustics before proceeding to the development of a method for including lacunae. In this regard, it is possible to conceive of a hypothetical model atmosphere in which caustics occur but lacunae do not. This would be a model in which there is no ground, the sound speed has a single minimum but no maxima. This is admittedly

SOUND SPEED

Figure 3.

Examples of the occurrence of lacunae in the propagation of rays from a source in a stratified atmosphere. The lacuna A occurs because of the splitting of ray paths at the height of a sound speed maximum, lacuna B occurs because of the presence of the ground and the fact that the sound speed initially decreases with height. -10-

not a realistic model,but it nevertheless should serve as a vehicle for checking out the computational method. The present report does not give a complete computer program for the prediction of acoustic waveforms via the ray acoustic model. Such a program is still under development. However, we do include in Appendix B a number of Fortran subroutines which have been developed to date, which nay be incorporated into such a program, and which exemplify the computational techniques. The emphasis in our discussion is on these techniques.

II. SOUND SPEED PROFILE

Sound speed data typically supplied in any computation scheme takes the form of individual values c (i=1,2,....,NCS) at heights i zi (i=1,2,..., NCS). However, in the types of calculations pertinent to geometrical acoustical predictions, one needs to know values of 2 2 c(z), dc/dz, and d c/dz at heights not necessarily coinciding with one of the z . To this purpose, we use an interpolation scheme known i as cubic splines and which was recently introduced into the under- 16 water sound propagation literature by Moler and Solomon . In these authors' notation, one lets

i - z (2.1a) Lzi = z i-1

Lc = (c. - c )/Az i=1,...,NCS (2.1b) i-1

= (z - z i_1)/Az i (2.1c)

= 1 - w (2.1d) and takes the sound speed c(z) for z between zi and z. to be of the 7-1 form of a. cubic polynomia3

2 6 2 c(z) = i -1 4 we... 4. (Az.) ra. (T, - ;)+a.(3w 1 L 1- 1 - 1)1 (2.2)

where the coefficients a i are constants chosen as described below. and z = z , this automatically reduces to c and c When z =zi-1 i i-1 respectively, so continuity of sound speed is automatically provided. The first, second, third, derivatives of sound speed according to the Noler-Solomon equation above are

2 dc/dz=Ac.+Az.La. i (3w 1 1 1-1 (3w - 1) + a - 13 (2.3a)

2 2 d c/dz = 6(7a1-1 + wa i) (2.3b)

3 3 d c/dz = 6(a. - a. )/Az. 1 1- 1 (2.3c)

dc/dz = Ac. - Az. (a. + 2a ) at z 1 1 1 i-1 1-1 (2.4a)

= Ac i+ Az.(2a. + a ) 1 1 i-1 at z.1 (2.4b)

d2c/dz2 = 6a 1-1 at z1-1 (2.5a)

= 6a. 1 at z.1 (2.5b)

2 2 Thus continuity of d c/dz is automatically insured while continuity of dc/dz requires

Ac + Az.(2a. +a ) = Ac -z. (a 1 1 i -1 1+1 1+1 i+1 +2a ii) (2.6)

for all values of i. Continuity of the third derivative is not imposed on the function. To determine appropriate values of the a i which insure continuity of the first derivative T.7e rote that Eq. (2.6) above implies

a = f(Aci_44-Aci)/Azi.0.-2a1+ Az11 i+1 1 dAz.+1 I

- a i-1 Az /Az i+1 (2.7)

and that, given a and a2, one could in principle generate all of the 1 succeeding a i 's. The linear 'nature of these difference equations implies furthermore that

a = K + L +M a (2.8) i " i i a2 i 1

for i >2, where

= A. - B K. - C.K (2.9a) KK. 1 il 1i-1

L. = -3iLi - C iLi_i (2.9b)

(2.9c) M i+1 = -314 i - C iN i-1

A = (1c iia- Ac i)/Az i+1 (2.10a) i

= , Az./Az ] (2.10b) B 1 i 1+1

C. = !z ihzi+1 (2.10c) 1

K 3 = A K = A -B A (2.11a) = 0; 2' 4 3 3 2

L = 1; L3 =_3 L = B B -C (2.11b) 2 4 3 2 3

(2.11c) M = ; = - • M = 2 3 C2 ' 4 2

Thus, if one starts kith the values K and K 2 3 given above, he may generate all of the successive K etc. i' Boundaryconditionsonthea.may be taken as =0. These araNCS are some•hat arbitrary but imply that the sound speed profile should be linear above z_!,CS a-' below z 1 . With this choice, one has

(2.12) 'ACS and the a. for i=3, , NCS are then computed according to Eq. (2.7). may be computed. In this manner all of the a i The computation just described is realized by a computer subroutine DASOL whose deck listing is given in Appendix B. The c i and z. are presumed stored in CO:.DION when this subroutine is called and the computed a (clenched ASOL) are stored in COMMON after this subroutine i returns. The number cf points is denoted by NCS (number of c's). The sound speed at an arbitrary value of_z is computed by a function subroutine CS?(Z). Given the value of z, this uses the values of the ,a., the c i and the z i (stored in COMMON) in Eq. (2.2) to compute the sound speed. (The deck listing is also given in Appendix B.) Analogous function subroutines are DCDZ(Z) and DCDZS(Z) 2 2 which compute the dc/dz and d c/dz at a given value of z according to Eqs. (2.3a) and (2.3b).

III. RAY PARAMETERS

For a height stratified atmosphere without winds, the ray equations of geometrical acoustics predict that

2 2 1/2 dx/dz = c/(v - c ) (3.1) where x is horizontal distance of the ray, z is vertical distance. Here, v , the horizontal phase velocity of the ray, is a constant for any given ray. Smell's law (a corollary of the ray equations) predicts that

v = c/(sine) = constant (3.2) • P where c is the local soumd speed, e is the angle between the momentary ray direction and the vertical. The choice of sign in Eq. (3.1) above depends on whether the ray is presently moving obliquely upwards or obliquely downwards.

Pit•c -14--

In a similar manner, the ray tracing equations predict that the rate of change of net travel time t along a ray with respect to height is

dt/dz = + (vp/c)/(17 - c2)1/2 (3.3)

The magnitudes Idx/dzI and Idt/dzi are computed by function subroutines R3KDZ(Z) and RDIDZ(Z). Both of these use the subroutine CSP(Z) to find the sound speed at height z. The phase velocity v p is assumed to be stored in COMMON. A turning point for a ray is a value of z at which c(z) = v p . In general if the sound speed profile has a minimum then there is an upper zu and a lower turning point zL. These are found by calling a subroutine TNPNT. This subroutine takes as inputs the phase velocity VP and the lower and upper bounds 23L and ZBU for the search. The search proceeds by dividing the interval (ZBU,ZBL) into NCS+4 intervals, each of width

= (Z3 t - ZBL) (NSCAN + 1) (3.4)

It successively p,rP ,-, ines the sign of the function CMVP(Z) = CSP(Z)-VP at points ZBU, ZBU 4- L, ZBU + 2A, etc., until an interval is found at which the signs at the two intervals are opposite, suggesting that a root is bracketed in that interval. The actual value of the root is found by a library subroutine ZREAL2. The search then goes on to succeeding intervals until a maximum of two roots is found. Output is NRTS the number of roots (0,1, or 2) and the values ZA and ZB of the roots; ZA is the first root (smallest z) and ZB is the second root (larger z). Typically, we would expect ZA to correspond to the lower turning point, Z3 to the upper turning point. In successive applications of integration between limits, one or both of which are turning points, it is important that one not over- shoot a turning point since then the square root in the denominator in Eqs. (3.1)and (3.3) would be imaginary. For this reason we have devised another subroutine called SHIFT which adjusts the values -15-

ZLOW and ZUP corresponding to a numerical approximation for the actual turning points to values which are in the immediate neighborhood of the input values but which are such that CSP(ZLOW) < VP and CSP(ZUP) < VP 8 The adjustments are carried out in units of 10 until these criteria , are satisfied. Integrals of Idx/dzI and Idt/dzI (or of any other z dependent quantity) between arbitrary values ZLOW and ZUP (not necessarily turning points) are accomplished by an integration function subroutine - RAINT. This performs such that

cZIP RAINT(RDUZ,ZLOW,ZIP) = Idx/dzI dz ( 3.5) JZLOW cZIP RAINT(RDIDZ,ZLOW,ZIP) = Idt/dzI dz (3 . 6) J. ZLOW

In the execution of this integration, the range of integration is broken into integrals from ZLOW to ZAVE and from ZAVE to ZUP where ZAVE = (1/2)(ZLOW + ZUP), i.e.

ZA\E ZA P INTE RAL = ( INTIG RAND) dz - (INTEL RAND) dz. ZLOW Z 1P

(3.7) The reason for this is that the library subroutine QUAD used to perform the integration is most efficient when it integrates away from a singularity and we anticipate the possibility that the integrand may be singular at either ZLOW or ZUP; these could be ray turning points. The integrals of Idx/dzI and Idt/dzI between lower and upper turning points are performed by a subroutine named RANG. The values of z corresponding to the turning point values are supplied as inputs, the other information needed is presumed stored in COMMON. Outputs are RTINE and RIM for the integrals over Idt/dzI and Idx/dzI respectively. The significance of these parameters is that the rays are periodic in path. The time required to go N half ray cycles is just

(N)(RTIME) while the horizontal distance traveled is (N)(RLNTH). Ray paths going from a given source location to a far field point may be characterized by (1) the horizontal phase velocity VP, (2) an index parameter IT which is 1 if the ray is proceding initially obliquely upwards, -1 if preceding initially obliquely downwards,

(3) another index parameter JT whose values +1 or -1 give the sign of dx/dz at the final point on the ray, (4) the number NUP of upper turning points which the ray passes through, (5) the number NDOWN of lower turning points, (6) the initial height ZSC of the ray, and (7) the final height ZLIS of the ray. These parameters are further explained in Fig. 4. One should note tha lo,if IT=JT, then NUP=NDOWN, if IT=1, JT=-1,then NDOWN=NUP-1; if IT=-1,JT=1 then NUP=NDOWN-1. The total horizontal distance R which the ray travels is

R = (N) (RINTH) + RST + REND (3.8)

where N is the number of complete half cycles the ray makes, given by

N = NIP + DOWN - 1 (3.9)

while !;ZIP RST = Idx/dzI dz IT = 1 (3.10a) ZSC

ZSC Idx/dzI dz IT =-1 (3.10b) SZLOW

NIP REND Idx/dzI dz JT =-1 (3.11a) ° ZLIS

ZLIS Idx/dzI dz JT = 1 (3.11b) ZLOW

2 NUP ZUP

ZSC IT= I ZLIS JT=-I ZLOW NLOW RLNTH RST REND

Figure 4.

Parameters describing a guided ray's path through the atmosphere; RLNTH is the half cycle ray repetition length, IT=1 or -l'if the ray is initially proceeding obliquely upwards or obliquely downwards, respectively, ..TT=1 or -1 describes slope at end point, ZUP and ZLOW are heights of upper and lower turning points, NUP is the number of upper turning points, NLOW is the number of lower turning points, RST is horizontal distance to first turning point, REND is corresponding distance from last turning point to receiver, ZSC is height of source, ZLIS is height of receiver.

•

-18-

The above formulas hold even should both NUP and NDOWN be zero, the computation giving for, say, IT=JT=1

rtZ IP ZLIS ZIP R = I Idx/dzi dz 14ZSC ZLOW ZLOW

fZLIS ldx/dzl dz (3.12) ZSC The computation of total range with the above listed inputs is accomplished by a subroutine named TOTRAN. It calls TNPNT first to find the turning points, then SHIFT to adjust the turning points so that the integrands exist throughout the integration range, then-- RANG to determine the ray half cycle length RLNTH and uses the library subroutine QUAD to find the initial and final integrals RST and REND. The above computation algorithms implicitly assume the lower point on any given ray is a lower turning point rather than the ground. The method may be easily extended to include ground reflections although we have not yet done so.

IV. RAYS CONNECTING SOURCE AND LISTENER

-Of pertinent interest in any ray acoustic calculation is the tabtilation of rays which connect given source and listener (receiver) locations. Let us denote source and listener heights by ZSC and ZLIS, the horizontal distance of listener from source by RANGE. Then, given a ray type denoted by parameters IT, JT, NUP, NDOWN as defined previously, and given a phase velocity VP we may define a function RMRkYD(VP) as the difference between actual range R and the range which would correspond to the given values VP, ZSC, ZLIS, IT, JT, NUP, and NDOWN. If this function is zero, then the ray being considered does pass through the listener location. Otherwise, it does not. The function subroutine RMRAYD computes this difference, VP is an input, • the remaining necessary parameters are stored in COMMON. -19-

To find the values of VP at which

Ri RAYD ( = 0 (4.1)

given fixed ZSC, ZLIS, IT, JT, NUP, and NDOWN, a subroutine MVP is used. This scans values of VP between VPHST and VPHEND at intervals of SDELT until an interval is bracketed within which RMRAYD changes sign. Once such an interval is found, a library subroutine ZREAL2 is used to find the precise value of the root. Up to NMAX such roots are found, the number actually found is denoted NFND, the roots being

denoted V?1,0(1), WW(2) NPED(NI•D). By use of FNDVP, one can, in principle, find all rays of a given type which connect source and listener. A systematic variation of ray types (IT, JT, NUP, and NDOWN) will in this manner give all the rays connecting source and listener.

V. RAY SPREADING

Two coplanar rays, both proceeding initially either obliquely upwards or obliquely downwards, may be characterized by phase velocities vP1 and v Assuming that v is arbitrarily close (but not identi* p2. p2 cally equal to) v we may characterize the separation of the rays pl by a parameter As which (see Fig. 5) is the perpendicular distance from a point on the first ray to the second ray. We consider As as positive if the second ray lies above the first, negative if below the first. The parameter Asmaylbe considered a function of horizontal distance x and also of the phase velocity. The limit

lim As/ _v ds/dv = p2 pl 171324.vp1

may be considered a uniquely defined function of range x, phase velocity vP' ray type (IT=1 or -1) and ray initial height ZSC. We term this

derivative the ray spreadinf function. One may note that within any RAY 2

Figure*5.

Definition of parameter As characterizing two adjacent rays with horizontal phase velocities v p1 and vp2. Note that As changes sign when the rays cross.

-21-

ray segment (i.e. between turning points)

21/2 ds/dv p = -±(dx/dv )/{1 + (dx/dz)

2 1/2 = ± (dx/dv )fl - (c/v ) 1 (5.2)

where the plus sign applies if the ray is proceeding obliquely down-, wards (JT=-1), the minus sign if it is proceeding obliquely upwards (JT=1), dx/dvp is the rate of change of horizontal distance traveled with respect to phase velocity at fixed z and fixed ray initial position. The derivative dx/dv may in turn be calculated if one knows the general ray type. For a ray proceeding initially upwards (IT=1) and going through NUP upper turning points and NDOWN=NUP lower turning points and ending with direction obliquely upwards, one has, for example,

Z LP sZIP Idx/dzI dz + N Idx/dzl dz. + jdx/dz 4ZSC ZLOW ZLOW

(5.3)

where N = NUP + NDOWN -1 = 2(NUP) -1. Here the integrand Idx/dzl is given by Eq. (3.1). To differentiate this expression with respect one must take into account the fact that ZLOW and ZUP as well to v 13, as Idx/dzI depend on vp. A formal application of the rules for diff- erentiating an integral with respect to a parameter leads to singularities and some tricks are required to avoid this. In particular, it is convenient to rewrite the above as •

= i(zsc,zta) + (N+1) i(zu,zip) + (Nu.) I(ZLOW,ZLI)

+ (N+1) I(ZLOW,ZLI) + (N) I(ZLI,Z1I) + I(ZLI,Z)

(5.4) where I(Z1,Z2) represents the integral of Idx/dzl between the indicated limits, ZUI is a fixed (v independent) value of z slightly less than P ZUP, ZLI is slightly larger than ZLOW. (See Fig.G.) One may also note

that RANGE

Figure 6.

Definition of parameters ZUI (slightly below upper turning point ZUP) and ZLI (slightly above lower turning point ZLOW) used in the calculation of ray spreading parameter ds/dvp. I (Z11,2 1P) = 1(ZIP-z) d dz (5.5) Z11

-1 2 Idx/dzi -(dc/dz) (didz)(v - (5.6) so an integration by parts gives

-1 2 2 1/2 I(ZII,ZIP) = f(dc/dz) (v - c ) ) Z II

2 c2)1/2 (v - IKZIP- )(didz)(dc/dz) -1 dz Z11 (5.7) and, consequently, one has

(d/dv )I(Z11,Z12) = f(v /c)(dcidz) -i ldxidz I ) 211 ZIP + (v ic)Idxidd(didz)(dcidz) -1 dz P Z1I (5.8)

Providing dc/dz does not vanish in the interval between ZIT and ZIP,. both c these terms should be finite. In a similar manner, one can show that

(didv )I(ZLOW,ZLI) = - 1 (v /c)(dc/dz) -1 !dxidzI) ZLI

ZIP (v /c)Idx/dzl(didz)(dc/dz) -1 dz ZLOW

( 5.9)

Int• I-

-24-

The derivatives of the remaining terms in the expression (5.4) are relatively simple since the integration limits are independent of,vp. In particular one has

ZII 2 2 -3/2 (d/dv )I(ZSC,ZII)= - S c)(v -c ) dz (5.10) ZSC P

Thus one obtains the expression (IT=1, JT=1)

dx/dv = Il(ZSC,ZII) + (N+1)J1(ZII) + (N+1)I2(Z1I,Z1P)

- (N+1)J1(ZLI) + (N+1)I2(ZLOW,ZLI) + (N)II(ZLI,ZII)

+ I1(ZLI,Z) (5.11)

where we have abbveviated

ZB 2 2 -3/2 Il(ZA,ZB) = -cv (v -c ) dz (5.12a) ZA P P

J1(ZA) = f (v /c)(dc/dz) -1 1dx/dzil (5.12a) z=ZA

ZB I2(ZA,ZB) = (v /c)Idx/dzi(d/dz)(dc/dz) -1 dz (5.12c) ZA

In general, for a ray of specified type (IT, „TT, NUP,NDOWN), the corresponding expression for dx/dv is p

Il(ZSC,ZII) dx/dv p= • + (2)(NIP)J1(ZII) + (2)(N1P)I2(ZII,ZIP) Il(ZLI,ZSC)

-.(2)(NDOWN)J1(ZLI) + (2)(NDOWN)I2(ZLOW,ZLI)

+ (;g1P+NDOWN-1)I1(ZLI,Z1I) + (5.13) Il(Z,Z1I) The two possibilities for the first term correspond to IT=1 and -1, respectively, while two possibilities for the second term correspond to JT=1 and -1, respectively. The integrand for the integrals of type Il is computed by a function subroutine FIRM(Z), while twice the values of those of type 12 are computed by a function subroutine FIRMUL(Z), i.e.

Il(ZA,ZB) = RAYINT(FIRM ZA,ZB)

Il(ZA,ZB) = RAYINT(FIRMUL,iA;ZB)/2

Also the quantity 2[J1(Z)] is denoted in the program by TRNPT(Z), i.e.

-1 2 2 -1/2 TRNPT(z) = 2v (dc/d (v -c ) (5.15) so the expression for dx/dv p becomes

EICD = TERM ST + (NIP)TRNPT(Z 11) + (N1P)RAYINT ( 11R1 IL ,Z 11,Z IP)

- (NDOWN)TRNPT(ZLI) + (NDOWN)RAYINT(FIRAIL,ZLOW,ZLI)

+ (N1P+NDOWN-1)RAYINT( FIRq,ZLI,Z1I) + TE1 LT (5.16) where the first and last terms are

TER4ST = RAYINT(IIRA,ZSC,Z1I) IT = 1 (5.17a) = RAYINT(ELR4,ZLI,ZSC) IT =-1 (5.17b) TERALT=RAYINT(FIRM,Z,Z11) JT =-1 (5.18b) = RAYINT(IIR4,ZLI,Z) JT = 1 (5.18b)

One may then calculate ds/dv from Eq. (5.2), i.e.

2 1/2 DSD%11 = - SEN(JT)(EKDIP)(1-(c/v 1 ) (5.19) -26-

The sequence of computations just described is carried out by a subroutine CDSDVP. The parameters VP, ZSC, Z, IT, JT, NUP, AND NDOWN are inputs, the output is DSDVP. The parameters ZLI and ZUI are computed internally and set to

ZLI = ZLOW + .ol(zIp-npw) (5.20a)

Z u =- Z IP - . 01 (Z IP-ZLOW) (5.20b)

The choice of .01 is of course arbitrary. The chief constraint is that dc/dz should not vanish between ZLOW and ZLI and between ZUI and ZUP. If one considers the variation of ds/dv with x along a single ray (say with IT=1) it is apparent that up to the first upper turning point ds/dv p should be positive since FIRM(Z) is negative; JT is positive. At the turning point one has

ds/dv = limit u_iciv 1 2) 1/2 37 -3/2 p z +ZIP f cv 2 -c 2 ) ZSCP P = { 1/ (dc/dz)} 211, (5.21) which, interestingly, is independent of ZSC. This follows if one breaks the integral above into integrals from ZSC to ZUI and from ZUI to Z, given ZUI < Z< ZUP, and expands c in a power series about its value v at z=ZIP. Between the first upper turning point and the first lower turning point the function ds/dv is given by

ds/dv l_(civ )2}1/2 RAYINT ( FIR1 ,ZSC , Z II)

+ TRNPT (Z 11) + RAYINT(FIRI TL , ZLI , Z IP)

+ RAYINT ( FIRM ,Z ,Z II)/ (5.22) A brief analysis indicates that this can be put in a form independent of ZUI, i.e.

-27-

1/2 3/2 1/2- 3/2 ds {v /2 ) /a (v /2) /a „ p {1-(c/v p)2}1/2 1/2 dv (ZIP - ZSC) (zip 2

ZIP EP (1) (1) Arg (zo ,ZIP)dzo - r Arg (z ,ZIP)dz SZSC o

(5.23) where

2 cv v (1 p Arg )-(z,ZIP) 2 2 3/2 3/2 3/2 (v - c ) (ZIP-z) (2av )

(5.24) and we have abbreviated --afor dc/dz at ZUP. The subtracted term in the arguments insures that the integrals exist. Also, as Z+ ZUP the second term in the brackets dominates and one has

'7 , 1/2 1/2 1/2 - (c/v (2a/v ) (ZIP-z) (5.25)

and ds/dv o 3 V a in accordance with Eq. (5.21). On this basis, we may conclude that the __amity in braces in Eq. (5.22) starts out large and positive for Z close to ZUP,decreases monotonically (since FIRM(Z) is always negative) and eventually goes to - 03 when Z ZDOWN. Thus there is one and only one point on the ray between the first turning point and the second turning point at which ds/d/r0. This point is identified as a point on a caustic (where adjacent rays intercept).

At the second turn inz point (first lower turning point) the same sort of limiting process described above gives

ds/dvp = .1/(dc/dz)} ZLOW (5.26)

which as mentioned above is a negative number. Between the first lower (second overall) and second upper (third overall) turning points, one may similarly argue that ds/dv goes P to zero at one and only one point, etc., before that point ds/dvp is -28-

negative, after that point it is positive, it approaches it/(dc/dz)l ZUP at the next upper turning point, etc. The general situation is as sketched in Fig. 7. The number of times ds/dvp goes to zero along a ray path (i.e., the number of caustics encountered) is just

Number of caustics = (Number of complete half ray cycles)

+ (zero or one) (5.27)

The second term is zero if JT=l (upgoing ray) and the current value of ds/dv is negative or if JT=r-1 (downgoing ray) and the current value of ds/dv Fis positive. Otherwise, it is one. The number of complete half ray cycles, one may note, is just NUP + NDOWN -1 if either NUP or NDOWN are greater than one. Thus, it is a simple matter to determine, at a given point on a ray, just how many caustics the ray has encountered in passing from source to that point.

VI. RAY AMPLITUDES

Given that the acoustic pressure in the immediate vicinity of the source is of the form implied by Eq. (1.4), the Fourier transform -3- ; p(w,r))defined such that

CO pa,t) = Re Io(wMe-itut dw 4, (6.1) So of the acoustic pressure may be inferred from the geometrical acoustics 7 model to be (in first approximation) given by a sum over rays. The contribution from any particular ray connecting source and receiver is simply /

= fluOP1/2-„ (zSC )(Atmosphere factor}{ Spreading factor] Pray o N iwt e ray k { c} (6.2) -29-

180

FIRST SECOND 100 UPPER UPPER

■••■• V a 20 RANGE 20 (km) 0 -20

-100 FIRST SECOND LOWER LOWER

-180

Figure 7.

Values of ds/dv p along two adjacent guided rays, illustrating the conclusion that the number of caustics encountered is the number of complete half ray cycles traversed plus 0 or 1.

-30-

where N is the number of times the ray has touched (tangentially) c is net travel time along the ray. The atmosphere a caustic, tray factor is given by

{Atmosphere factor} = f (o0c) zi(Poc)SC/1/2 (6.3)

while the spreading factor is the inverse square root of the ray tube area normalized such that this factor reduces to 1/R near the source (i.e., at the beginning of the ray). The criterion for determination of these factors is that

f 12/o 'ray oclfray tube area} = constant (6.4) . - along a ray, that the limit (1.4) be realized and that the net phase ubange from source to receiver be nUt + u/2. ray c A consideration of a cylindrically symmetric bundle of rays leaving the source at angles between 0 and 0-1-d0 with respect to the vertical leads one to the conclusion that the ray tube area should be a constant times i(ds/dvdruor l where ds/av is the quantity dis- P cussed in the previous section, rHGr is horizontal distance from source to listener. One can also show, by considering a medium in which the sound speed is constant, that near the source

2 2 3 R c /v Ids/dv I = rHor (6.5) fl-(c/v ) 2 1 1/2 so one identifies the spreading factor as the square root of

2 3 2 c /v {Spreading factor} = 2 1/2 (6.6) f1-(c/v ) 1 p rHor lds/dv pl where c is here taken as the sound speed of the source. One may note that the spreading factor goes to co whenever ds/dv p goes to zero, i.e., at a caustic. This is one indication that the general formula may not be applicable everywhere. The modification -31-

of the method to take into account proximities to caustics is discussed in the remaining portions of the report.

