An applicationof the spheroidal-coordinate-basedtransition matrix: The acoustic scattering from high aspect ratio solids RogerH. Hackmanand Douglas G. Todoroff NavalCoastal Systems Center, Panama City, Florida 32407 (Received29 March1984; accepted for publication22 April 1984} In a previouspaper [Roger H. Hackman,J. Aeonst.Soe. Am. 75, 35-45 (1984}], a spheroidal- coordinate-basedtransition matrix formalism was establishedfor acousticand elasticwave scattering.In thispaper, the acoustic scattering by a solidelastic cylinder with hemispherical endcapsand a length-to-diameter ratioof 10 is considered. Numerical results are presented for the backscatteredform function as a functionof frequencyfor variousangles of incidence.These resultsare compared with experimental measurements taken at the Naval Coastal Systems Center andgiven a physicalinterpretation. PACS numbers:43.20.Fn, 43.30.Gv

INTRODUCTION Thisprocedure has a distinctadvantage over competing In 1968,Waterman' introduced a newtechnique for approachesin that it appearsto havethe strongest theoreti- systematicallysolving the acousticdiffraction problem for calfoundation? sThe approach is formallyexact and com- an arbitrarilyshaped body, the transitionmatrix approach. putationallyefficient (where applicable), and both the For the sakeof completenesss,we brieflyouthne the devel- uniqueness3and the convergence 4'•of theprocedure can be opmentof thisformalism below. We assumethat theacous- provenfor sound-softand sound-hard scatterers for appro- tic field satisfiesthe standardHelmholtz equation priatechoices of basisfunctions. This is in markedcontrast to the usualintegral equation approach, 6 which leads to equationswhich are singular at certaindiscrete frequencies •'(r')-I-f, ds h.[•+VG (kIr -- r'l) --G(k Ir -- correspondingto the interior eigenvalueproblem, and to = thoseapproaches embodying the ad hocRayleigh hypothe- 10, r'•v, sis.? The T-matrix formalism has been extended to elastic where•3 is the total acoustic field, •b i is the incident field, G is wavescattering by Waterman sand Pao and Varatharajulu 9 thefree space Green's function and the integral is overthe andto acoustic(i.e., scalar wave) scattering from an elastic surfaceof the scatterer.The subscript( -I- ) denotesa bound- targetby Bostr•m.•ø aryvalue as the surface is approachedfrom its exterior and It hasbeen subsequently determined that thisapproach, thetwo different values of theright-hand side of theequation althoughformally exact, suffers from a severepractical nu- referto points exterior and interior to the scattering volume, merical limitation. For scattererswhich deviate strongly v, respectively.We notethat the latter case, i.e., when the froma sphericalshape the number of termsrequired in the fieldof the reradiating sources induced onthe surface ofthe expansionof the surfacefields increases dramatically and scattererexactly cancels the incident field, is oftenreferred the matrices tend to become ill conditioned. This behavior toas the "extinction theorem. "• Waterman'sprocedure isto had beenpredicted by Lewin,• who suggestedthat the expandthe incident and scattered waves, •3•and •b' ---- • -- •, sourceof this difficultyis a hypersensitivityof the surface and the Ctreen'sfunction G, in regularand outgoingsolu- field to minute errors of the field on the inscribedsphere. tionsto the scalarHelmholtz equation in sphericalcoordi- That is, assumethat the extinction theorem is satisfiedto a nates.Since the Green'sfunction is singularwhen the source givenorder of approximationon the inscribedsphere with a andfield points coincide, for theextinction theorem the ex- givenexpansion. The further this sphericalsurface must be pansionprocedure can be implemented only for field points deformed to conform to the surface of the scatterer, the whichare interiorto the largestsphere which can be in- greaterthe errorin the extrapolatedvalue of the fieldand its scribedin the scattererand centeredon the origin. However, derivativeson the surfaceand hence,the greaterthe induced • isregular throughout the interior of the scatterer. Thus, by error in the outgoingfield. theprocess of analyticcontinuation, it follows that • must Batesand Wall •2 were the first to suggestthat these dif- vanishnot just in the interiorof the inscribingsphere, but ficultiescould be alleviatedby formulatingthe scattering throughoutthe interior of the scatterer. An infinite,coupled problemin alternativeexpansion functions which "better setof equations isdeveloped from these results by expanding fit" the scatterer.These authors examined the efficacyof a the unknown surface fields in a suitable basisset. A formal two-dimensionalelliptic-cx•ordinate-bascd transition matrix solutionto this set of linear algebraicequations can be for the scatteringof scalarwaves from largeaspect ratio scat- straightforwardlyobtained, giving the expansioncoeffi- terers.They observedthat choosinga coordinatesystem, cientsof theoutgoing wave in termsof thoseof theincident suchthat the "radial coordinate= constant"surface lay as field. closeto the surfaceof the scattereras possible, alleviated the

