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NASACONTRACTOR REPORT
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DIFFRACTION OF A PULSE BY A THREE-DIMENSIONALCORNER
by Lu Ting and Fanny K'wg
Prepared by I. NEW YORK UNIVERSITY New York, N. Y. 10453 for
NATIONALAERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MARCH 1971 ". TECH LIBRARY KAFB. NM
1. Report No. I 2. GovernmentAccession No. 1 3. Recipient'sCatalog No. r NASA CR-1728 4. Titleand Subtitle
DIFFRACTION OF A PULSE BY A ~-CORNER 7. Author(.) 8. PerformingOrganization Re ort No. Lu Tingand Fanny Kung NYU ReDort"70-66
9. Performing Or anization Name andAddress 10. Work UnitNo. New Y0r% University University HeLghts 11. Contract or Grant No. New York, N.Y. 10453 NGL-33-016-119 13. Type of Report and Period Covered 2. SponsoringAgency Name and Address CONTRACTOR REPORT National Aeronautics & Space Administration Washington, D. C. 20546 14.Sponsoring Agency Code
" 5. SupplementaryNotes
6. Abstract Forthe diffraction ofa pulse bya three-dimensionalcorner, the solution is conicalin three variables 5 = r/(Ct), 8 and so. The three-dimensionaleffect is confinedinside the unit sphere 5 = 1, aiththe vertex of thecorner as the center. Theboundary data on the unitsphere is provided by theappropriate solutions for the diffraction of a pulse bya two-dimensionalwedge. The solution exterior to the Zornerand inside the unit sphere is constructed by theseparation of the variable 5 from 8 and q. The associatedeigenvalue problem is subjected tothe differential equation in potential theory with an irregular boundaryin 8 - n plane. A systematicprocedure is presentedsuch that theeigenvalue problem is reducedto that ofa sys ternof linearalgebraic 2quations.Numerical results for the eigenvalues and functions are obtainedand applied to constructthe conical solution for the diffractio of a planepulse. For the diffraction ofa generalincident waveby zooa'ners oredges, two theorems are presented so thatthe value at the rrertexof the corner or along the edges can be determinedwithout the Zonstruction of thethree-dimensional non-conical diffraction solutions. Relevantnumerical programs for the analysis are presented in the appendix. 17. Key Words (Selected by Author(s)) 18. DistributionStatement Sonic Boom Diffraction Unclassified - Unlimited planepulse
I 19. SecurityClassif. (of this report) 20. SecurityClassif. (of this page) 21- No. of Pager Unclassified Unclassified 91
For sale by the National Technical InformationService, Springfield, Virginia 22151
TABLE OF CONTEXTS
SUMMARY 1
INTRODUCTION 2
FORMULATION 3
THEEIGENVALUE PROBLEM 5
NUMERICALSOLUTIONS OF THEEIGENVALUE PROBUX 10
CONSTRUCTION OF THECONICAL SOLUTION 3.6
ANUMERIC& EXAMPLE 19
GENERALINCIDENT WAVE 21
"MEAN-VALUE"THEOREMS FOR DIFFRACTION BY ACONICAL, SURFACE AND APPLICATIONS 24
CONCLUSION 31
ACKNOWLEDGMENT 32
REFERENCES 33
FIGURES 34
APPENDIX 46
PRINTOUT 50
TABLES 84
iii LIST OF FIGURES FIGURE -PAGE 1. Incidenceof a planepulse on a corner. 34
2. Eigenvalueproblem for the corner. 35
3. A two-dimensionalcorner. 36
4. Plane'pulse incidence head on to face OAD, the reflectedplane pulse and the diffracted fronts around edges OA, OD, and vertex 0. 37
5. Two-dimensionalconical solutions around edges OA and OD. 38
6. Over pressurealong constant 8 lineson face OAD (cp=3rr /4). 39
7. Overpressure along constant cp lines onface OAB (9~/2) andalong constant 8 linesonface OBD (cp=3rr/4). 40
8. sonic Typical boom41 signature.
9. Diffraction by a cone. 42
C om parison of two cones. two 10. of Comparison 43
11. Incidentof a planepulse on a block at instant t > L/C (OB=L, OA=OD;W; also shown are thediffracted wave fronts around edges OA, OD, BA' , BD' and around vertices 0 and B). 44
12. Incidentof a planepulse on a bl.ockwith extended fictitioussurfaces at instant t > L/C (also shown are thediffracted wave frontsaround edges OA,OD and vertex 0). 45
iv DIFFRACTION OF A PULSE BY A THREE-DIMENSIONAL CORNER
by Lu Tingand Fanny Kung New York University, New York, N.Y.
SUMMARY
Forthe diffraction of a pulseby a three-dimensionalcorner, e.g., the cornerof a cube,the solution is conicalin three variables 5 = r/(Ct), 8 and to. The three-dimensionaleffect is confinedinside the unit sphere 5 = 1, or the sonic sphere r = Ct withthe vertex of the corner as thecenter. Theboundary data on the unit sphere is providedby the appropriate solutions forthe diffraction of a pulse by a two-dimensional wedge.The solution exterior to thecorner and inside the unit sphere is constructedby the separationof the variable 5 from 8 and q. The associatedeigenvalue problem is subjectedto the same differentialequation in potential theory for the sphericalangle variables 8 and cp, butwith an irregular boundary in 8 - Cp plane. A systematicprocedure is presentedsuch that the eigenvalue problem is reducedto that of a system of linearalgebraic equations. Numerical resultsfor the eigenvalues and functions are obtainedand are appliedto constructthe conical solution €or the diffraction of a planepulse. For the diffraction of a generalincident wave bycorners or edges, the solutions are no longerconical. Two theoremsare presented so thatthe value at the vertex of the corneror along the edges can be determined without the con- structionof the three-dimensional non-conical diffraction solutions. Relevantnumerical programs for the analysis are presentedin the appendix. 1. Introduction
The problem of diffraction and reflection of acoustic waves or electro- magnetic waves bywedges, corners and other two-dimensional or axially sym- metric obstacleshas received extensive investigations. A surveyof these investigationscan be found in Reference 1.
The presentinvestigation is motivatedby the study of the effect of sonic boom onstructures. The pressure wave createdby a supersonicairplane is three-dimensionalin nature. However, theradius of curvature of the wave front is usually much largerthan the length scale of a structure. Therefore,the incident waves canbe approximated by progressing plane waves withthe wave formusually in the shape of the letter N and are referred to as N-waves.
Fortwo-dimensional structures in the shape of a rectangularblock, thediffraction of a planepulse by the first corner is givenexplicitly bythe two-dimensional conical solution of Keller andBlank r2]. The solutionfor each subsequent diffraction by the next corners can be obtained bythe use of Green's function for a wedge [l, pp. 108-1151. Forright angledcorners, the diffraction. solutions can be obtained by the solution ofan Abel typeintegral equation [ 31. By means ofthe integral of Duhamel, thesolution for the diffraction of a planepulse by a two-dimensional structure was employedto construct the solution for the diffraction of an N-wave bythe same structure [4].
Forthe three-dimensional problem of the diffraction of sonic boom by structures,the first step is theconstruction of the solution for the diffractionof a planepulse by a cornerof a structure. The formulation ofthe differential equation and the appropriate boundary conditions for theproblem in three-dimensional conical variables is presentedin the next section.
2 2. Formulation
Forthe acoustic disturbance pressure p, thegoverning differential equation is thesimple wave equation,
in the region outside a trihedronsimulating the corner of a structure. As shown in Fig. 1, two edges, OA and OB, ofthe trihedron are inthe x-yplane and are bisected by the negative x-axis with half angle QI and thethird edge OD, which is thenegative z-axis with the vertex 0 as theorigin. Let t=O bethe instant when thepulse front hits the vertex. Theboundary condition on the three faces of the trihedron is
ap/an = 0 (2 1
The three-dimensionaldisturbance due to the vertex is confinedinside thesonic sphere r = Ct where r = (2 + + z2)%. Outsidethe sonic sphere,the pressure distribution is giveneither by the regular reflection ofthe plane pulse from the surfaces of the trihedron or by thediffraction of thepulse by the edges. When thepulse front is parallelto the edge, thediffracted wave is given by a two-dimensionalunsteady conical solution. When thepulse front is not parallel tothe edge, the diffracted wave is givenby a steadythree-dimensional conical solution. For both cases, solutions are givenin two conicalvariables in [Z] bymeans ofBusemann’s conicalflow method.
Due tothe absence of a time scaleand a lengthscale, the disturbance pressure p nondimensionalizedby the strength of the incident pulse should be a functionof the three-dimensional conical variables, x/(Ct), y/(Ct) andz/(Ct) or in terms ofthe spherical coordinates by 5 = r/(Ct), 9 and.cp. The simple wave equationfor p(5,8,Cp) becomes,
insidethe unit sphere, 5 = 1, andexterior to thetrihedron. Theboundary conditions are:
ap/a = 0 onsurface OAB, 8 = n/2, - (TT-a)< Cp < n-a (41
ap/acp= o surfaceon OBD, cp = -nW, n/2 < 8 < n (5b 1 and p = F(8,cp) onunit sphere 5 = 1 outsideof the trihedron.
The jump acorssthe sonic sphere is inverselyproportional to the squareroot of the area ofthe ray tube, (dS)’. Since all the raysreach thesphere come fromthe origin where dS = 0, the jump acorss thesonic 0 sphere is zero. The pressure is continuousacross the sphere 5 = 1; and F(8,cp) is definedby the solutions outside the sonic sphere in two conica variablesgiven by [Z].
3 To construct the solution by the method of separation of variables, the usual trial substitution p(g,e ,cp) = 2 (c) G(p,rp) is introducedwhere p = cos0and eq. (3) becomes
62 (1-C2) Z"(5) + 2(l-C2) Z'(C) - XQ+l) Z(C) = 0 (6 1
for 1 > 5 B 0, and
for the domain in p - cp planewith lcp[ < n-a when 0 2 p 2 -1, andwith I@[s l-r when 1 2 p > 0. Sincethe pressure p andalso G are singlevalued, G should be periodic in cp when 1 2 p > 0, i. e.,
Therange of cp is thereforerestricted from -n to TT andthe two ends are connectedby the periodicity condition.
Thereplacement of the variable 0 by p and theuse of the constant of separation Xa+l) are motivated by the intention of representing G(p,B) bythe spherical harmonics.