VII. GEOMETRY NEAR CAUSTICS

When viewed in a vertical plane containing the source, caustic surfaces appear locally as arcs of circles, the rays which touch it also appear locally as arcs of circles; the situation is as sketched in 8. Each caustic has a shadow side and an illuminated side. If a receiver is on the illuminated side, then one may expect in general that two rays touching the caustic tangentially will also pass through a point A on the illuminated side, both of these rays will have approximately the same radius of curvature R and will touch ray the caustic at points B and C, such as indicated in Fig. 9. Para- meters of interest here are (1) the radius R of curvature of the c caustic, (2) the distance 6 from point A to the caustic, (3) the arc length (A6)Re = £ along the caustic between points B and C; and (4) the angle 4, between the two rays at point A; as well as (5) the radius R of curvature of the two rays. These parameters are related and it ray is a challenging exercise in analytical geometry to determine their interrelationships. Fortunately, the end results are relatively simple the case of interest where 6<< R 6<< One finds, in parti- in c' Rray. cular

1 1 6 = (1/8)(R R )1.2 c ra y (7.1)

-1 1 0 = (R +R )g ray (7.2) Another quantity of interest is the separation As between two rays which touch the caustic at points 6 = -A0/2 and A6/2 9). Tf we irtPrnret As as positive if the second ray lies above the first, then

AO 1 As/(RAO) = -E(R 1+ R ) (7.3) c c r ay ILLUMINATED SIDE

SHADOW SIDE

Figure 8.

Sketch of rays in the vicinity of a cau stic. The caustic is approximately an arc of a circle, the rays are also locally arcs of circles. Note that the caustic has an illuminated side and a shadow side. Figure 9.

Detailed sketch of two rays which cross on the illuminated side of a caustic at a point A and which touch the caustic at points B and C respectively; R c is the radius of curvature of the caustic, st ray is the radius of curvature of either ray; d is the distance of A from the caustic, ¢ is the angle between the two rays where they cross, £ is arc distance along caustic between points B and C, E is arc length along either ray, As is the separation distance between the two rays.

where is distance along either ray in the positive sense from the caustic. Thus, if the upgoing ray in Fig. 9 is characterized by phase velocity vpl, the do;ingoing ray by phase velocity v p2 9we may characterize their respective ds/dv at the point A by

-1 1 (ds/dv ) = -(di/dv )(t/2)(R +R ) (7.4a) p 1 ray

1 (ds/dv ) = (d2/dv )(Q12)(R 1 +R ) (7.4b) p 2 c ray

where

-v ) (7.4c) dk/dv = Z/(v p2 pl

It should be noted that (ds/dvp)1 is equal and opposite to (ls/dv p) In typical applications, such as are discussed in the next section, it may be presumed that the point A is known, the phase velocities and slopes:15f the two rays and therefore ¢ aue known, the ray radius R ray is known, the parameters (ds/dv p) 1 and (ds/dvp) 2 are known and are equal and opposite, but R c , 6, and R are not known. A successive . solution of Eqs. (7.1-4) for the unknowns in terms of the knowns gives

= -(1/4)(v p..- )(ds/dvp) 1 = (1/4)(vp2-vp1)(ds/dv z p) 2

= (1/8)(vp2 -v )[(ds/dv pl p ) 2 - (ds/dvp) 1 ] (7.5a)

= • (v -v )[(Es/dv ) -(ds/dv p2 p1 p 2 (7.5b)

1 -1 R +R = f( -v )[(ds/dvp ) -(ds/dv ) 11 (7.5c) c ray vp2 pl 2 p 1

If we wish to characterize the distance of the point A from the caustic by a relevant dimensionless parameter, the natural choice (as explained subsequently) is the caustic proximity parameter whose definition may be taken to be

1/3 2/3 1/3 n-2 (/c) )+(l/R )] (7.6) [(1/R ray c -35-

This is negative on the illuminated side and, as may be noted, depends on the angular frequency w. In terms of the ray parameters described above, one may state that n for the point A on the illuminated side is

1/3 2/3 n = -2 (w/c) (1/8)f(v -v )[(ds/dv -(d /dv ) 11 2/3 p2 pl p 1

(7.7) which should always be negative (i.e.,[-IfD 2/3=[Ifl] 2/3).

VIII. THE SEARCH FOR CAUSTICS

To explore the possibility of the receiver being near but on the illuminated side of a- - caustic, all of the rays connecting source and receiver are ordered according to increasing phase velocity, those initially going obliquely upwards and obliquely downwards being considered as separate groups. For each successive pair of rays (i,i+l), one computes the corresponding values of ds/dv and determines the number of times each ray has touched a caustic according to the prescription in Sec. V. If the signs are the same or if the N c 's differ by a quantity other than one, no action is taken and one proceeds to the next pair (i+1-.)-i, ii-2.+1+1). Once the above criteria are satisfied, one terms the two rays as a possible caustic pair. They are tempo- rarily reordered such that the one with the larger N c is called "the first ray" the one with N c being 1 less is called "the second ray". The slopes of the two rays are determined from Eq. (3.1) and the angle ¢ (which could be negative) is computed in accordance with the corres- pondence in Fig. 9. One also computes 6 from Eq. (7.5a). Then one checks to see if t, and 6 have the same sign. If not, the process starts over with the next pair. If they do have the same sign, then one computes the caustic proximity parameter n according to Eq. (7.7). If Inl>4, one would decide that the caustic is too far away for any special modifications. However? if one finds Ini.14, the contribution

-36-

to the sum over rays from those two rays is deleted from the sum and replaced by a new composite term involving Airy functions. (The method of doing this is explained in the next section.) The second possibility is that the receiver lies near a caustic but on its shadow side. The following type of search is contemplated. First one examines the function RMRAYD(VP) described in Sec. IV. If the absolute value of this function has a local minimum (not zero) for some value of the phase velocity, then the possibility of the receiver being near the caustic is indicated. The search for such local minima is similar to that described in the discussion of FNDVP: one scans successive values of RMRAYD until one finds three successive

phase velocities such that (1) all three wRAYn's have the same slap and (2) the magnitude of the middle one is less than either of the two end ones. One then breaks this bracketed interval down into, say, 20 subintervals, calls FNDVP to see if . there are any roots in the

interval. If FNDVP finds two roots 2 these are considered as rays connecting source and listener and the process stops. If FNDVP finds only one, the subdivision is made progressively smaller until two roots are found (if there is one, there must be two) and these roots are added to the overall group of rays connecting source and listener. If FNDVP finds no roots, then the local minimum is found by the above scanning process and one continues this iteration until the location of the minimum is accurately bracketed. Its precise location is found by fitting a parabola to the final triplet of points and then finding the minimum of this parabola. The parameters IT, JT, VP, MP, NDOWN are then considered as defining a near miss ray. To locate the point on the considered caustic which is closest to the actual receiver location, one considers the two equationd

x = x(v ,z) (8.1a)

ds/dv = r(v ,z) (8.1b) P -37-

with IT, JT, NUP, NDOWN considered fixed. The two indicated functions may be considered as defined by subroutines TOTRAN and CDSDVP. The caustic is the locus of points at which dx/dvp=0. The scheme outlined above gives one such point. Successive points are determined from

3 Flav = -(3 P/3z)(dz/dv )

dx/dv = (3,x/av ) + (3x/3z)(dz/dv ) or

dz/dv p = -(3 F/3v )/(3F/3z) (8.2a)

dx/chr = ( ax/av) — (ax/az) F/av )/(a vaz) (8.2b)

One may note that these two functions on the right hand side are easily programed. One now simply numerically integrates these differential equations until he reaches a point at which the distance of (x,z) from the actual receiver location is a minimum. The scanning regime must, however, be restricted to points at which 31P/3z is nonzero, the other quantities on the right hand sides should be finite. The minimum distance is that corresponding to the allowable scanning region. Once this minimum distance point has been found, one varies x and z until a neighboring point is found at which two rays pass through with approximately the same value of v p as that corresponding to the caustic point. Parameters corresponding to these two rays at this new point are tabulated and one determines the approximate circle which describes the caustic in their vicinity according to the equations given in Sec. VII. The caustic proximity parameter corresponding to the receiver location is then computed according to Eq. (7.6) only with (l/Rc) + (l/Rray) replaced by Eq. (7.5c), 6 is replaced by the negative of the distance of the receiver location to the caustic circle. The parameter n so computed should be positive, otherwise the search in this instance stops. If n is greater than, say, 5, the presence of the caustic is disregarded.

ink: a -38-

Otherwise, it is taken into account by the method described in the next section.

IX. FIELD NEAR A CAUSTIC

The method we adopt for incorporating caustics into the computation 6 is based on results derived by Haskell in 1951. While Haskell was primarily concerned with the nature of guided modes near turning points, his analysis may easily be reinterpreted as implying that, near a caustic, the contribution due to the two rays which intersect at a point A on the illuminated side (see fig. 9) should be replaced by

p = (G)expr icot e Mi( n) (9.1)

where

2/3 = (3/2 d6 (9.2)

17 Here Ai(n)is the Airy function defined by

Ai(n) = (1/1 cos[ (s 3/3)+ ns] ds (9.3) o

Also, t c is ray travel time from the source to the point on the caustic - ' closest to the receiver point; is the component, normal to caustic, of either ray's wave number vector (w/c times unit vector in ray direction); S is the perpendicular distance from the caustic. The function G is a slowly 7arying function of position chosen such that Eq. (9.1) matches on to the corresponding ray theory expression when n « —1.

As regards the matching on, one may note that, if n « -1, the Airy function approaches an asymptotic limit17

7T -1/2 1n -1/4 2/84T sin[(2/3)In1 /4] (9.4)

so Eq. (9.1) above approaches

d in11/4] r ; .+N/(2Oileiu/4/ eXPL iWt k,dd] - i expj iwt +i k,dd] 1 c ' o (9.5)

the first term is identified, with k1>0, as the contribution from the ray which has not yet touched the caustic, the second from the ray which has already touched the caustic. This follows since the

tray . = t c T r(ki/w)dd (9.6)

correspond to the travel times of the two rays, respectively, to the point under consideration on the illuminated side. A verification of this latter statement may be given from consideration of the fact that the tray for rays coming into the caustic may be considered as 7 a continuous function of position which satisfies the eikonal equation

(Vt )2 = 1/c2 ray (9.7)

where tray reduces to t c at the caustic. Consequently, if the component of normal to the caustic is -k Vtrav i/w (wave number vector divided by w is gradient of the eikonal function) then .

6 t 7 t Vt dd ray c ray (9.8) 0

which is just Eq. (9.7) with the minus sign. Similar considerations apply for the eikonal function tray of rays leaving the caustic and the identification corresponding to the plus sign is recovered. In the vicinity of the caustic, given the respective geometry sketched in Fig. 9, the value of Ici may be readily shown to be approximately

= (w/c)V2i(1/R 1/2 1/2 ray) + (1/Rd] ,5 (9.9

. _ . this holding to a high relative appvoximation very close - to the caustic. Consequently, the value of ri is given by

2/3 1/3 1/3 rt = -WO 2 [(1/Rray)q-(1/Re)] d (9.10)

which as might be expected is exactly the same as given in Eq. (7.6) for the caustic proximity parameter Also, one should note, on eliminating .4, from Eqs. (7.1): and (7.4), that

1/2 51/2 (ds/dv ) = -(dt/dv )V2 [(1/R c)+(1/R )1 (9.11a) p 1 ray

= -(dsidvp) 2 (9.11b)

SO 1/4 1/6 1/1711 = (2c/w) [(1/1(c)+(1/Rray)] dk/dv 1/2 / Idsidv 1 1/2 1/61 P 1 P (9.1• •

The fact that the two individual terms in Eq. (9.$) must correspond to Eq. (6.2) allows us to identify the parameter G in the former as

1/2, f.(m).o ;Atmosphere factorlf Spreading factor with Ids/d1-1/2omitted} o V -1/6 1 -1/2 (2/0ein/4 (_i) pc (2c/f.:.:) [(1/Rc )+(liR W14-61 dtidv ray I J P ■ (9.13) -41-

where Npc is the number of prior caustics encountered by the two rays. These formulas developed above give one a straightforward method for incorporating caustic corrections when the receiver lies on the illuminated side of the caustic. Given the parameters describing two rays which touch the caustic, these parameters being appropriate to the receiver location, one first computes n according to Eq. (7.7), computes 1/Re + 1/Rray according to Eq. (7.5c), computes 2: according to Eq. (7.5b), then di/dvp from Eq. (7.4c). These numbers are then used to calculate the factor G in Eq. (9.13). The parameter t e is just the average travel time of the two rays from the source to the receiver location. As regards the calculation of the Airy function Ai(n), subroutines 10 capable of evaluating this' function are given by Posey in his thesis, so there is no real computation problem involved. If the receiver is on the shadow side:of the caustic, the process is similar, but one must first find two rays passing through a point (on a line from the receiver normal to the caustic) on the illuminated side in order to determine dk/dvp , [(1/Re) + (1/Rray)], and t . Once c this is done, the parameter n is computed from Eq. (9.10), only with replaced by the negative of the distance from the receiver to the caustic. The function G is computed just as described previously. Since the Airy function decreases as

Ai(n) = (1/2)n-1/2n-1/4 e-(2/3)n3/2 (9.14) for large positive n, we may anticipate the contribution from the caustic on the shadow side to decrease relatively rapidly. Since Ai(0) = .355, 4 Ai(5)Z1.1x10 , one can certainly ignore values when n is greater than 5.

X. CONCLUDING REMARKS

The computational method outlined here is still under development and, at present, computer subroutines are available for performing only part of the steps envisioned for the overall waveform synthesis.

33•3^:. 3. -42-

The computer subroutines presently available are given in Appendix B along with a sample MAIN program which calls them and which may be used in studying acoustic propagation with the use of these subroutines. The project is being continued as a Ph.D. dissertation by Mr. Kinney and it is expected that an operational and comprehensive computer program based on the computation method should be available by summer 1976. It should also be stressed that the overall method described here is expected to avoid many of the limitations one customarily associates with ray theory computations. The fact that the method produces amplitudes and phases rather than merely finding ray paths and travel times is significant. Also the fact that it allows for the possibility of more than one ray connecting source and receiver is important for realistic infrasound applications. The method of taking the presence of caustics into account should extend the applicability of the geometrical acoustics theory down to frequencies formerly considered to be the sole domain of guided mode theory and should be regarded as an important extension of the geometrical acoustics theory. There are still some unsatisfactory features in the theory which might be given additional attention. One of these is the neglect of lacunae previously mentioned in the Introduction. While some work has been done on propagation into a shadow zone, e.g. by Pekeris 18 19 the results are difficult to inter- and by Ingard and Pridmore-Brown, pret in the generalized sense required for incorporation into a computation scheme such as described here. Thus, some considerable intellectual effort probably remains to be exerted before one may satisfactorily handle lacunae. Closely related to the lacunae problem is the coupling of two adjacent sound channels. The present theory assumes, in particular, that energy trapped in one channel stays in that channel. In reality, there is always some penetration of energy from one channel to the other and one may envision that a satisfactory description may be found by using an extended WKB approximation, matching at turning points on both sides of the barrier comprised of the region where the sound speed is higher than the horizontal phase velocity. There is also the problem of aretes 20formed by the meeting and termination of caustic surfaces. Here the idealization of a caustic having a radius of curvature much larger than a wavelength breaks down and the theory developed here becomes inapplicable. However, we believe aretes to be so isolated in occurrence that the possibility of a random receiver location being close to an arete or of lying on a ray which touched a caustic close to an arete is relatively small. Thus, there would seem to be little urgency in taking such phenomena into account. The incorporation of winds, additional dispersion due to gravity, earth curvature, sound absorption due to dissipative processes, and of phase shift on ground reflection would seem to be relatively minor problems since the theory for doing so is relatively well developed and is discussed in particular in previous reports written under this project. We have chosen not to include such effects in the discussion here primarily because of the premise that one may make faster progress in the long run if he first starts out with a simpler model, checks this model out thoroughly, and then adds the embellishments needed for a more nearly accurate simulation of nature in a sequential fashion. -44-

REFERENCES

1. Thomas, J. E., A. D. Pierce, E. A. Flim, and L. B. Craine, "Bibliography on Infrasonic Waves", Geophys. J. Ray. Astr. Soc. 26, 399-426 (1971). (This reference contains an extensive list of papers published prior to 1971. The following references are noted in particular since they describe work done on the same Air Force project of which the present report is a part).

2. Pierce, A. D., and C. A. Moo,"Theoretical Study of the Propaga- tion of Infrasonic Waves in the Atmosphere': Report AFCRL-67- 0172, Air Force Cambridge Research Laboratories, Bedford, Mass. (1967).

3. Pierce, A. D., and J. W. Posey,"Theoretical Prediction of Acoustic- Gravity Pressure Waveforms Generated by Large Explosions in the Atmosphere' Report AFCRL-70-0134, Air Force Cambridge Research Laboratories, Bedford, Mass. (April, 1970).

4. Pierce, A. D., Charles A. Moo, and Joe W. Posey,"Generation and Propagation of Infrasonic Waves': Report AFCRL-TR-73-0135, Air Force Cambridge Research Laboratories, Bedford, Mass. (April, 1973).

5. Pierce, A. D., Wayne A. Kinney, and Christopher Y. Kapper, "Atmospheric Acoustic Gravity Modes at Frequencies Near and Below Low Frequency Cutoff Imposed by Upper Boundary Conditions': Report AFCRL-TR-75-0639, Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. (March, 1976).

6. Haskell, N. A., "Asymptotic Approximation for the Normal.Modes in Sound Channel Wave Propagation", J. Appl. Phys. 22, No. 2, 157-168 (1951).

7. Pierce, A. D.,"Geometric Acoustics' Theory of Waves from a uPoint Source in a Temperature - and Wind - Stratified Atmosphere, Report AFCRL - 66 - 454, Air Force Cambridge Research Laboratories, Bedford, Mass. (August, 1966).

8. Pierce, A. D., J. W. Posey, and E. F. Iliff, "Variation of Nuclear Explosion Generated Acoustic - Gravity Waveforms with Burst Height and with Energy Yield", J. Geophys. Res. 76, 5025-5042(1971).

9. Pierce, A. D., and J. W. Posey,"Theory of the Excitation and Propagation of Lamb's Atmospheric Edge Mode from Nuclear Explo- sions", Geophys. J. Roy. Astron. Soc. 26, 341-368 (1971).

10. Posey, J. W.,"Application of Lamb Edge Mode Theory in the Analysis of Explosively Generated Infrasoune: Ph.D. Thesis, Dept. of Mech. Eng., Mass. Inst. of Tech., (August, 1971).

11. Pierce, A. D. "Propagation of Acoustic -Gravity Waves from a Small Source above the Ground in an Isothermal Atmosphere", J. Acoust. Soc. Amer. 35. 1798-1807 (1963). -45-

12. Sommerfield, A., and J . Runge, Ann. der. Physik 35, 277-298 (1911).

13. Blokhintzev, D., "The Propagation of Sound in an Inhomogeneous and Moving Medium", I., J. Acoust. Soc. Amer. 18, 322-328 (1946), II., J. Acoust. Soc. Amer. 18, 329-334 (1946).

14. Poincare, H., Theorie Analvtique de laLumiere (Georges Carre, Paris, 1889).

15. Tolstoy, I., "Phase Changes and Pulse Deformation in Acoustics," J. Acoust. Soc. Amer. 44, 675-683 (1968).

16. Moler, C. B., and L. P. Solomon, "Use of Splines and Numerical Integration in Geometrical Acoustics," J. Acoust. Soc. Amer. 48, 739-744 (1970).

17. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965) (see in particular Chapter 10).

18. Pekeris, C. L., "Theory of Propagation of Sound in a Half-Space of Variable Sound Velocity under Conditions of Formation of a Shadow Zone", J. Acoust. Soc. Amer. 18, 295-315 (1946).

19. Pridmore-Brown, D. C. and U. Ingard, "Sound Ptopagation into the Shadow Zone in a Temperature-Stratified Atmosphere above a Plane Boundary", J. Acoust. Soc. Amer. 27, 36-42 (1955).

20. Pierce, A. D., "Maximum Overpressures of Sonic Booms Near the Cusp of Caustics", Purdue Noise Control Conference Proceedings, July 14-16, 1971, 478-485. • APPENDIX A

BIBLIOGRAPHY OF RELATED WORKS

PERTAINING TO GEOMETRICAL ACOUSTICS - 47 -

Albers, V. M., Underwater Sound (Dowden, Hutchinson and Ross, Inc., Stroudsburg, Pa., 1972).

Barnes, A. and L. P. Sol=on, "Some Curious Analytical Ray Paths for Some Interesting Velocity Profiles in Geometrical Acoustics", J. Acoust. Soc. Am., 53, 140 (1973).

Barry, - G., "Ray Tracings of Acoustic Waves in the Upper Atmosphere", J. Atmos. Terrest. Phys., 25, No. 11, 621 (1963).

Bergman, P. G., "The We, :e Eruation in a Medium with a Variable Index of Refraction", J. Acoust. Soc. Am., 17, 329 (1946).

Brekhovskikh, L. M., "A Limiting Case of Sound Propagation in Natural Wavelengths", Sov. Phys. Acoust., 10, 89 (1964).

Brekhovskikh, L. M., "The Average Field in an Underwater Sound Channel", Soy. Phys. Acoust., 11, 126 (1965).

Brekhovskikh, L. M., Waves in Layered Media (Academic Press, New York, 1960).

Brekhovskikh, L. M., "Possible Role of Acoustics.in the Exploring of the Ocean", Raoports du 5e Congres International d'Acoustique, Vol. II: Conferences Generales, Liege (1965).

Bucker, H. P., "Sound Propagation in a Channel with Lossy Boundaries", J. Acoust. Soc. Am., 48, 1187 (1970).

Budden, K. G., The Wavecuide Mode Theory of Wave Propagation (Academic Press, Inc., New York, 1961).

Chen, K. C. and D. Ludwig, "Calculation of Wave AmPlitudeS by Ray Tracing", J. Acoust. Soc. Am., 54, 431 (1973).

Clark, R. H., "Sound Propagation in a Variable Ocean", J. Sound Vib., 34, (4) , 457 (1974).

Clark, R. H., "Theory of Acoustic Propagation in a Variable Ocean", NATO' SACLANTCEN Memorandum SM28 (1973).

Davis, J. A., "Extended Modified Ray Theory Field in Bounded and Unbounded Inhornogenious Media", J. Acoust. Soc. Am., 57, 276 (1975).

Deakin, A. S., "Asymptotic Solution of the Wave Equation with Variable Velocity and Boundary Conditions", SLAM J.Appl. Math., 23, No. 1, (1972), and Appl. Mech.. Rev., 5417 (No. 7, 1974).

Denham, R. N., "Asymptotic Solutions for the Sound Field in Shallow Water with a Negative Sound Velocity Gradient", J. Acoust. Soc. Am., 45, 365 (1969).

Eby, E. S., "Frenet Formulation of Three-Dimensional Ray Tracing", J. Acoust. Soc. Am., 42, 1287 (1967). -48-

Eby, E. S., "Geometric :hecry of Ray Acoustics", J. Acoust. Soc. Am., 47, 273 (1970).

Lby, R. K., A. C. Williams, R. P. Ryan and P. Tamarkin, "Study of Acoustic Pror_agatIon in a Two-Layered Model", J. Acoust. Soc. Am., 32, 89 (11,7..60).

Ecart, C., F. ,.drcdvna7.ios of 0:eans and Atmospheres (Pergamon Press, Na;: York, 1960).

Ewing, W. M., W. S. fardetsy and F. Press, Elastic WaveS in Layered Media (McGraw7Hill Ecck Co., New York, 1957).

Fitzgerald, R. n, A. N. Guthrie, D. A. Nutile, and J. D. Shaffer, "Influence of the Subsurface Sound Channel on Long-Range Propagation Paths and Travel Tiues", J. Acoust. Soc. Am., 55, 47 (1974).

Gcssard, E. E. and W. H. Hooke, Waves in the Atmosphere (Elsevier. Scientific Publ. Co., New York, 1975).

Gutenberg, B., "Prooagation of Sound Waves in the Atmosphere", 14, 151 (1942).

Guthri e, K. M., "Wave Theory of SOFAR Signal Shape", J. Acoust. Soc. Am., 56, 827 (1974).

Guthrie, K. M., "The Connection Between Normal Modes and Rays in Under- Water Sound", J. Sound Vib., 32, No. 2, 289 (1974).

Hale, F. E., "Long-Range Procagation in the Deep Ocean", J. Acoust. Soc. Am., 33, 456 (1961).

Hirsh, P:, "Acoustic Field of a Pulsed Source in the Underwater Sound Channel", J. Accust. Zoo. Am., 38, 1018 (1965).

Jacobson, M. J., W. L. Siecoan, N. L. Weinberg and J. G. .Clark, "Perturbation Method for Determining Acoustic Ray in Two-Dimensional Sound-Speed Medium", J. Acoust. Soc, Am., 57, 843 (1975).

Joist, W. J., "An application of Poisson Process Models to •ultipath Sound Propagation of Sinusoidal Signals", J. Acoust. Soc. Am., 57, 1409 (1975).

Katz, E. J., "Effects of the Propagation of Internal Water Waves on Underwater Sound Transmission", J. Acoust. Soc. Am., 42, - 83 (1967).

Erol, H. R., "Irt=.nsil-. ” 0. 7-- ' ,... -4 ons along a Single Ray", J. Acoust. Soc. An., 53, 264 (1973.

Y-01, H. R., "Sane Ray and Intensity Solutions in the Compley Plane", Acoust. Soc. Am., 54, 96 (1973).

Lysancv, V. P., "Averace Decay in a Surface Sound Channel with an Uneven Boundary", Sov. Phys. Accust. 12, 425 (1967).

-49-

M,1%cchel- son, J. D. an J Daintith, "Practical Model of Shallow-Water Acoustic Procacation", J. Acoust. Soc. Am., 41, 850 (1966).

McKinnon, R. F., J. S. Partridce and S. H. Tobe, "Calculation of Ray-Acoustic Intensity", 5. Acoust . Soc. Am., 52, 1471 (1972).