1058 J. Acoust.Soc. Am. 78 (3), September 1985 1058 ill-ctmdititmingproblems. In laterwork, Wall l• attempted There havebeen a numberof previousformulations of to developa transitionmatrix in prolatespheroidal coordi- acousticscattering problems in spheroidalcoordinates. Ko- natesfor vectorwave scatter•g, but wasunable to construct tani,•? who is perhapsthe earliest example, considered the a Green'sfunction exposion suitable for generalwaves. He acousticdiffraction by a circulardisk in oblatespheroidal did succeedin constructinga Green's function for axisym- coordinates.Bowkamp, •s and later Spence,•9 gavemore metricwaves, •4 and later applied this formalism to a cylin- thorough,rigorous treatments of this problemand of the dricalantenna problem. •z While otherauthors •c'-2• have diffractionby a circularaperture. Spence and Granger • are suggestedtechniques for improvingthe numericalstability generallyaccredited with formulatingthe scatteringfrom a of the spherical-coord•inzte-basedT matrix, noneof these rigid prolatespheroid as a modalharmonic sum in spheroi- approachesholds the promise or the intuitiveappeal of the dal coordinates.These authors presented numerical results spheroidal-coordinate-basedformalism for largeaspect ra- in the form of beampatterns for variousaspect ratios and tio scatterers. anglesof incidencefor low frequencies.Senior •! gavenu- The difficultyin constructinga Green'sfunction in mericalresults for rigid and pressurerelease spheroids for spheroidalcoordinates lies in thenonseparable nature of the axially incident acousticwaves, and Kl'•schchevand vectorHelmholtz equation in this coordinatesystem. As a Sheiba• extendedthese results to obliqueincidence. There consequenceof this nonseparability,the vectorwave func- hav•been a numberof solutions for the diffraction ofplane titms are not orthogonaland the standardtechniques for wavesin thehigh frequency regime? -z7 The more predomi- constructingthe Green's function become intractable. nantapproach z•-z• has been to usean extension of theWat- WalP4 succeeds inconstructing an orthogonal set of spheroi- son transformation to convert the modal harmonic series for dalvector basis functions by considerin^g grad(•l for the lon- sound-hardand sound-softspheroidal scatterers into a resi- gitudinaldegrees of freedom,and curl(•v) for the transverse dueseries which converges more rapidly at highfrequencies. degreesof freedom,where • and;• aresolutions to thescalar Lauchle,z? however, directly evaluated the harmonic series Helmholtz equation.However, he succeedsonly at the cost by introducinghigh-frequency asymptotic expansions for of generality---asindicated above, these basis functions can- the spheroidalfunctions. not describeoff-axis scattering processes. Silbiger•s wasthe first to considerthe acoustic scatter- AsWaterman has pointed out, 2• Huygen's principle is ing by a penetrablespheroid in spheroidalcoordinates. His the funclamcntalconcept underlying the T-matrix formal- treatmentwas incomplete in that the resultwas formulated ism, and the Green'sfunction technique outlined above is in termsof a surfaceimpedance operator Z, whichis in gen- simplyone way of obtainingthe T-matrixequations. Water- eral unknown. While Z was evaluatedfor severalspecial man23 and Pao • havedeveloped a simple alternative to the cases,no attemptwas made to incorporatethe full complex- abovedevelopment for displayingthe mathematicalcontent ity of an elasticprolate spheroidal scatterer. Yeh 3ø consid- of Huygen'sprinciple for vectorwaves in sphericalcoordi- eredthe scatteringfrom penetrableliquid (i.e.,no shearde- nates.This technique is basedon Betti'sidentity greesof freedom)prolate and oblatespheroids and gave - = o, numericalresults for the specialcase that modecoupling is whereu andvare solutions to thevector Helmholtz equation not present,i.e., when the soundspeed of the scattereris and where • is the assaciated stress tensor. In an earlier identical to that of water. paper•z (paperI), thecovariant generalization of this tech- All of the abovecalculations involve only sc•_!or fields niquewas used to establisha transitionmatrix formalism for and the boundaryconditions can be implementedin a natu- the scatteringof generalscalar or vectorwaves in prolate ral fashiondue to the orthogonalityof the solutionsto the spheroidalcoordinates. For vectorwaves, this approachis scalarHelmholtz equationin spheroidalcoordinates. For- far moreconvenient than that originallyadopted by Water- marty, the acousticscattering from elastictargets and the man, especiallyfor coordinatesystems in whichthe vector scatteringof elasticwaves (in elasticmedia) are far more Helmholtzequation is not separable,as it obviatesthe need difficult,as the transversepolarization degrees of freedom for constructingthe vector Grecn's function. Thus the orth- requirethe introductionof the prolatespheroidal vector ogonalityof thevector basis states is nolonger an important wavefunctions.As a consequenceof the nonorthogonality of consideration.In theabove work, the spheroidal vector basis the vectorwavefunctions, the boundaryconditions involv- functionswere generated by consideringgrad(•) for thelon- ingvector waves become relatively intractable in thiscoordi- gitudinaldegrees of freedomand curl{r• ) for thetransverse natesystem. Although a numberof authorshave dealt with degreesof freedom.Here again, •b and yg are solutions to the elasticwaves in spheroidalshells and solids, •-• it wasnot appropriatescalar Helmholtz equations.We shah refer to until after 1970that the acousticscattering from an elastic this choice of basis functions as the "standard'" choice.26 bodywas treated. In a seriesof papers,Grossman et al.,•6 Likethe standard spherical-coordinate-based approach, this Gutman and Kl'•shcbev,•? and Kl'•shehev•s consideredthe formalismis suitablefor scatteringproblems involving arbi- acousticscattering from a spheroidalelastic shell. These trarily shapedbodies. Now however,we havethe capability authors make the same choice for the vector basis set as the to explicifiytailor the formalismto the aspectratio require- presentauthor, in paperI. mentsof the scatterer.In the presentpaper, we considera To a considerableextent, the historyof the formulation specificapplication of this approachto acousticscattering of electromagneticscattering in spheroidalcoordinates par- from a finite,solid elastic cylinder to determineits efficiency allels that of the acousticscattering problem. The earliest in dealingwith slender,elongated scatterers. attemptwas in 1927by MSglich,•9 whoconsidered the dif-