Theboundary conditions on the surfaces of the trihedron, eq. (4), become
The conditionthat p is bounded in particular at the two poles, 8=0 and 0=~,yields the condition,
IG~ The periodicitycondition, eq. (8), suppliesthe condition along the remaining boundaries, G(p,-n) = G(p,n) and G (p,-n) = G (p,n) for 1 2 p > 0 (9d 1 CP CP The differentialequation, (7), andthe boundary conditions, (9a-d), definean eigenvalue problem. The determinationof the eigenvalues, x's, andthe associated eigenfunctions, G (p,cp), are describedin the nextsection. x 4 3. The EigenvalueProblem Forthe eigenvalue problem formulated in the preceding section, two edgesof the trihedron, OA and OB, are assumed tobe normal to the third edge.This restriction is imposeddue to two consideration: 1) thesurfaces of the corner can bedefined by surfaces of constant 8 or constant cp's and 2) thesolutions outside the sonic sphere for the diffraction problem can beconstructed as solutionsof two-dimensional problems [2]. Forthe solution of theeigenvalue problem itself thesecond consideration is irrelevant. The eigenvalueproblem in this section will, therefore,be formulatedfor a wider class ofcorners as shown in Fig. 2. The surface OAB is a conicalsurface with 8 = f3 andthe two surfaces OAD and OBD are planeswith cp = n-a and cp = - n + a respectively. Theboundary conditions forthe solution of eq. (7) are aG/ap = 0 along p.=p =cospwith n-a < lcpl < TT 0 (loa) The eigenvalueproblem is now definedby the differential eq. (7) andthe boundary conditions of eqs. (loa - d). In orderto reduce the problem tothat €or a set of algebraicequations, two representations of theeigenfunction G (p,cp) associatedwith ihe eigenvalue will besought: A X onefor the region, R+, with p. > CJ- andthe other for R- with p. < p0 (Fig. 2). 0 These two so1ution.sand their normal derivatives will bematched across thedividing line t~, = p. for lcpl < n - a. 0 + + Forthe upper region R , theeigenfunction G (v,cp) which is periodic A in cp onaccount of eq. (loa)or eq. (8) canbe represented by the Fourier series in cp withperiod of 277, Foreach m, eq. (7) yieldsthe Legendre equation for ~-~(p.) h 5 p=1. P-mA is identified as thegeneralized Legendre function [5] and defined by where F denotesthe Gaussian hypergeometric function. The factor l/J?(l-tm) is omittedon the right side of eq. (13) since it is automaticallyabosrbed intocoefficients A and B with a netsaving of programming and computing m m t ime. Forthe lower region R-, i. e. , -1 Ip. < p. theeigenfunction G- which 0 x’ fulfillsthe boundary conditions of eqs. (lob and c), canbe expressed as a cosine series in (cp - n + a) withhalf period of 2(n - a) i.e., n=O, 1,2.. where @ = 2(n - a). Similarlyfor each n, eq. (7) yieldsthe Legendre equationfor the generalized Legendre function Px -m’Q(-p). It is finite for -1 5 p. I p < 1. 0 Bothexpressions (12) and (14) fulfillthe differential eq. (7) and + theboundary conditions (lob-d) appropriate for regions R and R- respectively. + Acrossthe dividing line of R and R-, where p = t~, and lcol < I-T - a, 0 thematching conditions are thecontinuity of the eigenfunction Gx and its normalderivative aGx /ap. The continuityof the second derivatives are thenassured since both and G- fulfill the same differentialeq. (7). c;f h Thematching conditions and the remaining boundary condition, eq. (loa), can be written as ++ G (p.,cP> = (<,SI)) for lcPl In - a (15) x0 a~i(p-,cp)/ap. for lcpl < n-a ++ a~ (po,u)/ap= io for TT-a< lcpl 2 TT (16 1 Equations(15) and (16) canbe reduced to a systemof linear homogeneous algebraicequations for the unknown constants A B and E byFourier m’ m n analysisor by fulfilling eqs. (15,16) at a numberof cp’s inthe appropriate intervals.Before carrying out the reduction, it shouldbe noted that the differential eq. (7) andthe boundary conditions (loa-d) are symmetric with theplane cp = 0. The eigenfunctioncan, therefore, be expressed either as evenfunctions or odd functions of cp. Indeed, inthe expression (11) for 6 G+ thecosine terms representthe even solution and the sine terms represent L the odd solution,In the expression (14) for $, the terms with even and oddvalues of n are theeven and odd solutionsof cp respectively. They can be rewritten as j=O, 1.. . j=l,2.. . where I, = 2jl-r/@, Gj = (2j-lh/@, C.=E2j (-1)' and D.=E (-1)' . j J J 2j-1 It is now obviousthat the cosine terms in eq. (17) are even andthe sine terms are odd in cp. Sincethe odd solutionand the even solution are + orthogonalto each other, the conditions of eqs. (15) and (16) for G and h G-x shouldbe fulfilled by their even terms and alsoby their odd terms respectively. The eigenvaluesand the associated even eigenfunction can bedetermined separately from the eigenvalues and the associated odd eigenfunction. By thisuncoupling, the rank of thecharacteristic de- terminant is reducedby a half,and a doubleroot of the characteristic equation of themixed solutions may now be split toone single root for theeven and one for the odd solutionsrespectively. In actual numerical computations, it is much easier tolocate the eigenvalue and construct theeigenfunction for a singleroot than for a doubleroot of the charac- teristicequation. The solutionof the eigenvalue problem defined by eqs. (15) and (16) canbe carried out by collocation method, i.e., totruncate the series of eqs. (11) a.nd (17) andto impose eqs. (15) and (16) at a finite numberof value of cp's inthe appropriate intervals. In this method, it is necessary touncouple the solutions first and then impose eqs. (15) and (16) tothe evenand to the odd solutionsseparately. When thesolution of theeigenvalue problem is tobe carried out by Fourieranalysis, the uncoupling of theeven and odd solutions will come about automatically as follows : Equation (16) holdsfor the interval -n < cp < IT, it will bemultiplied bycos ncp and will beintegrated over the interval. Equation (16) then becomes thealgebraic equations for the coefficients ofthe even terms, where 7 and (' ) means the first derivativewith respect to its argument. Similarlyeq. (15), which holds for the interval lcpl s - a/2 = $12, will bemultiplied bycos[2kn (cp"$/2)/@] or cos v kcp with V k = 2b/@ and will beintegrated over the interval -@/2 < cp .= 112. The result is Sincecos ncp andcos v cp are evenfunctions of eqs. (18) and(20) k V, containonly the coefficients A and C. associatedwith the even parts in + m J Gx and G-x respectively.Similarly the algebraic equations for the coefficients B and D. associatedwith the odd parts are obtainedfrom eqs. (15) and (16). m J Equation (16) will bemultiplied by sin nq and integrated over the interval -n < cp < 7~ andeq. (15) will be multipliedby sin kcp andintegrated over theinterval -$/2 < cp < @I2with Gk = (2k-l)n/@. The results are, j=1,2,. . . for n = 1,2,..... and where 8 Equations (18) and (20) definethe eigenvalue problem forthe even solution,while eqs. (21) and (22) forthe odd solution. The numerical method forthe solution of the problem will bedescribed in detail in thenext section. 9 4. Numerical Solutionsof the Eigenvalue Problem To obtainthe numerical solutions, the infinite cosine series in + eq. (ll), theeven part of Gx, will betruncated with the maximum of m equalto Mc. Similarlythe cosine series in eq. theeven part of - (17), G will betruncated with the maximum of j equalto J Likewise, A' C' MC and J will bethe maxima of n andk, respectively, so thateqs. (18) and C (20) yield M + J + 2 linear homogeneous equationsfor the M + J + 2 C C C C constants A 5,.. . . % and Co, C1.. . . Thesesolutions are nontrivial 0' cJ C C if A is a rootof the characteristic equation, where a = ~-~+l(p~)[1+61il hhi for h, i=l,2. . .Mc+l , hi X a hi = <"j(-p0)X Iji for h=l, 2. . .Mc+l, i=M +2,. . . , .M +J +2, and j=i-M -2 C cc C - i+l a = (Po) 'ki for h=M +2,. ,Mc+Jc+2, hi X c . . i=l,2,. . . ,Mc+l, and k=h-M -2 C and forh, i = M +2, . . ,M +J +2, with k=i-14 - 2. C . cc C * Forgiven values of M and JcJ theroots of A (1 ) = 0 in a given C C intervalof ), canbe located by numerical evaluation of A (X) as a function C of 1. When thederivative of A at a root is non-zero,the coefficients C 6) A,, C. are proportional to thecofactors of the determinant, i. e., 3 for M=O, 1,2.. . . , M C -1 (25) C =N for j=O, 1,2.. . , J j %, Mc+l+j C * where Ahi is thecofactor of theelement a ofthe determinant with = hi 1 X . The constant N is definedby the normalization condition, 10 0 -1 0 LO The equation for N is po m=O 1 JC A numericalprogram is describedin the appendix so thatfor a given pairof M and J thedeterminant A canbe evaluated as a function C CY C (X) of X. The searchfor the roots can be restricted to X > -4 becauseof the -V -V -V symnetryof the function P (p) about ?, = i.e., PA (p) = P-x-l(p). X -4, Forthe given range of X, say -% I X < 3, theeigenvalues for the even solutions are locatedand the first derivative of A at eachroot LJc C (X) are computed. Inthe examples considered in this paper, thederivative doesnot vanish at Age, therefore,the numerical program in the appendix carries outthe determinations of the coefficients A and C inthe m j eigenfunctionsby means ofeqs. (25) and (27). Similarlyfor the odd solutions,the series ineqs. (11) and (17) will betruncated with the maxima of m and j equalto M and Js respectively. S There will be Ms equationsfor eq. (21) and M. equationsfor eq. (22). The J roots of thecharacteristic equation yield the eigenvalues €or the odd solutions. The ratios of thecofactors and the normalization condition of eq.