•ezzin,D, M. J., "Ray Acc ._:.stics Model of the Ocean Incorporating a Sound Velocity Profile with a Continuous Second Derivative", J. Acoust. Soc. Am. 53, r-S 1 1 1=.7",1

Milder, D. M., "Ray and Wave Cr:variant for SOFAR Channel Propagation", J. Acoust. Soc. Am., 46, 1259 (1969).

Miller, M. K., "Calculation of Horizontal Ranges and Sound IntensitieS by Use of Numerical Integration Techniques", J. Acoust. Soc. Am., 44, 1690 (1933).

Munk, W. H., "Sound Channel in an Exponentially Stratified Ocean, with Applications to SO=", J. Acoust. Soc. Am., 55, 220 (1974).

Murphy, E. L., "Modified Ray Theory for Two Turning.-Point Problem", J. 'Accust. Soc. Am., 47, 899 (1970).

• Murphy, E.- L., "Modified Ray Theory for Bounded Media", J. Acoust. Soc. Am., 56, 1747 (1974).

Neubert, J. A., "Multipath Sua-mability in Ray Theory Intensity Calculations in the Real Ccean", J. Acoust. Soc. Am., 51, 310 (1972).

Nicholas, N. C., "Perturbation Calculations of Propagation Loss in the Deep Ocean", J. Acoust. Soc. Am., 49, 1621 (1971).

Nomady, V. G. and H. - ',I:Derail, "Sound Propagation and Attenuation in the Deep Ocean at Very Tor: == - 7 ,=, s", J. Acoust. Soc. Am., 320 (1975).

Officer, C. B., Sound Trar=lission, (McGraw7Hill, New York, 1958).

Pedersen, M. A., "Theory of the Axial Ray", J. Acoust. Soc. Am., 45, 157 (1969).

Pedersen, M. A. and DerSavne White, "Ray Theory for Source and Receiver on an Axis of Minimum Velocity", J. Acoust. Soc. Am., 48, 1219 (1970)..

Pedersen, n. A. and DeWavne White, "Ray Theory of the General Epstein Profile", J. Accust. Soc. Am., 44, 765 (1968).

Pedersen, M. A. and D. F. Gordon, "Comparison of Curvilinear and Linear Profile•Aporoxination in the Calculation of Underwater Sound • Intensities by Ray Theory", J. Acoust. Soc. Am., 41, 419 (1967).

Pedersen, M. A., "Acoustic Intensity Anomalies Introduced by Constant Velocity Gradients", J. Acoust. Soc. Am., 33, 465 (1961). -50-

Pedersen, M. A., and D. F. Gordon, "Normal-Mode and Ray Theory Applied to Underwater Acoustio Conditions of Extreme Downward Refraction", J. Acoust. Soc. .:7 17!., El, 232 (1972).

Pedersen, M. A. and D. F. :;crd:,n, "Theoretical Investigations of a Double lamily of ::cam a1 nodes in an Underwater Acoustic Surface Duct", J. Acoust. Soc. Am., 47, 304 (1970).

Pedersen, M. A., "Ray Theory ___lied to a Wide Class of Velocity Functions", J. Acoust. Soc. Am., 43, =,l9 (1968).

Pekeris, C. L., "Theory cf =r-c - -cation of Sound in a Half-Space of Variable Sound Velocity C.der Conditions of Formation of a Shadow Zone", J. Acoust. Soc. _ 18, 295 (1946).

Pekeris, C. L., "Theory cf Protagation of ExplosiVe Sound in Shallow Water," Geol. Soc. An. Me=., '27, 1 (1948).

Potter, D. S. and S. R. Murthy, "Solution of the Wave Equation in a Medium with a Particular Velocity Variation", J. Acoust. Soc. 34, 963 (1962).

Raphael, D. T., 'New Approach to the Determination of Acquiring Rays in Singly and Doubly Layered Oceans", J. Acoust. Soc. Am., 48, 1249 (1970).

Raphael, D. T., "Closed-For= Solutions for SOFAR Ray Acoustics in Media with Bilinear Sound-Steed Profiles", J. Acoust, Soc. Am., 56, 80 (1974).

Shuby, M. T. and R. Halley, "Measurement of the Attenuation of Low-Frequency Underwater Sound", J. Aooust. Soc. Am., 29, 464 (1957).

Silbiger, A., "Phase Shift at Caustics and Turning Points", J. Acoust. Soc. Am., 44, 653 (1967).

Solomon, L. P., D. K. Y. Ai and G. Haven, "Acoustic Propagation in a Continuously Refracting nedium", J. Acoust. Soc. Am., 44, 1121 (1968).

Solomon, L. P., A. Barnes S. Port, "Fitting Velocity Profiles with Two-Dimensional Cubic Solines", J. Acoust. Soc. Am., 56, 1389 (1974).

Solomon, L. P., W. C. Merx, "Technique for Investigating the Sensitivity of Ray Theory to Small Changes in Environmental Data", J. Acoust. Soc. Am., 56, 1126 (1974).

Solomon, L. P., "Geometric Aocustics with Frequency Dependence", J. Acoust. Soc. Am., 44, 1115

Solomon, L. P. and L. Ar7if:::, "Intensity Differential Equation in Ray Acoustics", J. Acoust. Soc. Am., 50, 960 (1971).

Solomon, L. P. and C. Comstock, "Two-Time Methods Applied to Underwater Acoustics", J. Acoust. Sc:. Am., 54, 110 (1973). -51-

Stewart, K. R., " ,- dr= 1 of the Ocean Using a Depth/Sound-:Sp2ed • Profile with a C2.7:tin=s First Derivative", J. Acoust. Soc. Am., 36, 339 ( 1 969.

Stickler, D. C., "::c-mal-e Prc::ram with Both the Discrete and Branch Line Contril:utions", Accust. Soc. Am., 57, 856 (1975).

Tolstoy, 1., Wave Pro7a7atf::n ::.:oGraw-Hill Book Co, New York, 1973)-

Tolstoy, I. and C. S. Clay, Ccean Acoustics (McGraw-Hill, New York, 1966).

Tolstoy, I., "W.K.B. Attroximation, Turning Points, and the Measurement of Phase Velocities". J. Acoust. Soc. Am., 52, 356 (1972).

Ugencius, P., "Intensity ^s in Ray Acoustics. I.", J. Acoust. Soc. Am., 45, laa

Ugencius, P., "Intensit7 Ic -lations in Ray.Acoustics. II.", J. Acoust. Soc. Am., 45, 2O (1E. :=9.).

Ugencius, P., "Intensity l-Lations in Ray Acoustics. III. Exact Two-Dimensional Fcr=lation", J. Acoust. Soc. Am., 47, 339 (1970).

Uride, R. J., "Intensity S7=ation of Modes and Images in Shallow-Water Sound Trans:nissicn", Acoust. Soc. Am., 46, 780 (1969).

Warfield, J. T.and !A. Jacobson, "Invariance of Geometric Spreading Loss with Chances f_r_ 7..av Parameterization", J. Acoust. Soc. Am., 50, 24.2 ( 1 971).

Weinberg, H. and R. E-.:niEce, "Hcrizontal Ray Theory for Ocean Acoustics", J. Acoust. Soc. Am., 63 (1974).

Weinberg, H., "Cor.t.'7 -- -ig,nt Curve-Fitting Technique for Acoustic Ray Analysis", J. Accust. Soc. Am., 50, 975 (1971).

Weinberg, N. L. and T. Dunardale, "Shallow Water Ray Tracing with Nonlinear Veloc 4-v J. Acoust. Soc. Am., 52, 1000 (1972).

Weston, D. E., "Guido 7)-- :,=-77-i- ion in a Slowly Varying Medium", Proceedings of tie P1-_ysicaa. Society LXXIII, 3.

White, DeWayne, "Velocity Profiles that Produce Acoustic Focal Points on an Axis of Mint.= -.-elooity", J. Acoust. Soc. Am., 46, 1318' (1969)..

Williams, A. O. and W. Horns, "Axial Focusing of Sound in the SOFAR Channel", J. Acc•st. Soc. Am., 41, 189 (1967).

Wood, D. H., "Parameterless Lzamnles of Wave Propagation", J. Acoust. Soc. Am., 54, 1727' IE..-73).

Wood, D. H., "Refraction Ocrrection in Constant Gradient Media", J. Acoust1 . Soc. Am., 47, 1448 (1970). -52-

Wood -, D. H., "Green's for Unbounded Constant Gradient Media", J. Acoust. _cc. 1233 (1969).

Yeh, K. C. en:: C. H. of Ionospheric Waves (Academic Press, New Yor:::, 1972). - APPENDIX B

DECK LISTING OF FORTRAN SUBROUTINES

FOR GEC!!ETRICAL ACOUSTICS COMPUTATIONS

IN A MEDIUM WHERE SOUND SPEED

VARIES WITH HEIGHT

-54—

PPOSRAM 14A /N (INPUT, GUT PU r. TAPE5= INPUT .TA2E6=OUTPUT) MAIN 1 COMYON VP, Il OJOS t 7I (100) .CI (ICO),ASOL(100) MAIN 2 DIMENSION Z TS (1r) ) MAIN 3 REAC(5, 4 )NCS, (ZI ( I) ,I=I I NCS). (CM), I=1,NCS) t MAIN 4 1ITtil,NUP.N1OWN,7C.7.LIL,NmEX,RIN::E.V.PHST,VPMENO,S9ELIA,VF MAIN 5 ,I=1,`P MAIN 6' lITtJT,NUF,NJOWN,7SC,ZLIS00-1EX,RANCEtVPHSTIVFMENO,SOELTA,VP MAIN 7 REA0(5,*)(2T3(I),I=1,i5) MAIN 8 • wRIrE(6,*)(7.Ts(I),I=1,1) MAIN 9 CELL DASOL - MAIN 10 DO 5 1=1,19 MAIN 11 ZC = ZTS(I) MAIN 12 CALL COSCVP(Vt7.C.ZSC,IT,JT,NUR,NCOWNtOSOVP) MAIN. 13 5 PPINT 4 ,"OSOVP=",OSOVP MAIN 14 CALL EXIT - MAIN i5 END MAIN 16

SUBROUTINE TOTRN(VP,IT,JJ,NUP,NCCWN,ZSCIZLIS,P) TOTRAN COMMON VPT TOTRAN 2 rXT=RNAL PDXDZ TOTPAN 3 CALL INFNT(VP,7.2L,Z0U.N.SCEN.N-ITS,2LOWIZUP) TOTPAN .4 CELL SHIFT(ZLOWtZU?) TOTRAN 5 CELL PANG(RTIYE,PLNTStZLO:4,7Uc) TOTRAN 0 = 1.E-6 TOTRAN 7 Ir (IT ,LT. 0) GO TO 5 TOTRAN 8 CALL GUE0(ZUZSC,D,REL,1tANS1tRCYOZ1NERR 7 0) TOTRAN 9 RST = —ANSi TOTRAN 10 GO TO 10 TOTRAN 11 5 CONTINUE TOMAH 12 CALL OUAD(ZLOWtZSCO,RELt1,ANS2,RCXCZ t NERR I O) TOTRAN 13 RST = A%S"? TOTRAN 14 10 IF (JT .LT. 0) CO TO 20 TOTRAN 15 CALL OUAC(ZLOWsZLISICirEL,ItANS3,FOXOZ I NERR,0) TOTRAN 16 RENO = ANS.? TOTRAN 17 - GO TO 30 TOTRAN 18 20 conrinu: TOTRAN 19 CALL CUAC(ZUR,ZLIS,OtREL,1tANS41PCX0Z,NERPt0) TOTRAN 2C REND = TOTRAN 21 30 N -= NUIP + NCOWN — 1 TOTRAN 22 R . = N*RLNTH + RST + RFNC TOTRAN 23 RETURN TOTRAN 24 ENO TOTRAN 25

SUBPOUTIN= FNOV c (KMAX,ZSCtZLIS.FANGE,IT,JT,NUF.NOCHNOPqS1t FNOVP 1 11/FNNO,SCELTA,NFNC I VPFNC) FNOVP 2 CCMPON V c rviltNCS.ZI(10C),CI(:3C),ASOL(100), FNOVP -3 1ZSCCtZLISC,R:.•SECtITC1JTC.NUPC,NCCWNC FIOVP 4 oimixsr:N V=FNO(1),X(1) FNOVP . 5 eXT=RNAL FtlAY0 . FNOVP 6 ZSCC = ZSC FNOVP 7 ZLI .3C = ?LIS - Fnnvo 8 RANSEC = PANG:: FNOVP 9 ITC = IT FNOVP . 10 JTC = JT FNOVF 11 %UPC = NN 0 FNOVP 12 'NCOWNC = KDOWN FNOVP 13 NFNO = 0 FNOVF 14 V 0 1 = VP"ST 'FNOVP 15. Fl = c.1•=4YO(V1) FNOVP 16 • 3 v;t2 = Itz, + SOELTA FNOVP 17 -55--

F2 = 0 M 7 4Y0(Vn2) FNDVP 18 IF (F1*F2) 10,5,5 FNDVP 19 5 IF (VP2 .GT.-VPHEK0) FETURN FMCVP 20 VP1 = V 0 2 FNInvP 21 FS = F2- 22 GO TO 3 F tPg:i 23 10 GZ = VF1 - F1*SZELTA/(F2 - =1) FNDVP 24 X(1) = GZ FNDVP 25 CALL ZPEAL2(RMRAYC,I.E-5,.7.:1,SDEL 1 4,5,1,X,101IFR) FNDVP 26 NFtil = KFND + FNDVP 27 VPFNO(NPNC) = X(1) FNDVP 28 IF (NFNO .E1. NyAx) PETCR FNDVP 29 GC TO 5 - FMCVP 30 END FNDVF 3i

FUNCTION FNRAY1tV=I7- RMRAYD 1 . CCmMON VPP,I1,::CS.Z:(1 1:70,Citl::),ASOL(100), RMRAYD 2 1ZSCC.ZLISC,RANGEC,ITC,JTC,NURC,NCCWNC RHRAYO 3 - ZSC = ZSCC ;MAYO 4 ZLIS =' ZLISC RMRAYD 5 Rcoe = RAt:GFC RMRAYO 6 IT = ITC RHRAYD 7 JT = JTC RMRAYD 8 NUP = NU'C RMRAYD 9 4.CoWN = NEoWNe RMRAYD 10 • CALL TOTR44(VPI•IT,JT,NLF,NC:.w4,ZSC,ZLIS,R) RMRAYD 11 RmP.AY0 = PCOM - R RMRAYD 12 - RETURN RMRAYD 13 ENO 4MRAYD 14

SUSROUTINE SHIFT(ZLOw,ZUP) SHIFT 1 SU34OUTINE SHI=T MOVES THE VALUES OF Z (ZLOW,ZUP) FOUND - SHIFT 2 FoR TUE ruRNIN; PCINTS (FY TNPNT) S: AS TO AVOID INTEGRATION SHIFT 3 THROUSH SINr3ULARITIES, AS C:ULE cJA;;EN IN THE CALCULATION • SHIFT 4 OF ALM:ST ALL THE CUANTITIES I\CLOCEC IN THIS PROGRAM. . SHIFT 5 SHIFT IS CALLED EY VIIN ONLY, :43 AFTER TNPNT IS CALLED. SHIFT 6 n = 0 SHIFT 7 CALCULATE THE )IFFEPENCE EETHEE', THE SOUND SPEED AT THE LOWER SHIFT 8 TURNIN5 POINT AND THE PAS_ VEL:CITY. SHIFT 9 5 C'KL = C!'"IP(ZLOw) SHIFT 10 IF THE SCUNO Sf.I EED IS LESS TH1% :.7 P, 1-;=.'- RE SAFE, AND WE GO ON TO SHIFT 11 CHECK IHE UPPER. TUR.NINS PCIA.T. :-.:ThEPWISE, WE 4C0 A TINY ViCUNT SHIFT 12 TO ZLOW . ANC CONTP.LE DDING SC 1..\TIL THE SOUNC SPEED IS LESS THAN 9P. SHIFT 13 . IF(CHKL .LE. ::.:3) G3 TO :0 : SHIFT 14 7Lo4 = ZLCW + L.E-3 SHIFT 15 4 = N+1 SHIFT 16 IF SHIFT IS UNSUCC=SSFUL :\ A ;C.::.! TRIES, WE WANT IT TC STOP. SHIFT 17 IF(N .GE. 113;.:) .7-. TUPN SHIFT . 18 GO TO 5 SHIFT 19 WE TRY Tu► = sl ,•.:: =0= THE UP=E --Z TU-:NINS POINT, AND AGAIN, AS LONG AS SHIFT 20 THE SOUND S=E-1 :S LF$S TH:\ V:, WE'RE SAFE,. SHIFT 21 13 CHKU = Cmv 0 (Z!:n) ' SHIFT 22 IF(CH4U .LE. t;.C) ::ETURN SHIFT 23 ZUP = ZUR - 1.E- SHIFT 24 N = N+1 SHIFT 25 IF(N .GE. 13C2) RETU=N SHIFT 26 GO TO 10 . SHIFT 27 cm] SHIFT 28

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FUNCTION C1VP(2) CMVP THIS FUNCTICN ROUTHE SII'FLY CALCULATES THY. DIFFERENCE CMVF 2 CAS A FUNCTICN OP NE/GHT 7) r._7;47pN TH= PHASE VELOCITY •CMVP 3 (WHICH IS INPUT) ANC THE Seer:3 SPEED (WHICH IS A FUNCTION CNVP 4 OF HEIGHT Z). CNVP 5 COMMON V= CMVP 6 CMV° = CSF(7) - V= CMVP 7 RETURN CdVP END CMVF 9

sUARcurINE Th 0 NT(VP.ZBL,2E, U,N2CAN I NRTS,ZA,29) TNPNT 1 SUBROUTINE TNPNT FINES THE TURN:NG PCINTS (VALUES OF 2 _TNPNT 2 AT WHICH THE OIFFECENCE FIE'reE• THE SOUND SPEED ANO THE PHASE TNPNT 3 VELOCITY VANISiES) GIVEN THE P=AS_ VELCCITY (VP). ZBL AND "ZnU TNPNT 4 . ARE THE LO•Eq AND UPFER BCUNCS, RESFrCTIVELY, BETWEEN NHICH THE TNPNT 5 SEARC4 FOR I'VE TUR/sING FCINTS IS CCNCUOTED. NSCAN + 1 IS THE TNPNT 6 NUM2ER OF SUBINTERVALS INTc WA/C4 THE INTERVAL OF SEARCH IS TNPNT 7 SU3DIVIDEB. NRTS IS THE Nt.:-3ER OP TURNING PCINTS FOUND (WE TNPNT 8 NORMALLY EXPECT TWO). ZA IS THE LOWER TURNING POINT (IF FOUND) TNPNT 9 ANO Z3 IS THE UP =EP CNE (IF FOUND). /0 EXTERNAL cmVP 11 DIMENSION X(1) ET 12 . 60N40N VrC,I1INCS,ZI(160) TNPNT 13 VPC = V 0 TNPNT 14 ZPL = ZI(1) TNPNT 15 ZeU = zi(Ncs) TNPNT 16 NSCAN = NCS + 3 TNPNT 17 CALCULATE THE WICTH CF THE SU2InTERVALS TNDNIT 18 OELTA = (23U - ZPL)/(NSCAN + 1) TNPNT 19 - CALCULATE CSF(29L) - VP . , TNPNT 20 F1 = CMVP(ZBL) TNPNT 21 START. THE SEARCH AT 28L TNPNT - 22 21 = Z8L TNPNT 23 NRTS = 0 TNPNT •24 FIND THE UPPER LIMIT OF THE SUBINTERVAL TNPNT 25 10 Z2 = 2 1 * DELTA TNPNT 26 CALCULATE CSF(72) - VP TNPNT 27 F2 = CMVP(22) TNPNT 25 TAXE THE P-;ccUDT OF =1 ANC F2, 4NO IF IT IS POSITIVE, WE HAVEN'T TNPNT 29 FOUND THE SUFINTERVAL WITH A TURNING POINT IN IT YET. SO WE GO TO 15 T(.1°NT 30 AND START AT THE 2CTTC1 CF THE NEXT SUBINTERVAL. ' TNPNT 31 TEST = F1l'F-2 TNPNT 32 IF(TEST .GT. G.0) GO TO I": TNPNT 33 IF F1*F2 IS NET,ATIVE. WE'VE GOT A SU2INTERVAL WITH A TURNING - TNPNT 34 POINT IN IT. AT THIS POINT, WE N4

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• GO ON To_THF n:xT SUIATERVAL. TNPNT 52 IF(ZOU .GE. ZI) GC TO 10 TNPNT 53 20 RETURN TNPNT 54 "ENO . TNPNT 55

SUCROUTINE RANG (FTImE,RLNrH,7Low,zu 0 ) RANG • .1 SU:ROUTINE PANG PERFCRNS THE FIt.:aL STEF IN THE CALCULATICN RANG 2 ('Y INTFGRATION OF OT/OZ ANC OX/W. OFTWE=N THE TURNING POINTS' RANG 3 ZLO4 t ZUP) OF THE RAY REcETITIcN TIME AND LENGTH, RTI9E AND RANG 4 RLN14, ,4ES=ECTIVELY. RANG 5 EXT=RNAL RGTOZePOYOZ RANG 6 RT/ME = PAINT(POTCZ,7LOW,ZU=) RANG 7 ?,LOTH = RAINT(RCXCZ,ZLOw,ZU=) RANG 8 RETURN RANG 9 END RANG 10

. SUBROUTINE DASOL DASOL 1 SU9ROUTINE DASOL CALCULATES THE COEFFICIENTS OF - THE CUOIC OASOL 2 SPLIhE USEG TO APaROxIMAIE THE SC NC-SPEED PROFILE, ANC AS OAS OL 3 OEFINFC SY DASOL 4 nELzci)*:.scL(r-1) I. 2* CCELZ(I) - DELZ(I+1ASOL(I)W 4- DASOL 5 + DELZ(I41) 4 ASCLII+1) = GELC(I41) - QELC(I) OASOL 6 WHERE CELZ(I) = Z(/) - Z(I-1) DASOL 7 CELCII) = (C(I) - C(I-1))/DELZ(I). DASOL 8 COMMON vo,ri,Ncs,w ic6),crcioo),AsoLcioo) DASOL 9 N = 1 OASCL 10 OELZ = 1.0 DASOL ii DELC = o.a OASOL i2 AKM2 = 0.0 , DASOL 13 ALM2 = 0.0 OASOL 14 AKM1 = 0.0 DASOL 15 ALM1 = 1.0 OASCL 16 ' NSTP = ACS - 1 DASOL 17 10 0E1.2 0 = ZI(N+1) - ZI(N) OASOL 18 DELCI = CI(N+1) - CI(N) 04501 i9 ALPHA = OELZ DASOL 20 GAMMA = OELZP 21 OFTA = 2..C*IALPHA + GL gal= 22 OEE = (CELC 0 /CELZ°) - (CELC/CELZ) DASOL 23 IF(N .E1. 1) GO TO 30 DASOL 24 AK = (OEE - ALPHA*AK•2 - 2ETA•.0'1)/GAMMA OASOL . 25 AL = I - ALPHA* ALP2 - eETA*ALmil/GAMmA DASOL 26 IF('l .=1. NSTP) GC TO 1G.3 DASOL 27 AKM2 = 4X''11 DASOL 28 ALM2 = ALM1 29 AKM1 = AK if)ggti. 30 ALM1 = AL DASOL 31 30 N = N + 1 OASOL 32 , OELZ = CELZP 04501 33 OELC = DELCc DASOL 34 GO TO 10 04501 35 100 ASOL(1) = 0.0 OASOL 36 ASOL(Z) = -AK/AL DASOL 37 OELZ = 1.0 OASOL 38 OELC = 0.0 DASOL 39 N = 1 DASOL 40 110 0Eiz3 = zI(4+1) - ZI(N) 41 DELCP = CIC1+11 - CI(A) g= 42 ' A1 0:44 = '7.P'LZ DASUL 43 GAMMA = GELD' DASOL 44

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?;,TA = 2.ce(lLpHA 4 GArwA) OASOL 45 :EE = (::LC?/DEL7P) - (CELC/OELZ) DASOL 46 1c- ( N s Ef:. 1) GO TC 1.3 13 DASOL 47 M =N+1 DASOL 48 AOL(m) = ( 7)E.E - ALPHA'ASOL(N-1) - BETA‘ASOL(N))/GANMA DASOL 49 :F(N .E':. NITP) GC TO 2C3 DASOL 50 133 N = (N + 1 DASOL 51 CELZ = CILZ° DASOL ' 52 :_LC = CEL:P DASOL 53 SO T3 11: DASOL 54 2:0 RETU-N DASOL . 55 ENO DASOL 56