1059 J. Acoust.Soc. Am.,Vol. 78, No. 3, September1985 R.H. Hackmanand D. G. Todoroff:Scattering from ratiosolids 1059 fractionof a planepolarized wave incident on a perfectly tensorin prolatespheroidal coordinates and in AppendixB, ßconducting circulor disk. His treatmentwas incorrect, how- we derivethe particularvariant of the Moore-Penrosein- ever,and the problemwas not properly dealt with until 1948 verseused in the presentwork. by Meixner.•ø Meixner z• andFlammer s2 later formulated the scatteringof an electromagneticwave by a conducting I. THEvECTOR BASIS FUNCTIONS prolatespheroid in spheroidalcoordinates for the special The prolatespheroidal, vector basis functions •., used casethat thesource is a dipoleoriented along the axis of the here were definedin paperI. We remindthe readerthat spheroid.Both of theabove problems can be reduced to sca- r = 1,2refers to transversedegrees of freedom,r ----3, to the lar form and the vectorcharacter of the electromagnetic longitudinaldegree of freedom,and that n refersto the set waveneed not be confronted. The moredemanding problem (•,m,l) requiredto specifythe particular solution to thesca- of the scatteringof a planepolarized electromagnetic wave lar Helmholtz equation.In that paper,it was shownthat by a conductingprolate spheroid, which required the useof thesefunctions satisfy the relations the prolate spheroidalvector wavefunctions, was first for- mulatedby Schultz,sz for end-onincidence, and Rauchs4 extendedthis work to oblate spheroids. Siegel et al? -•6per- •dsIt(Re •)-Re •,•,. --t(Re •.).Re •0,] =0, (1) formedcalculations based on Schultz'sformalism and pre- sentedresults for several{low) frequencies. These authors did not make the "standard" choice for the vector basis func- .ds [t(½.,)-•be., --t(•e., ).½.,] = 0, (2) tions,i.e., thatof paperI, but insteadchose to representthe vectorfields in rectangularcoordinates with each of the •ds[t(•.)-Re ),,..--t(Re ½,,.).•.] componentsgiven a representationin termsof thesolutions = (3) to the scalarHeimholtz equation. There was little further activityon this scatteringproblem until 1975,when Asano where 0 •. is a real, symmetricmatrix. In easeswbe• the and Yamamoto•? formulatedthe generalproblem of the meaningwill not be obscured,we shalldrop the referenceto scatteringof a planeelectromagnetic wave with an arbitrary 7'. angleof incidenceon a dielectricspheroid. These authors' Theserelations were usedin conjunctionwith Betifs treatmentof theproblem differs from that of Schultzprimar- third identity to derive a mathematicalrepresentation of ily in the useof the standardvector basis functions to repre- Huygen'sprinciple, exterior to sentthe E and H fields.Numerical results were presented in the formof angulardistributions for bothoblate and prolate spheroidalscatterers for severaldifferent aspect ratios for --i • O..,,a.,=• ds[t+-•. --t(½.ku+ ], 14) lowfrequencies. In a subsequentseries of papers,Asano, zs Kotlarchyket al., •9 Asano and Sato, •ø and Asano 6• applied iZo...f..=f.ds [t+-Re ,. --t(Re ½.).u+]. (Z) the model to an ensembleof randomlyoriented scatterers. and interior to This brief review would not be completewithout men- tioningthe pioneering work of Van Burenet al.,• Kinget (6) al.,•s and King and Van Buren •4 on the generation ofspher- 0---- • ds [t_.Re •o._t(Re •)-u_ ], oidalfunctions, and the later improvements of Patzand Van Buren.• The avai•bilityof reliable,efficient codes for the - '2, b.= - 1, (7) calculationof thespheroidal functions has had a significant the scatteringregion. Here, {a. }, { f. }, and {b. } are the impacton the presentwork. expansioncoefficients of the incident,scattered, and refract- The firstimportant result of the presentwork is the de- edwaves, and the superscript0 denotes quantities pertaining monstrationthat the "standard"set of prolate spheroidal, to the scatteringregion. The -I-(--) subscriptsrefer to vectorbasis functions is overcomplete. The (•r,m;l} = (e,O,O), r ----1,2 transversevector basisstates can be written as linear boundaryvalues taken at the exterior(interior) of thebound- ary of the scatterer,and s denotesthat boundary. combinationsof the remaining(•r----e,m-----0,/•0) trans- The matrix O •, is the cornerstoneof our approachand versevector basis states. It followsthat all previouscalcula- tionsutilizing these functions are suspect.This point is dis- itsdetailed properties are of someinterest. For r = 3, O t• is cnssed further in Sec. I. diagonalin I and independentof m, i.e., In Sec.II, wediscuss the application of ourapproach to 0 •!.m = [ (• -J-2•)/kL ] •l,'. (8) the acousticscattering from elasticsolids. To validatethe For the transversedegrees of freedom,O •.. takesthe form theoreticalprocedures and computercodes, we havemade a O ],."'= O•.' ----(p/kr)/2 •'.(hr), (9) detailedexperimental study of theacoustic scattering from a finite, 10:1aluminum cylinder with hemi.•phericalendcaps where• •, is a parityconserving angular integral which is (Sec.III). SectionIV is devotedto a comparisonof the pre- easilycalculable and may be expressed in termsof theexpan- dictionsof our approachwith the experimentalresults and sioncoefficients of the angularspheroidal functions. In the to an analysisof the elasticexcitations underlying the more sphericallimit (f--,O,•--•oo with f•-•kr), prominentfeatures in the backscatteredform function. lira•2 ;.(hr) ---- 6n', (10) In AppendixA wegive explicit expressions for thestress

1060 J. Acoust.Soc. Am., Vol. 78, No. 3, Septem.ber 1985 R.H. Hackmanand D. G. Todoroff: Scattering from ratio solids 1060 andEq. (3)is simply an expression of theorthogenality of the a,,=i• R,,,.bn., 117) vectorspherical basis functions. In thepresent case, where •r is,/n genera/,nonzero and the f•nctionsare nonortho- f,, = -i Z R,,,,. b,,., (18) gonal,O t•' isnonetheless intimately involved with the linear independenceof the basis functions [•, }. Thatis, if with 6. =0, thenby Eq. (5),the necessary and sufficient condition for the - xv*. .Re ], (lO) v_anlshingof the { f, } isthat O t•' beinvertible. We haveexamined the conditionof O t•. by performing •, = R,, (•,•Re •,). 120) eigenvaluedecomposition of/2 •t, under varying levelsof To ob• a fo•! •lufion • •s. {17)and {18),we must truncationof theexpression 66 now im•se the • X t = 0 •unda• conditionto r•ucc the N numar of deg• of f•dom of the eldtic field. It w• •o• byB•m •øthat t•s may• a•mpHsh• byusag- l' --0 e. (6) m •!a• the eldtic displa•men•, •. (16},to •e with r --=1,2 and 0coordinate system for a particularscatterer by II. THEORETICAL CONSIDERATIONS We beginby out!ini•gthe derivationof the spheroidal- f= [(aspectratio) • -- l]m/{aspectratio). coordinate-based transition matrix for the acoustic scatter- ing from an elasticsolid immersed in an inviscidfluid of With thischoice, the spheroid• = l/fexacfiy inscribesthe infiniteextent. We assumethe usual boundary conditions scatterer. The integrationsin Eqs. (19), (20), (23),.and(24) were •-u+ = •-u_, (13) performedusing Gaussian quadrature. The matrixinversion •.t+ = •-t_, (14) impliedin Eq. (25)was implemented by writing •xt_ =o, apply at the surfaceof the scatterer.To accountfor the (asuperscript T denotesthe matrix transpose) and then using acousticpenetration of the scatterer,we introducethe ex- •Gausselimination tosolve for T r, afterscaling both Q and pansionof the elasticdisplacement Q. Thisprocedure tends to benumerically more stable than a direct invcraionof Q followedby matrix multipUcation. The remaininginversion P -•M, (thesuperscript 0 denotes quantities pertaining to the scat- terer)and Eqs. (4) and (5) become in Eqs.(26} and (27) requiressomewhat greater care. A de-