(26) define the coefficients B l.. . . BM and D~.. . . inthe eigenfunctions. S DJ S It shouldbe pointed out here that the convergence of the eigenvalues andeigenfunctions of the truncated problem has been assumed and the equivalencebetween the matching conditions of eqs. (15) and (16) and matchingof the Fourier coefficients is also assumed. Some confidencein theseassumptions will beoffered by the following numerical results. Forthe special case of a two-dimensionalcorner, namely a = 1712, = n/2 in Fig. 2, theeigenvalue problem can be solved exactly by choosingthe edge of thecorner as theaxis (Fig. 3). Inspherical coordinates 0 and @,the two sides are @ = 0 and cp = %/2. The eigenfunction can be written as 11 - - where p = cos 8 and j = 0,1,2,...... The eigenvalue h is defined by the conditionthat €"2j13 shouldbe finite at 8 = or = -1. Insteadof eq. (13), A -V anequivalent express ionfor P is A (L) [5]. where v=2j/3 and X=(1-;)/2 Thepower serie? representationfor the hypergeometric function is divergent at X=l or p = -1 unlessthe series hasonly finite number of terms,say k + 1 terms. The conditionfor 1 is v-x+k = 0 or 1. = k+2j13 jk * For j = 0,1,2,3. .. .. , theeigenvalues are X = 0,1,2,3 ... .. , 0, k i1, = 213, 513, 813.. , X2,k = 413,713,. , = 2,3 ...... 13, k ,...., x~,~= 813,. . . The numericalprogram in the appendix developed for a three-dimensional corner will now betested by setting B = r(12 and cx = n/2. The eigenvalues between -0.5 and 3.0 forthe even and the odd solutionsrespectively are shown inTable I forvarious combinations of El M and Js. The numerical c' Jc'c' s resultsfor each combination yield the same numberof eigenvalues in the same intervalof X as theexact solution. For the combination of M =8, C J =4, M =9, J =5, theeigenvalues given by the numerical program are in C S S theagreement with the corresponding exact values within 0.2%. This is a confirmationof the procedures developed in this paper a.nd thenumerical programfor the eigenvalue problem. A fewwords should now besaid about how topick the numbers M and J C +c (also M and J,), i.e., how many terms inthe series representationsof G S and G- shouldbe employed. Sincethe eigenvalues of the truncated problem are assumed toconverge as M and J increasesimultaneously, their limits shouldbe independent of the differences between M and J where M and J stand for M Jc, or M C' s' Js' Thisfact is alsoconfirmed by all ofour numerical results. Nevertheless, it is relevantto point out the implications of the difference between M and J, i. e., thedifference between the numberof terms inthe series solutionfor G+ and that for G-. The conditionsof eqs. (15) and (16) call forthe matching of G+ and G- and of their normalderivatives G+ and G- EL EL * Note that G(8, r~.)= constant is aneigensolution, therefore, A=O is an eigenvaluefor any combinations of @ and CX. 12 + across = po with lcpl < n-a andthe vanishing of G at p = po with tJ n-a < I n. Intuitively,the two matchingconditions would require the same numberof terms in both series whilethe latter conditionon + + G is handledby the extra M-J terns in G . P The qnalitative relationship between M and J can be established clearly if eqs. (15) and (16) are fulfilledby the collocation method, i.e., eq. (15) and (16) will beenforced at thegrid points uniformly distributedin the intervals for cp. The gridsizes in these intervals are 2(h-a)/J and(2a/(M-J) respectively. The conditionfor these two gridsizes to be equal or almostthe same is m/M (m-a)/J It means thatfor given J, the integer M should be so chosenthat the ratio M/J will bethe nearest rational number to l-r/(n-a). Forthe special cornerconsidered in the diffraction problem, it is thecorner of a cubewith a = n/4, @ = n/2. Some ofthe eigenvalues can befound by inspection,namely, all the even integers. The first few exacteigensolutions are grouped as even and odd solutionsof cp and listedin Table 11. Inthe truncation of the series for G+ and G-, it is, desirablethat these exact eigensolutions with A. I: M shouldbe reproduced. This implies thatthe factor of q? inthe last term ofthe truncated series of G+ shouldbe equal to or be the nearest one to that of G-. Forthe evensolutions, the condition is and forthe odd solution it is Condition(32a) is identicalto eq. (31) fromcollocation method while condition(32b) will beequivalent to (31) if the grid points in the collocationmethod are shifted by a half of thegrid size. Listedin Table I are thenumerical results for the eigenvalues corre- spondingto various Combinations of M , J and Ms, J . It is clear that cc S forthe same total number of terms M + J, thecombination of M and J which fulfilleqs. (32aand b) are inbetter agreement with the exact solution thanthose with M equal to J. With therelationship between M and J established,the next step is tochoose the integer J forthe truncation of the series. To give a measure of theaccuracy of the eigenvalue 1 andthe corresponding + (J) eigenfunctions G and G- thefollowing norms are introduced (J) (J)’ 13 1, and I2 measurethe convergence of the eigenvalues and the eigenfunctions as the numberof the term increases. & and I., measurethe degree of ac- curacyof the approximations to the matching conditions and the boundary con- dition at p, = p, i.e.,eqs. (15) and (16). 0) It shouldbe pointed out here that although the eigenfunctions are continuous, its derivatives may notexist along the edges. The singularity alongthe edge 8 = rr is builtin by therepresentation for G- (p,,cp). The singularity along the edges, 8 = p and cp = n-a or cp = -n+a, canbe ascertained by investigatingthe behavior of the solutions of the differential equations for G(~,cp), eq.- (7). Inthe neighborhood of anedge, i. e. , IT! << 1 and << 1 with p, = t~, -pJ @=n-a-cp eq. (7) canbe approximated by the Laplace e quat ion 0 inthe first three quadrants of the - $ planesubjected to the boundary conditionthat the normal derivative of G shouldvanish along the positive N p,-axisand along the negative @-axis. The potentialsolution G nearthe " -1 - 213 origin p, = cl, = 0 shouldbehave as the real partof [T(l-p,:) + icn] [6]. The same resultcan be obtained by observing the three-dimensional corner directly: the surfaces of cp = rr-a and 0 = /3 intersect at rightangle and in theneighborhood of the edge away fromthe vertex, the solution behaves as that of a two-dimensionalconvex right corner. Alongthe interface of G+ and G-, i. e. , p, = and > 0, thenormal -1/3 derivative of G behaves as the real part of @i) i. e., 14 G is singularnear the edge but its productwith cosine or sine functions P of ep is integrableand itself are squareintegrable. Therefore, eqs. (18) and (21) are meaningfuland the norm 4 exists. Table 111 shows theeigenvalues between -1/2 and 3 forthe corner of a cube, i. e. , 2a = n/2and 8 = n/2. The series representations are truncatedwith Mc = 8 and Jc = 6 forthe even solutions and with M = 10 S and Js = 8 for th.eodd solutions.The coefficients in the series repre- sentationsof the eigenfunctions are alsolisted. The norms, % and &, are computed foreach eigenvalue. It is foundthat all the 5's are-less than 1%and all the h 's are less than 10%. Thematching of the normal f derivativeof G and G- as measuredby is less satisfactory as ex- pecteddue to the singularity at theedge and due to the fact that the exactvalue of aG/ap, i.e. zero,has been used in the denominator for 4 in the interval p f2 < cp < IT. Calculationshave been performed for various values of J and Js. It C is foundthat when J = 6 and J2c = 9 andalso J = 8 and J2s = 11, all IC Is the I, Is and & 's cosrespondingto the eigenvalues between -112 and 3 are less than .lX Forthe circular cone, the eigenfunctions are P-n(p)x exp (incp). The eigenvalues x Is are definedby the condition of dP-n(p)/dp=Ox at p=-cos@. For eachinteger n z 1, theeigenvalue X has a multiplicityof 2 andonly for n = 0 theeigenvalue x has a multiplicityof unity. Table IV shows chatthe eigenvalues for the three-dimensional corner canbe approximated by those of a circularcone with the same solidangle. It also shows that when aneigenvalue for the circular cone has a multi- plicityof two, i.e. ~ for ml, thecorresponding values are found for the cornerwith even and odd eigensolucionsand when the multiplicity is unity, i.e., n=O, thecorresponding value for the corner has only an even eigen- solution. It. shouldbe noted that when a cornsrdegenerates to a plane, it is thenidentical with the equivalent circular cone of solid angle 2n. The resultsin Table IV notonly serves as a indirectconfirmation ofthe numerical program but also suggests the use of anequivalent circular cone for a quickapproximate determination of theeigenvalues of a three-dimensionalcorner. 