FUnCTION CSPCZ) CSP - CSP 2 THIS FUNCTICk: R:UTINF CALCULATES INTERMEDIAT7 VALUES OF CSP • 3 THE SOUND SPEED 7 R. C=ILE ACCC•CING TO THE EQUATION CSP 4 CSP(?) = 4.:1 4R'C(I-1) + N=C(I) + CSF 5 cu:ELzcil**2)4,- LtscL(i-1) , (:i5,IR4.:E - WBAR + ASCL(I)* CSP 6 *CW 44'3 - H)). CSP 7 CSP 8 ccw-tcr: VP,II,NCS,ZI(1.00),CI(10C).ASOL(100) CSP 9 CEFIE - THE LC D _? 4NC UPPER ECUNDS CF THE SOUND-SPEED PRCFILE. CSP 10 ZL = ZI(1) CSP 11 .• ZP = zir%:s) . CSP 12 CUTS:OF OF THESE 1"?.DU%0S, LET THE SOUND SPEED 8E CONSTANT AND EQUAL CSP 13 TO 1. 6.- CORRESPDNDI5 ADJACENT IALUES. CSP 14 17 (Z .LT. ZL) GC TO 50 CSP 15 IF (Z .GT. Z?) G.O TO 60 CSP 16 I = NCS CSP 17 10 J = I-1 CSP 18. FOR 1NY '1 L"= Z, .;:r 144!:T I SUCH THAT Z IS BETWEEN ZI(I-1) AND Z1(I). CSP 19 W= START WIT!-, TH= iiIGHEST VALU= 70R I ANO WORK DONNWARC UNTIL WE CS° 20 FIND Ills" TNT 7 P1: 1.. TH4T CONTAFIS Z. CSP 21 ZTEST = 71(J) - CSP 22 IF 7 IS 9ETEN Z:(I-1) ANC ZI(I), WE GO TO 40 AND CALCULATE CSF(Z). CSP 23 IF (Z .GT. ZTEST) GC TO 40 CSP 24 IF Z IS ?.OT '7ETW=EN 7I(I-1) AN ZI(I)• WE CHOSE THE NEXT VALUE LCWER CSP 25 FCR I AND C:ATINUE THE SEARCH. CSP 26 I =. J CSF - 27 50 TO 10 CSP 28 4C CV:TV:U.7 CSP 29 Z IS 2=TWEEN ZI(I-11 AC Z/(I) CSP '30 DZLZ = ZI(2) - ZI(J) CSP 31• 4 -''' (Z "' .71(.1))/C 1L-Z - CSP . . 32 W=A? = 1.: - W CSP 33 T=Rmi = wr-. .,,R-::cJ) 4- w*c/cI) CS° 34 GUT1 = W?4P•'.3 - W34= CSP . 35 G"T = 4•• 3.- - 4 - CSP - 36 T=RM2 = (CELZ"2)s(ASCL(J)*GUT1 + ASOL(I) 4 GUT2) • CSP 37 4.... ‹. p , Tr...7.w! 4 T=Pv2 CS° ' 38 RETURN . CSP '-.39 :: : 7 P = ci(I) CSP 40 R=TURN CSP 41 CSP = CI ('.CSI CSP 42 RETURN CSP 43 2%0 CS? 44

OCOZ THE =U%:TIC:, 0:0'(7) CALCULATES TM: FIPST DERIVATIVE or OCOZ -59-

THE SOUNO SPEED WITH R=SPECT TO HEIGHT Z, ANC ACCORDING 0c07 3 To THE ECU4TION (ICOZ 4 OCOZ 5 OCOZ(Z) = DELC(I) + CELZ(1)4 (-40L(I-1) 4 t3 4 WDAP: 43+2 - 1) + OCOZ 6 + ASOLCI1*(3*W"2 -1)) DCOZ 7 .OtOZ a PLEASE SEE FUNCTICi CSF(Z) FCR A mCPE DETAILED EXPLANATICN O THE OCUZ .9 CALCULATIONAL 0 ROCECURE THAT FCLLOWS, AS THE TWC PROCEDURES ARE OCOZ 10 NEARLY IDENTIC1L. OCOZ 11 COMmCN Vo.I1$NCS.7"I(1CC),CI(10J)..•SOL(110) DCOZ . 12 DEFINE THE LOWER 4+C UPPEP OCUNDS CF THE SOUND-SPEED PRCFILE. OCCZ 13 ZL = ?ICI) OCOZ ' 14 ZP = ZIPICS) .DCOZ 15 OUTSIDE OP THESE BOUNDS. LET DC/OZ = 0. OCOZ 16 IF(7. .LT. ZL) GC TO 50 new 17 IF(z .GT. 2.) GC TO 50 OCOZ 18 I = NCS ()COL 19 • 10 J = I-1 . • OCCZ 20 ZTEST = ZI(J) bcoz 21 'F(7_ .GT. zrEsT) .Go To 40 ocnz 22 1 = J DCOZ 23 GO TO 10 ncoz 24 .0 CONTINUE ocoz 25 .-: z Is BETWEEN ZICI-1) 4hC Zit') DCOZ 26 OPLZ = 7I (I) - ZI(J) DCOZ 27 .• DELCI = ICI(I) - CI(J))/CELZ DCOZ 28 W = (Z - ZI(J))/CTLZ OCOZ 29 WPAR = 1.0 - W DCOZ 30 Tp.:43,%• = ASOL(I)*((3.0*(."2)) - 1.0) DCOZ 31 TRM38 = ASOL(J).*((3.0*(:E,IR,B2)) - 1.0) DCOZ 32 TRi3 = CELZ*(TR;34 - TR:, 39.) DCOZ 33 3CD2 = DELCI + TP?3 DCOZ 34_ RETURN OCOZ 35 50 CCOZ = 0.0 DCOZ 36 RETURN DCOZ 37 END . DCOZ 38

SUBROUTINE COSOVPCVPaC,ZEC,IT T JT.NUP,NOCWN,CSOVP) COSOVP 1 COMMON V°T . COSCVP 2 EXTERNAL FTRMUL,FTRM COSDVF 3 VPT-= Vo COSOVP 4 CALL TN=NT(VR,Z9LTZPU,NSCAN,NRTS9ZLOW,ZUP) CDSCVP ' 5 CALL S4IFTIZL04,ZUP) COSCVF 6- 3 = 1.E-6 CDSOVP 7 - ZIU = ZUP - 0.51 4 fZUF - ZLOW) COSOVP 6 n1 = ZLO).: + 1.C1*(ZUF - ZLCW) COSO'IP 9 CALL OLIAC(ZIL,ZIO,O,REL,),TRP), ,FTcM,NFRR,O) COSOVP to -, IF •(IT .LT. 0) GO TO le CDSCIIIF ' 11 CALL CUAC(ZSC,Z1U,O.RZL,J.TR?'/,FTP4,N=PR,O) • CQSDVP 12 SC 13 15 COSCVP 13 • 10 OP, LL OUA7(ZIL,Z5C,O,REL,J,TI,FTFM#NERRIpa) CDSOVP 14: 15 IF (JT .LT. 0) GO TO 20 COSCVP ' 15 CALL CUA.I(ZIL,ZC.O9PEL,OITR:'.F,FTRP.NERRO) COSUIP 16 SO TO 25 17 20 CALL W ACCZC.ZIU,C,REL,0. 1"RmF,FTR:-,NERR.0) C=171° 18 25 CCNTINUE CDSOVP 19- , TomUl = TPN-3 T(Z/U) COSCVP 20. TRmL1 = TRNPT(ZIL) . 21- CALL rIU ,10C ,U P,ZIV.O,REL+1.TqVU2,FT 0 PULIN E.0)7tR ("OlS4 V I3P 22 1:A.E. r"Jaf7C:LC.:.7.1L.:;,P=:-.1.7.',=T-ZMUL,N:NR,8) 23. I. + t:L(TR).LI. 1 - i' lL1')) I. (NUia•+.NOOWN - 1) 4`TRPM + gt= 24 Cill(C,!? = r!-)1 1 NOOWN 4 (-TRML1 - + TRPL2) f TPMF COSCVP 25:-

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%in t./ VF 40 2 COSOVP 26 CSPZC = CEPtZC) COSOV 0 27 csPzsn = csP(zc)**2 COSOVP 28 IF tJT .LT. 0) GO TO 30 COSOVP. 29 OSOvp = -tCSPZC 4 (SORT (VFS: - CSPZSQ) /VPSO)) 4 COX0vF COSCVP 30 GO TO 35 CDSOVP 31 30 OSO• P = tCSPZC*(SCRTtVPSC csgzsni/vPsn))icoxovp COSCV! 32 35 CONTINUE CDSOVP 33 RETURN COSCVP 34 ENO MOW, 35

, FUNCTION FT?M(Z) FIRM FTRM 2 -CSPFVF FTRM 3 FIR4tZ) = FIRM 4 tVP**2 - CSF' 4 21 4 *1.5 FIRM 5 FTRM 6 COMMON VD,K - FTRM 7 VPS1 = 1/c X 4 2 FIRM 8 CSPSQ = CSP(Z)**2 V FTRM 9 IF (VPSC .GE. CSPSO) GO TO 20 FIRM 10 C = 1 V FIRM it 10 TPH1 = 1.E-50 FTRM 12 GO TO 30 FIRM 13 20 C = 0 FTRM 14 TF-1.1 = (SORT(VPSC1 - CSPSC))**3 FIRM 15 IF tTR41 .LT. 1.E-50) GO TO 10 FIRM 16 TRM2 = CSP(Z) *VP FTRM 17 30 FIRm = -TFM2/1-ZP1 FTRM 18 RETURN FTRM 19 ENO FTRM 20

FUNCTION OCOZS(Z) - DCOZS 1 DCDZS 2 FUNCTICN OCCZS(Z) CALCULATES rHE SECCNC OERIVATIVE OF THE 'OCCzS . 3 SOUND SPEED C KWITP RESPECT TO HEIGHT z, AHD ACCORtING TO THE DCOZS 4 E2UATION OCOZS 5 DCDZS 6 OCOZS(Z) = E 4 (W8A5.:*ASOL(I-1) + W*ASCL(I)) DCOZS 7 DCOZS - a PLEASE SEE FUNCTICN CSP(Z) FC- Ja 4CRE DETAILED EXPLANATION OF THE "DCDZS 9 CALCULATIONAL DROCEO(;RE THAT FOLLOWS. AS THE TWC PROCECURES ARE DCDZS 10 NEARLY IDENTICAL. OCOZS 11 commoN vri,II,N3S.7It100).CI(19.0).ASOL(100) • OCOZS 12 DEFINE THE UPPER ANC LCwE EOUNCS CF THE SOUND -SFEE0 PROFILE. DCOZS 13 ZL = ZItl) OCOZS 14

Z= = ZI(KCS) OCOZS . 15 - OUTSID= OF THESE 20UNOS, LET DCIOZS(Z) = O. OCOZS 16 IFIZ .LT. ZL) G,C 10 53 OCOZS 17

IF(7 - .GT. Zp) GC TO 50 DCOZS 18 I = NCS DCOZS 19 10 J = I-1 DCDZS 20 ZTEST = ZI(j) OCOZS 21 IF(Z .GT. ZTEST) GO TC 40 DCDZS 22 I-= J OCOZS 23 GO TO 10 . DCDZS 24 40 CONTINUE DCOZS 25

Z IS 9FTwFEN ZI(T - 1) ANC ?I(J). OCOZS 26 ort., = Z' (I) - zI(J) DCOZS 27- W = (Z - ZI(J))/CF_LZ . OCOZS 28

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WEAR =- 1.0 - OCOZS 29 OCOZS = 6.0 4 (C..N24 04 Ascl(J)) + (14 4 ASOL(I))) OCOZS 30 RETURN OCOZS 31 50 OCOZS = 0.0 OCOZS 32 RETURN DCOZS 33

. EhO DCO7S 34

FUNCTION FTRMUL(7) FTPMUL 1 FTRMUL 2

- 2.*V P *CCO7S FTRMUL 3 FTRMUL(Z) - FTRMUL 4 (DCOZ**2)*(VP* 4 2 -CSP* 4 2)**0.5 FTRMUL 5 FTRMUL 6 CCMMCN Vv,K FTRMUL 7 CSPSO = CsP(Z)* 4 2 FTRMUL 6 vcsri = VP 4 *2 FTRMUL 9 OCOZSO = CCOz(Z)**2 FTRMUL 10 IF(JPSD .GE. CS 0 SP.) GO TC 50 FTRMUL 11 K = 1 FTRPUL 12 40 ON = 1.E-50 FTRMUL 13 GO TO 60 FTRMUL 14 50 K = 0 FTRMUL 15 Oh = CCOZSC*(S'PT(VPsC - CS=s1)) FTRMUL 16 IF(1N .LT. 1.E-5C) (",0 TC 43 FTRMUL 17 60 FTRPUL = -2.*(VP*CCOZS(7.))/CN FTRMUL 18 RETURN FTRMUL 19 END • FTRMUL 20

FUNCTION TRNPT(Z) TRNPT 1 CCmmOn V 0 0( TRNPT 2 CS 0 S0 = CSP(Z)**2 TRNPT 3 VPS1 = V 0 **2 TRNPT 4 IF (VPSC .GE. CSPSO) GO TO 50 'TRNPT 5 X = 1 TRNPT 6 40 ON = i.E-50 TRNPT 7 GC TO 60 TRNPT 8 50 K = 0 TRNPT 9 ON = 0007(2)*(SCPT(V 0 SC CSPSQ)) TRNPT 10 IF (.1i3S(n) .LT. 1.E-5G) Go TO 40 TRNPT 11 60 TRNT = (2.*vP)/Dk TRNPT 12 RETURN TRNPT 13 ENO TRNPT 14

FUNCTION FOXCZ(Z) RDXCZ 1 ROX02 2 FUNCTION RCACZCZ) C!LCULATES THE INTcGRAND USED VT SUBROUTINES ROXOZ 3 RA%G ANC ?TNT TC cr.LcULATE THEE =yi PFPETITIO`l LENGTH,LEN RLNTH. ROX127 4 THE ECUATIoN FOR PCCZ(Z) IS R•X02 5 RDXDZ 6 1/V° ROXC2 7 ROx0Z(Z) - ROXOZ 8 (1/CSP ,"Z - 1/VP**2)**;1.5 RDXDZ 9 ROXDZ 10 CCMMOH VP,K RDXOZ 11 CSPSC = CSPC7 )**2 ROXCZ 12 VPS"! = V=**? R1XOZ 13 IF (CSPil .LF. 4°'+7)ll GC TO 10 ROX02 14 K = 1 . ROXCZ 15

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5 OS' = 1.E-50 ROXCZ 16 GO TO 20 ROXOZ 17 LO K = 0 ROXOZ 18 DSO: = 1./CSPSO ROXDZ 19 OSOV = 1./VPSI ROXOZ 20 OSO = CSCC - OSCV ROXOZ 21 IF (OSO .LT. 1.E-EJ) GO TO 5 ROXCZ 22 ?0 RCXJZ = (l./VP)/SCRT(OSC) ROXCZ 23 RETUR1 ROXOZ 24 ENO RDXCZ 25

FUNCTION ROTOZ(Z) ROTOZ 1 ROTCZ 2 FUNCTION RCTCZ(Z) CtLCULATES THE I ► TEGRANO USEO BY SUOROUTINES ROTOZ 3 RANG AND PAINT TO CALCULATE THE PAY FEFETITION TIME, RTIME. ROTOZ 4 THE EQUATION FOR ROTCZ(Z) IS ROTCZ 5

ROTOZ 6 • 1/CSP* 4 2 ROTOZ . 7 ROTOZ(Z). - ROTCZ 8 (I/CSP**2 - i/V" 4 2)**0.5 ROTOZ q ROTCZ. 10 COMMON 1/ 2 ,K ROTOZ 11 .CSPS3 = CSP(Z)**2 ROTOZ 12 . VPS3 = V° - 2 ROTC? 13 IF (OSPSC .LE. VF50) GO TO 30 ROTC? 14 K = 1 RDTCZ 15 20 .CS() = 1.E-50 ROTOZ 16 GO TO 40 ROTC? 17 !O K = 0 ROTCZ 18 • DSCC = 1./CSPSO ROTC? " 19 OSCV = 1./V 0 SI ROTOZ 20 OSO = OSCC - CSIV ROTOZ 21 IF COSI .LT. 1.E-5C) GO TO 2C ROTOZ 22 0 RCTDZ = (1./CSP50)/SCRT(CSI) - ROTCZ 23 RETURN ROTOZ 24 ENO ROTC? 25

FUNf;TION FAINT(CSCZR,ZLOW1ZUP) RAINT 1 RAINT 2 UNCTION RAINT PERFOcHS THE INTEGRATION'OF ROXOZ ANO RCTOZ 3 ;ECESSAY TO 0.1TAI• THE FAY REPETITION LENGTH ANO TINE, RLNTH 4 ■ NO PT-E.v RESnEOTIVELY. # - RAINT 5 RAINT EXTERNAL CSCZR RAINT 7 ZAVE = CZUP 4- ZLO0/2.0 RAINT 8 0 = 1.E-6 RAINT 9 CALL OUAO(ZLOw.Z.:7Z,C.P.U.I.ANS1,CSOZR.NER 0 ,0) RAINT 10 CALL OUAC(ZUn,ZAVE,D,SEL,1,4xS2,CSOZR.NERRIO) RAINT 11 RAINT = (ANS:. - ANS2) RAINT 12 RETURN RAINT 13 ENO RAINT 14

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SUEROUTINE 7•EAL2 (F,EPS,EPS2,E1A,NSIG,N0,1TMAX,1ER) 2FELC010 C 2RELCC20 C-ZFEAL2 --S 3- 2FELCC33 C 2FELE640 C FUNCTION - ZREAL2 FINES THE REAL ZERCS OF A REAL FUNCTION2FELCu50 C -- USED 1,HEN INITIAL GUESSES ARE GCCC 2RELOCEJ C LSAGE - CALL 2RrAL2(F,EFS,EPS2,ETA,NtIG,N,X,17MAX,IER)2FEL007(1 C FARAVETEFS F - - A FU•CTICN F(X) SUEPEiCGRAF WRITTEN EY THE LSEF2RELCCE0 C EFS - 21: .;- STCFPING CRITEFICN. A KCOT X IS ACCEPTED 2RELG0S0 C IF TkE AESOLLTE VALLE OF F(X) .LE. FPS 2REL01CC • C (INPUT) 2REL0110 C EFS2 - SPREA: CRITERIA FCR MULTIPLE RCCTS. IF THE 2RELC120 C ETA ITH RCCT (X(I)) HAS BEEN COMFUTED ANC IT IS 2RE10133 C FCIN:. TILT THE'ABSCLUTE VALUE CF 2RELC140 C _X(I)-x(J) .LT. EFS2 WHERE X(J) IS A 2RELC153 C FREVICUSLY CCMPUTEC RCOT, THEN THE 2RELL1b0 C COMFUTATICN IS RESTARTEC WITH A GUESS EQUAL 2RELC170 .0 TO XII) 4 ETA. (INPUT) • 2REL0180 C NSIG - 1ST STCPPING CFITERICN. A RCOT IS ACCEPTED IF ZRELC190 C TWC SUCCESSIVE AFPFOXIMATICNS TO A GIVEN 2RELC2G0 C RCCT AGREE IN THE FIRST NSIG CIGITS. (INPUT)2AELG213 C N - ThE bUMSER CF ROOTS TO CE FCUNC (INPUT) 2RELC22C C X - CU It.FCT X IS AN N-VECTOR OF INITIAL GIESSES 2RELC230 C FCR N RCCTS. ON CUIPUT, X CONTAINS THE 2REL024J C COMPUTED RCOTS. 2REL•250 C IT•AX - Ch :NFLT = THE MAXIMUM ALLO4AELE NLMEER CF 2RELC2E0 C ITERATICNS PER RCOT AND ON OUTPUT = THE 2FELC270 .0 _ TiUki!EF OF ITERATIONS USED ON THE LAST ROCT. 2FELE,2tG C .- TER - ERROR P;RAPETER (OUTPUT) 2REL.0293 C WARNING ERRCR = 32 + N 2FELL300 C N = 1 INCICA1ES A SINGLE ROOT WAS ONFASSED 2FELC313 C EECALSE ITMAX WAS EYCEEDEC FOR THIS RCCT. 2FELC320 C X(I) FCR THIS FCCT IS SET TO 111111. 2FEL0330 . C N = 2 INCICATES A SINGLE FOOT WAS EYFASSEC 2RELL340 C EECf-USE THE CERIVATIVE CF F FOR THIS 2RELC350 Z ROOT BECOMES TCC SMALL. X(I) FCR THIS 2FELC360 C FOCI IS SET TO 222222. NOTE THAT THIS 2RELC37C C ERROR CCNCITION MAV CAUSE AN OVERFLCW. 2RELC38C C N = 3 INEICATES THAT SEVERAL CF THE ABCVE 2RELC3E) C ERRCR CCNDITIONS OCLURRED. EACH'X(I) IS 2RELC4OU C SET TO EITWER 111111. OR 222222. AS ABCVE 2RELC415 C PRECISION - cINSL= 2REL0420 C FEQU. IMSL RCL1INES - LERTST 2FECC433 . C LANGUAGE - FCRTR;h 1i0 2REL04 2FELL'.5C C LATEST REVISICN - CC10EER E t 1973 2RELL4EJ C 2RELC470 DIMENSION X(11 2RELC480 DATA Pl,P•ClIZERC,ONE,TEN/.1,.00120.0,1.0,10.0/. 2RELC4SC IEF = C 2FEL0500 I R=0 2REL051° CRIT1 = TEJ (-NS1G) 2RELC520 DO 30 I=1,N 7FELC53J IC = 1 2RELL.540 XI = 2RELC550 5 AXI = AESIXI) 2FEL0560 IF .EC. 1) GC IC 15 2REL0570 NM1=1-1 2RELC580 CO 1C J = 1,NM1 2RELE550 IF tA0S(xI -.X(J1) .LT. EP521 XI = Xi + ETA 2RE10600 10 CONTINUE 2RELE61d 15 FxI = FIXI) 2RELC620 AFXI = AESIFXI) 2RELE63J C TEST FCR CONVERGENCE 2RELE64a IF (AFXI .LE. EFS) GC TO 25 2RELL65J CI = .C201 2RELC6E0 IF TAXI .GE. P1) DI = PS21*AxI 2REL067.1 hI = AMIh1(AFXI1DI) 2REL0680 FXIPNI = F(XI + 2REL.06S.J DER = IFxIPHI - FXI)/ ■,I 2RELC7OG IF ICER .EC. ZERO) GO 70 20 . 2RELG716 XIP1=FXT/DER 2RELL720 IF (LEGv.5R(xIPI) .NE. 0) CO 10 2C 2REL0720 XIPI=XI-XIPI 2RE1.674.1 ERR = ABSIXIFI - XII 2REL0750 XI = XIPI 2RELC7E0 C TEST FOR CONVERGENCE 2REL0770 IF(AXI.EC.ZERO) AxI=CNE 2RELLf780 ERR1=ERR/AxI 2RELO7S0 IF tLEGVAP(EFR1) .NE. 2) ERF1 = ERR 2RELC800 IFIERR1.LE.CRIT1) GO IC 25 2REL0810 IC = IC + 1 2REL082B IF (IC ITHAX) GC TO 5 2REL0830 C RCCT NOT FOLNC, NO CONVERGENCE 2RELC842 XII) = 111111. 2REL085J IR=I;f1 2RCLO8E2 IER=23 2REL0870 GO 7C 33 2RELL880 C RCCT NOT FOUND, DERIVATIVE = E. 2RELu890 20 YID = 222222. 2fiEL0903 IR=IF+i 2REL0910 IER=34 2RELC.922 CC TC 30 2RELG933 25 XII)=XI 2FEL0940 2C CONTINUE 2RELL952 I1- PAX = IC 2RELC96C, IFIIER.EC.J) GC TO 9.;15 2RELE970 IFIIR.LE.1) L,0 TO 9& C3 . 2RELE98G IER=35 2RELCEISO '000 CO.TINUE 2REL1.03 CALL UE;ISTIIER.614ZRELL2) 2REL101;) SOE5 RETURN 2REL1E20 ENC 2REL1030

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SUBROUTINE OLAC(4,2,C,REL,NrANS,FLN,NERR,IMAP) QUAD 2 A = Lti, ER LIVIT CF INTEGR:,TICN (INPUT) QUAD 4 0 = UFFER LIPIT CF I;;TEG;:::.TICN (INPUT) QUAD 5 D = RECUIREC RELATIVE TOLtRA:.CE (INPUT) . 6 REL = ESTIt'ATE OF RESULTING RELATIVE TCLERANCE (OUTPUT) (OlitiJ AAg 7 N = SII■ GULA:iIII FLAG. SET N=0 6HEN NO SINGULARITY ALONG PATH. QUAD 8 SE7 N=1 ilHEN ONE CR !:DRE SINGULARITIES LIE CA PATH QUAD • 9 ANS = CO•FLTEC 1121LE-CF :.TEGRAL (OUTPUT) QUAD 10 FUN = NAME CF FW.CTION GENERATING THE INTEGRANT] QUAD 11 NERR = ERFCR FLAG (OUTPUT) ttg 12 NEFA = -1 STEP SIZE C:N NCI BE NAGE SMALL ENOUGH QUAD 13 NERR = -2 CURD INCOPLETE IN LIM (200) TRIES QUAD 14 NEiR = -•..7 G FAS 2EEN SET TOO SMALL QUAD 15 NERR .GT. C --SUCCESS--GIVES NUMEER OF TRIES HECUIRED QUAD 16 IMAP = PROGRESS VAP FLAG. SETIMAP=1 V1HEN MAP IS DESIRED. QUAD 17 SET IMAF=J AHEN NOT DESIRED QUAD 18 CIMENSION 104(2),i;3(4).0., 12(E), Z4(2),2044), Z12(6) QUAD 19 couaLE PRECISION YCOLE QUAD 20 DATA )+411),114(2),(im3(1),1=1,4),CW12(I),I=1,61/.652145154862546, QUAD 21 1.347P.54845137454,.2E2Ed27EZ37a362,.3137C6645877887,.22228103445337 QUAD 22 15, .1C122e..F.ZE. 29Z376,.24 41753134::3,.23349.2536538355, . ..,.,_.„.:.,:.,,.... _ QUAD 23 1.2631Ei42E722CE6,.1ECC7832e5:12345 1 .1C6S3B325995318, - -' ' QUAD 24 1.04717E.33E""/2/ QUAD 25 LIM. CAN 2E CHANGED IF EITHER MCRE OR LESS TRIES ARE DESIRED • QUAD 26 LIM=2CC QUAD 27 C=D QUAD 28 IS C SET TCO SMALL QUAD 29 IF (C.LT. 1.E-13) CO TC 2S0 • QUAD 30 10 IF (IGP.EC.M 1) PRINT 1 QUAD 31 1 FORMAT ( 2X,14HLEFT ENC PCINT,23X16HLENGTH.26X$12H8.-P7. RESULT QUAD 32 1 11X,19HREL.ERROR IN 3-PI. TI1X,4H1000 ) QUAD 33 MCP = C.0 QUAD 34 K = 0 QUAD 35 ?CNSEK = 0 QUAD 36 NCL7 = I QUAD 37 ANS = C. QUAD 38 F2 = O. QUAD 39 NERR=0 QUAD 40 Y = A QUAD 41 YOBLE = DELE(Y) QUAD 42 F = C/200. QUAD 43 E = 0. / QUAD 44 FO.ii****441 .11.4.******* ******** ****4 ********* ****44#4.414******44***44444** QUAD' 45- - FIRST TRY CN FULL SPAN AND ALSO LAST STEP GO THROUGH HERE QUAD 46 20 H = (9-Y)/2. QUAD 47 EGN =SIGCI.,),)N QUAD 48 H=A2S(H) QUAD- 49 LAST = 1 QUAD 50 ALL INIERMEOITE STEPS BEGIN HERE QUAD .51 30 X = Y • H*SGN 52 IS H TOO ShALL IC BE SENSE: RELATIVE TO X QUAD . 53 IF(X+.1*H.E7..x) GO TO 27C QUAD 54 IF(K.G1.LI1) GC TO 28 C55QUAD .....,*,44- ************ V*******441 4 ****** 41.41480#4, QUAD 56 4 FCINT !:iSCI.SS:E. 57 Z4(1)=.33ii!1C43.-“I a5E*H QUAD 58 14(2)=.86112E311:34053*H QUAD 59