1061 J. Acoust.Soc. Am., Vol. 78, No. 3, September 1985 R.H. Hackman and D. G. Todoroff:Scattering from ratio solids 1061 tailedexamination of P for solidspheroids and finite cylin- Poe•"•' = 4•rPo• itSo•l(h,cos0•o) derswith hemisphericalendcaps reveals that this matrix tendsto beill conditioned.A typicaleigenanalysis of P yields X S•t( h,•',•') je•, (h,• '), (31) eigenvaluesranging in sizefrom 10 -26 to 10+2. Holding the •d p, is the (•gle•d•t) •plitude of •e •t• physicaldetails of the scatteringproblem fixed, we find that wave the exactspectrum of eigenvaluesdepends upon the number of expansionterms. Invariably, however, after some level of truncation,increasing the number of termsin theexpansion furthermerely tends to increasethe numberof eigenvalues roughlyconsistent with zero.The nonzeroelgenvalues (and eigenvectors)are stable, to machineaccuracy. Interestinglyenough, the stability of thecalculation of xs,. the backscatteredform functionis to someextent, indepen- Here,(Bo,•o) and (B• } •e thesphe•c• anD• wMchde•e dentof the conditionof P. For spheroidalbodies, the form the dir•tion of the incident•d scatteredwav•, r•ctive- functionconverges quite quickly and is stableunder an al- ly, andthe ex•nsion •fficients of •e •t• wave•ve mostarbitrarily large increase in the numberof expansion • expr• in te•s of the T mat• (T is diagonal terms.The finitecylinder displays a greatersensitivity to the and m for a•ymmetfic obj•m). ExpUcifiyintr•g conditionof P andat highaspect ratios, when the condition •dpo into•. (30)•d ut•zing the•p•tic r• for- of P has deterioratedsufficiently, the calculationbecomes mula for he•t •v• unstable.This latter behavior is probablydue to the greater relianceof the cylindricalcalculation on off-diagonalele- ments in P, M, and R. Thereis a simplerationale for the relativeinsensitivity of T to the condition of P. Note that those' elastic states whichare weakly coupled to thenearfield can be n,.o more than weakly coupledto the farfield.Thus, R and R must In g•e•, we ch•e • = •o = 0. •s dc•ifion is •ns•- annihilatethe samecombinations of the expansioncoeffi- •nt wi• •e g•eral pr•fi• • the field and r• in a cients{b, } asP, andit followsthat Tis independentof the fo• •cti• of unityfor a •d sphefic• •tte•r of •di• existenceof thesestates. The difficultywith the finite cylin- L/2 in the •-fr•uency •it. deris its greater reliance on off-diagonal matrix elements. In We have chos• • t•t our th•retic• t•h•ques general,the far off-diagonal matrix elements are intrinsically a•st a •i• cyHnd• g•met• for two r•ns. F•t, lessaccurate than the diagonalelements and this annihila- thef• wav• w•ch • pm•gate ona solid,i•te cylin- tion processis not accuratelyperformed. der have•n ext•sivelystudi• •d •e well •o•. • Sinceit is presumablythe presenceof the uncoupled(or Sin• a 10:1cy•der is sufficientlylong •at •e at most,weakly coupled} [b, } which is the sourceof the •ndifions on •e md•ps shouldnot •ve • appr•ble limited numericalstability for the finite cylinder, it should eff•t on th• wav•, t•s fa• hel• •nsiderably in the an•- provenumerically advantageous to removethese states from ys•. In the pr•mt •, •c •y two •ntfibu•ng wav• our space.Defining A asthe operator which projects into the (1} thelower l•tu&n• m•e, w•ch • • char• spacein whichthe matrix P is well defined,and usingA to • • •i•y sy•etfic wave with disp•ent •m•- consistentlyobtain the admissibleinterior elasticstates in nen• onlyin the radi• •d •ial •r•ions, •d (2)the low- Eqs.(17) through (24), we write Q in the form •t flexuralm•e, which• g•l de• u•n Q = RA (A rPA )-•A rM. (29) dir•tions. •e dis•ion c• for th• wav• In AppendixB we giveour procedurefor constructingA and Fig. 1. •e •gher !on•tudi• •d fiexur• m• havecut- implementingthe projection process in ourcalculations. The off fr•um&• w•ch •e beyondthe •ge •side• h•e method is a variant of the Moore-Penrosepseudoinverse and •e •ion• m• •not • excit• a•mti•y. technique?One immediate result of thisprojection proce- •nd, this g•met• pm•d• a mores•g•t •t of dure is that the T matrix (for the cylinder)becomes much the fo•sm t• woulda spheroidor a su•r spheroid.A morenearly symmetric. We findthat [Tn. -- T•.•]is consis- spheroidis a s• g•met• for which• the sphefi• tentlysmaller than 10-3 or 10-n timesthe magnitude of the and spheroi• b•is functionsare ve• well suit•. B•- largestdiagonal transition matrix element.There is essen- t '• h• no• t•t evens•l deviationsfrom a spheroi- tially no effecton calculationsfor spheroidalbodies. d• g•met• 1• to a sight det•oration • the nu- To compareour theoretical predictions with experimen- mefi• •o• of •e s•d•d sphefi•- •a•-b• T mat•. In •e next s•tion we tal measurements,we introduce the farfield form function our •m•l p•ur•.

,-• {L/2} I Po I' III. EXPERIMENTAL whereL is the lengthof the target,Po is the amplitudeof the Backscatteredacoustic waveforms were acquiredfrom incidentacoustic plane wave a 10:1solid aluminum cylinder with hemisphericalendcaps

1062 J. Acoust.Soc. Am., Vol. 78, No. 3, September1985 R.H. Hackmanand D. G. Todoroff:Scattering from ratiosolids 1062 4.0 enceof a receiverhydrophone situated on a straightline betweenthe sourcetransducer and the scatteringtarget. LONGITUDINAL WAVE The driveand receive electronics are depicted in Fig. 2. 3.0 A singlecycle sine wave, amplified by a poweramplifier (Kronhitemodel DCA-50), provided the drive signal for the piezoelectrictransducer (USRD type F-33). The acoustic 2.0 signalsreceived by the hydrophone were preamplified and bandpassfiltered prior to digitizing by the HP-5180A wave- FLEXURAL WAVE / form recorder.