15 5. Constructionof the Conical Solution In 2 thegeneral formulation of theconical problem, the method of separationof variables was introduced. The disturbancepressure p(c,8,v) can be written as The eigenvalues 1's andthe eigenfunction G x are determinedin 4. The function Zx (17)is a solutionof the differential eq. (6), i. e., c2 (1-5" Z" (5 + 2 (If25 z I (c - x (h+l) z (c = 0 (35 The boundaryconditions are Z (0) < m and 2 (1) = 1. The condition on theunit sphere [=1 permits thedetermination of the coefficients 5. from theboundary data on the unit sphere, eq. (5), andthe eigenfunction Gx (p,cp) independentof Z x (5). The equationfor the coefficient 5 is 0 -i-r i-r n-a The factor1/2 is dueto the normalization condition of eq.(12b) for Gx inwhich the integration with respect to cp is carriedover only half theinterval. In the numerical determination of 5. thedouble integral canbe reduced to a lineintegral by splitting the boundary data to even andodd solutions and then to a cosineand a sine series in cp respectively. Forthe even solution the results are: (37 where 16 I Correspondingto the even boundary data, eq. (37) holdsfor all the eigenvaluesof the even solutions and Am and C, are the coefficients de fined by eq.defined by (25)each for 1. J Similarlythe coefficients 5 for odd eigensolutions are obtained as follows : where Forthe determination of the pressure distribution inside the unit sphere, it is necessaryto construct the solution Zx (c) ofthe differ- entia1 eq. (35), subjectedto the boundary conditions at [=O and 5=1. The solution is obtainednumerically by the "shooting method".For 5 to be finite at 5=0 thesolution can be represented by the power series, By setting a =1, the.differentia1equation can be integrated numerically 0 Ar from a small 5, (say 0.001) with initial data 'z (5 )=c ~1+1(X+l)c:/ (41+6)] x0 0 and yi([o) = Act-' [l+(h+l)(h+2) 5:/(41,+6)]; Thenumerical integration for 2 (5) is continueduntil 5 is close to 1 A (say 0.99). The value of 2x (1) is obtainedby patching the numerical solution with the series solution of 5-1, i. e., The appropriatevalue for a or lim Zc-' is l/"x (1). The numericalprogram O 5-0 inthe appendix computes a autohticallyand generates the function Z (c). 0 h Thus for a givenboundary data F(p,~p)on the unit sphere, the conical solutionof eq. (3) is defined. 18 6. A Numerical Example Figure 4 shows theincidence of a planepulse normal to a facesay OAD ofthe corner of a cube. Inorder to demonstrate the use of both the evenand the odd eigensolutions,one of the two edgesparallel to the plane pulse is chosen as thenegative z-axis as shown inthe figure. After the incidence, t > 0, partof the plane pulse is reflectedby the face OAD of thecube with double the intensity. The remaining part advancingover thecorner is undisturbed. The diffractionby either one of the edges OA and OD, is givenby a two dimensionalconical solution. It is confinedin a circularcylindrical surface with radius Ct andthe edge as the axis provided it is outsidethe domainof influence of the vertex. Figure 5 shows theincident plane pulse, the reflected pulse and the sonic circle in a cross-sectionnormal to the edge OA at a distance greater than Ct fromthe vertex. In terms of polar coordinates p and T, the boundarycondition on the sonic circle p = Ct for the pressure variation fromthat behind the incident pulse is PC = 1 The strength of theincident pulse is chosen as unity.For the two- dimensionalconical solution, the Busemann conicaltransformation is introduced [ 21, *- * p = p / {1+[1-p2 ]'} , = (Ct) and 7 = 7 In the new variables, thedisturbance pressure becomes a potentialsolution pj: TJ; andthe region - inside the circle < 1 with 0 < < 3n/2 is mapped~~ to a halfcircle with 'i; ex.p (iy) = [pj'exp (i~")]~'~.The solutionin terms of -N PJT is L2] (1-z2)312 p,(p,T) = arctan (1+7)/2+2: cos 7 wherethe arctangent lies inthe first and the second quadrants. The circular cylinder with axis OA andradius Ct intersects thesonic sphere aboutthe vertex along a greater circle andcovers the spherical surface onthe right side of plane OBD. Similarly,the diffraction by the other edge, OD, created-another two-dimensional conical solution given by eq. (42) with p thedistance from the edge OD and T theangle measured from face OBD. Again,the circular cylinder with axis OD andradius Ct intersectsthe sonic sphere about the vertex along another great circle andcovers the partof the spherical surface below plme OAB, i. e., p < 0. The surface of thesonic sphere can be divided into four regions: 1) notcovered by either one of these circular cylinders 2) coveredonly by the cylinder with axis OA 3) coveredonly by the cylinder with axis OD and 4) coveredby both cylinders. Withthe plane cp=O defined as the plane bisecting the exterior angle ofthe two vertical surface OAD and OBD, thefour regions and the boundary data are defined as Withthe boundary data F(p,cp) defined by eq.(43) the coefficients in eq.(34) forthe pressure variations inside the sonic circle are com- 5 putedfor all theeigenvalues between -% and 3. The pressuredistribution onthe three faces ofthe corner are shown inFigs. 6 & 7. The pressure distributionsfor r/(Ct) < 1 are obtainedfrom the present analysis. They matchwith the corresponding two-dimensional values across the sonic sphere with a discontinuityin slope. They confirm,with less than 1% variation) the symmetryof the solution with respect to the plane bisecting the faces OAB and OAD andthe value of 817 at the vertex as predicted byTheorem I in 2. When thecalculation is performedwith only eigenvalues less or equalto 2, theresults differ from those in Figs. 6, 7 within 6%. 20 7. GeneralIncident Wave Thepreceding concial solutions for an incident plane pulse can be employed to construct diffraction solutions for an incident wave of more generaltype by superposition of plane pulses. In particular, when the incident wave is a plane wave ofthe type with n = Ct - (q x+n,y+% z) (45 1 wherethe wave form 11, is an arbitrary function of its phase q and nl ,n, and 5 are thedirection cosines of the normal to the plane wave. The incident wave dueto the sonic boom canbe locally represented as a plane wave sincethe length of a structure is usually much smaller thanthe radius of curvature of the wave front. Thewave form $ is in general a sequenceof a weak shock wave and anexpansive wave or a com- pression wave as shown in Fig. 8. Ifthe wave form is a Heavisidefunction, the diffraction due to the three-dimensionalcorner is given by thepreceding conical solution and will bedesignated as p+(r/(Ct),€l,v). The solutionof the diffracted wave correspondingto a general waveform of a sonic boom is where n is thephase of the i-th shock wave withstrength i - dl (qi-0).Note that q increasesin the direction opposite to the normal ofthe plane wave. Anothermethod for the construction of nonconical diffraction so- lutionsshould be pointed out here. When theboundary data on the sonic spherec=r/(Ct)=l around the vertex of a three-dimensionalcorner, which is now a functionof t inaddition to the spherical angles 8 and v, can bewritten as a power series with respect to t, i. e., F(t,8 ,cp) = C t' F (0,cp) for t > 0 (47) V Y wherethe summation is carriedover a sequenceof positive numbers, y 2 0. The solutioninside the sonic sphere can likewise be written as 21 . .. . p(v) is subjectedto the boundary condition on thesphere c=1, andthe boundary conditions on the surfaces of the corner remain the same as eqs. (3,4). Sincethe differential operators with respect to 0 and cp are independent of Y, thecorresponding eigenvalue problem in 8-cp planeafter the separation ofthe variable 5 is the same as that for V=O, i. e., theconical problem. The solutionp(y) can therefore be written as, where p = cos 8. The eigenvalues A's andthe eigenfunctions G (p,cp) are A identicalwith those obtained for y = 0 in ,2,&. LJith theboundary con- dition Z(V'(1) = 0 imposedon .Z(y), theconstants (VI are also related A x 5, tothe boundary data F (p,8) by the same set ofequations, eqs. (37) and v (38), andcan be determined by the same numericalprogram for y = 0. The appearanceof y occursonly in the differential equation for Z(y)(c), which is x d" Z 