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C 8 FCINT AESCISSAE QUAD 60 28111=.183434642495E50 4 H QUAD 61 28(2)=.52552240991E329 4 H QUAD 62 ZE(3)=.79EEE6477413627 4 H QUAD 63 Z8(4)=.966E89856497536 4 14- QUAD 64 C EVALUATE FLNCTION AND FERFCRP WEIGHTED SUM QUAD 65 G4=14 4 (1%4(1)*(FLN(X+24(1))+FUNCX-24(1)))4 QUAD 66 1W4(2) 4 (FUN(X424(2))+FUN(X-24(2)))) QUAD 67 G8=0. QUAD 68 CO 40 1=1,4 QUAD 69 21=FUN(X+20(I)) QUAD 70 22=FUN(X—Z8(I)) QUAD 71 40 GE=GE41.8(I) 4 (21+22) QUAD 72 G8=G8 4 1, QUAD 73 C 4-44744,-*******4 * *** *44 ***** 4114*******44,4*******IL.OUAO 74 ABG=AEZ(G8)41.E-260 QUAD 75 TE=ABS(G8—G4)+1.E-14 4 AEG QUAO 76 C RE IS THE RELATIVE ERRCR IN THE SUBINTERVAL THE 4 PT. RESULT RAKES QUAD 77 C IF THE 8 PT. RESULT IS EXACT QUAD 78 RE = 1.E-14 + TE/AEG QUAD 79 IF(K.EC.3) F=A8G .- QUAD 80 C P IS T1-E MAX AES VALUE OF ENTIRE INTEGRAL AS HE KNOW IT UP TO HERE QUAD 81 C K IS THE CCLNTER OF THE NUMBER OF ATIEVPIS QUAD 82 50 K = K 4. 1 QUAD 83 EH = F 4 P QUAD E4 ER = TE*RE QUAD 85 Q= EW/ER QUAD 86 IF(IMPF.NE.1) GO TO 70 QUAD 87 60 XLONTH=2 4 H QUAD 88 ERR=RE442 QUAD 89 ' G100=0 4 100.0 ' QUAD 90 PRINT 2 ,Y,XLGNTH.G8 ,ERR,C100 _ QUAD 91 2-FORMAT (E22.15, 2E30.15, 2E22.5) QUA 92 70 01E = G4*.CE25 93 Cl = H/2./5E 44 .125 QUAD 94 D2 = HiD1 4 C1E QUAD 55 C Cl IS THE ESTIMATE CF THE DISTANCE - A - TO THE SINGULARITY QUAD 96 C 02 IS AN IPFCRTANCE FACTOR WHICH NORMALLY RANGES FROM AEOUT 10. QUAD 97 C TO C.1 . WHEN THE RESULT IS UNIMPORTANT, 02 IS LARGE. QUAD 98 - QUAD 99 THE MAGIC GC—GC CR NO—GO QUANTITY IS 100Q FCUND AS FCLLQWS. QUAD 100 WE RECLIRE THAT THE RELATIVE ERROR IN THE 8 PT. SUBINTERVAL QUAD 101 VALUE. (RE 44 2) TIMES THE ImPQRTANCE OF THE SUDINTEGRAL (ABG/P) QUAD 102 BE LESS THAN HALF THE REQUIRED TOLERANCE C . QUAD 103 ALTERNATIVELY, (C/2)*(F/AEG)./(RE* 4 2) RUST BE GREATER THAN 1.0 QUAD 104 THE AECVE EXPRESSICN, t•.HEN MULTIPLIED CUT, IS 1000.- QUAD 105 IF(Q.LE. 0.C1) GO TO 120 QUAD 106 COMPARISON CF 4 FT. ANC 8 PT. LOOKS GCCD. QUAD 107 80 ES = O. QUAD 108 IF(N.NE.1) CC TO 200' QUAD 109 :44-* * ***** .*********************4*** QUAD 110 CHECK THE 12 PCINT RESULT QUAD 111 12 FOINT ABSCISSAE QUAD' 112 212(1)=.122334085114E9 4LH QUAD 113 212(2)=.3E782149e958180 4 H QUAD 114 212(3)=.58731754226617 4 H QUAD 115 212(4)=.76S5C2E7194305 4 H QUAD 116

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212(5)=.90411725E370475+H QUAD 117 212(6)=.9815E0634246719*H QUAD 118 C EVALLATE FLNCTION ANC PERFCRi4 WEIGHTED SUM QUAD 119 G12=0 QUAD 120 DO 100 I=1,6 QUAD 121 13 G12=G129-W12tI)*(FUN(X*Z12(I))+FUN(X212(I))) QUAD 122 Ci2=G12 4 H QUAD 123 ES=AES(G12•-GE) QUAD 124 G8=G12 QUAD 125 ER=ES QUAD 126 IF(ES ■. 100. 4 EW) 200,200,110 QUAD 127 C NCT GCCD ENCUGH. TRY AGAIN. QUAD 128 110 H=H/4.0 QUAD 129 F1 = 0.25 QUAD 130 GO TC 190 .QUAD 131 c ***** ******4*****44** ***** * ****** ************************************** QUAD 132 C QUAD 133 C THIS REGICN CF THE PROGRAM MCCIFIES THE STEP LENGTH WHEN QUAD 134 C SUBINTERVAL IS NOT SHALL ENCUGH QUAD 135 120 IF (NCLT .NE. 1) GO TO 130 QUAD 136 C FIRST CUTBACK QUAD 137- F1 = C16 QUAD 138 • H=AHIN1(.754N,C14016) QUAD 139 GO TO 190 QUAD 140 C SUBSECLENT CUT2ACKS IN THIS SERIES. QUAD 141 130 Fl = F1*016 QUAD 142 H = F1*H QUAD 143 190 NCNSEK = 0 QUAD 144 NCUT = 0 QUAD 145 LAST = 0 QUAD 146 GO TO 30 QUAD 147 C***** *** ****** ******************************************** 1***** QUAD 148 C QUAD 149 C SUCCESSFUL SLEINTERVAL INTEGRATION QUAD 150 C INCREASE STEP AS INCICATEC QUAD 151 200 ANS=ANZ*G8 QUAD 152 E = E -, AMAX1(ER, ES,1.E•14*ABG) QUAD 153 IF(LAST.EC.1) GO IC 300 QUAD 154 C HCP IS AN CLC SUCCESSFLL STEP QUAD 155 210 IF(HCF) 220,220,230 QUAD 156 220 HCP = H QUAD 157 230 F2 = 0.50*F2 * ALOG(H/HCP) QUAD 158 HCP = H QUAD 159 YOBLE = YDELE + DBLE(2.0*H'SGN) QUAD 160 Y = YDELE QUAD 161 NCNSEK = NCNSEK + 1 QUAD 162 IF(NCt, ZEK .GT. 4 ) GO TO 250 QUAD 163 IF(F2) 24C,250,250 QUAD 164 F2 .LT. O. SAYS IT- HAS NOT FCRGOTTEN THE PAST FAILURES Y'7 1. QUAD 165 240 P, C = C1*021(1.+2.*D2) QUAD 166 GO IC 260 QUAD. 167 F2 *GE. O. SAYS THE HISTORY hAS SEEN SUCCESSFUL . QUAD 168 250 HC = C2*(C1+2.*H) 4 016 QUAD 169 260 H = HG QUAD 170 NCLT = 1 . QUAD 171 P = AMtX1(F,AEG) cluAn 172 IF(SGN'Y + 2.C'H •• SGM*8) 30120920 QUAD 173

• ***** . • 6.... • • • • - C ************ 4**4444, 414 QUAD 174 C QUAD 175 C ERROR EXITS OLAC 176 270 hERR=-1 QUAD 177 WRITE(€, 3 ) H,Y . QUAD 17O 3 FORMATC53H GLAD FAILURE, STEF SIZE CANNOT BE MADE SMALL ENCUGH./ QUAD 179 1E6H IF YOU 'AISH TO CONTIN1,E ;•OvE SINGULARITY TO THE ORIGIN./ QUAD 180 211H STEP SI2E=,E24.1E, luX,15:-LEFT ENO PCINT=,E24.1E) . QUAD ial GO TO 300 QUAD 182 280 hERR=-:2 QUAD 183 WRITE(E, 4 ) LIM,Y,H - QUAD 184 4 FORMAILIAH1CLAD INCOHDLETE IN 14, 7H TRIES.,17H LEFT ENO POINT= QUAD 185 1E24.1E,10X,11H STEF SIZE=,E2 ,:..16), QUAD 186 GO TO 300 QUAD 187 290 NERR=-Z . - QUAD 183 PRINT 5 . \ CUA0 189 5 FCRMAT (68h RECUESTED TOLERANCE TCO SMALL, ROUTINE WILL FROCEEU US QUAD 190 ZING 10.0E-14 J QUAD 191 C=10.CE-14 - QUAD 192 GO TO 10 QUAD 153 C QUAD 194 C HERE WE RETURN TC THE MAIN; PFCGRAM WITH OR WITHOUT AN ANSWER QUAD 195 360•REL= 2.*E1(.5ES(ANS)+1,E-29C) QUAD 196 IF(NERR.GE.G.) NERR=K QUAD 1.97 IFt8-A.LT.O.J,ANS=-ANS QUAD 198 RETURN QUAD • 199 END QUAD 200 • • AFCRL -TR -76 -

COMPUTATIONAL TECHNIQUES FOR THE STUDY OF INFRASOUND PROPAGATION IN THE ATMOSPHERE

Allan D. Pierce and Wayne A. Kinney

School of Mechanical Engineering Georgia Institute of Technology Atlanta, Georgia 30332

FINAL REPORT 15 October 1973 to 31 December 1975

13 March 1976

Prepared for

AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSPACE RESEARCH UNITED STATES AIR FORCE HANSCOM AFB, MASSACHUSETTS 01731 ABSTRACT

A discussion is given of theoretical studies on infrasound propagation through the atmosphere which were carried out under the contract. Topics discussed include (1) the modification and adaptation of a computer program for the prediction of pressure signatures at large distances from nuclear explosions to include leaking guided modes, (2) the nature of guided infrasonic modes at higher infrasonic frequencies and the methods of extending waveform synthesis procedures to include higher frequencies, and (3) the propagation of infrasonic pressure pulses past the antipodes (over halfway around the globe). Summaries are included of all papers, theses, and reports written under the contract and conclusions and recommendations for future studies are given. An updated version of the computer program INFRASONIC l•VETORMS originally given by Pierce and Posey in the report AFCRL-70-0134 is included as an appendix. Chapter I

INTRODUCTION

1.1 SCOPE OF THE REPORT

The present report summarizes investigations carried out by the authors during the years 1973-1976 on the propagation of low frequency pressure disturbances under Air Force Contract No. F19628-74-C-0085 with the Air Force Cambridge Research laboratories, Bedford, Massachusetts. The study performed was theoretical in nature.

The central topic of this study was the generation and propagation of infrasonic waves in the atmosphere. The principal emphasis was on waves from man made nuclear explosions although certain aspects of the study pertain to waves generated by natural phenomena including, in particular, severe weather.

Specific topics considered during the study include the following:

1.) The adaptation of the computer program INFRASONIC WAVEFORMS to include leaking modes and to improve its accuracy in synthesizing early long period arrivals. (INFRASONIC WAVEFORMS is a digital computer program for the prediction of pressure signatures as would be detected at large horizontal distances following the detonation of a nuclear device in the atmosphere. The original version of this program was developed by Pierce 1 and Posey under a previous Air Force Contract [F19628-67-C-0217].) The developed theory for this adaptation has already been explained 2 in

Scientific Report No. 1 of the present contract; the present report describes the numerical implementation of this theory (Chapter III), and gives some specific numerical examples. The complete current version of INFRASONIC WAVEFORMS is included here as Appendix A.

2.) The development of a ray acoustic model for the synthesis of higher frequency portions of infrasonic waveforms. The theory developed during this study is given3 in some detail in Scientific Report No. 2 and a discussion of this phase of the work is accordingly not repeated here.

3.) The modification of the multi-modal synthesis method to avoid truncation of upper limits on frequency integration. The method developed is presented here in Chapter IV and represents an extension of the W.K.B,J. technique to the case when the atmosphere has two sound channels. The resulting theory clarifies the problem of selection of modes for in-

elusion into the synthesis and leads to a relatively simple method for revising the synthesis program. (This revision, however, has not yet been carried out.)

4.) Study of infrasonic waveform synthesis for propagation near and past the antipodes. The method for doing this was briefly mentioned in the 4 1973 AFCRL report (pages 25 and 26) by Pierce, Moo, and Posey . In Chapter V of the present report the theory underlying this is given and •some numerical examples are given.

In Chapter II, we list all of the reports, papers, and theses which were written during the course of this study. The abstracts given there plus the abstract of the present report should be considered as a comprehensive summary.of the accomplishments during the contracting period. In subsequent chapters of the present report, detailed discussions are given of some of the topics described above. In Chapter VI, some recommendations are made for future work in the field.

1.2 BACKGROUND OF THE REPORT

The general topics of infrasonic wave propagation, generation, and detection have been of considerable interest to a large segment published bibliography (the existence of which allows us to omit extensive 5 citations here) lists [Thomas, Pierce, Flinn, and Craine, 1971] over 600 titles, most of which are directly cncerned with infrasound. Litera- ture pertaining to the infrasonic detection of nuclear explosions con- stitutes a considerable portion of these. Earlier work by Rayleigh 6 7 8 [1890] , Lamb [1908,1910] , G. I. Taylor [1929,1936] , Pekeris [1939, 1938] 9 and Scorer [1950] 10 , among others, which was concerned with waves from the Krakatoa eruption [Symond, 1888] 11 and from the great Siberian meteorite [Whipple, 1930] 12 is also directly applicable to the understanding and interpretation of nuclear explosion waves.

The present report thus merely summarizes a continuation of a small number of facets of a lengthy pattern of research which has been carried on by a large number of investigators in the past. In a more restricted sense, the work reported here represents a continuation of work done in three previous studies performed under contract for Air Force Cambridge Research Laboratories. The first of these was Air Force Contract No. AF19(628)-3891 with Avco Corporation during 1964-1966; the second was Air Force Contract No. AF19628-67-C-0217 with the Massachusetts Institute of Technology during 1967-1969, the third was AF19628-70-C-0008 (also with M.I.T) during 1970-1972. Summaries of the earlier work may be found 13 in the appropriate final reports by Pierce and Moo . [1967] ,,by.Pierce and Posey [1970) 1 , and by Pierce, Moo, and Posey [1973) 4 .

One of -thd principal results of the first two aforementioned pre- .vious contracts was a canputer program INFRASONIC WAVEFORMS; the deck listing of the then current version of which is given in the report by Pierce and Posey [1970] 1 . This program enables one to compute the pressure waveform at a distant point following the detonation of a nuclear explosion in the atmosphere. The primary limitation on the program's applicability to realistic situations is that the atmosphere is assumed to be perfectly stratified. However, the temperature and wind profiles may be arbitrarily specified. The general theory underlying this program is somewhat similar to that developed by Harkrider [ 1964] 14 but differs from his in that it incorporates background winds and in that it has a different source model for a nuclear explosion. Chapter II

PAPERS, THESES AND REPORTS

The following gives author, title, and abstract of papers, theses, and reports written during the course of this project.

2.1 A. D. Pierce, "Theory of Infrasound Generated by Explosions," Colloque International sur les Infra-Sons, Proceedings (Centre National de la Recherche Scientifique (CNRS) 15, quai Anatole Prance, 75700 Paris, September, 1973). A review is given of recent studies by the author and his colleagues on infrasound generation by explosions and the subsequent propagation through the atmosphere. These studies include (i) development of computer programs for the prediction of pressure signatures at large distances from nuclear explosions, (ii) development of an alternative approximate model for waveform synthesis based on Lamb's edge mode, (iii)development of a geometrical acoustics' theory incorporating nonlinear effects, dispersion, and wave distortion at caustics, and (iv) theoretical models for the mechanisms of wave generation by explosions. The basic theory is briefly outlined in each case and some of the more significant results are explained in terms of simplified physical models. Such results include the predicted dependence of far field waveforms on energy yield and burst height, suggested techniques for estimating energy yield from waveforms, and an explanation of amplitude anomalies in terms of focusing and defocusing of horizontal ray paths.

2.2 W. A. Kinney, C. Y. Rapper, and A. D. Pierce, "Acoustic Gravity Wave Propagation Post the Antipode," J. Acoust. Soc. Amer. 55, S75 (A) (1974). The previous theoretical formulations and numerical computations of pressure waveforms (such as described by Harkrider, Pierce, and Posey, and others) apply only to atmospheric traveling waves which have traveled less than 1/2 the distance around the earth. In the present paper, a technique resembling that previously introduced by Brune, Nafe, and Alsop [Bull. Seismol. Soc. Am. 51, 247-257 (1961)] for elastic surface waves on the earth is discussed and applied to the acoustic-gravity wave propagation past the antipode problem. The principal modification to the older theory is a shift in phase of n/2 to the Fourier transform of the wave after it has traveled over halfway round the globe from the source. The source of the wave is presumed to be a nuclear explosion of given energy E. Numerically synthesized waveforms of antipodal arrivals are exhibited and compared with those for direct arrivals. The necessary modifications to the Lambmode model theory of Pierce and Posey [Geophys. J. Roy. Astron. Soc. 26, 341-368 (1971)] are also described.

2.3 C. Y. Kapper, "Leaky Infrasonic Guided Waves in the Atmosphere," J. Acoust. Soc. Amer. 56, S2 (A) (1974). Prior theoretical formulations and computational techniques for the prediction of pressure waveforms generated by large explosions in the atmosphere have considered only fully ducted modes. In the present paper, a technique for including weakly leaking guided modes in concert with fully ducted modes is developed. Modification of previous theory includes the extension of the boundary condition at the upper halfspace to include a complex horizontal wavenumber. The major alterations to the computer program infrasonic Waveforms (as described in report by Pierce and Posey, 1970) incurred consist of the computation of the imaginary part of the newly incorporated complex wavenumber, extension of the normal-mode dispersion function to lower frequencies, and a second-order correction factor to the phase velocity.

2.4 W. A. Kinney, "Asymptotic High-Frequency Behavior of Guided Infrasonic Modes in the Atmosphere," J. Acoust. Soc. Amer. 56, S2 '(A) ( 1974). Refinement of previous theoretical formulations and numerical computations of pressure waveforms as applied to atmospheric traveling infrasonic waves could include a description of their asymptotic behavior at high frequencies. In the present paper, calculations based on the W.K.B.J. approxirlation and similar to those introduced by Haskell [J. Appl. Phys. 22, 157-167 (1951)] are performed to describe the asymptotic behavior of infrasonic guided modes as generated by a nuclear explosion in the atmosphere. The results of these calculations are then matched onto numerical solutions which have been given by Harkrider, Pierce and Posey, and others. It is demonstrated that the use of these asymptotic formulas in conjunction with a computer program which synthesizes infrasonic pressure waveforms has enabled the elimination of problems associated with high- frequency truncation of numerical integration over frequency. In this way, small spurious high-frequency oscillations in the computer solutions have been avoided.

2.5 C. Y. Kapper, Computational Techniques in Infrasound Waveform Synthesis, M. S. Thesis, School of Mechanical Engineering, Georgia Institute of Technology (December, 1974). This thesis is concerned with two major theoretical and programming modifications to the digital computer program INFRASONIC WAVEFORMS for the synthesization of acoustic-gravity pressure waveforms generated by large explosions in the atmosphere. The first modification involves the extension of the guided mode approximation for pressure waveforms in the atmosphere into leaking mode regions and a conse- quent search for the imaginary part of the complex horizontal wave number. Particular results include a plot of phase velocity versus angular frequency showing the extension of the normal mode dispersion function into a leaky mode region for a multilayer atmosphere and a report on the search for the imaginary part of the complex horizontal wave number of a leaky mode for a two layer atmosphere. The second modification involves the extension of the systhesis of acoustic- gravity pressure waveforms to distances beyond the antipode. A phase shift is noted for waves passing through the antipode and a comparison of pre and post antipodal waveforms is presented.

2.6 W. A. Kinney, A. D. Pierce, and C. Y. Kapper, "Atmospheric Acoustic . Gravity Modes Near and Below Low Frequency Cutoff Imposed by Upper Boundary Conditions," J. Acoust. Soc. Amer. 58, S1 (A) (1975). Perturbation techniques are described for the computation of the imaginary part of the horizontal wavenumber (kI) for modes of propagation. Numerical studies were carried out for a model atmosphere terminated by a constant sound-speed (478 m/sec) half space above an altitude of 125 km. The GR 0 and GR1 modes have lower-frequency cutoffs. It was found that for frequencies less than 0.0125 rad/sec, the CR, mode has complex phase velocity; lc, varying from near zero up to a maximum of 3 X 10 -4 km-1 with analogous results for the GR 0 mode. There is an extremely small frequency gap for each mode for which no poles in the complex k plane corresponding to that mode exist. These mark the transition from undamped propagation to damped propagation. In the complete Fourier synthesis, branch line contributions compensate for the absence of poles in these gaps. Computational procedures are described which facilitate the inclusion of the low-frequency portions of these modes in the waveform sysnthesis. •

2.7 A. D. Pierce, and W. A. Kinney, Atmospheric Acoustic Gravity Modes at Frequencies Near and Below Low Frequency Cutoff Imposed by Upper Boundary Conditions, Report AFCRL-TR-75-0639, Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. (March, 1976). Perturbation techniques are described for the computation of the Imaginary part of the horizontal wavenumber (k I) for modes of propagation. Numerical studies were carried out for a model atmosphere terminated by a constant sound-speed (478 m/sec) half space above an altitude of 125 km. The GR and GR modes have lower-frequency 0 1 cutoffs. It was found that for frequencies less than 0.0125 rad/sec, the GR mode has complex phase velocity. 1 ' k 1 varying from near zero up to a maximum of 3 X 10 -4 km-1 with analogous results for the GR 0 mode. There is an extremely small frequency gap for each mode for which no poles in the complex k plane corresponding to that mode exist. These mark the transition from undamped propagation to damped propagation. In the complete Fourier synthesis, branch line contributions compensate for the absence of poles in these gaps. Computational procedures are described which facilitate the inclusion of the low- frequency portions of these modes in the waveform sysnthesis.

2.8 A. D. Pierce, and W. A. Kinney, Geometric Acoustics Techniques in Far Field Infrasonic Waveform Synthesis, Report AFCRL-TR-76- , Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. (1976). A ray acoustic computational model for the prediction of long range infrasound propagation in the atmosphere is described. A cubic spline technique is used to approximate the sound speed versus height profile when values of sound speed are input for discrete height intervals. Techniques for finding ray paths, travel times, ray turning points, and rays connecting source and receiver are described. A parameter characterizing the spreading of adjacent rays (or ray tube area) is defined and methods for its computation are given. A method of determining the number of times a given ray touches a caustic is also described. Formulas are given for the computation of acoustic amplitudes and waveforms which involve a superposition of contributions from individual rays connecting source and receiver and which incorporate phase shifts at caustics. The possibility of a receiver being in the proximity of a caustic is considered in some detail and distinction is made between cases where the receiver is on the illuminated or shadow sides of a caustic.- It is shown that a knowledge of parameters characterizing two rays at a point in the vicinity of a caustic provides sufficient information concerning the caustic to allow one to give a relatively accurate description of the acoustic field in its vicinity. The resulting theory involves Airy functions and uses concepts extrapolated from a theory published in 1951 by Haskell. The net result is a detailed computational scheme which should accurately cover the contingency of the receiver being near a caustic in the calculation of amplitudes and waveforms. A number of FORTRAN subroutines illustrating the method are given in an appendix. Limitations of the theory and suggestions for future developments are also given. Chapter III

NUMERICAL SYNTHESIS OF WAVEFORMS

INCLUDING LEAKING MODES

3.1 INTRODUCTION

The computer program INFRASONIC WAVEFORMS has been modified to allow inclusion of the contribution at low frequencies from leaking modes (specifically the GR 0 and GR1 modes) to numerically synthesized infrasonic pressure waveforms. The procedure incorporated in this modification involves a partly manual calculation of the imaginary and real parts of the horizontal wavenumber, k I and kR, respectively) as discussed in Scientific Report No. 1. 2 That calculation is outlined in more detail here. The numbers presented for illustration are appropriate to the case of observations at 15,000 km distance from a 50 megaton explosion, where the explosion is at 3 km altitude, and where the atmosphere is assumed to contain no winds. (This restriction is just for illustrative purposes, but is not a limitation on the method.)