1.C Dataacquisition was under the control of a HP-9826 microcomputer.Digitized representation of the analog waveformswas transferred to themicrocomputer for off-line analysis.The data analysis routine allowed for signal averag- .5 2'.o ing,convolution, and Fourier transformation of the digi- ka tizedwaveforms, along with graphic displays of the resultant FIG. 1. The dispersioncurves for the lowestlongitudinal and flexural operations.All recordedwaveforms were repetitively aver- modeson an infinitealuminum cylinder of circularcross section. agedfor 30 scans. Waveformswere sampled and digitized such that the Nyquistrate was always exceeded. An 8192-pointdiscrete fastFourier transform was implemented with a rectangular asa functionof aspectangle O (theangle the acousticaxis window(no weighting) to obtainintermediate frequency do- makeswith the longsymmetry axis of the target).The ex- maindata. In orderto spanthe kL/2 regionof interest(5- perimentalfarfield form function was determined using the 20),several different center frequencies were required. The expression69 3-dBdown points were chosen as the cutoff points for the contributionof eachfrequency band to thecomputed form function.In this context,we notethat manyof the form f. (• ,e) =œ72 2rg,(kL/2,t•) functionplots represent averages ofnumerous independent wherer •sthe distance from the center of thescattering target measurements. tothe measurement point, g, (kœ/2,8 ) isthe transform of the In viewingthe individual backscattered waveforms for backscatteredwaveform, ps(t ), andg•(kL/2) is the trans- analysis,it became apparent that some of the returns persist- formof theincident pulse at thescattering target, p,(t ). ed wellbeyond the availabletime window. The effectwas Acousticmeasurements were performed in a 10-X 10- particularlysignificant inregions where the theory predict- X 7-ftreinforced concrete test pool filled with a suitablevol- ed highQ resonantfeatures. These observations were not umeof freshwateras the acousticmedium. The sourcetrans- surprisinginthat any experimental measurement hasan in- ducer,receive hydrophone, and scatteringtarget were herentlimitation on the resolutionof suchfeatures. Assum- placedalong a horizontal line, held by a systemofprecision inga decaymodulus (time required for the amplitude tode- positioningdevices. Lateral, longitudinal, andvertical align- creaseto 1/e of its initial value)equal to the available mentof the components wasaccomplished optically using a sightingtelescope. The respective positions were such as to maximizethe shortest reverberant path, thus optimizing the time domainmeasurement window. A 6-in.-longtarget was MICROCOMPUTER selectedbased on therequirements that a sufficientback- X-YPLOTTER HP-g826 scatteredreturn would be observed end on, and that a suffi- cientnumber of returnsfrom surface waves traveling around GENERATOR WAVEFORMHP-5180A RECORDER thetarget would be acquired toyield a representativemea- I I sureof the form function. This latter criteria was not fulfilled GENERATORFREQUENCY BAND.PASSFILTER incertain cases, as will be discussed atthe end of this section. I Theinitial angular orientation (90' off axis) of the scattering targetwas determined acoustically bymaximizing theback- AMPLIFIERPOWER i AMPLIFIERPRE- scatteredsignal. Throughout the measurements, theposi- tionsof the transducerand hydrophoneremained fixed, I whilethe aspect angle of thetarget was varied. Therelative pressure amplitudes po(t) (of the incident waveform),and p,(t) weremeasured using a commonhy- drophone(B& K model8203}, placed inthe farfield region of the transducer,a distancer from the scatteringtarget. -• 24' Appropriatecorrections forspreading and absorption were HYOROPHONE madeto yield an effective pressure wave front, p•(t ), inci- TRANSDUCER TARGET denton the target. No attempt was made to correct for the FIG. 2. A schematicrepresentation ofthe experimental geometry and appa- slightdistortion (estimated tobe less than 1%) by the pres- ratus.Here, 8 definesthe experimentalaspect angle. 1063 J.Acoust. Sec. Am., Vol. 78, No. 3, September1985 R.H.Hackman and D. G. Todoroff: Scattering from ratio solids 1063 windowsize (minus the specularreturn}, we estimatethat thatfor kL/2• 20,a changeAL•,,,= 2 resultedin a change thesemeasurements allow for the correctrepresentation of in the form function of less than 2%. featureswith a maximumQ value of five. The particular The calculationwas relatively insensitive to the number theoreticalresonant features of interestclearly exceedthis of significantfigures retained in the orthogonalizationpro- value. cess,provided No, was sufficientlylarge, and we chose In order to determine the effect such artificial termina- N•h ----18--20. Increasing/V• by two generallyresulted in tion had on the resultantform function,a simplesubtraction a changein the form functionof muchless than 0.5%. was employedfor the 30ø off-axismeasurement. Surface Noattempt was made to measure thedensity and sound multipathswere removedby subtractingthe multipath speedsof the aluminumcylinder. Instead, the genericvalues waveform(without target) from the backscattered target re- p -- 2700kg/m 3, C• = 6420m/s, and C, = 3040m/s were turn.Although a residualsignal was clearly present (giving used.Thus some chscrepancybetween the measuredand riseto an interferencepattern) and someof the signalstill calculatedform functionis expecteddue to the mismatchin residedoutside of the "expanded"measurement window, the elasticparameters. To obtainsome idea of the sensitivity theagreement with theory was excellent, as will bediscussed of the calculationto reasonablevariations in theseparam- in the followingsection. eters,we haveexamined the positionof the lowestresonance for end-on incidence under a 10% variation in the Lame' parameter/t(the bar speeddepends strongly on thisparam- IV. DISCUSSION eter).TM We findthat the peak shifts by no more than 5%. A series of calculations of the monostatic backscattered In Fig. 3 weexhibit a collectionof calculatedmonostatic form functionwas performed in the regionkL/2 < 20 for beampatterns which have been selected to displaythe more comparisonwith experiment.There are two parametersto prominentfeatures/n the backscatteredform function.In fix in the calculationof the m = 0 T matrix at a givenfre- particular,the single, strong, high/2 lobeat d = 56ø and kL / quency,the numberof termsincluded in the truncationof 2 = 5.4 and the "ridge" of lobeswith d•30 øbeginning at the infinitedimensional matrix equations, Lm• in Eqs.(25)- kL/2 = 6.6 representsignificant departures from rigid be- (27), and the numberof significantfigures retained in the havior.It is worth notingthat with two exceptions,at • / orthogonalizationof P, i.e., No• (seeAppendix B). In the 2 = 5.4 and 10.8,there is no ancillarystructure (in the beam frequencyrange kL/2<16, the numberof expansionterms pattern}associated with theseoff-axis lobes. The two excep- wasfixed by increasingL•,• until the form functionvaried tionsdisplay on-axis resonances. We alsonote for futureref- by 0.5% or lessfor a changeAL,• = 2. The numberof erencethat the peakamplitude of theridge, which has large, termsrequired for convergencewith this criterionvaried high {2 peaksat kœ/2 = 6.7, 9.3, and 12, showsa general betweenten at the low-frequencyend and 30 at the upper tendencyto movein the directionof increasing8 with in- frequencies.For kL/2> 16,L• wasfixed at 30. We note creasingfrequency. These features are particularly revealing