5" (1-5") - + 2 [l+(y-1) 523 - [y(y-1) 5" + x (x+1)] z = 0 (51 dC 2 dS dC2 The boundaryconditions are the same, i. e., Z(1) = 1 and Z(0) is finite. Z'y'x (C) canbe determined by the same procedure described in 5 for Z(O)x (5). Now it is clear thatthe extension of the proceeding conical so- lutionsto the diffraction of a generalincident wave by a three- dimensionalcorner can be carried out provided that the boundary data F(t,B,cp)on the sonic sphere can be found. This suggests that the next step is tosolve the diffraction problem of a three-dimensional waveby a two-dimensionalcorner. For a two dimensionalcorner, the three dimensional diffracted waves canbe constructed by superpositionof solutions for incident plane waves with different wave formsand directions of incidence or bythe method of separationof variables in cylindrical coordinates and time. Theseschemes ingeneral are tediousfor the actual construction of the solution. By makinguse of the special geometry of the 90" convexcorner, an integral 22 equationcan be set up andsolved by an iteration scheme similar to theprocedure for a two dimensionalincident wave [3]. Thismethod of approach is presentlyunder investigation. Inthe next section, two theorems are presented so thatthe value alongthe edge or the vertex of a corner can be related directly to an incident wave without the cmstruction of the diffraction solution. The incident wave canbe completely arbitrary and hence can also be a diffracted wave froman adjacent corner. 8. "Mean-Value"Theorems forDiffraction by a ConicalSurface and Applications In a previous article [7], a mean value theorem for the diffraction ofan incident wave by a cone was derived. It states thatthe value at thevertex of the cone is equalto 4rr/(4n-R) times thevalue of the incident wave at the vertex as if thevalue is averagedlocally over theexterior solid angle of the cone, 4n-R insteadof over the whole space, 4~.This relationship was obtainedunder the restriction that theincident wave doesnot hit any portion of the conical surface prior to its encounterwith the vertex of the cone. In this section, the derivationof the theorem is partiallyrepeated with modifications so thatthe theorem is validunder a weakerrestriction, that is, the incident wave may hitthe conical surface within a finite time interval T aheadof the epoch of its incidenceon the vertex. , In case such a finite time interval T cannotbe found, a secondtheorem is introduced. Ex- tensionof the results to solutions of inhomogeneous wave equations is alsopresented in this section. Let t=O be the instant when an acoustic incident wave cp(i) hits thevertex of the cone located at theorigin. Let T bethe finite time interval such that for t I -T theincident wavedoes not hit the conical surface. Let D(t) denotethe domain outside of which the incident wave vanishes at theinstant t,e.g. , q~(~)(x,y,z,-T)=Ofor (x,y,z) not in D(-T). Domain D(=-T)does not intersectthe cone G withsolid angle R, as shown in Fig. 9; Inthe absence of the cone, the incident wave cp(i) at the origin can be related to the initial data at an instant t=t <-T bythe PoissonFormula [SI, 0 S s where r is distancefrom the origin, and S is thesphere with radius R=C (t+t ) > C(t+T) andwith its center 0 at theorigin. The Poissonformula can be identified as a special form of the Kirchhoff's formula [ 81, S where a/& means differentiationalong the outward normal to the surface S and rcp] is theretarded value of cp. On thesphere S with R=a(t+t ), it 0 (i1 follows rcp(i) (x, y,z, t)] = cp (x,y, z,- to). Similarly,the retarded values of 24 (i)and cp(i) are thecorresponding values at t = -t . All of them vanish 'n t 0 outsidethe domain D(-t Let S denotethe part ofthe sphere outside 0 ). C ofthe cone G, and S" denotethe part insidethe domain D(- to). Since D(-to)with -to < -T doesnot intersect G, S contains S*. The integrand C is non-zeroonly inside S * therefore,the domain ofthe surface integration canbe reduced from S to S' or to Sc. To obtain the resultant wave cp at thevertex of the cone, the passages inthe derivation of the Kirchhoff formula [7,9-J fromthe Green's theorem will berepeated. The Green'stheorem is now appliedto the volumebounded by the two concentricspheres S" and S withradii r=R and r- respectively andthe conical surface aG yields (5 where aG* is the part of aG lyingbetween SJc and S 0. On thesurface of the cone aG, ar/an=O.With the boundary condition of acp/an=O, theintegral over aG" vanishes. As 0 -, 0, the integral over S approaches - (4n-a) cp(o, o,o, t). Equation (53) reducesto 0 On the spherical surface SJc, [cp] = cp(x,y, 2,-to) = cp'i)(x,y, z,-to) = [cp (i), and similar relationshipholds for the t- and n- derivatives. Eqs. (52) and (54) yield: T (55 1 Thisresult can be stated as follows: Theorem I The resultant value at the vertex of a conewith solid angle fl is equal to 47-r/(4n-n) times the incident wave at thevertex, if a finite T canbe foundsuch that the incident wave hitsthe conical surface within the time interval T prior to its encounterwith the vertex of the cone. In a practicalproblem, the conical surface which forms a part ofthe surface of anobstacle, has a finitelength, say L. Inapplying the theorem, it is essentialthat the part ofthe obstacle inside the sphere S withradius C(t + T) is conical, i. e. , C(t + T) < L (56 1 It definesan upper bound for T. On theother hand, for a given T with CT < L, theinequality (56) definesan upper bound for t forwhich eq. (55) holds. When anincident wave is diffracted first by a part of the surface of theobstacle other than the conical surface of finite length L, con- dition (56) cannotbe fulfilled. Even for a coneof infinite length, the incidentpulse may bein contact with a partof the conical surface all the time, consequently,the finite time interval T assumed in Theorem I doesnot exist. For both examples, the value at thevertex of the cone cannotbe related directly to the incident wave byTheorem 1. A theorem relating the value at thevertex of one cone to that of another cone will bepresented. Figure 10 shows therelative orientations of two cones C, and ($, withsolid angles nl and &. Theirvertices coincide at theorigin and theirboundaries aq and a($ have a common region r, i. e., Let Q (x,y, z, t) denote the resultant of an incident wave cp(i) and 1 its reflectedand diffracted waves bycone C, alonewith &,/an=Oon thesurface of the cone 3%. 9 (t) denotesthe domain outside of which vl=O at theinstant t. At t=O, theorigin lies onthe boundary of 4 (t) and 4 (t) doesnot contain the origin for t < 0. It will beassumed that there exists a finite time interval T such thatthe domain D(t I -T) is inpartial contact with the cone C, only over the surface r, i. e. , The nextstep is to examinethe solution cp, which is theresultant of the same incident wave cq(i) and its reflectedand diffracted wavesby cone C, alone. The initial data for cp2 are prescribedby cpl (x,y, z, -to) and , (x,y, z, -t ) for - t < -T. If cone C, doesnot intersect the Cp, Cp, 0 0 domain 4 (t) for t < -T, it is evidentfrom conditions (58) that The derivation of eq. (54) fromthe Green's theorem will berepeated forthe volumebounded by two concentricspheres S and S and by theconic 0 surface 3% or 3% respectively. The two spheres, S and S are centered 0 at theorigin and their radii are C(t+t ) and u. The followingresult equivalentto eq. (54) is obtained: 0 26 ". . S* j for j = 1,2 t 2 - T On the sphere S with radius C(t+t ), it is evident [cp .] = cp. (x,y, 2,- to). 0 J J Equation (59) shows that [cpi], [&./an], [&&/at) and D.