3.2 CALCULATION OF COMPLEX WAVENUMBERS

The first step in the calculation is to obtain values for the phase velocities v (w), v (0, and v (0 for the GR and GR modes, and to n a b 0 1 obtain values for the elements R (w,v) and R v) of the transmission 11 12 (w , matrix [R]. These calculations should be done, in particular, for all frequencies extending below the mode's nominal lower cutoff frequency. 2 As mentioned in the previous report , R and R depend on the atmos- 11 12 pheric properties only in the altitude range 0 to z T (the bottom of the upper half space), and these are independent of what is assumed for the upper half space. Also, v n (w) is the phase velocity for a given (n-th) mode for values of w greater than the lower cutoff frequency coL ; here va (w) and vb (w) are values of the phase velocity w/k at which the functions $NAM1 NSTART=1, NPRNT=1, NPNCH=-1, NCMPL=-1 $END $NAM2 IMAX=24, ZI=1. ,2.,4.,6.,8.,10.,12. ,14.,16.,18.,20.,25. 30.,35.,40.,45.,55., 65.,75.,85.,95.,103.,115.,125., T=292.,288.,270.,260.,249.,236.,225.,215.,205.,198.,205.,215.,227., 237.,249.,265.,260.,240.,205.,185.,184.,200.,250.,400.,570., LANGLE=1, WINDY=25*0.0, WANGLE=25*0.0 SEND $NAM4 THETKD =35., VI = 0.143, V2 = 0.3318, CM1 = 0.001, 0M2 = 0.031, = 30, NVPI = 80, NAXMJD = 10 SEND $NAM1. NSTART=6, NPRNT=1 , NPNCH=-1, NCMPL=-1 $END

Figure 1. Listing of input data required to generate tabulations of Rll and R versus phase velocity and angular frequency in the 12 vicinity of the dispersion curves for the G R and G R modes. 0 1 ti

260 340 . .420 500 SOUND SPEED (m/sec)

Figure 2. Model atmosphere showing sound speed versus altitude for numer- .ical example treated in the present chapter. The atmosphere is bounded by an isothermal upper half space beginning at 125 km altitude. and R12, respectively, vanish. For a given mode, the values of v R11 a chosen are those from the curves v (w) and vb (w) which lie the and vb a closest of all such curves to the curve v (0 for w>wL . n

As regards the calculation of R and R12, the computer program 11 INFRASONIC WAVEFORMS may be used, only with an alternate version of the subroutine TABLE. A copy of subroutine TABLE with the appropriate modifications incorporated and indicated is given in Appendix B. A deck listing of all of the input data that is required to obtain R11 and R12, and that is appropriate to the running example, follows in Fig. 1. Values for R il and R12 need only be calculated for phase velocities between, say, 0.143 and 0.3318 km/sec, and for frequencies between 0.001 rad/sec (as close to zero as would seem necessary and corresponding to a period of 6,283 sec or 1.75 hr) and the value of w B for the upper half space (.0128 rad/sec in our numerical example). In the calculations reported here, the upper frequency was taken as .031 rad/sec in order to confirm the continuity of the dispersion curves. A sample portion of the printout of Rli and R12 corresponding to the model atmosphere of Fig. 2 is given in Fig. 3 . The same set of output from a computer run which lists the R 11 and R12 also includes the v (0 for the GR and GR modes. n 0 1

Values of v (0 and v (w) for these modes are obtained by two a b successive runs of INFRASONIC WAVEFORMS using in sequence two modified versions of the subroutine NMDFN. These modifications are so minor that the deck listing is omitted and we describe here the nature of the modifications.

To obtain v (0, one need only change the third from end execut- a able FORTRAN statement of subroutine NMDFN from

FPP = RPP(1,1)*A(1,2) - RPP(1,2)*(GU + A(1,1)) (3.1) to

FPP = RPP(1,1). (3.2) R v 11 R P 12 ' OME6A -= .30928-02 .14300+00 .21671+01 -.65152+02 .14539+00 -.72963-01 -.22523+02 • 14778+00 -.19992+01 .16898+02 .15017+00 -.34415+01 .493364-02 .15256+00 -•43203+01 . 1 .72532+02 54 95 + 130 -.46324+01 .85619+02 .15734+00 -.44356+01 .88883+02 0 0 • 15973+ -.38270+01 .83475+02 • .6212+00 -.29260+01 .71114+02 .16451+00 -.18579+0i .53814+02 .16890+00 -.74204+00 .33657+02 .16929+00 .31761+00 .12611+02 .17168+00 .12 3 76+01 -.75995+01 • 1 7407+ 0 0 .19579+01 -.25568+02 .17646+00 -.24418+01 -.40247+02 • 17885+00 .26 746+01 -.50952+02 .18124+00 .26605+01 -.57340+02 .18363+00 .24195+01 -.59371+02 18602+00 • -.19834+01 -.57261+02 .18641+00 .13917+01 -.51424+02 .19080+00 -.68860+00 -.42421+02 .19319+00 ...,80574-01 -930906+02 .19558+00 -.87165+00 -.17582+02 .19797+00 -.16447+01 -.315614-01 • 20036+00 -.23637+01 .116904-02 • 20275+00 -.29996+01 .26326+02 .20514+00 -.35295+01 ,40198+02 .20753+00 -.39379+01 .20992+00 .52832+02 -.421531-01 .63849+02 _ • _ 3

Figure 3. Sample printout of RH and R12 versus phase velocity for various fixed values of angular frequency. Output generated with the input data of Fig. 1. To obtain vb (w), one need only change the same statement to

.FPP = RPP(1,2). (3.3)

The same limits for phase velocity and angular frequency as are used for the calculation of R and R should be used in the calculations for v 11 12 n' v , and v In our example, when these limits are used, the GR mode a b. 1 corresponds to mode #3, and the GR 0 mode corresponds to mode #4 for the (w) is calculated. For the cases when v (w) and vb (w) are case when vn a calculated, the GR mode corresponds to mode #4 and the GR mode corre- 1 0 sponds to mode #6. A sample output listing of vn (w), va (w) and vb (w) for the two modes is given in Fig. 4. An additional listing of vn (w), va (w), and vb (w) for the two modes versus various values of w is given in Table 1.

3.3 CALCULATION OF a AND a

The next step in the -procedure is to manually calculate values for the variables a and $ which enter into an approximate version [Eq. (9) in Scientific Report No. 1] of the eigenmode dispersion function. These parameters represent the partial derivatives of R 11 and R12 , respectively, with respect to phase velocity v evaluated at v=v a and v=vb , respectively. Since R and R also depend on w, a and $ may be considered as functions 11 12 of angular frequency (but not of phase velocity).

It may be recalled that va (w) and vb (w) are values for the phase velocity at which R 11 and R12 , respectively, vanish. From the listing versus v and w, let the adjacent values R211 , R311 and of, say, R11 R111, R for R corresponding to the values for phase velocity v 11 , v21 , v31 411 11 and v41 , respectively (for same chosen w), such that v 21 and v31 brackett • R and R would then be of opposite sign. In the a value for v a' 211 311 listing of v, for various co, the values for v should all turn R11 , R12 out to be equally spaced. Given this fact, it is possible to reasonably approximate a from the listings of R 11 by the formula

i)([5/6]ell+[1/12]fil+[1/41g d = (1/Av lihil) (3.4) Table 1. Tabulation of frequency dependent parameters for the GP ° I modes. Tabulation is for frequencies below and GP cutoff, definitions of the various quantities are given in the text and in Scientific Report No. 1.

I '

GR0 MODE . GRI MODE

V Va w vb . w V v n a Vb L2375 .31185608.001030 .31205939 .001030 .31209836 '.013407 .22781499 .001030 .24434330.001030 0 .2 5 C734, L3407- .31181806.002 61 .31205552 . 0 02061 .31209447 .013624 .22664568 .002061 .24409612 .001738 .250544, .3117759 .4 0 38 7.003093,31204906 .003093 .31208799 .014040 .22425580 .003 0 93 .24367787 .002061 .25u424, 546 0 .31172882 .004124 .31204001 .004124 .31207393 .014424 .22186593 .003655 .24337478 . 0 03093 •2459,30 ; 6 5 01 .31167509 .005156 .31202834 .005156 .31i!06727 .01 1443/1 .22177526 .004124 .24307887 .004124 1 .24(,11,1,0 7 5 32 .31161209 .006187 .31201405 .006187 .31205303 .014778 .21947606 .005156 .24228453 .00515 6 •24/3j 59, 6 5 63 .31153394 .007218 .31199710 .00 7 218 .3 1 203620 .015107 .21708619 .006187 .24127431 .005160 .246154, 9070 .31148610 .008250 .31197748 .000250 .31201679 .015413 .21469631 .005 11 45 .24098491 . 0 0618 7 .246acp, 9079 .31148516 .009281 .31195515 .00928 1 .31199478 .015469 .21423833 .007210 .24001984 .006963 .2 4 5764/ 0 595 .31142505 .010312 .31193006 .010312 .31197010 .015699 .21230644 .008181 .23859504-00 7 218 .2 4 5350; 9053 .31138841 .011344 . .31190215 ,011344 .31194291 .015 066 .20991657 .008750 .23848240,000250 .2 4 3461 0 1 11 .31134515 .012375 .31167139 .012375 . ► 31 191302 .016717 .20752670 .009 281 .23660913%00829 3 .2 43 37 4- 0 626 .31122460.013 4 07 .31183768 .013407 .3 1 188045 .016453 .20513682 .009 4 79 .23620517, 0 0 9 28 1 .241. 183: 1655 .310295291.014 4 38 .31180003,.014436 .31184510 .016501 .20463309 .010312 .23432748 009362 .2 4 0964' 1 650 .310291161.015469 .311761041.015469.311007/4 .016675 .20274695 .010 5 010 26 0 .230595/ 2005 18 .23341529% .30790129 .016 5 01 .31171786:.016501 .31176630 .016586 .20035708 .01.1344 .23153728 % 010312 .2344013( 5 213° .30551142 .017 32 .31167120 .017532 .31172258 .017085 . 1 97 9 6 7 21 . .011381 .23142542 .011034 .2 3.5 2r:5; 2173 .30475278 .018 5 63 .31162087 .018563 .31167591 .017274 .19557733%012115 .22903555 .011344 .235140; 2 2 40 .30312155 .019595 .31156653 .019595 .31162620 017 4 54 .19318746 .012375 .22809942 .011712 .233815 232 0 .30073168 .020F26 .31150781 .020626 . .31157334 017532 .19211887 .012 752 .22664568 .012314 231 42 5 .' 2 4 12 .29834181 .021658 .3114441.5 .021658 .31151721 .017626 .19079759 .013311 .231168f z490 .22425580 .01237 .29595194 .022689 .31137478 .022680 .3 1 145 763 017790 .18840772 .013407 .22381942 .012455 .229035t; ? 5 66 .29356207 .023720 .31129855 .023720 .31139444 017 0 46 .18601784 .013A0 0 .22186593 .013345 .220645( !63 0 .29117220 .024 7 52 .31121368 .024752 .31132738.018096 .13362797 .014255 .21947606 .013407 .2 2 6325F f)80 .28948366 .025783 .31111721' .025783 .311ns619.01 82 40 .13123810 .014 4 38 .21842295 .013790 .224255f ! 7 10 .28878233 .026514 .31100382 .026814 . 31 118049 .018 3 711 .17884823 .014659 .21708619 .014199 .22j865r, !779 5 .26639246 :027 46.31086276 .027846 .3 1 109984 .018 5 10 .17645836 .015027 .21469631 .014438 .22:-./366; !A46 .28400259 .028877 .31066848 .028877 .31101364 .018563..17547997 .0157164 .21230644 .014575 .219476( 0 12 .28161272 .029°09.31034189 .029909 . 31 092114 .018638 .17406848 .015 4 69 .21151653 .014922 .2170660

•

Figure 4. A sample output listing of vn (w), va(w), and vb (w) for the G Ro and G R modes. 1 Table 1. Tabulation of frequency dependent parameters for the GR 0 4 modes. Tabulation is for frequencies below and GR, cutoff, definitions of the various quantities are given in the text and in Scientific Report No. 1. where

(3.5a) v41 - v31 = 1131 V21 v21 - V11

e = R - R (3.5b) 11 311 211

f = R - R + R - R (3.5c) 11 411 311 211 111

(3.5d) gll = (R211 R311)/ell

h = R + R - R - R (3.5e) 11 311 211 111 411

In like manner, from the listing of R 12 versus v and w, if one lets the adjacent values R112 , R212 , R312 , and R for R correspond to the 412 12 values for phase velocity , and v42, respectively (for some Y12' v22 , v32 chosen w), such that v and v bracket a value for v then one can 22 32 b' approximate 8 by the formula

= (1/Av + [1/12]f (3.6) 8 2)([5/6]e12 12 + [1/4]g12h12) where Av g and h are defined by equations analogous to 2' e12 , 12' 12 Eqs. (3.5) (last aubscript changed from 1 to 2).

Because we use a numerical method (i.e., that described above) to calculate a derivative (it would be preferable to have an explicit formula), there is a small amount of numerical noise in the tabulation versus w of a and 8 computed in the above manner. This noise is noticable only for the GR mode and may for all practical purposes be filtered out by plotting a 1 and 8 versus w and then drawing smooth curves through the respective sets of points. (See Figs. 5 and 6.) While this procedure is somewhat labori- ous, it circumvents doing additional runs of the program to get values of R and R at more closely spaced values of phase velocity. It also cir- II 12 cumvents a somewhat elaborate computer programming chore which would do Figure 5. A plot of the parameter a versus to for the G R 1 mode. The parameter a is 2R /av evaluated at the phase velocity where 11 p R 150. 11 Figure 6. A plot of the parameter 13 versus w for the G Ri mode. The parameter (3 is alt. /av evaluated at the phase velocity where p R z.O. 12 such steps automatically. (We suspect that the programming time would surpass all time which would ever actually be spent on manual circula- tions such as described above.) In any event, in view of the relatively small values of k I which are actually obtained (as described further below) and in view of the recommendations (also given further below) concerning the use of the same lc, in many different types of calculations, the accuracy of the a and 13 so obtained is more than sufficient.

3.4- CALCULATION OF COMPLEX PHASE VELOCITY

The applicable expression for calculation of a mode's phase velocity (real above cutoff frequency, complex below) is Eq. (10a) in Scientific Report 2 No. 1 (which for brevity is not repeated here). This involves parameters v and v (whose computation is described in Sec. 3.1), and a b X, which may be considered as a function of w and which is defined by Eq. (10b) in the prior report. This latter quantity X depends on a/a, A11, G and A 12. The latter three are computed by taking the phase velocity as v and using Eqs. (4), (7a), and (7b) of the prior report. a These calculations are straight forward, and do not require detailed explanation. Listings of G, A , Al2 , and X for various values of w 11 and for the GR and GR modes are given in Table 1. 1 0

As explained in the prior report, below cutoff (that is, below w L = 0.0125 rad/sec for GR 1 and below wL = 0.0118 rad/sec for GR o , in the running example) the real part k R of the horizontal wavenumber is the real (1) part of w/v , and the imaginary part k I is of course zero. Finally, the extension by first iteration of the normal mode dispersion curves (1) below cutoff is obtained by simply calculating w/k R. Listing of v , kR , and w/kR for various w for the GR and GR modes are given in 0 1 Table 1. Plots of kI and w/kR are given in Fig. 7.

3.5 INPUT DATA FOR GR AND GR o 1

The present version of INFRASONIC WAVEFORMS allows for the possibility of phase velocity w/kR , imaginary component kI , and source free amplitude AMP to be input as functions of angular frequency w for any given C)0.32

E >- G R0 CUTOFF 0 0.26 o w

•zx(f) 0.24 a.

0.002 0.004 0.006 0.008 0.010 . 0.012 ANGULAR FREQUENCY (radian/sec)

0.002 „ 0.004 0.00G 0.008 0.010 0.012 ANGULAR FREQUENCY (radian/sec)

Figure 7. Numerically derived plots of phase velocity w/kR and of the imaginary part k, of the complex horizontal wavenumber k versus angular frequency w for theG Ro andGRi modes. Nominal lower frequency cutoffs for these modes are as indicated. Note that lc, is identically zero above the cutoff frequency. mode. The only modes for which this is necessary are GR and GR This 0 1. input data is partly obtained by the procedure described above. Here we describe how the remaining portion of the input data is obtained.

To obtain values of phase velocity and source free amplitude at frequencies above cutoff one uses the current version of INFRASONIC WAVEFORMS with the variable NCMPL of NAMELIST NAM51 set less than zero. This gives an output essentially identical to what would be obtained with the original version of the program. The input data for this run would be the same as if one were computing waveforms without consideration of leaky modes. A sample listing of such input data is given in Fig. 8. The run will give mode numbers and tabulations of phase velocity VPHSE and amplitude AMP versus angular frequency OMEGA for the GR0 modes at frequencies,above cutoff. The only output which need and GR 1 be retained for future use are the tabulations of VPHSE versus OMEGA for these two modes, since amplitudes at frequencies above cutoff are computed automatically in the run which utilizes this information as input data. A sample tabulation of the pertinent output (for the running example considered here) is given in Fig. 9.

Input data of phase velocity VPHSE and amplitude AMP for frequencies below cutoff are obtained by a second run of the program, again with NCMPI, < 0, only with the original model atmosphere replaced by one which has a thick intermediate layer plus on upper half space replacing the original upper half space. Thus, in the NAM2 input list, MAX is increased by one, the original ZI and T are unchanged, but one adds a ZI for the new value of IMAX which is, say 100 km larger than the largest ZI for the original model atmosphere; the temperature T for the new MAX + 1 layer (i.e. for the new upper half space) is set equal to an arbitrarily very large value (say, 2x10 7 ° K). Doing this will artificially shift the cutoff frequencies for GR0 and CR1 down to values which are, for all practical purposes, equal to zero. The input data for this run should include choices of angular frequency and phase velocity limits (V1, V2, OM1, and 0M2 of NAM4) which are appropriate for an exploration of the properties of GR0 and GR1 at frequencies below their original cutoff frequencies. It is imperative that 0M2 not be too large since INFRASONIC WAVEFORMS will $NAM1 NSTART=1, NPRNT=1, NPNCH=-1, NCMPL=-1 SEND $NAM2 IMAX=24, ZI=1.,2.,4.,6.,8.,10.,12.,14.,16.,18. 20.,25.,30.,35.,40.,45.,55., 65.,75.,85.,95.,105.,115.,125., T=292.,288.,270.,260.,249.,236.,225.,215.,205.,198.,205.,215.,217., 237.,249.,265.,260.,240.,205.,185.,184.,200.,250.,400.,570., LANGLE = 1, WINDY = 25*0.0, hANGLE = 25*0.0 SEND SNAM4 THETKD = 35., V1 = 0.15, V2 = 0.495, GNI = 0.005, OM: = 0.1, NOY1 = 30, NA,PI = 30, MAr.10D = 8 SEND SNAM5 ZSCRCE = 3.0, :OBS = 0.0 SEND $NAM8 YIELD = 50.E3 SEND SNAN10 ROBS = 15000., TFIRST = 46.2E3, TEND = 52.2E3, DELTT = 15., IOPT = 11, SEND SNAMEI NSTART=6 SEND

Figure 8. Input data to obtain phase velocity versus angular frequency above cutoff frequency for theGR ancIGR modes. 0 1 GR0 MODE GR 1 MODE MEGA V n OMEGA. Vn . 0148275'; .31E4t.)1;52 .61tia)755 .1117 ) :43 ., 0161253 .2054827t d181334 .01E46552 .265CC28.5 • .01711538 .19758621 .C1723448 .01933:52 .31145/5C • 155446E1 .01756053 3;.974 ..3:3- • 3114..?492 .15163793 d2137 J31 • 01796558 . 18568566 . 32 1 516 7 9 • 31.,6c.345 .31810345 .18351434 .J217379 .016323E9 .17974136 .32232362 • .3 76L- 931 .018o525-2 . 17379313 ∎ .31892241 .32211..: .159 • 30E14224 .1684474E .32214436 .3[539871 .018-35156 .1E78448:: .02216121 305C,2694 .01939212 .16487Z165 .0221.7751 • 3046.517 .01922762 161896:;.:5 • 32219823 • 3CL1E532 .01933150 .155.53747 .0222Z:376 • 30391164 .01e-t85S4 .15594828 .02223357 .,zo21681c .:11S73352 • 1533303:4E .02229554 .301681C..2 • 02239972 • 25.8706SCI .32259355 .25275362 .02293273 2E0862C7 .023:11724 277716E6_ .023242E:6 263965:-.- 2 .02353165 .25706897 .023.333E9 .24517241 .024G6701 • 23327586 024325 Z.8 .22137531 • 324533ES .20548276 .02$5551.7 • 2E622217 • 02484741 • 15756621 02438335 .15163793 .02512335 .1656895E .02526dE2 .17974136 .025420 E2 173793_C .02558111 . .1678448.3 4025E6520 .1E4870E5 .32575227 • 1E18965.5 .02593679 . .1559462E .02613807 • 1503331'C

Figure 9. Sample output of phase velocity versus angular frequency at frequencies above cutoff for the G R and G R modes corresponding to the input data of Fig. 8.0 1 encounter numerical difficulties at higher frequencies when the height of the upper halfspace is as high as considered here. (If it were not for this fact, this run could be used to generate essentially the same information as in the previous run.) For comparison, Fig. 10 indicates the types of atmospheric profiles used in the two runs with NCMPL < O.

The second run gives values for the source free amplitudes AMP and phase velocities VPHSE for the GR and GR modes for frequencies below 0 1 cutoff. The latter of these are expected to be virtually identical to the w/kR which are obtained by the method described in Sec. 3.4. Also, the source free amplitudes are expected to match on smoothly to those obtained from the prior run for high frequencies even though the two model atmospheres are not identically the same. (This is because the energy transported by the GR0 and Gill modes is predominantly in the lower atmosphere.) Furthermore, we expect these amplitudes to be virtually the same as would be obtained by the modified residue method described in Scientific Report No. 1 for the original model atmosphere. The actual amplitudes should have a small imaginary part, but in view of the rela- -3 tively small values of the k (less than 10 nepers/km) obtained, we I are confident that this imaginary part may be neglected to an excellent approximation. The only aspect of the leaking phenomena which conceivably could be of significance is the accumulative exponential decay represented by the factor exp(-k ir), which is retained in subsequent calculations.

Sample input data for this second run with NCMPL < 0 are given in Fig. 11; a listing of the output values for OMEGA, VPHSE, and AMP below the original cutoff frequencies for the GR 0 and GR modes of the running 1 example is given in Fig. 12.

3.6 WAVEFORM SYNTHESIS

The final step in the waveform synthesis is to run the program INFRASONIC WAVEFORMS with input data including the information concerning the GR and GR modes computed as described in the preceding two 0 1 sections. The essential difference between this run and the first such 4

260 ' 340 420 600 260 340 420 600 SOUND . SPEED (mdsec) 'SOUND SPEED (misec)

•

Figure 10. Two model atmosphere profiles; the first is the same as in . Fig. 2; the second has the original upper halfspace replaced by a layer of finite but large thickness with a halfspace above it of extremely high temperature and sound speed. Second atmosphere is used to generate phase velocities and source free amplitudes at frequencies below nominal cutoff frequencies. $NAM1 NSTART=1, NPRNT=1, NPNGH=-1, NCNPLF-1 $END $NAM2 IMAX=25, 21=1.,2.,4.,6.,8.,10.,12.,14.,16.,18.,20.,25.,30.,35.,40.,45.,55., 65.,75.,85.,95.,105.,115.,125.,225., T=292.,288.,270.,260.,249.,236.,225.,215.,205.,198.,205.,215.,227., 237.,249.,265.,260.,240.,205.,185.,184.,200.,250.,400.,570. 1 2.E7, LANGLE=1, WINDY=26*0.0, WANGLE=26*0.0 $END $NAM4 THETKD= 35., V1 = 0.18, V2 = 0.34, ON1 = 0.001, 0M2 = 0.02, NCOff = 30, NCPI = 30, MANYOD = 8 $END $NAM1 NSTART=6 $EXD

Figure 11. Input data to obtain phase velocity and source free amplitudes theGR and G R below nominal cutoff frequencies for 0 1 modes. n;;F6A vr-!Sr. AhP - n . nmr64 vpHsF AmP S1 rei :05102934 .0-071-00 -2P:3-0--(3- .L/ 0 10 -.031019h8 .0 0 1P, * 11 237 .2 -.00003722 .:5 1 e 0 L, -. 0 3100520 .00e -C1 .2 14 129 -.00003b31 -Goe r) / .41eot, -. 0 509A5R9 -0 0 2 9 1 .27983 -.00004009 .512 0 4 13096170 '..TiO3-1--7 -:?-7-%7T1- .104.7 6 .3 1 en3 -.0309321, 0 .ti0S1s2 .271`1 1 -.0u00/1 2 0 5 .V1 4 0 5 -.03089855 .0 11, 06 .e7567 -.0UOUn7C4 .Un c)9Y .3 1 2n2 -.030859c1 .i0/173 .27379 -.00f1 05235 :0-1 W/11 .51 2 ► 1 ivra- 15mo 0-0-14r7 3' :7.72P:4 -=,7:-0-01M5510 .0 0 0 0 u ..51enu -.0307663/ .0 0 5c9 .2A,9C6 -.00006619 .0 0 /t;5 .31) 0 0 -.03071222 .0 0 5P2 .26621$ -.0000750/ .u(M21 .11 0 1 -.03065299 .0002 ,4 .2650,9 -.0000q2q1

.51A 06- .0-n -Eka - ..-,I•TA-27-0 -;01017320 .0 0 6c, t) .31100 -.0305A8 A5 .unociu eAllb -.0001n67e .311 0 4 -.03051919 .t07/1i) .es/24 --nun2h511 .0101 .311 0 2 -.030444S7 .on/c5 -esY4 6 -.0002 0 4?,! _.;73 -.73-11-01-1- -0-W751 .- 00-605 ----25172 -=;f- 1-V(1b374c .01 11 6 .311Ro ...03027970 .0 0 621 .e5uhu -.001 8/19P9 .V1214 .311 0.0 -.03 0 1S9361 .0 11 6c3 -eillAu -.0())51,bn5 .V 1 e79 .S11g4 -.030093651 -7„.00?4'SLI3b . 4i1-371 5- ---01 294Pza9- .0nc:Po ..- .)4 5 7 1 - . 11 (Je'./i/1 1 =110 .u .31179 -.029135741 .0."), ■ ..7""0 - . 0 01iCO25 .v1470 .31170 -.029773741 .unyc,2 .c-"eot? 20)15:11 .31173 -.02 0 b5474• .olilli .e 0 07•) - .• (1 0k0 e‘3n9. --:31--1711- -.r129-52-998 .0,09 .e. ,!0A9 - .NR0 5 S‘1,-) - 3 1 1 Flo -.0293 0 8 0 9' .003 .eA?-.A-0 .Q1/16 .3 1 1 1 2 -.02425/1 6 .010).% .>?6 , 6 .!)105 . s1 I Cb -. 0 2q10932 .01)70 .el'Ili - . 41 kRbqA4 LI 184,9- s1 TS2. .1) 1 21.i - .(? -Ki 7 2 - .n03i46C0 o19 7,q i11/1 0 -.028/65s7 .01e79 .eiii- -.ni133I) 7o .tJ 7 0 11 0 .3 1 1 AO -.02R54 (41 .u1:04 .e2 .../ho - . (1 01 0 6 1 5.. .11 1 5'1 5 .r.;/5:J - .nU121275 .0406 .2?-41 ,4 . 6 0AuCtrcS --; -01.610 -.22547 ".-0 03037A0 .01'47t .eivhi .01400 .eiii;,.! -.0v2/11409 .01541 .e11A9 -. 0 u2591 1J1 - .71:,--15-6-i ----- 2 1. 51-11. .0100/ .eo09..1 -.0u230706 .o1071 .en/c9 -. 0 0223902 .ult,72 .2ne?u - . 0 109 6 9 () 8 :0167 ► ---.2n2j).7- .01i?u .1 0 655 -.0016P7?2 .01/3b .1c4 4?o -.0u1569Q2i __.01ihi .1 0 1 11 3 -.0u1412 0 t .1.07 9 & T1A5ri2 -,, 0 -01 - 1412s1' 00 0 - .1 0 462 -.0o10qqaf .0101 .1Rv001 -.nor1879Sii

Iture 12. Sample output of phase velocity and source free amplitude at frequencies beim,- cutoff for the s R and G R modes correspon- 0 1 ding to the input data of Fig. 11. run described in Sec. 3.5 is that one sets NCMPL > 0, and that one supplies P values for the par ameters in the input list NAM51. A listing of the input data for the run, allowing for the leaking modes, and appropriate . to our running example is given in Fig. 13. The phase velocities input for the GR and GR modes are those derived from the two computer runs describ- 0 1 ed in Sec. 3.5. The source free amplitudes for these modes are supplied only for frequencies below cutoff and these are derived from the second run of Sec. 3.5. The inlPginary parts of the wave number are the numbers whose computation is described in Sec. 3.5. The reason we use the phase velocities below cutoff as computed in Sec. 3.5, rather than as in Sec. 3.4, is that both calculations agree to the same order of accuracy as would be expected for the approximations inherent in the method of Sec. 3.4. Consequently, we expect the values from the computer run to be the more nearly accurate. Of course, the values of k I have to be computed by the method of Sec. 3.4 since the computer program in its present form does not compute these directly.