12.8

12.4

12.0 FIG. 3. The calculated monostatic beam 11.6 patternsfor a 10:1finite aluminum cylinder

11.2 with hemisphericalendcaps. .9 10.8

10.4 Z I 102

9.8

*• 9.3

• • 9.0 •.3 u. 8.6 6.7

5.4

5.0 .3 .6 .9 FORM FUNCTION

1064 J. Acoust.Soc. Am., Vol. 78, No. 3, September 1985 R.H. Hackmanand D. G. Todoroff:Scattering from ratiosolids 1064 of theelastic response of thescatterer and we shall explore resonancepeaks is still missing. An examinationof thetime this in detail below. domainwaveform shows that evenwith theexpanded win- In Figs.4 and 5, the calculatedbackscattered form func- dow, only a portionof the energyof thesepeaks has been tion (solidpoints) is comparedwith the experimentaldata captured,i.e., the waveformhas been artificially and prema- (solidlines) for aspectangles 0 rangingfrom 0' to 90', in 10' turely truncated. In the nonresonantregion, the flexural increments.In regionswhere the experimentalfrequency waveis still excited,due to the strongcoupling mechanism, bandsoverlap, both setsof data are presented.The dashed and the elasticcylinder acts as a phasesteered array, firinga linesin Fig. 5 for 0 = 30', representmeasurements taken highly directionaland hence,strong signal back at the withthe expanded window discussed in theprevious section. source. This mechanismalso explains the surprising We note that the small scale oscillations in the data taken strengthof the resonantresponse. with theexpanded window are an artifactof the subtraction The singlestrong lobe at 56ø (see the 0 ----50. and0 ----60 ø procedureand do not haveany underlyingphysical sign/i- aspectorientations in Fig. 5) is associatedwith thelongitudi- cance.In thisfigure, it isclear that thelonger window gives a nal (i.e.,m ----0) responseofthe cylinder.An examinationof considerableimprovement in the agreementbetween the the elasticdisplacements reveals a stationarylongitudinal dataand the calculations. It isalso clear that the very high Q wave with,• --- L/2 almost identical to that excited at end-on featuresin thecalculation (e.g., at kL/2 = 6.7 and0 = 30') incidenceat this frequency,but approximately50% stron- arestill beyond the capabilities of theexpanded window. ger.At thisangle of incidencel0 --- 56ø) and frequency, there Considerfirst Fig. 4, i.e.,end-on incidence. Since T-ma- isexactly one water wavelength along the projected length of trix calculationsfor end-onincidence require only the m = 0 the cylinderand it is the coherentshaking of the two ends termin Eq. {33)and thus tend to beless time cons-ming than whichis responsiblefor the relativelystrong excitation. Un- off-axis orientations, a somewhat more ambitious calcula- fortunately,due to the veryhigh Q natureof thislobe and the tionalprogram was undertaken here than for moregeneral smallsignal strength, even with theexpanded window only a anglesof orientation;hence, the expandedfrequency scale small enhancement ofthe form function was observed at this and greaterdensity of points,as comparedto Fig. 5. The angle.Therefore the data and calculations were not included signalstrength from the targetat thisangle was too small to in the figure. reliablysubtract the multipathsso no attemptwas made to Considerfinally, the broadsideform function.The fre- obtainthe high Q featurespresent in the data. Given this quencyregion considered here is too low for anysignificant qualification,the calculationis in reasonablygood agree- elasticactivity in the broadsideor nearbroadside direction mentwith the data. The three peaks at kL/2 -- 5.4, 10.8,and (70ø,•90ø). It is interestingto notethat the off-axisbehav- 16.2are clearlythe,• = L/2, L, and 3L/2 resonancesasso- ior in this angularregion differs markedly from that ob- ciatedwith the largelynondispersive, lowest longitudinal tained in studiesof the scatteringof obliquely incident modeof an infiniteelastic cylinder. We haveexamined the acousticwaves by an infinitecylinder by Flaxet al.?• How- elasticdisplacements at the surface of thescatterer, and both ever,the lobe structure in Fig. 3 in thisangular region can be thecalculated phase speed and the displacementsare in ac- simplyexplained'by treating a finiterigid cylinder of length cordwith thispicture. L as a linear phasesteered array excitedby the incident Considernext the prominent ridge of lobesin the back- acousticwave. The primary,secondary, and tertiarylobes scatteredform function depicted in Fig. 3 andin the 0 = 30' areall presentand their position, angular width and relative portionof Fig. 5. This ridgeis associatedwith the excitation magnitudeare consistent with thissimple picture, although of the lowestflexural wave of an infinitecylinder due to the the third lobe is somewhatlarger than expected.We note matchingof the trace velocityof the incidentacoustic wave that thereis sufficient experimental sensitivity to the angular with the velocityof propagationof the flexuralwave (see Fig. alignmentin this region,due to the relativelynarrow (in 0 ) 2). An examinationof the elasticdisplacements reveals that lobes,to accountfor the small discrepanciesbetween the the three clearly identifiableresonance peaks at 0 = 30. in data and calculations. Fig.5 areassociatedwiththeA ---- 2L/5, L/$,and 2L/7 reso- Wenote in closing'that, inrelated work, Suet al.?2 and nantwavelengths. The agreementof thecalculation with the Numrichet al.?3 have compared with experiment the predic- data,particularly that data obtained with theexpan.ded win- tionsof the standard,spherical-coordinate-based T matrix dow,is quitegood. However, the very high Q portionof the for a finitealuminum cylinder of smallaspect ratio (2:1).