(-t ) are invariant J J JO withrespect to j. S? is defined as thepart of the spherical surface S J lying in D. (-t ) and is alsoindependent of j. Withthe right side of JO Eq. (60) invariantwith respect to j, it yields So far theproblem was formulated as propagationand diffraction of waves. It can alsobe formulated as aninitial and boundary value problem and restated as follows : Theorem I1 For two conicalsurfaces 3% and 3% withthe same locationfor the vertices and with the same initial data cp=f (x,y, z) and cp =g(x, y, z) at t theinstant t=-t <0, theresultant disturbance at thevertex for each 0 conealone is inverselyproportional to the exterior solid angle of the coneprovided that the support D for f and g doesnot intersect either oneof the cones, and that the part ofthe boundary aD in common with oneof the conical surface is alsowith the other, i. e. , (a. n 3%) =(an r) as)c(ac, 3%) (62) It is obviousthat when the wave equationhas an inhomogeneous term h(x,y, z, t) with support E(t) similar conclusionscan be obtained: Corollary Theorem I and I1 will a120- be valid for solutions of the inhomogeneous wave equation & - C = h(x, y, z, t) provided that the %t support E(t) of h doesnot intersect the cone or both cones respectively. An interesting application for Theorem I andthe Corollary will be theinitial and boundary ,value p’roblem for an inhomogeneous wave equation inthe interior of a conicalsurface aG withsolid angle (or the C C exterior of a cone G withsolid angle fi=4rr - oc > 2n). The mathematical problem is : -2 D. E. &-C cptt= h(x,y, z, t) insidethe cone G for t > 0, C From thedefinition of the problem both the support E of h andthe support D of f and g lie in G and, therefore,do not intersect G. Theorem I and thecorollary state &at: The value at the vertex, cp(o, o,o, t) is related to the solution, CP~~(O,O,O,t), ofthe initial value problem for the inhomogeneous wave equationin three-dimensional space with the removal of the conical surfaceby an amplification factor equal to the local enlargement in solidangle, i. e., 4n/nC. Thisstatement is expressedby the following equation, Ct (6 3) 0 s;r where S* is the part ofthe spherical surface inside the support D of f Ct and g and SJc is the part ofthe spherical surface inside the support E r of h at theretarded instant t-r/C. The spheres are centered at the originand the subscript denotes the radius. It is ofinterest to note that the value at thevertex of the cone is independentof the geometry of thecone G nordoes it dependon the C distributionof f,g, and h andtheir supports D and E with respect to thespherical angles 8 and Anotherexample is thediffraction problem being analyzed in this paper. As a matter offact, the investigations in this section are motivated by thedesire to determine the values along an edge or a corner of a rectangular block or structure due to the arrival of a conical so- lution originated from anadjacent edge or corner without the construction ofthe non-conical solution. Figure 11 showsan incident plane pulse of strength E impingingon a blockwith normal parallel to an edge OB. Afterthe instant t = 0, thepulse is reflected bythe face OAD anddiffracted around the edges OA, OD andthe corner 0. Let OA, OB and OD tobe designated as thex-, y-, z-, axisrespectively. From Theorem I, thevalue at thecorner 0 of solid angle n/2 is p1 (0,o,o, t) = e[4n/(4n-n/2)] = (8/7) E for W/C > t > 0. (64) This is inagreement with the numerical result in 6 and is valid until the arrival at t = W/C of the diffracted wavefrom the adjacent corners. 28 Forthe convenience of description, it is assumed that the height OD is equalto the width W(=OA) and is relatedto the length L(=OB) bythe inequality 2L> W>L (65) For a point (x o,o) onthe edge of OA withsolid angle rr, Theorem I gives 0' pI (xo, 0,0, t) = E[~TT/ (4rr-rr)I = (4/3) E for x /C > t > 0 0 Due to symmetry, x canbe assumed to be less than W/2 andthe upper limit for t is theOepoch forthe arrival ofthe diffracted wave fromthe vertex 0. For thedisturbance pressure elsewhere such as onthe surfaces of theblock it is necessaryto construct the diffracted waves. Behindthe incidentfront the disturbance pressure is p1 = E outsidethe envelopes ofsonic spheres along the edges OA and OD. Insidethe sonic sphere around the vertex 0 the pressure is given by the- three-dimensionalconical solution constructed in 2, i. e. , P1= E + EP3c (x,y,l) with x, G, z equal to x/(Ct),y/ (ct), z/(Ct) respectively. Outside the sonic sphere but inside oneof the cylindrical envelopes, the solutions are conicalin two dimensions i. e., p1 - E + cpZc(x,$) and p1 = E + tpZc(z,G) for edges OA and OD respectively. Of coursethere are alsodiffracted wavesfrom vertices A, D, E, andedges AD and DE. When t > L/C, the wave frontadvances over the edges BA', BD' and thevertex B. Ifthe edges BA', BD' were absentand the planes OBA and OBD were extendedbeyond BA' and BD', point B canbe considered as the vertex of a cone, C,, withsolid angle R1=n andthe solution p1 at B is validfor t > L/C (Fig. 12). Inthe presence of the edges BA' andBD', point B canbe considered as thevertex of a cone C,, withsolid angle Rz=n/2. Cone C, andcone C, have faces OBAB' and ODD' in common and time interval T canbe equated to zero. Theorem I1 yieldsthe resultant value at B forcone C,, i.e., includingthe diffraction of p1 by corner B, = (6/7) E[ l+~~~(o,L/Ct,o)] for L/C > t-L/C > 0 (67) The upper limit of t = 2L/C is dueto the finite length L ofboth cone C, andcone &. In case of (# + L2)% < 2L, theconical solutions originated at the vertices A and D arrive at corneqBbefore t = 2L/C, additional terms shouldbe added for t > (ti! + L2 )yC. They canagain be defined by- - " Theorem 11. The incoming wave fromcorner A is EP~~(X',y, z) withx'=(W-x)/ (Ct). Point B can beconsidered as the vertex of cone c; defined as beforeand as thevertex of cone C, withthe extension of face OABA' overthe edges OB and BA'. The solidangle for C; is 2n andthe ratio of solid angles exteriorto C, and & .is (4~~-2n)/(4rr-n/2)= 4/7. Theorem I1 yieldsthe extra term (4/7) ~p~~ (w/ (ct), L/ (Ct), 0). Similarly a term (4/7) EP3c (O,L/ (Ct), W/ (Ct)) shouldbe added to Eq. (67) to take into accountthe incoming wave from vertex D for t > (w2 + Lz)'/C. The disturbancepressure at a pointP(x L, 0) onthe edge BA' can 0' also be related to the diffracted waves fromedge OA andfrom vertices 0 and A by means of Theorem 11. Point p canbe considered as the vertex ofcone C, withsolid angle & = n (Fig. 11). When the face OABA' is extendedover the edge BA', point P becomesthe vertex of cone 6 with solidangle fll = 217 (Fig.12). The resultantpressure at point P due to diffraction is equalto 2/3 of thevalue of theincoming wave alone. The factor2/3 is theratio of exterior solid angles of C, and C,. Thisresult is validuntil the arrival of thenon-conical diffracted wave fromvertex B' or A'. It hasbeen demonstrated that the theorems are usefulto extend the knowledge of conicalsolutions to the adjacent edges and vertices for limited time intervals. To obtainthe pressure distribution on the surfacesof the rectangular block or structure, it is still necessary toconstruct three-dimensional non-conical solutions for diffractions byedges and corners. I Conclusion In thispaper the following results are obtained: the conical solution for the diffraction of a planeacoustic pulseby a three-dimensionalcorner of a cube is obtainedby separation of variables. thesolutions of the eigenvalue problem in spherical angles for theconical problem remain the same forthe generalized conical solutions described in 2 and also for potential or unsteady heat conductionproblems, so long as theboundary conditions in 8,cp plane are the same. thetechnique for the solution of the eigenvalue problem is applicable to a moregeneral shape of boundaryon the unit sphere formedby two great circles ofgiven longitudes and a horizontal circle of givenlatitude. Theboundary conditions can be generalizedto the type ap+b ap/& = 0 withboth a and b as non-negativeconstants. thenumerical results suggest that the eigenvalues for corners canbe approximated, one by one, by theeigenvalues for circular conesof the same solidangle. "mean-value"theorems in 2 are derivedfor solutions of wave equations so thatthe resultant wave at thevertex of a cone canbe related to the incident wave orthe value at thevertex of a different cone.These theorems are usefulto extend the knowledgeof the conical solutions to the adjacent corners or edges. applications of the Theorems in 8 yieldinteresting and simple results: for aninitial value problem of an inhomogeneous wave equationinside a conicalsurface with vanishing normal derivative on thesurface, the value at thevertex of the cone can be relateddirectly to the same initialvalue problem in three-dimensionalspace. The value at thevertex depends only on thevalue of the solid angle of thecone but not on the geometryof the cone. It dependson the radial distribution of theinitial data and the inhomogeneous term butnot on their distributionwith respect to the spherical angles. Acknowledgment The authorswish to acknowledgeProfessor Antonio Ferri, Director ofthe Aerospace Laboratory and Professor J.B. Keller, Head ofthe Deprtmentof Mathematics, New York University, Bronx, New York, for their valuable discussions. REFERE3CES Friedlander, F. G., "Sound Pulse,Cambridge University Press, New York, New York,1958. Keller, J.B. andBlank, A., "Diffractionand Reflection of Pulses by Wedges andCorners, I' Communicationon Pure and Applied Mechanics, Vol. 4, No. 1, pp. 75-94, June1951. 3) Ting, L, "Diffraction of Disturbances Around a Convex RightCorner with Applications in Acoustics and Wing-body Interference, 'I Journal ofAerospace Sciences, Vol. 24, No. 11, pp. 821-831, November 1957. 