In Fig.14 we show CAICOMP plots of modal and total waveforms obtained before and after the inclusion of leaking modes. (This is for our running example, 15,000 km from a 50 megaton burst at 3 km altitude, the receiver being on the ground.) One may note that the inclusion of the leaking modes eliminates the spurious precursor in the waveform and raises the amplitude of the first peak. It is also important to note that the waveform with leaking modes included begins with a pressure rise. This is what one would probably expect from intuition alone, and would also appear to be more realistic.

3.7 FURTHER EXAMPLE (HOUSATONIC)

To further explore the effects of inclusion of leaking modes, we chose the case of waveforms observed by Berkeley, California, following the Hausatonic detonation at Johnson Island on October 30, 1962. A previous comparison of theoretical and observed waveforms for this event is given in the Geophysical Journal article by Pierce and Posey 1 5 This case is also the central example in the 1970 AFCRL report by Pierce and Figure 13. Sample input data for synthesis of infrasonic waveform including leaking modes. The data for the NA151 input list is as derived from previous computations described in the present Chapter. Figure.14. CALCG1P plots of modal and total waveforms before and after inclusion of leaking modes. Example is for the case of a 50 megaton burst at 3 km altitude in the atmosphere of Fig. 2; receiver is at distance of 15,000 km. Poseyl , and is discussed within the Lamb edge mode theory context in some detail in Posey's thesis. 16

The model atmosphere assumed for the computation is exactly the same as in Fig. 3-12 of the 1970 report, only we let the upper half space begin at 125 km (IMAX = 24). Rather than repeat the tedious calculations of the k for the GR and GR modes for this model atmosphere, we assumed that I 0 1 they would be essentially the same as for the running example in the previous section. Thus the steps in Secs. 3.5 and 3.6 needed only to be carried out to obtain a waveform sysnthesis.

In Fig. 15, we give comparisons of the CALCOMP plots for this event before and after the inclusion of leaking modes. One may note that the first of these does not agree with the comparable CALCOMP plots in Fig. 3-10 of the 1970 AFCRL report. This is of course because we have here taken the upper halfspace to begin at a lower altitude. This choice of where the upper halfspace begins is of little consequence when leaking modes are included, and consequently the agreement of the old computation with the leaking mode included case is quite substantial. Further, the new computation is regarded as an improvement in that the spurious initial pressure drop has been eliminated.

On the basis of the calculations described above, we have redrawn the Fig. 7 in the Geophysical Journal article which compares observed and theoretical pressure waveforms for the Housatonic-Berkeley event. This revised figure is given here as Fig. 16. The only difference is in the center waveform. The precursor is now absent and the first peak to trough amplitude has been changed from 157 pbar to 170 pbar (less than 10% increase); the remanider of the waveform is virtually unchanged. The discrepancy with the edge mode synthesis hasn't been diminished and remains .a topic for future study. (It was not addressed during the present study.) NON- LEAKI NG LEAKING

fit'llivvvv‘Aevv-s...

NAVAgivvi

AV*

1 . .280 300 320 2.80 300 320 TIME AFTER BUR ST ' (min)

Figure 15. CALCOHP plots of modal and total waveforms before and after .the inclusion of leaking modes. The eventis observations at Berkeley, California, following the Housatonic detonation at Johnson Island on 30 October 1962. The energy yield assumed in the theoretical computations was 10 megaton. The model atmosphere is as previously used by Pierce and Posey in AICRL-70-0134, only the upper halfapace begins at 125 km. OBSERVED WAVEFORM

75/.2.bor

REVISED MULTI -MODE SYNTHESIS

co 50 0

111 bars

285 290 295 300 305 TIME AFTER BLAST (min.) . Figure 16. Observed and theoretical pressure . waveforms at Berkeley, California, following the Housatonic detonation at Johnson Island on 30 October 1962. The observed waveform is taken from Donn and Shaw (1967). The energy'yield assumed in the theoretical computations was 10 megatons. This is a revised version of the Fig. 7 in the 1971 paper by Pierce and Posey Oeophys. J. Roy. Astron. Soc. 26, 341-368). The original multi-mode synthesis figure has been replaced by one including leaking modes. Chapter IV

AS PTOTIC HIGH-FREQUENCY BEHAVIOR

OF GUIDED MODES

4.1 INTRODUCTION

Due to temperature and wind stratification, the earth's atmosphere pos- . sesses sound speed channels with associated relative sound speed minima. Fig.17 shows a standard reference atmosphere wherein two such sound speed channels are indicated; one with a minimum occurring at approximately 16 km altitude and the second with a minimum occurring at approximately 86 km altitude. Given the presence of such a channel, an acoustic ducting phenomenon can occur, as is demonstrated in Fig.18, wherein the energy associated with an acoustic disturbance can become trapped in the region of a relative sound speed minimum. It is this mechanism of ducting only that is of interest here.

In the computer program INFRASONIC WAVEFORMS, the computation of modal waveforms involves the numerical integration over angular frequency of a Fourier transform of acoustic pressure where this integration is truncated at the high-frequency end. It has been speculated that this abrupt truncation leads to the generation of what might be called "numerical noise" in the computer output. It was felt useful, therefore, to extend this integration beyond the heretofore upper angular frequency limit by means of some appropriate high-frequency approximation. In the case of an atmosphere with just one sound channel, the technique for doing this is well known and dates back to a paper published by N. Haskell 1951. Haskell's method is the W.K.B.J. (Wentzel, Kraners, Brillouin, Jeffreys) method, then in common use in quantum mechanics, although its invention dates back to Carlini- 8 19 Green in the early 19th century.

The approximations associated with the W.K.B.J. method of solution apply to the analytical model on which the computer program is based at 150 SUBTROPICAL SUMMER ARCTIC SUMMER . ARCTIC WINTER •

100

--.1 4 50

200 400 600 800 1000 -50 0 50 TEMPERATURE (•K) WEST-TO-EAST WIND (m/sec) .

Figure 17. Temperature and wind speed versus height profiles for standard reference atmospheres. Calculations in present chapter are for U. S. Standard Atmosphere 1962 without winds. The presence of two temperature minima indicates two sound speed channels. r.

WORN M.1.0 - ow.. ammo. - 411•••• M.110

TURNING POINTS

■ nlIONIO OEM •■••■•• MEOW am... anoint gnomon • .1 11• •••• 1-

Vp ACOUSTIC' PRESSURE 1- SOIJND SPEED PROFILE •

Figure 18. Sketches of sound speed versus height and acoustic pressure amplitude versus height for a guided mode illustrating the mechanism of acoustic ducting in a sound speed channel centered at a region of minimum sound speed. The energy of the disturbance may be considered as concentrated in the height region between turning points. frequencies above approximately 0.05 radian/sec (periods less than 2 minutes). Below that limit, effects due to density stratification in the atmosphere and gravitational forces cannot be neglected. Such effects therefore are not germane to the discussion here.

The application of the W.K.B.J. method of solution to the problem of describing propagation of acoustic disturbances in an atmosphere that contains two adjacent sound speed channels has previously been discussed in the literature by EckartP who invented the simple method of seeking a W.K.B.J. model for each of the sound speed channels spearately, then combining the results rather than treating the problem with a single model. In the present chapter, Eckart's method is applied and numerically verified for the case of infrasonic waves in the atmosphere.

4.2 THE W.K.B.J. MODEL

The W.K.B.J. model for propagation of acoustic disturbances in a single sound speed channel may be considered as an approximation for the acoustic pressure divided by the square root of the ambient density, which in general may be expressed as

P_ *(z ) e-iwteikx (4.1) 0 where w is angular frequency, k is the wave number associated with the horizontal dimension x, z is altitude. Here 4(z) satisfies the reduced wave equation,

d 20 (4 2 (4.2) c2( z) k2 = biz2 where c(z) is sound speed as a function of altitude. The W.K.B.J. approximation applies in general to all differential equations of this type if the coeffieient of IP is sufficiently "slowly varying." It would appear in particular to be valid in the present context provided

(4.3) where A is some representative wavelength of interest. This approximation states that substantial changes in sound speed should not occur within distances corresponding to a typical wavelength of interest if the model is to apply.

A particular sesult of the W.K.B.J. approximation is that dispersion curves (v vs. w) of guided modes are given by the equation

z top -2 v 2 zdz = (2n+l)w Ec P 2w (4.4) CC; Zb0 t tOM where v is phase velocity, n = 0, 1, 2, 3, and where zbottom and ztop identify the lower and upper bounds of the sound speed channel, respectively. This integral is a direct result of the W.K.B.J. method of solution 21 , and its numerical solution enables the plotting of dispersion curves.

4.3 COMPARISON OF DISPERSION CURVES

Particular insight into the high-frequency behavior of guided infrasonic modes was gained when the above integral was solved numerically by computer for both the upper and lower channels, the model atmosphere being that given in Fig.17. The resulting dispersion curves computed in this manner are shown in the lower portion of Fig 19. One set of curves (the dashed curves) is appropriate to the W.K.B.J. model for the lower channel and the other set (the solid curves) is appropriate to the W.K.B.J. model for the upper channel. In the upper portion of the same figure are shown again dispersion curves as generated by the computer model INFRASONIC WAVEFORMS. It should be mentioned that the computer model solves a more complex problem in the sense that the simplifications inherent in the W.K.B.J. model are not present.

As is illustrated in the lower portion of Fig.19, the two sets of dispersion curves generated by the W.K.B.J. models intersect with one another at various points. A comparison of the dispersion curves shown in both the upper and lower portions of Fig. 19 reveals that these points U) COMPUTER MODEL . 2 DISPERSION CURVES

I 0.34 VELOC La .

a. 0.27 0 0.1 L642. ANGULAR FREQUENCY (SEC -I )

W.KB.J. MODEL DISPERSION CURVE SETS

0.3 ca 0

1.1.1

- td LOWER (1) CHANNEL UPPE R a. - CHANNEL 0.27- 0 0.1: ANGULAR FREQUENCY (SEC-1 )

FigUre 19. A comparison of theoretical guided mode dispersion curves for the U. S. Standard Atmosphere 1962. The upper set of curves were generated by full wave calculations with the multi-modal synthesis program INFRASONIC WAVEFORIS. The lower sets were obtained by applying the W.K.B.3. method to the upper sound channel (solid lines) and the lower sound -channel (dashed lines), respectively. of intersection mark regions of resonant interaction in the phase velocity-angular frequency plane between adjacent modes of the computer model. To better illustrate this observation, in the right hand portion of Fig. 20 is shown one such region of interaction with its corresponding point of intersection between two dispersion curves of the W.K.B.J. models shown to the left. It should be mentioned that the dispersion curves of the computer model never intersect with one another. An analytical explanation of this fact has previously been given by Pierce 22 .

4.4 INFERENCES CONCERNING ENERGY VERSUS HEIGHT DISTRIBUTION

The above observation may be stated differently by saying that, for relatively high angular frequencies, the dispersion curve corresponding to a given mode of the computer model is comprised of portions of dispersion curves from both sets of the curves generated by the W.K.B.J. models. Two important inferences about the asymptotic high-frequency behavior of guided infrasonic modes can be drawn from this statement. First, for some frequency ranges, and depending on how dispersion curve portions match between curves of the computer model and the W.K.B.J. models, it can be inferred that the acoustic energy associated with a given mode is comprised of energy associated more with propagation of acoustic disturbances in one sound speed channel than in the other. Also, as frequency increases, this association alternates back and forth between channels. To illustrate, if, for a small range of frequencies, a portion of a dispersion curve of the computer model matches (in the phase velocity-angular frequency plane) a portion of one of the W.K.B.J. model curves for the upper channel, then that implies that, for that mode and for that small frequency range, the acoustic energy density associated with that mode is greater in the upper channel than in the lower channel. Secondly, in the standard reference atmosphere, the sound speed minimum for the upper channel is less in magnitude than the sound speed minimum for the lower channel. It can be inferred, therefore, that those acoustic disturbances for which phase velocities are less in magnitude than the sound speed minimum for the lower channel are associated more with acoustic energy trapped in the upper channel than in the lower channel, and thus, for this reason, do not contribute significantly to the acoustic energy at the ground. This inference implies that care must W.KE3.1 APPROXIMATION COMPUTER ::MODEL 34- 0.34

Z2 UPPER CHANNEL. 0.32

0.30

LOWER.' CHANNEL RESONANT 1 NTERACTioh' BETWEEN ADJACENT- MODE

0.28-4 0.06 0.08 0.10 0.12 0.0G 0.08 0.10 ( ANGULAR FREQUENCY ANGULAR FREQUENCY CSEC - I) (SECT) •

Iigure 20. A detailed (blown-up) plot of a section of E.g. 19 showing a region of resonant interaction between two modes, one ducted in the upper channel, the other ducted in the lower channel. The full wave calculation (computer model) indicates that the two modes interact such that the actual dispersion curves do not cross, but indicates that the W.K.B.J. and computer model curves are nearly the same except in the region of resonant interaction. be taken as to which nodes are chosen to superpose in the attainment of the final pressure waveform at the ground, as some may not contribute.

4.5 IMPLICATIONS FOR WAVEFORM SYNTHESIS

In the previous synthesis of guided pressure waveforms at long distances, the acoustic modes were numbered in order of increasing phase velocity (i.e., SO, Sl, S2,..., etc.) and the sum over modes was truncated at a finite maximum number of modes. The analysis given here indicates that this may be a very poor approximation for synthesizing high frequency portions of waveforms observed near the ground since there is always some frequency above which the first, say, N modes all correspond to channelling in the upper sound speed channel.

The preferable alternative would appear to be (for synthesis of ground level arrivals from sources below 50 km altitude) to ignore the upper sound speed channel completely for frequencies above, say, at least 0.2 rad/sec (possibly 0.1 rad/sec) corresponding to periods below at most 30 sec (possibly 1 min). - The dispersion curves could then be taken as given by the W.K.B.J. approximation and the mode amplitude versus height profiles could be computed by the method' outlined by Haskell. The Dis- persion curves and amplitudes so computed would fit directly into the general scheme outlined by Pierce and Posey" which forms the theoretical basis for the current version of INFRASONIC WAVEFORMS. Chapter V

EXTENSION OF INFRASONIC WAVEFORMS TO INCLUDE

DISTANCES BEYOND THE ANTIPODE

5.1 INTRODUCTION

Previous theoretical considerations incorporated into the digital computer program INFRASONIC WAVEFORMS restricted synthesis to waves that had traveled less than one-half the distance around the earth. The purpose of this chapter is to further exemplify techniques to enable computer synthesis of acoustic-gravity pressure waveforms at points whose distances are greater than halfway around the world from a nuclear explosion. Extension of prior theory shows that for wave propagation past a point on a spherical earth, one-half the great circle distance away from the point of detonation (i.e., the antipode), a phase shift of w/2 radians to the Fourier transforms of each modal wave is incurred. Modification to the computer program necessitates the reinterpretation of the great circle distance r, the inclusion of the 7112 phase shift, and a modification to the earth curvature correction factor. Computations are presented for pre and post antipodal waveforms.

5.2 THEORETICAL CONSIDERATIONS FOR POST ANTIPODAL WAVEFORMS

In considering acoustic-gravity waves that have passed beyond the antipode, certain specific definitions for the various waveforms must be adopted. To an observer located on the surface of a spherical earth between the source and the antipode the pressure waveform that is first observed is the direct arrival or A l arrival. The Al arrival has traveled the shortest great circle distance r to reach the observation point. The next waveform observed at the above observation point is the A or antipo- 2 dal arrival. The A arrival has traveled the longer great circle distance 2 from the explosion point around the glove passing through the antipode to reach the observation point. The A3 arrival is the A l pressure waveform that has traveled completely around the globe with respect to the observation point. Further arrivals exist but are not considered here. The distance r is measured in kilometers and is the great circle distance measured from the detonation point to the final observation point. Figure 21 shows some typical pressure waveforms recorded in suburban New York for the Russian explosion of 58 megatons at Novaya Zemlya 23 on 30 October 1961.

Previous numerical syntheses of acoustic-gravity waveforms have only considered direct arrivals. The extension of this theory to include waveform prediction for antipodal arrivals is described here. An investigation of a small region of the earth's surface in the vicinity of the antipode where prior theory breaks down yields certain waveform charac- teristics that enable waveform synthesis to be extended to ranges past the antipode. By taking the antipodal region small in area than say 1/100th of the earth's area as a whole we can consider this region to be flat. Then the equation governing propagation of any frequency in any guided mode near the antipode is the cylindrical wave equation in the form of

2 2 2 2 2 a F/arA + wridaivarA (11vp )a Vat = 0 (5.1) where F would represent the rA and t dependent part of the integration kernal for synthesization (i.e., integration over frequency of any given modal waveform where the height dependent part is omitted here)• The quantity V is the corresponding phase velocity. The assumed cir- P cular symmetry of the wave about the antipode is inherent in the absence of the angular derivative terms in the above equation. The distance r A is measured positive out from the antipode. The wave solution to Eq. (5.1) for the total acoustic pressure p and small r A can be written for time t as

Ao F = DJ (krA )cos(wt+e) (5.2)

For the above, k = w/V represents the horizontal wave number, w the angular frequency, and e some phase angle. The quantity D is some arbitrary constant while J o (krA ) is the Bessel function of zero order. I t Direct Arrival A l r = 6,630 km

350 pbars

Antipodal Arrival A2

• I t 3313'60 km

L tn (LI - 240 pbars a. .

2nd Antipodal Arrival A3

r = 46)720 km

190 ?bars

Time (15 minutes between marks )

Figure 21. Infrasonic "pressure waveforms recorded in suburban New York following the detonation of a 58 megaton yield nuclear device in Novaya Zemlya LISSR on 30 October 1961. [Extracted from

. Donn and Shaw, Rev. of Geophys. 5, 53-82 (1967).] When r is suffienciently large (i.e., greater than three wavelengths) A a solution for the total acoustic pressure p can be considered as a sum of ingoing and outgoing waves with respect to the antipodal region. The asymptotic solution for large krA can be written for time t as

F = A(r )-1/2cos(wt+kr + D A A in) (5. 3) + B(rA)-1/2cos(wt-krA+ (Pout)

In Eq. (5.3) 0 is some phase angle while w and k are as previously defined. The plus sign in the argument of the cosine denotes an ingoing wave. Equation (5.3) is not defined at rA = 0 and,as rA approaches zero, wave amplification is predicted. Figure 22illustrates waveform amplification approaching the antipode for three different values of r for a ten megaton nuclear explosion. The antipode is reached when r = 20,000 km.

Realizing that Eqs. (5.2) and (5.3) should represent the same pre- sure waveform at large rA we can now show the existence of a phase difference between waveforms approaching and leaving the antipode. For large rA , the Bessel function J o (krA ) can be represented by its asymptotic approximation such that Eq. (5.2) becomes

1/2 F = D(2/wr k) cos(kr -n/4)cos(wt+e) (5.4) A A or with the aid of trigonometric identities as

1/2 F = -2-D(2/7r Ak) [cos(wt+e+kr -114) A (5.5) + cos (wt+e-krA +7/4)]

Equating (5.3) to (5.5) then requires that

A=B D/(2irk 1/2 (5.6a)

E it/4 (5.6b)

out w /4 (5.6c) .15.1 Hours After r=17,000 km Detonation

• t 0 5420.00 5480.00 5540.00 5600 .00 560.00 5720.00 5780:00' Time (sec), after detonation tr) - r.18,000 km , raL- 16.0 Hours After • .0 Detonation

140 ?bars * *10 i 5750.00 5810.00 sa70.00 5930.00 5990.00 6050.00 6110.00 Time (sec), after detonation r=19,060 km 0 16.9 Hours After oD Detonat ion

190 ?bars. $10 1 6070.00 6130.00 190.00 6250.00 '310.00 6370.00 6430.0.0 I I I I _ I I . Time (sec), after detonation

Figure 22. Theoretical pressure waveforms of a pulse propagating towards the antipode (corresponding to a great circle distance r of 20,000 km). Computations presented are for a 10 megaton burst in a standard atmosphere without winds. Note the amplification in amplitude for values of r successively closer to 20,000 km. SO

(5.7) flout . in 1T/ 2

The latter shows that a pressure waveform undergoes a phase shift of 90 degrees. Based on this knowledge the computer program has been altered to synthesize pressure waveforms for the A2 arrival that passes through the antipode.

5.3 MODIFICATIONS TO INFRASONIC WAVEFORMS FOR POST ANTIPODAL WAVEFORMS

Waveform synthesis for ranges beyond the antipode necessitates only minor adjustments to the computer program. By considering the theoretical development of Brune, Nafe, and Alsop (1961) 24for circular spreading of waves over a spherical surface of radius r e (i.e., r e = 6374 km for earth) the amplitude correction factor for the curvature of a spherical earth, appearing in subroutine TMPT, is altered for post antipodal waveforms by replacing the term sin(rir e) by its absolute magnitude, where . r is inter- preted as the total distance the wave has traveled from the point of detonation. For post antipodal arrivals considered here r would be between "e and 2wr e kilometers. The earth curvature correction factor in subroutine TMPT appearing as

CF = (1.1(6374. * SIN(RAD)))**0.5 (5.8)

is replaced for post antipodal waveforms by

CF = (1./(6374.*ABS(SIN(RAD))))**0.5 (5.9) where ROBS = r and

RAD = ROBS/6374. (5.10)

To accomodate the change in phase as the waveforms pass through the antipode two computer cards of the form

PH2 = PH2 + 1.570796 (5.11) are inserted in the deck listing of subroutine TMPT after lines 160 and 177.

After incorporating the above modifications into subroutine TMPT the computer program was then utilized to synthesize various theoretical waveforms. Using the Soviet shot of 30 October 1961 as the source, a phase shift upon passing through the antipode is exhibited in Fig. 23 for two observation ranges of a synthesized pressure waveform. Further dispersion beyond the antipode of the pressure waveform is shown in Fig. 24 for a ten megaton explosion. A comparision of antipodal arrivals for a computer synthesized pressure waveform and a microbarograph recorded by Donn and Shaw in suburban New York 5 for the 58 megaton Soviet test is presented in Fig. 25. Considering the scattering in waveforms that can occur at such large arrival distances, it is not unreasonable to say that the amplitudes and typical periods of the two plots are of the same order of magnitude.

• 4

Expected First Before Antipode Peak r = 19)000 km Arrival 1000 pbars .

*I0 6030.00 6090.00 6150.0 6210.00, 6270.00 6330.00 6390 ' Time (sec) after detonation tl) L

61/ Expected First After Antipode 1- Peak Arrival r= 21000 km 1100 pbars

\/. *101 6650.00 6710.00 . 6770.00 6830.00 6890.00 6950.00 7plo.00 Time (sec ) after detonation;

Figure 23. Theoretical pressure waveforms just before (great circle distance r of 19,000 km) and just after (r of 21,000 km) passing through the antipode (20,000 km). The 112 phase shift after the antipodal passage is evidenced by the second figure. Time of expected first peak arrival derived from linear extrapolation of computed time of first peak arrival versus great circle distance for•r<20,000 km to case of r>20,000 km. Source is the 58 megaton

. Soviet test in Novaya Zemlya. 18.6 Hours After • t r= 21,000 km Detonation

• It L 200 libars .a *10 1 Z.. 6710.00 6770.00 6830.00 6890.00 7010.00. Time (sec), after detonation (1) 22.2 Hours After =25,000 km Detonation

85 mbars ••

100 1 0000.00 8060.00 8120.06 6 .180.00 8240.00 8300.00 Time (sec),after detonation 26.7 Hours After Detonation r = 30,000 km

• 70 pbars

1 *10 9600.00 9660.00 9720.'00 9780.00 91340.00 9900.00 Time (sec), after detonation .