FIG. 4. The backscattered farfield form functionsfor a 10:! finitealuminum cyl- _o inder with hemisphericalendcaps for i- end-onincidence, plotted as a function of the dimensionlessfrequency/eL/2. The heavydot• repr•ent calculat•l vnl= • .1,3 uesand thesolid lines, experimental val- ue• Note that in someregions, several frequencybands overlap. 2 4 6 8 10 12 14 16 18'

1065 J. Acoust.Soc. Am., VoL 78, No.3, September1985 R.H. Hackmanand D. G. Todoroff:Scattering from ratio solids 1065 2.0 10 • OFF AXiS 20' OFF AXIS z

1.0

.o 2.0-- 30øOFF AXIS 40 ø OFF AXIS z

1.0 ß :.! FIG. 5. The ba=kscattcred faff•ld form function for a 10:1 finke alumi- num cylinder with hemispherical endcapsfor off-axis ;n•. The 7 9 11 13 15 7 9 11 13 15 ßKLJ2 KL/2 notationis th= sameas in Fi& 4. The additional dashed fin•s in the 0- 30' graph representexperlink- .9 OFF AXIS OFF AXIS

z

.0

.9 OFF AXI,• OFF AXIS

•o 3

7 9 11 13 15 7 9 11 13 15 KL/2 KL/2

.9 90• OFFAXIS ß . ß

7 9 11 13 15 KL•

ACKNOWLEDGMEHTS APPENDIX A The authors are indebted to D. H. Trivett for stimulat- In applicationsof thespheroidal-coordinate-based tran- ing and productiveconversations throughout the courseof sitionmatrix, it is necessaryto haveexplicit expressions for this work. One of us (RHH) would alsolike to thank A. L. the vector basis functions and their associated stress tensors. Van Burenfor discussionspertaining to thegeneration of the Expressionsfor thebasis functions were given in paperI; we prolatespheroidal functions and for copiesof hisspheroidal givethe stresstensors in thisAppendix. functioncodes. In particular,the eleganteigenvalue routine FollowingMorse and Feshbach, TM we intr6ducethe in- 0fPatzand Van Buren • servedas the heart of our eigenfunc- dependentvariables tion routine. RHH would also like to thank I. Dobeck for = ',V,cos ). discussionson the Moore-Penrosepseudoinverse and T. Racketsfor producingthe dispersioncurves in the text. Fin- The associated unit vectors ally, the authorswould like to thank L. Flax for his contin- ued encouragementand support.

1066 J. Acoust.Sec. Am.,VoL 78, No. 3, September1985 R.H. Hackmanand D. G. Todoroff:Scatter'ng from ratio solids 1066 definea right-handedcoordinate system. This differs from ha= (f/sin •b)[(1 - •/2)(•2 _ 1)1,/2. thedefinitions adopted in paperI whichutilized a left-hand- The stresstensor is definedby ed coordinatesystem. Thus, there are severalminus sign differencesbetween the vector basis functions given in paper •(½,•)=ZV4•i + •(V,• + •,•V), I andthose adopted here, and these differences are noted in where Eq.(A4). For calculational purposes, it is convenient towork witha representationof the vector basis functions in which the"physical" components are utilized, i.e., is the unit dyadicand where 3 • __-• (•.),•,. (A3) The relationbetween these physical components and the co- =2 • I•x••,-•/] +•b•.•(ln vnziantexpressions given in paper! [the(•.)• ], is (•), = + (h,)-'(•),, x(a,a•+ 6a,). {A7) wherethe upper sign is for ½ -- 1,3 andthe lower, for •- -- 2. •t Z• = R•z(fir,••t(hr,•,• ) denotea pro•rly no•- • •lution to thescal• He•holtz equationwith k = kT, The h• denotethe scale factors •d let•t = •z(hT) denotethe eigenvMue. For the trans- h, =f[(• 2_ •/2)/(•2 _ venedegr• of freedom,the physical •m•nents of the - (AS) str•s tensor(•u) •e $venby

1067 J.Acoust. Sec. Am., Vol. 78, No. 3, September 1985 R.H.Hackman and D. G. Todoroff: Scattering from ratio solids 1067 In theabove expressions, wehave eliminated all reference too•x/v•õ :, c9:X/o•q:,and c9 :•y/•2 throughthe use of the defining differentialequations for R.,• and Using•. = R,,,t(h•.,g}S•t(ht.,•I,q•)to denotea properlynormali7ed solution to the scalarHelmholtz equation with k = k,., andA•,t =.4,,,t(hL)tO denote the eigenvalue,for the irmtational degreeof freedom,we find

(AIS)

(AI9)

1068 J. Acoust.So(=. Am., VoL 78, No.3, September1985 R.H. Hackmanand D. G. Todoroff:Scattering from ratio solids 1068 (A20)

= f• [(1--•7:)• -- •7:)] m • •'e--1&b • ' (A21) = (A22)