4) Ting, L., andPan, Y. S., "Incidentof N-Waves onStructures, 'I NASA SecondConference on Sonic Boom Research, NASA SP-180, pp. 89-98, May 1968. Hobson, E. W., "The Theory of Sphericaland Ellipsoidal Harmonics, ChelseaPublishing Company, New York, New York, pp. 178-190,1955. Churchill, R.V., "Complex Variablesand Applications, 2nd Ed. McGraw-Hill Company, New York, New York, pp. 218-229,1960. Ting, L., "On theDiffraction of an Arbitrary Pulse by a Wedge in a Cone,"Quarterly of Applied Mathematics, XVIII, No. 1, pp. 89-92, April 1960. Tychonow, A. N. andSamarski, A. A., "PartialDifferential Equations ofMathematical Physics, Vol. 11, Holden-Day,Inc., San Francisco, California, pp. 381-392,1967. 9) Baker, B.B. and Copson, E.T., "MathematicalTheory of Huygen's Principle, I' 2ndEd., Oxford UniversityPress, New York, New York, pp.36-38, 1950. 33 Fig. 1 Incidence of a planepulse on a corner Y I EO (IOcl 1 p=~~~e E E E EQ (IO dl- I -EQ (IOd l R+ F B."". A" A F 4 ' EQ(IOa) pc,=cosp r t *+ a R- -a- -I D EQ (IO b) EQ (IOc) Fig. 2 Eigenvalueproblem for the corner 35 0 Fig, 3 .4 two-dimensional corner 36 /" SONIC I CICLE SONIC CICLE REFLECTED PLANE PULSE Fig. 4 Planepulse incidence head on toface OAD, the reflected planepxlse and the diffracted fronts around edges OA, OD, 37 and vertex 0 C 8' pc =I D' SECTION NORMAL TO EDGE OA SECTION NORMAL TO EDGE OD Fig. 5 Two-dimensionalcogical solutiors aroundedges OA and OD P-Po c 1.1 I I .c -----Y"- 3 - DIMENSIONAL D- IIMENSIONAL CONICAL SOLUTION :0 UICAL io ,UTION If(Ct 1 " I 0 0.5 1.0 1.5 Fig. 6 Over pressure along coilstant 0 lines face 04Dki,=3~/4) 39 " 1.8 1.6 I.4 1.2 1.0 0.8 0.6 0.4 3- DIMENSIONAL 2 -DIMENSIONAL CONICAL SOLUTION CONICALSOLUTION - ~___.~ - ~ 0 . 0 0.5 I .o 1.5 Fig, 7 Over pressurealong constant r,J lines on face OAB ((3~72) andalong constant 8 lines on face OBD (t3=3rr/4) OVER 1 ' . PRESSURE Fig. 8 Typical sonic boom signature FRONT -AT" t=-to, / Fig. 9 Diffraction by a cone 42 Fig. 10 Comparison of two cones 43 I FRONT EDGE FRONTOF FRONTOF EDGE OB EDGE BD' J Fig. 11 Incident of a plane pulse on a block at instant t >L/C 44 (OB=L, OA=OD=W; also shown are thediffracted wave fronts aroundedges OA,OD,BA' ,BD' andaround vertices 0 andB). \ IY 0 b\ z\ FRONT OF EDGE OB Fig. 12 Incident of a plane pulse on a blockwith extended fictitious surfaces instant t >L/C (also thediffracted \ at shown are 45 wave frontsaround edges OA,OD andvertex 0) APPENDIX NUMERICAL PROGRAMS Fourcomplete sets oflisting programs for CDC-6600 are givenin theappendix. They are called first program (A), first program (B), secondprogram and third program. Their input and output formats and theoperating instructions are presented as follows: FirstProgram (A): determinationof eigenvalues and eigenfunctions for odd function. First Program (B): determinationof eigenvalues and eigenfunctions for evenfunction. InputDefinition IIPP . a controlconstant To calculatedeterminant and search for eigenvalues; let IIPPequal to any negative or zero integer. To calculatedeterminant and coefficients of eigen- function; let IIPPequal to any positive integer. NMAX : for odd function NMAX=M foreven function NMAX=M SY C LMAX : for odd function LMAX=J foreven function LMAX=J S’ C XLAM :x IfIIPP 5 1; XLAM is only a searchingeigenvalue. More accuratevalue for the roots of @(h.)=O canbe obtained by interpolation. Be carefulnot to overlookmultiple rootacross which AQ) may notchange sign. IfIIPP > 1; XLAM is aneigenvalue as locatedby the precedingmethod, and ready to calculate the coef- ficientsof eigenfunctions. Input Card 1; IIPP, NMAX, LMAX (Format315) Card 2; XLAM (FormatF15.0) Outputand Definition : inputdata previously described LMAX”””) IIMAX : numberof linear homogeneous equations,for the odd solutions; IIMAX=LMAX+NMAX; forthe even solution: IIMAx=LMAX+NMAx+2 ALPHA : 2a 46 Outputand Definition PPHI : iP=2(rr-a) XNUI : v = Trh 0 =cos(@)where @ is thelatitude of upper surface XMU : Po XLAM : previouslydescribed DET : thevalue of determinant for a given X; Ac(h)=(a. .I 1J. BMIN(J), : coefficientsin the series representation of eigen- DMIN(L) functions, foreven function First M +1 output are Am's C Next J +1 output are Cj 's C for odd functionFirst Ms are Bm's Next J are Dj ' s S These Am's, Cj's, Bm's, Dj's are definedby eq. (25). SecondProgram: Determination of the coefficient IS, ofthe eigen- functionexpansion from the initial data. InputDefinition I1 : controlconstant II=1; tocalculate 5 for odd function I1 equalto any integer other than one; to calculate K foreven function 1 : previouslydescribed XLAM : theseare the coefficients calculated from program """ one (A) for odd functionor program one (B) foreven f unc t ion Input II=1, NMAX, LMAX (3 15) XLAM (F15.0) BMIN(J) } (5F15.0) DMIN (L) (5F15.0) END FILE 47 Input II=2, NMAX, LMAX END FILE Note thatthe bracket indicates a set ofdata composed of a given 1 and its correspondingcoeffcients BMIN(J) and DMIN(L). Notealso thatthese sets are thenlisted into a groupof odd functionsand a groupof even functions separated by an end file card. Thisuse of the end file card facilitates the orderly calculations of all odd functionsfirst and then all evenfunctions. Outputand Definition XLAM These are NMAX, LMAX, IIMAX, ALPHA, inputdata yhich PPHI, XNUI, XMU are listed BMIN (J) identification for DMIN (L) verification and K (Lamda ) 5 thecoefficient of the eigenfunctionexpansion from theinitial data Remark: thesubroutine FSF andsubroutine CON2D prepare theboundary data on thesonic sphere €or the problem described in 5. For a differentproblem only these two subroutinesshould bechanged. ThirdProgram: determination of the pressure distribution on the surfaces. Input same as thesecond program, except the first card of each set of inputdata contains both and its corresponding 5 which was calculatedfrom the second program. Outputand Definition XLAM : x AAO : definedeq.in (39) K(Lamda) : 5 48 :cp where 8 and cp are sphericalcoordinates p=cos8 forpoints on thesurfaces Pl : pressure on surface OAB onfigure (4) where 8 = j3, p=cosp; cpN = Tr - a+-4 (N-l) , N = 1,2,. ..5 3rr : pressure on surface OAD where cp = -4' = (N-1) where N = 1,2,3,4,5 'N 2 +: -3rr : pressure on surface OBD where cp = * -4' ON = +: .(N-l) , N = 1,2,.. .5 Note: PI, P2, P3 are definedin eq. (34) OutputFormat Odd function NMAX, LMAX, IIMAX, AZPHA, PPHI, XNLJI, XMU XLAM, AAO, K(Lamda) Set 1 BMIN(J) { DMIN(J) Set 2 { } Even function NMAX, LMAX, IIMAX, ALPHA, PPHI, XNUI, XMU Sect. 1 { } Sect. 2 { } At the last pageof the output, a tableof PHI, PI, XMU, Pa, P3 is tabulatedin five columnswith Format (2E16.5,15X, 3316.5). Remark: Inthis program, there are 10 Z(c)'s where c's are 0.111, 0.211, to 0.9989. Incase of a differentincrement of 5, thesubroutine T102 mustthen use a different value of XTEST. 49 C FIRST PROGRAM (A) C ODD FUNCTION TO CALCULATEDETERMINANT AND COEFFICIENTS DIMENSIONAEL(50r50)r8(50,50) DIMENSIONPDET(50950)9EEEE(50,50) DIMENSION SUMA~101~~SUM~~1@1~r~MIN~~O~,DMIN~3~~~S~MIN~~O~~ lSDMIN(30) LLMAX=50 READ(595)IIPP,NMAX,LMAX 5 FORMAT(315) IIMAX=LMAX+NVAX I TTEM= I I MAX PIz301415926 ALPHA=PI/2e PPHI=Zor(PI-ALPHA/Z.) XNUP=PI /PPHI INTERV=30 LEP=LMAX JEP=NMAX uo=o. XMU=O XMUFIX=XMU WRITE(69502) NMAX,LMAX~IIMPXIALPHAIPPHI,XNU~,XMU 502 FORMAT ( * NWkX=*,I5,5X,*LMAX=+~I5~5X9*IlMAX=x,I5/5X9 * IALPHA=*rE12~5~5X~~PPHI~*~El2o5~5X,”XNU1~*~El2~5~5X~*X~U~*~E12~5) 400 READ(Lr31) XLAY 31 FORMAT(F14.0) IF(ENDF1LE 5) 5559,5556 5556 CONTINUE C CALCULATION OF ELEMENTS OF DETERMINANT DO 303 J=?rIIMAX DO 300 K=l,IIMAX IF(J-NMAX 1 10~10,200 IO IF(Y-NMAX 1 11911r3C 11 CONTINUE FEL=1 14 IF(J-K) 16915916 15 DELTAzl. GO TO 17 16 DELTA=O. 17 SS=K CALL PPDD(SS,XLAW,XMUILLMAX~PP,DP) AEL(JI#)=DP*EFL*DELTA GO TO 300 30 SS=(2w(K-NMAX)-l)*XNUl CALL PPDD(SS,XLAM,XMUtLLMAX,PPIDP) LH=K-NMAX LI=J CALL XLLLLILH~LIIALPHAIXNU~~XII) AEL(JIK)=DP*XII GO TO 300 200 IF(K-NMAX21292129230 2 12LH=J-NMAX LI=K CALL XLLLL(LH9LI rALPHA,XNUl*XII 1 5S=K CALL PPDD(SStXLAM*XMU9LLMAX,PP,DP) AEL(J,#)=PC*XII GO TO 300 230 ZF(J-K) 23?9232~233 2 32 DELTA=l GO TO 234 233 DELTA=O 234CONTINUE EEL=1 e '237 SS=(2 *(K-NMAX)-l)*XNUl CALL PPDD(SS,XLAM,XMU*LLMAX,PP,DP) AEL(J,K)=-(10dALPHA/(20*PIl)*EEL*PP*DELTA 300 CONTINUE 303CONTINUE DO 307I=l,ITTEM DO 306JZ1,ITTEM B( I,J)=Oe EEEE(IqJ)=AEL(I,J) 306CONTINUE 307CONTINUE CALL LEQ(AEL~B,IIMAXIO,~O,~~~DE~) WRITE(6r32) XLAM rDET 32 FORMAT ( * XLAM=*~E~~~B~~XI+DET=**E~~~~) IF (IIPP)400r400t1002 1002CONTINUE C CALCULATION OF COFACTORS DO 903I=l, ITTEM DO 902 J=l,ITTEM ADET(I,J)=EEEE(T*J) 902 CONTINUE 903 CONTINUE ICORD=l DO 511 J=lr JEP JCORD= J CALL COFACT(ADET,ITTEM,ICORD~JCORD~DETCOE) BMIN(J)=-(-le)**JCORD*DETCOF 5 11 CONT INUE DO 512L=lrLEP JCORD=L+JEP CALL COFACT(ADETIITTEM,ICORD,JCORDIDETCOE) DMIN(L)=-(-l.)*+JCORD*DETCO€ 512CONTINUE DELU=(l.-UC)/INTERV ISTEP=INTERV+l DO 611 I=l,ISTEP XMM=(I-l)*DELU SUM1 O=O 0 DO 601J=lrJ€P SS= J CALL PPDD(SS,XLAM*XMM,LLMAXrPPrDP) SUMlO=SUMlO+(BMIN(J)*PP)**Z 601 CONTINUE SUMA(I)=SUMlO 611 CONTINUE DELV=Il.