Figure 24. Theoretical pressure waveform for a pulse propagating away from the antipode. Decrease of amplitude and increased frequency dispersion occurs with increasing great circle distance r. The source is a 10 megaton nuclear explosion in a standard atmosphere without winds. Theory. r.33,360 km 380 /Abars

0'1 cr1 L ro SD

29.7 Hours After Detonation Data r=33,360 km

J 0

Time After Detonation (15 minutes between marks)

Figure 25. A comparisoh of theoretical and observed antipodal (A 2) arrivals for pressure wave recorded in suburban New York following.the detonation of a 58 megaton yield nuclear device in Novaya Zemlya USSR on 30 October 1961. Note that the amplitude scales for the two records are not the same. Observed waveform taken from Donn and Shaw, Revs. of(hophys. 5, 53-82 (1967). Chapter VI.

CONCLUSIONS AND RECOMMENDATIONS

4.1 REMARKS CONCERNING INFRASONIC WAVEFORMS

The new version of INFRASONIC WAVEFORMS contained in this report (Appendix A) allows for the computation of waveforms which have propagated past the antipode and for the computation of waveforms including leaking modes. Our re-74rks here concentrate on the latter modification.

If one chooses a model atmosphere in which the sound speed is constant above some arbitrary large height, it is inevitable that the GR 0 and GR modes should have lower cutoff frequencies and be leaking below 1 that altitude. Beyond a certain point, one would expect that the computations should be independent of this choice of height, provided the analysis were carried through with some degree of exactitude. If there were a genuine sensitivity, this would indicate that these modes carry an appreciable fraction of their energies at high altitudes and this would in turn suggest that the neglect of physical dissipative mechanisms (such as viscosity and thermal conduction, Joule heating, etc.), which increase dramatically at extremely large heights for the frequencies of interest here, is not a valid approximation.

The reason we cannot take the bottom of our upper halfspace to be arbitrarily large is that some modal height-amplitudes decrease exponentially at large altitudes. This exponential decrease implies that, if one attempts to calculate the transmission matrix [R] connecting variables at the bottom of the upper halfspace to those at the ground, then the elements of [R] are going to be extremely large and the mathematical theorem that the determinant of [R] be 1, while true in principle, is not going to be satisfied for the actual numerical values computed because of the loss of significant figures. The net result is such large fluctuations in the eigenmode dispersion function due to round-off errors that it is impossible to determine its roots. This problem always arises at sufficiently high frequencies when the upper halfspace bottom is taken too high.

In Chapter III, a simple expedient for circumventing this difficulty is implicitly described. One uses one atmosphere for low frequencies, another atmosphere for higher frequencies. The atmosphere for the higher frequency calculations has its halfspace beginning at, say, 125 km altitude while the atmosphere for the lower frequency calculations has its upper halfspace beginning at, say, 225 km. Given the premise that, for the CR and GR modes (which appear to be the only modes for which we have 0 1 problems at low frequencies), the energy is ducted below 125 km, the temperature above 225 km can be made as large as one desires without changing the answers. Thus one simply chooses this temperature to be so large that the lower cutoff frequencies for the two modes are, for all practical purposes, zero. In this manner one can construct the phase velocities and source free amplitude functions versus frequency for these modes down to arbitrarily small frequencies.

Another question is whether or not the k I (imaginary part of wavenumber) for the leaking modes are physically meaningful. They obviously would be meaningful were the actual atmosphere terminated by an upper halfspace and were there no physical dissipation mechanisms. However, the actual atmosphere is more complicated than this model and one has to accept the fact that (1) an approximate atmosphere is going to give rise to approximate answers and (2) that the values of the k I are going to depend on the choice of the bottom height of the upper half space. Thus the k are really somewhat arbitrary. Fortunately, the values of I the k so derived are very small, at least for the example we have numeri- I cally carried out, that the computed waveforms are almost the same as if the k were identically zero. I

With the above remarks in mind, it is recommended that the calculations of the k for the CR and GR modes below cutoff not be carried out I 0 1 in the synthesizing of waveforms. Rather, one should either set the k I for frequencies below cutoff as given in our numerical example or to -10 2x10 (i.e., for all intents and purposes, zero). The reason the k I should not be set identically to zero is that the computer program uses the nonzeroness of k as a flag to decide whether to look for an input I value of AMP (source free amplitude) or to compute the number internally (it can't do this at frequencies below cutoff and will consequently return ANP = 0). While this may seem a rather simple thing to do, considering the elaborate mathematical theory developed 2 in Scientific Report No. 1, the analysis and computations which preceded the formulations of this recomMendation were necessary, if only to establish that the procedure has some rigorous mathematical basis.

In any event, it is evident that one must and should include contributions from the frequencies below the nominal low frequency cutoff (determined by the upper halfspace) if one is to adequately synthesize the initial portions of waveforms. The present report shows how this may be done. The procedure, although requiring several (three, in general) runs of the program rather than just one run to accomplish this, is relatively straightforward. It is obviously feasible to auto- mate this so that only one run is necessary, but the time limitations of the present study precluded our doing so.

6.2 DISCREPANCY WITH LAMB EDGE MODE THEORY

It was hoped that the inclusion of leaking modes into the multimode synthesis would eliminate the discrepancy between the numerical predictions of the Lamb edge mode theory and the multi-mode theory. It is evident, however, from Fig. 16 in the present report that this was not turned out to be the case. The cause of the discrepancy has not been resolved and time limitations precluded its resolution. There is always the possibility that either program may have a mistake. However, barring this, it should be pointed out that the modified multimode theory should be the more nearly correct. The Lamb edge mode theory 1S contains a number of approximations which the multi-mode theory does not contain. Consequently, it is recommended that the multi-mode model as modified here be used in preference to the Lamb edge mode model. The relative simplicity of the edge mode model still retains an intrinsic appeal and, consequently, it is recommended that some future effort be expended in revising the model (possibly by including higher order terms in the dispersion relation) such that the discrepancy is resolved.

6.3 GUIDED MODES AT HIGHER FREQUENCIES

The procedure outlined in Chapter IV for using a modified W.K.B.J. approximation to order the modes and to compute modal parameter at high frequencies looks eminently feasible and is recommended for inclusion into the multi-mode synthesis program INFRASONIC WAVEFORMS. Although, again, time limitations precluded this, we regret not having done so in the present study. The motivation for doing this, however, is not as strong as for the low frequency modifications because the commonly available data in the open literature is markedly poor as regards high frequency arrivals. If and when such a modification is carried out, one should ideally have appropriate data with which to compare the numerical predictions.

Another problem is that there is some question as to whether a mult- modal theory with a finite number of modes (even when judiciously selected) can ever adequately synthesize higher frequency arrivals. In many respects, we believe that an appropriate modification of a geometrical acoustics theory would-be preferable.

6.4 GEOMETRICAL ACOUSTICS MODEL

The geometrical acoustics model described3 in Scientific Report No. 2, although still incompletely developed, appears to hold considerable promise for the understanding of higher frequency arrivals. We know now how to take the edge mode into account and how to handle the problem of caustics. Problems of aretes, lacunae, and wave diffusion from channel to channel still remain, but we believe these can be overcome with only a modest amount of additional theoretical effort. The ultimate objective of the analysis should be to develop the simplest possible theory sufficient to explain and interpret available data. In this'respect, we would suggest that both the multi-mode and geometrical acoustical models. While perhaps more elaborate than should be ideally required, could be used as research tools to conduct numerical experiments which test simpler models. The statistical models developed by p. Smith25 for underwater acoustics appear especially attractive in this regard and we believe that one should be able to test his models using the geometrical acoustics model described in Scientific Report No. 2. Also, the types of numerical experiments envisioned should provide the inspiration and support required to refine Smith's models such that they be capable of a more nearly precise description of infrasonic waveforms. REFERENCES

1. A. D. Pierce and J. W. Posey, "Theoretical Prediction of Acoustic-Gravity Waveforms generated by Large Explosions in the Atmosphere", Report No. AFCRL-70-0134, Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. 01731 (30 April 1970).

2. A. D. Pierce, W. A. Kinney, and C. Y. Kapper, "Atmosphere Acoustic Gravity Modes at Frequencies near and below Low Frequency Cutoff Imposed by Upper Boundary Conditions", Report No. AFCRL-TR-75-0639, Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. 01731 (1 March 1976).

3. A. D. Pierce and W. A. Kinney, "Geometrical Acoustics Techniques in Far Field Infrasonic Waveform Syntheses", Report No. AFCRL-TR-76-XXXX, Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. 01731 (7 March 1976).

4. A. D. Pierce, C. A. Moo, and J. W. Posey, "Generation and Propagation of Infrasonic Waves", Report No. AFCRL-TR-73-0135, Air Force Cambridge Research Laboratories, Hanscom AFB, Mass. 01731 (30 April 1973).

5. J. E. Thomas, A. D. Pierce, E. A. Flinn, and L. B. Craine, "Bibliography on Infrasonic Waves", Geophys. J. Roy. Astron. Soc. 26, 399-426 (1971).

6. J. W. S. Rayleigh, "On the Vibrations of an Atmosphere", Phil. Mag. 29, 173-180 (1890).

7. H. Lamb, "On the Theory of Waves Propagated Vertically in the Atmosphere", Proc. London Math. Soc. 1, 122-141 (1908); "On Atmospheric Oscillations", Proc. Roy. Soc. London A84, 551-572 (1920).

8. G. I. Taylor "Waves and Tides in the Atmosphere", Proc. Roy. Soc. London Al26, 169-183 (1929); "The Oscillations of the Atmosphere", Proc. Roy. Soc. London A156, 318-326 (1936).

9. C. L. Pekeris, "The Propagation of a Pulse in the Atmosphere", Proc. Roy. Soc. London A171, 434-449 (1939); "The Propagation of a Pulse in the Atmosphere, Part II", Phys. Rev. 73, 145-154 (1948).

10. R. S. Scoren, "The Dispersion of a Pressure Pulse in the Atmosphere", Proc. Roy. Soc. London A201, 137-157 (1950).

11. G. J. Symond, The Eruption of Krakatoa and Subsequent Phenomena (Trubner and Co., London, 1888).

12. F. J. W. Whipple, "On Phenomena Related to the Great Siberian Meteor", Quart. J. Roy. Meteor. Soc. 60, 505-513 (1934).

13. A. D. Pierce and C. A. Moo, "Theoretical Study of the Propagation of Infrasonic Waves in the Atmosphere", Report No. AFCRL-67-0172, Air Force

. Cambridge Research Laboratories, Hanscom AFB, Mass. 01731 (1967). 14. D. G. Harkrider, "Theoretical and Observed Acoustic-Gravity Waves from Explosive Sources in the Atmosphere", J. Geophys. Res. 69, 5295-5321 (1964).

15. A. D. Pierce and J. W. Posey, "Theory of the Excitation and Propagation of Lamb's Atmospheric Edge Mode from Nuclear Explosions", Geophys. J. Roy. Astron. Soc. 26, 341-368 (1971).

16. J. W. Posey, "Application of Lamb Edge Mode Theory in the Analysis of Explosively Generated Infrasound", Ph.D. Thesis, Dept. of Mech. Engrg., Mass. Inst. of Tech. (August, 1971). 17. N. A. Haskell, "Asymptotic Approximation for the Normal Modes in Sound Channel Wave Propagation", J. Appl. Phys. 22, 157-168 (1951).

18. F. Carlini, Ricerche sulla convergenza della serie the serva alla soluzione del problema di Keplero, Milan (1917).

19. C. Green, "On the Motion of Waves in a Variable Canal of Small Depth and. Width", Trans. Camb. Phil. Soc. 6, 457-462 (1837).

20. C. Eckart, "Internal Waves in the Ocean", Phys. of Fluids 4, 791-799 (1961).

21. P. M. Morse and H. Feshbach, "Perturbation Methods for Scattering and Diffraction", Sec. 9.3 in Methods of Theoretical Physics, Vol. II (McGraw-Hill Book Co., New York, 1953) pp. 1092-1106.

22. A. D. Pierce, "Guided Infrasonic Modes in a Temperature and Wind-Stratified Atmosphere", J. Acoust. Soc. Amer. 41, 597-611 (1967).

23. W. L. Donn and D. M. Shaw, "Exploring the Atmosphere with Nuclear Explosions", Rev. of Geophys. 5, 53-82 (1967).

24. J. N. Brune, J. E. Nafe, and L. E. Alsop, "The Polar Phase Shift of Surface Waves on a Sphere", Bull. Seism. Soc. Amer. 51, 247-257 (1961).

25. P. W. Smith, "The Average Impulse, Responses of a Shallow-Water Channel", J. Acoust. Soc. Amer. 50, 332-336 (1971); "Averaged Sound Transmission in Range-Dependent Channels", J. Acoust. Soc. Amer. 55, 1197-1204 (1974). APPENDIX A

SOURCE DECK LISTING OF THE PRESENT VERSION OF INFRASONIC WAVEFORMS

This supercedes the source deck listing originally given by Pierce and Posey in AFCRL-70-0134. Changes incorporated include those described by Pierce, Moo, and Posey in AFCRL-TR-73-0135 and those described in the present report. APPENDIX B

SOURCE DECK LISTING OF

AN ALTERNATE VERSION OF SUBROUTINE TABLE

This version of SUBROUTINE TABLE is used, as described in Chapter III of the present report, to tabulate listings of R 11 and R12 versus angular frequency OMEGA and phase velocity VPHSE which are used in calculating the parameter a and a for the GR0 and GR1 modes which in turn are used in calculating the values of the imaginary component lc" of horizontal wavenumber for these modes at frequencies below cutoff. This version of TABLE should replace the version in Appendix A when a tabulation of R11 is desired. and R 12

74/74 CPT=1 FTN 4.4+8401 /5/094'69. 12.21.

SUBROUTINE TABLE(ON1,012,v1,ve,NOm,NvP,THETK,OM,V,INMODE,NuPT) C TABLE (SUBROUTINE) 7/19/68 LAST CARD IN DECK IS NO.. C

----AESTRACT----

C TITLE - TABLE C • GENERATION CF SUSPICILNLESS TABLE OF NORMAL MODE DISPERSION C FUNCTION SIGNS C TABLE CALLS SUDRCuTINE mPOuT TO CONSTRUCT THE MATRIX OF C NORMAL MODE DISPERSION FUNCTION SIGNS INMODE (STORED IN C VECTOR FORM CCLomN AFTER COLUMN) FOP,. REGION IN FREQUENCY- C PHASE VELOCITY PLANE (OMI.LE.OmEGA.LE.0M2.AND.V1.LE.VP.LE C .V2). SUBROUTINE stSFCT IS CALLED IC EVALUATE THE SUSPI- C INDEX ,ISUS, OF EACH INTERIOR ELEMENT IN THE MATRIX ' C SCANNING FROM LEFT TO RIGHT, TOP TO BOTTOM. IF ISUS .NE. C C , INMODE IS ALTERED AS FOLLOWS. C ISUS=1. ROA ADDEO ABOVE SUSPICIOUS ELEMENT AND COLUMN C ADJEc TO ITS LEFT C =2 COLUmN ADDED TO RIGHT OF SUSPICIOUS ELEMENT- AN3 RCW ADDED ABOVE IT C =3 RC.: ADDED BELOW SUSPICIOUS ELEMENT AND COLUMN C ADDED TO ITS RIGHT C =4 COLuN tiluEu TO LEFT OF SUSPICIOUS ELEMENT C AND ROW ADDED BELOW IT C HOWEVER, NEITHER THE NUMBER OF ROWS NVP NOR THE NUMBER OF C COLUMNS NOM WILL EE INCREASED BEYOND 160. IF ISUS CALLS FOR AN ADDITIoN.:L RCW WHEN NVP = 100 , THE MESSAGE (NVP = 1CO N = XX M = XX) WILL BE PRINTED. C N IS ROW NO. CF SUSPICIOUS ELEMENT. N IS COLUMN NO. IF C ISOS CALLS FOR AuDITILN OF A COLUMN WHEN NOM = 11:10, THE C MESSAGE (NOm = 1C.0 N = XX M = XX) IS PRINTED. C wHFN INMOD7:. HAS EEEN EXPANDED SCANNING IS RESUmEU AT THE C ELESIENT IN NEW !,.ATRix WITH SAME ROW AND COLUMN NOS. AS THOSE OF SUSPICIOUS ELEMENT IN OLD MATRIX. IF NUPT IS PCSITIVE ItimCCE WILL EE PRINTED AS IT IS RETURNED FROM mFouT AND IN ITS FINAL FORM.

LANGUAGE - FORTRAN IV (3E0, REFERENCE MANUAL C28-6515-4) AUTHOR - J.W.PCSEY, M.I.T., JUNE91958 C. r --USAGE-- C C SUBROUTINES MRCUT,SOSPCT,LNGTHN,WIDENtNMDFN ARE CALLED IN TABLE. C C FORTRAN USAGE CALL TAULE(0 ,11,0M2,V1,V2,NL,m,NvP,THETK,OM,V,INmODE,NoPT) C C INPUTS C 4;1!1 mlNIn.m VALUE OF FicEOLENCY TO BE CONSIDERED. C R 4 4 C 0m2 NAXIMLM VALUE CF FRE:..LENY- TO BE CONSIDERED C F*4 74/74 OPT=1 FTN 4.4.4-R401 75/09/09. 12.21

• V1 MINImum VALUE OF PHASE VELOCITY TO BE CONSIDERED . C R*4 C V2 MAXIMUM VALUE OF PHASE VELOCITY TO BE CONSIDERED C R. • NOM INITIAL NO. OF FREQUENCIES TO BE CONSIDERED C I*4 C NVP INITIAL NO. OF PHASE VELOCITIES TO - BE CONSIDERED C I*4 • C THETK PHASE VELOCITY DIRECTION (RADIANS) C R*4 - • NOPT PRINT OUT OPTION. IF NOPT = -1, NO PRINT. IF NOPT = 1, C I*4 INMOCE IS PRINTED IN ITS INITIAL FORM (GENERATED-BY MPOUT) C AND IN ITS FINAL FORM.

C OUTPUTS C • NOM . -TOTAL NO. CF FREQUENCIES CONSIDERED C . 1+4 C NVP TOTAL NO. OF PHASE VELOCITIES CONSIDERED C I*4 C OM VECTOR WHOSE ELEMENTS ARE THE VALUES OF ANGULAR FREQUENCY C R;4(D) CORRESPONDING TO THE COLUMNS OF THE INMOUE MATRIX

C V VECTOR WHOSE ELEmENTS ARE THE VALUES CF PHASE VELOCITY C R}4(D) CORRESPONDING TO THE ROWS OF THE INMOBE MATRIX C C INMODE EACH ELEMENT CF THIS MATRIX CORRESPONDS TO A POINT IN THE C I*4(D) FREQUENCY (O1 - PHASE VELOCITY (V) PLANE. IF THE NORMAL C MODE EISPERSION FUNCTION (FPP) IS POSITIVE AT THAT POINT, • C THE ELEMENT IS +1, IF FPP IS NEGATIVE, THE ELEMENT IS -1, C IF FPF COES NOT EXIST, THE ELEMENT IS 5. INMODE HAS'NVP • C ROWS AND NOM COLUMNS. MATRIX IS STORED AS A VECTOR, C COLUMN AFTER COLUMN. • C C C --EXAMPLE-- •C C LET INMODE = -1,5,5,5,1,-11-1,-1,1,11-17*1,101,1.1 C WITH NON = NVP = 4 C ANC OM = THETK = 3.14159 C V = 1.0,2.0,3.0,4.0 C. (VALUES NOT CORRECT, FOR ILLUSTRATION ONLY) C • C THEN THE TABLE WILL BE PRINTEC AS FOLLOWS. C C VPHASE MORHAL MODE DISPERSION FUNCTION SIGN C 1.00003 C 2.00030 X-++ C 3.00000 X--+ C 4.00000 x--+ C C OMEGA 1234 C PHASE VELOCITY DIRECTION IS 90.0000EOREES C C OMEGA = C 0.10002E 01 0.15000E 01 0.20000E 01 0.25000E 01 C 74/74 CPT=/ FTN 4.44- R401 75/09/09. 12.21

C --PROGRAM FOLLCWS BELOW-- C C C DIMENSICN 0 3 (10G),V(100),INMODE(10000),DORN(100),KORN(100) DIMENSION F??(2,2) COMMON IAAX,C1(105),VXI(1D0),VYI(100),HI(100) C C :)POUT IS CALLED TO PrWCUCE INMODE tiATRIX AND OM AND V VECTORS. CALL MROuT(OM1,0M2,V1,V2,NuM,NUP,INMODE,OM,V,THETK) C C IFLAG = 1 INDICATES FIRST TINE THRCUGH WRITE PROCEDURE - -• IFLAG = 1 C C INMODE IS PRINTED IF NOPT IS POSITIVE IF (Ni.iPT.GE.0) GO TO 123 5 IFLAG = C NOPER=0 C NCPFIR IS THE NUM3ER OF EXPANSION OPERATIONS PERFORMED IN THE PRESENT G SCAN OF Tri. MATRIX. THUS, NOPER IS THE NUMBER OF SUSPICIOUS POINTS C FOUND IN THE PRESENT SCAN. - C C BEGIN SCANNING OF INTERIOR ELEMENTS OF INMODE IN UPPER LEFT CORNER N = 2 M = 2 10 CALL SUSPCT(N,M,NVP,INMODE,ISUS) C C POINT (N,m) IS SUSFICIOJS IF ISUS.NE.0 IF(ISUS.NE.0) CO TO 60 C C CHECK FOR END CF RCH 20 IF (V.LT.(t.0-1)) GO TO 30 C C CHECK FOm LAST ROW IF (N.LT.(NVP-1)) GO TO 40 GO TO 121

C - MCVE ONE COLUMN TO RIGHT M = M+1 GO TO 10 C C ADVANCE ONE ROW AND START AT COLUMN TWO N = N+1 M = 2 GO TO 10 C C CHECK FOR MAXIMJA VALUE OF NVP 60 IF(NVP.LT.1Jj) GO TO 62 61 FORMAT (24H NVF = 100 N =,I3,814 H =.I3) wRITE (6,61) N,H GO TO 20 62 IF(NC.m .LT. 100) GO TO 70 63 FORm.tT(244N3M = 100 N=,I3, 8H M=,I3) 64 WRITE(6,3) N,M GC TO 20 70 IF(I.)US .NE. 1) GO TO 75 )UTINE TABLE 74/74 CPT=1 FIN 4.4+R401 75/09/09. 12.21

C C ADD ROW ABOVE SUSPICIOUS POINT N1=N-1 C C ADD A COLUMN TO LEFT OF SUSPICIOUS POINT M1=M-1 GO TO 100 75 IF(ISUS .NE. 2) GO TO 83 C C ADD A COLUMN TO RIGHT OF SUSPICIULS POINT M1=M C C ADD ROW ABOVE SUSPICIOUS POINT N1=N-1 GO TO 100 80 IF(ISUS .NE. 3) GO TO 85 C C ADD A COLUMN TO RIGHT OF SUSPICIOUS POINT M1=M C C ADO ROW BELOW SUSPICIOUS POINT N1=N GO TO 100 C C ADO ROW BELOW SUSPICIOUS POINT 85 N1=N C C ADD A COLUMN TO LEFT OF SUSPICIOUS POINT M1=M-1 100 CONTINUE CALL UNGTHN(OM,V,INMODE,NCM.NVF,NVPP,N1.1,THETK) CALL WICEN(OM,VgINMODE,NOM,NOMP,NVPP,M1,1.THETK) NVP=NVPP NOM=NOMP NOPER=NOPERfl GO TO 10 121 CONTINUE IF(NOFER .GT. 0 .AND. NVP .LT. 100 .AND. NOM .LT. 100) GO TO 5 C C DO NOT PRiNT INMODE IF NOPT IS NEGATIVE IF1NOPT .LT. 0) RETURN C C LABELING 122 FORMAT (6H1VPHSE,E0C,36HNORMAL MODE DISPERSION FUNCTION SIGN') 123 WRITE (6,122) DO 133 I=1,NVF DO 128 J=1.N0M J88=(J-1)*NVP+I J89=INMODE(Jd8)-1 IF (J89) 126,125,124 124 CONTINUE C C IF IhMODE = 5, DORN = 1HX CATA 01/1HX/ DORN(J) = 01 GO TO 127 125 CONTINUE ITINE TABLE 74/74 OPT=1 FIN 4,4 -1 R4(11 75/(19/09.

C C IF INMOUE 1, DORN = 1H4 DATA 02/1H+/ DORN(J) = 02 GO TO 127 126 CONTINUE C C IF INMODE = —1, DORN = DATA 03/1H.-./ DORN(J) = 03 127 CONTINUE 128 CONTINUE C C PRINT ROW I OF TABLE WRITE (5,132)VII),(DORN(J), J=1,N01) 130 FORMAT(1H ,Filie5,3X 1 1011A1) 133 CONTINUE. J1U = 10 DO 150 J=1,NOM - C • C NUMBER COLUMNS 150 KORN(J) = MOD1J,J10) WRITE (6,213) (KORN(J), ..1=1,NOM) 213 FORMAT (bHGOMEGA,6X,1G0I1) C - C CONVERT THETK FROM RADIANS TO DEGREES X = THETK*18J/3.14159 WRITE (6,413) )( 413 FORMAT (1H ,11X$27HPHASE VELCCITY DIRECTION ISIF9.3, 1 8HOEGREES ) WRITE (6,513)- 51$ FORMAT ( dHaOMEGA =I C C LIST VALUES OF OMEGA WHICH CORRESPCND TO COLUMNS CF TABLE WRITE (6,613) (UM(I),I=1,NOM) 613 FORMAT ( 1H ,5E14.5) C C IF SUSPICION ELIMINATION HAS NOT BEEN PERFORMED, BEGIN IT AT THIS TIME IF(IFLAG.EQ.1) GO TO 5 DOLVP=(V2-111.)/(NVP-1) OMEGK=0M1 DELOM=(OM2-0M1)/(NOM-1.) CO 9dS IAA=1,NOM WRITE (6,933) CMEGK 933 FORMAT (1H t3X16H0MEGA=,E14.5) DO 977 JAA=1,NVP VE=V1+(JAA-1)*COLVP AKX=OMEGK/VE AKY=6:C CALL RiiRR(OMEGK,AKX,AKY,RPP,KY) WRITE (6,944) VE,RPP(1,1),RPP(1,2) 944 FORMAT (1H ,E12.5,6X,E12.5,3X,E12.5) 977 CONTINUE OMEGK=OEGKI-DELOM 986. CONTINUE RETURN - ENO