APPENDIX B allof the remaining columns relative tothis choice. Clearly, Considerthe ill-conditionedproblem thisprocedure can be applied only N timesbefore we have exhaustedthe possibilities. In practice, we orthogonalize all PC=M, columns,then order the columnsby their lossof significant whereC andM aren X m rectangularmatrices and where P digits(through the subtraction process) and truncate at the is an n X n real, symmetricmatrix with one or more zero pointwhere more than 18 significant figures have been lost. eigenvalues.Here, C is to bedetermined. Let Z k bethe k th ThecolumnsofthenXNmatrixA thus formed, must be eigenvalueof P andlet Pi(k ) bethe corresponding normal- a linearcombination of theeigenvectors having I/tk [ > 0, i.e., ized eigenvector,i.e., N Aq= •__•p•(k )A• (B7) • Pi•p•lk ) = Akp,(k ). (B2) which in matrix notation is Then the transformationmatrix whichdiagonalizes P is A = UN.4. From theorthonormality of thecolumns of A it followsthat Weassume that P hasNnonzero eigenvalues, and that these the N XN matrixA is orthogonal.From F_x1.(BS), we thus have cigenvaluesareordered by decreasing magnitude. The in- consistenciesmay be removed from Eq. (B1) by eliminating thenull space of P. Thisis moststraightforwardly accom- Returningto Eq. (B4)and ut'dizing plishedby performing an eigenvalue decomposition ofP and projectinginto the space spanned by eigenvectorshaving A rPA =.4 r.(2N `4 (B10)• nonzeroeigenvalues; that is, we wish to cast Eq. (B 1) into the we find form {A•PA )A T½=A •M, (Bll) aN c = M, i.e., that A performsthe necessaryprojection process. Note that it is not necessaryto construct`4. where The exactrelation between the lossof significantdigits in the columnsof P duringthe orthogonalizationprocess and the s•.• of the magnitudesof the eigenvalucsin •2N is /2N= ,li 0 (BS) unclear.This is unimportant,however, since our real con- cernis thelinear independence of the columns (and rows) of 0 2N the truncated matrix. [A,I>O, n= 1,...• r, andwhere UN is thetruncated diagonalization transforma- tionmatrix. However, the explicit construction of theeigen- •P. C. Waterman,"New Formulationof AcousticScattering," 1. Acoust. valuesand eigenvectors tends to betime consuming, and we Soc.Am. 45, 1471-1429(1969). aThisterminology ismost generally accredited toC. W. Oseen,Ann. Phys. adopta somewhatdifferent procedure here. 48, 1 (1915);and P. P. Ewald,Ann. Phys. 49, I (1916). From the spectraldecomposition •P. A. Martin,"On the Null-Field Equations for the Exterior Problems of Acoustics,"Q. J.Mech. Appl. Math. XXXHI, 385-396(1980); P. A. Mar- P,2= i A•p,(k)p2(k), (B6) tin, "AcousticScattering and Radiation Problems and the Null-Field k=l Method," Wave Motion 4, 391-408 (1982). it is clearthat ifP is of rank N, then P hasexactly N linearly 4A.G. Ram,"Convergence of the T-Matrix Approach to ScatteringThe- ory," J. Math. Phys.23, 1123-1125(1982}. independentcolumns. To forma basisfor thisN-dimension- •G. Kristensson,A. O. Ramre,and S. Strom,"Convergence of the T-Ma- al space,we apply the modified Graham-Schmidt orthogon- t fix Approachin ScatteringTheory II," J. Math. Phys.24, 2619-2631 alizationprocedure to thecolumns of P, withpivoting. That (1983). SForrecent work utilizing this technique, see W.Tobocman, "Calculation is, at eachstep in the orthogonalizationprocedure, from of AcousticWave Scattering by Meansof HelmholtzIntegral Equation. amongthe non-normali7ed columns of P, wechoose the col- I," •. Acoust.Soc. Am. 76,599-607 ( 1984};W. Tobocman,"Calculation of umnwith the largestnorm, normalize it andorthogonalize AcousticWave Scattering by Meansof theHelmholtz Integral Equation.

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1070 J. Acoust.Sec. Am., Vol. 78, No. 3, September1985 R. H. Hackmanand D. G. Todoroff:Scattering from ratiosolids 1070 (•an. 1970). demie,New York. 1976),pp. 96-145. 6•B.J. King, R. V. Baler,and S. Hanish, "A FORTRANComputer Program a•Therelevant literature isreviewed in$. D. Achenbach,I•'ave Propagat/on for Calculatingthe Prolate Spheroidal Radial Functions of theFirst and in Elastic•!ids (North-Holland,New York. 1976),pp. 240-249. SecondKind and Their First Derivatives,"NRL Report 7012 (March ø•L.R. Dragonette,"Evaluation ofthe Relative Importance ofCircumfer- 1970). entialor CreepingWaves in theAcoustic Scattering from Rigid and Elas- taB.$. Kingand A. L. VanBuren, "A FORTRANComputer Program for tic SolidCylinders and from CylindricalShells," NRL ReportNo. 8216 Calculatingthe Prolate and Oblate Angie Functions of theFirst Kind and (197a}. Thdr First andSecond Derivatives," NRL Report7161 (Nov. 1970). •øForexample, see Richard K. Cook,"Acoustics," in.4merican Institute of aSgee,for example, B. J. Patz and A. L. VanBurcn, "A FORTRAN Computer PhysicsHandbook, edited by D. E. Gravy {McGraw-Hill, New York, Programfor Calculatingthe Prolate Spheroidal Angular Functions of the 1963},pp. 3-88.Cook gives a 10% variationin/.• and no variationin•. First Kind," NRL MemorandumReport 4414 (March 1981). ?lL.Flax, V. K. Varadan,and V. ¾.Vatarian, "Scattering of anObliquely •"I'hereare a number of measures ofthe condition of a matrix.To be specific IncidentAcoustic Wave by an Infinite Cylinder,"•I. Aeonst.Soc. in thispaper we adoptVon Neumannand C•oldstine'sP-condition num- 1832-1835 {1980}. nJ.-H. Su, V. V. Vnradan,V. K. Vatarian,and L. Flax, "AeousticWave Scatteringby a Finite ElasticCylinder in Water," J. Acoust.Soc. Am. 68, P•4 ) = •.•l/•, 686=691(1980). asthe standard for themeasure of condition.Here, A• and,{• arethe •S. K. Numrich,V. V. Vatarian,and V. K. Vatarian,"Scattering ofAeons- largeatand smallcat eigenvaluea of thematrix.4 in absolutemagnitude. A tic Wavesby a Finite ElasticCylinder Immersed in Water," J. Aeonst. goodgertcanal reference is L R. Westlake,.4 Handbookof NumericalMa- Soc.Am. 70, 1407-1411{1981). trixInversion and Solution of Linear Equations {W'dey, New York, 1968}. ?4p.M. Morseand H. Feshbaeh,Method• of TheoreticalPh3•ics (McGraw- •?See,for example, G. $tran&Linear •41gebra and Its Applications (Aca. Hill, New York, 1953),pp. 661.

1071 J. Acoust.Sec. Am.. Vol. 78, No. 3. September1985 R.H. Hackmanand D. G. Tedore,f:Scattering from ratiosolids 1071