+UO)/INTERV DO 711 f=lrISTEP XMM=(I-l)*DELV SUMll=Oe DO 603L=l,L€P SS=(Z+L-l)*XNUl CALLPPDD(SSrXLAM9 XMM,LLMAXrPPrDP) SUMll=SUMll+(DMIN(L)*PPI**2 603CONTINUE SUMB(I)=SUVll 604FORMAT(15XrI5,3E2O08) 7 11 CONT I NlJE TOSUMl=SUVA ( 1) TOSUMZ=SUMB(l) DO 622I=2rINTERV COEF=3o+(-lo)**I TOSUM;=TOSUMl+COEF+SUMA(I) TOSUM2=TOSUM2+COEF*SUMB(I) 622CONTINUE TOSUMl=(TO5UMl+SUMA.(ISTE!'))*DFLU/3. TOSUMZ=(TOSUVZ+SUMR( ISTEP)I*DELV/3o ODDN=TOSUMl*PI/2o+DPHI*TOSU~Z/4o 641FORMAT(9X*3E2008) DO 633IzlrJEP SBMIN(I)=BMIN(I)/SQRT(ODDN) BMIN(I)=SBMIN(I) 6 33 CONT I NUE C CALCULATION OF NORMALIZED COFACTORS "COEFFICIENT OF EIGENFUNCTION WRITE(6961 6 FORMAT(//*COEFFICIENTS OF ODD FUNCTION*) 304 FORMAT(~E~OO@/(~X~~E~O~~I) WRTTE(69304) ( Bb41N( I1 rI=lrJEP) DO 634LZlrLEP SDMIN(L)=DMIN(L)/S~RT(ODDN) DMIN(L)=SDVIN(L) 634 CONT I NlJE WRITE(6r304) ( DMIN(L)rL=lrLEP) GO TO 400 5559 STOP END 52 ."" ..._.. SUBROUTINE XLLLLILH,LI,ALPHA,XNUl~XII) XH=LH XI=LI EPSIL=l.E-09 PI=3.1415926 PPHI=PI/XMU1 VC=(2.*XH-l.)+XNU1 TEST=ABS(VC'XI )-(1.E-08 1 IF(TEST)20,20,10 10 VAL-XI/IP.*XNUl) VAA=VAL+Om5 J2VO=VAA+EPSIL IF(ABS(J2VO-VAA)-2.*EPSIL)14~11~11 11 JZ=LH+l XII~2.*~-1.)~*JZ+COSIVAL~PI~*~I/~PI*~VC**2~X1~*2~~ GO TO 70 14 xrr=o GO TO 30 20 IF(LI)21,21,23 21 EEL=2. GO TO 24 23 EEL=lo 24 XII=PPHI/(2.*PI)*EEL 30 RETURN END 53 SUBROUTINE PPDD(SSIXLAM~XMUILLMAX,PP,DP) XNU=SS DDZl TEMl=Oo TEMP-00 ZX=(l.-XMU)/2. IF(ABS(fX)-0~000001) 35935931 31CONTINUE DO 40 LxlrLLMAX ZL=L TEMl=TEMl+DD*ZX**(L-l) TEM~=TEM~+(ZL-~O)+DD*ZX**(L-Z) DD=DD*(ZL-XLAM-~O)*(ZL+XLAM~/(ZL**~+ZL*XNU) 40 CONTINUE ARTFL=((1.-XMU)/(lO+XMU))**(XNU/2.) PP=ARTFL*TEMl DP~-XNU*PP/(l.-XMU+*Z)-Oo5*ARTFL*TEM2 GO TO 41 35 PP=O. DP=Oo 41 CONTINUE RETURN END 54 SUBROUTINE LEQ(AIBINEOS~NSOLNS,IA~IB,DETI CLEQ EQUATIONSLINEARSOLUTIONS FORTRAN I1 VERSIONLEQF 0020 C SOLVE A SYSTEM OF LINEAREQUATIONS OFTHE FORM AX=BBY A MODIFIEDLEQF0030 C GAUSS ELIMINATION SCHEME LEQF0040 C LEQF0050 C NEQS = NUMBER OF EQUATIONSUNKNOWNS AND LEQF0060 C NSOLNb = NUMBER OF VECTOR SOLUTIONSDESIRED LEQFOOTO C IA = NUMBEROF ROWS OF A AS DEFINED BY DIMENSION STATEMENT ENTRY LEQFOO8O C IB = NUMBER OF ROWS OF B AS DEFINED BY DIVENSION STATEMENTENTRY LEQF0090 C ADET = DETFRMINANT OF A, AFTER EXIT FROMLEQFOlOO LEQ C LEUFOllO D IM EN S ION A(IA,IA)*B(IBIIB) LEQFOl2O A(IA,IA)*B(IBIIB) DIMENSION NSIZ = NEQS LEQF0130 NBSIZ = NSOLNS LEQF0140 C NORMALIZE EACH ROW BY ITS LARGESTELEMENT. FORM PARTIAL DETERNT LEQF0150 DET=I.O LEQFOl6O I= lrN SIZ LEQF0170 DO 1 I=lrNSIZ BIG=A( 1911 LEQF0180 IF(NSIZ-1)50~50,51 LEQF0190 J= Z,N SIZ LEQF0200 51 DO 2 J=Z,NSIZ IF(ABSF(BIG)-ABSF(A(IIJ)))) 3,292 LEQF0210 7 BIG=A(I,J) LEQF0220 2 CONT I NUE LEQF0230 BG=l.O/BIG LEQF0240 DO 4 J=l,NSIZ LEQF0250 4 A(IIJ)=A(IIJ)*BG LEQF0260 DO 41 J=lrNBSIZ LEQF0270 41 B(I9J)=B(I9J)*BG LEQF0280 DET=DET+BIG LEQF0290 1 CONT I NU€ LEQF0300 c STARTSYSTEM REDUCTION LEQF0310 NUMSYS=NSIZ-l LEQF0320 DO 14 I=lrNUMSYS LEQF0330 C SCAN FIRST COLUMNOF CURRENT SYSTEM FOR LARGESTELEMENT LEQF0340 C CALL THE ROW CONTAININGTHIS ELEMENT, ROW NBGRW LEQF0350 NN= I+1 LFQF0360 BIG=A(TII) LEQF0370 NBGRW= I LEQF03AO DO 5 J=NNIKSIZ LEQF0390 IF(ABSF(BIG)-ABSF(b(J*I))) 69595 LEQF0400 6 BIG=A(JII) LEQF0410 NBGRW=J LEQF0420 5 CONT I NUE LEQF0430 BG=l.O/BIC LEQF0440 C SWAP ROW I WITH ROWNBGRW UNLESSI=NBGRW LEQF0450 IF(NBCRW-I)7,1097 LEQF0460 c SWAP A-MATRIX ROWS LEQF0470 7 DO 8 J=I*NSIZ LEQF0480 TEMP=A(NBGPW*J) LEQF0490 A(NBGRW*J)=A(I*J) LEQF0500 8 A(I*J)=TEMP LEQF0510 DET = -DET LEQF0520 C SWAP B-MATRIX ROWS LEQF0530 DO 9 J=lrNBSIZ LEQF0540 TEMP=B(NBGRW,J) LEQF0550 B(NBGRW*J)=B(I,J) LEQF0560 9 B( IrJ)=TEMP LEQF0570 C ELIMINATE UNKNOWNSFROM FIRST COLUMNOF CURRENT SYSTEM LEQF0580 55 10 DO 13 KzNNvNSIt LEQF0590 C COMPUTE PIVOTALMULTIPLIER LEQF0600 PMULT=-A(K,I)*BG LEQF0610 C APPLYPMULT TO ALL COLUMNS OF THECURRENT A-MATRIX ROW LEQF0620 DO 11 JxNNqNSIZ LEQF0630. 11 A(K,J)=PMULT*A(IIJ)+A(K*J) LEQF0640 C APPLYPMULT TO ALL COLUMNS OF MATRIX B LEQF0650 DO 12L=l,NBSIZ LEQF0660 12 B(K,L)=PMULT*B(I*L)+B(K,L) LEQF0670 13 CONT INUE LEQF0680 14 CONT INUE LEQF0690 c DO BACK SUP ST I TUT I ON LEQF0700 C WITHB-MATRIX COLUMN = NCOLB LEQF0710 50 DO 15NCOLB=lrNBSIZ LEQF0720 C DO FOR ROW = NROW LEQF0730 DO 19 I=l,NSIZ LEQF0740 NROW=NSIZ+l-I LEQF0750 TEMP=O 0 LEQF0760 C NUMBEh OF PREVIOUSLY COMPUTEDUNKNOWNS = NXS LEQF0770 NXSzNSIZ-NROW LEQF0780 C ARE WE DOING THEBOTTOM ROW LEQF0790 IF(NXS)16r17916 LEQF0800 C NO LEQF0810 16 DO 18 K=l,NXS LEQF0820 KK=NS I Z+l-K LEQF0830 18 TEMP=TEMP+B(KK,NCOLB)*A(NROW~KK) LEQF0840 17 B(NROW,NCOLB)~(3~NROW~NCOLB)-TEMP)/A(NROW~N~OW~ LEQFO850 C HAVE WE FINISHEDALL ROWS FOR B-MATRIX COLUMN = NCOLB LEQF0860 19 CONT INUE LEQF0870 C YES LEQF0880 C HAVE WE JUSTFINISHED WITH B-MATRIX COLUMN NCOLB=NSIZ LEQF0890 15 CONT INUE LEQF0900 C YES LEQFO910 C NOW FINISH COMPUTINGTHE DETERMINANT LEQF0920 DO 20 I=l,NSIZ LEQF0930 20 DET=DET+A(IrI) LEQF0940 C WE ARE ALL DONE NOW LEQF0950 C WHEW. LEQF0960 RETURN LEQF0970 END LEQF0980 SUBROUTINE COFACT(ADETIIIMAX,ICORD,JCORD,DETCOE) DIMENSION ADET(50r50)r6DET150r50),CDET(5095~) M= 1 MM= 1 10 N=l MN= 1 IF(M-ICORD)20912r20 12 MM=M+l 20 IF(N-JCORD)23922923 22MN=N+I 23 BDET(M,N)=ADET(MM,MN) N=N+1 MN=MN+l IF(N-IIMAX+l)20920931 31 M=M+1 MM=MM+l IF(M-IIMAX+l)10910935 35 ITERM=IIMAX-l35 DO 37 I=l*ITERM DO 36J=l,ITERM CDET( I,J)=Om 36 CON T r NUE 37CONTINUE CALL LEQ(BGET~CDET,ITERMIO,~O~~O~DETCOE) RETURN END 57 C FIRSTPROGRAM(B) C EVENFUNCTION TO CALCULATEDETERMINANT AND COEFICIENTS DIMENSIONP.EL(50,50),6(50r50) DIMENSION ADET (50~50),EEEE(50,50) DIMENSION SUMA(101)~SUMB(101)~BMTN(30),DMIN(30),S3MIN~~O~~ lSDMIN(30) LLMAX=50 READ(5r5)IIPP*NMAX,LMAX 5 FORMAT(315) I I MAX=LMAX+NMAX+2 ITTEM=IIMAX PI=301415926 ALPHA=PI/2. PPHI=~o*(PI-ALPHA/~~) XNUl=PI/PPHI INTERV=30 LEP=LMAX+l JFP=NMnAX+l uo=oo XMU=O e XMUFIX=XMU WRITE(6,502) NMAX,LMAX~IIMAX,ALPHA~PPHI~XI'JU~~X~IU 502 FORMAT( * NKkX=*~I5,5X~+LMAX=*~I5~5X~*II~AX=*~I~/5X~ )i lALPHA=*,E12.5r5X,*PPHI=~rE12.595X,*XNUl=.X,E12e5r5XtK-XMU=YrE12.5) 400 READ(5r31) XLAM 31 FORMAT(Fl4.0) :F(ENDFILE5) 5559,5556 5 5 56 CONT I NllE C CALCULATION OF ELEMENTS OF DETERMINANT DO 703 J=l,ITTFM DQ 300 K=l, ITTFM IF(J-NPAX-1) 10*10,200 10 IF(K-NMAX-1) 11*11,70 11 IF(J-1) 1.2,12,13 12 FFL=2. GO TO 14 13 EEL=l. 14 IF(J-K) i4915916 15 DELTA=l. GO TO 17 16 DFL-TA=IJ. 17 SS=K-1 CALL PPDD(SS,XLA~,XM!J,LLMAX,PP,DP) AEL(J9K)=DP*EFL*DELTA GO TO 700 30SS=Z*(K-NMAX-Z) *XNU1 CALL PPDD(SS,XLAM,XMU,LLMAX,PPPDPI LH=Y-NMAX-2 LI=J-A CALL XLLLL(LHILIIALPHA~XNU~,XII) AEL(J,K)=DP*XII GO TO 300 ZOO IF(Y-t!MAX-i) 212,217,230 2 12 LE= J-NhdAX-2 LI=Y-1 CALL XLLLL(LH,LI,ALPHA,XNUlrXII) SS=K-l C.4LL PPDD(.~.~,XLA~F.XMIJ,LL~.:~X,P~,DP) AEL(J,K)=PP*XII GO TO 300 230 IF(J-K) 233,232,233 232 DELTA=1 GO TO 234 2 33 DEL TA=O 234IF(J-NMAX-2) 236,235,236 235EEL=2* GO TO 237 236 EEL=lo 737SS=2 +(K-NMAX-Z)+XNUl CALL PPDD(SS,XLAM,XMU,LLMAX,PP,DP) AEL(J~K)=-(~o-ALPHA/(~~*PI))*FEL*PP*DELTA 300CONTINUE 303CONTINUE DO 307I=l,ITTEM DO 306J=l,ITTEM B( I,J)=Oo EEEE(I,J)=AEL(I*J) 306CONTINUE 3 07 CONT I WE CALL LE~(AEL,B,ITTF~r0,50,5O,DET) WRITE (69 32 1 XLAM,PET 32 FORMAT(* 32 XLAM=*,E~~~~,~X~+DET=~,E~~O~) IF( IIPD) 4r09400,1n02 1002 CONTINIJE C CALCULATION OF COFACTORS DO 903 I=l,ITTEM DO 902J=l,ITTEM ADFT(I,JI=EEEE(I*J) 902 CONTINUE 903 CONTINUE I CORD= 1 DO 511 J=lrJEP JCORD= J CALL COFACT(ADET,ITT~M,ICOR~*JCORD*DETCO~) BMIN(JI=-(-lo)**JCORDanFTCOF 526 CONTINUE 511 CONTINUE DO 512 L=l?LFP JCORD=L+JfP CALL COFACT(AD€T,ITTEM,ICORD,JCORDIDETCOE) DMIN(L)=-(-lo)++JCORD*DETCO€ 5 36 CONT I NUE 512 CONTINUE DELU=(lo-UO)/IYTERV ISTEP=INTERV+l DO 611 I=l,ISTEP XMY=(I-l)*DELU SS=O. CALL PPDD(SS,XLAMIXMMILLMAX,PPPDP) SUM10=2o*(@.~INI1)9PP)*~2 DO 601 J=2,JEP SS= J- 1 CALL PPDD(S.~~XLAMIXMM,LLMAX,PP,DP) SUYlO=SUM10+IBMINIJ)*PP)~~2 601 CONTINUE SUYA(I)=SUM10 59 .611 CONT I NUE DELV=(l.+UO)/INTERV DO 711 I=l,TSTFP X”=( I-l)*DELV ss=0 CALL PPDD(SS,XLAM,XMM,LLMAX,PP,DP) SU~ll=Z.*(DMIN(l)*PP)*~Z DO 603L=Z,LEP SS=Z.+(L-1)*XMU1 CALLD”DD(SS,XLAM, XMM,LLMAX,PP*DP) SU~ll=.SUMll+(DMIN(L)*DP)**~ 603 CONT INtJE SUM5 ( I 1 =SUM1 1 711 CONTINUE TOSUMl=SUMA( 1) TOSUMZ=SUMB ( 1 1 DO 622I=Z,INTERV COFF=3.+(-1.)**1 TOSU~l=TOSU~l+COFF*SU~A(?) TOSUMZ=TOSUMZ+COEF*SUMR ( I ) 622 CONT I NIJE TOSUMl=(TCSUMl+SUM4(!STEP) )*DELU/3. TOSUMZ=(TOSUMZ+SUMP(ISTEF))*DFLV/3. ODDN=TOSU~l*~I/Z.+PPHI*T~SUM2/4. C CALCULATIONCF NORMALIZED COFACTORS --COEFFICIENT OF EIGENFUNCTION DO 633 I=l,JEP SBMTN(T)=B~IN(T)/SoRT(~~~~) BMIN(I)=SBVIM(!) 6 33 CONT INU€ WRITE(696) 6 FORMAT( /* COEFFICIENTS OF EVENFUNCTION*) 304 FO~MAT(5E20.R/(5X15EZ~.~)) WRITE(h9304)(BMIN(I)rI=19JEP) DO 63sL=l,LFP SDMIN(L)=D~IN(L)/SORT(ODDN) DMIN(L)=5.DYIN(L) 6 34 CONT I NUE WRITE(6,?04)iDMIN(L)rL=I,LEP) WQITE(69401 1 401 FORMAT(////) GO TO 400 5559 STOP EN D SUBROUTINE XLLLL(LHILIIALPHAIXNU~~XII) C C C THIS SUBROUTINE IS FOR tl*BFTA-tlf C C XH=LH XI=LI EPSIL=l.F-09 PI=3.14159?6 PDHI=PI/XWUl VC=Z.*XH+XNUl TEST=ABS(VC-XI)-(l.E-08) IF(TEST)20,20910 10 VAL=XI/(2.*XNUl) JZVO=VAL+EPSIL IF~ABS~J~VO-VAL~-~.*~PSIL~~~I~~~~~ 11 JZ=LH+l XII=~.*~-~.)~*JZ~SIN~VAL*PI~~XI/~PI*~VC**Z-XI**Z~~ GO TO 30 14 XII=O GO TO '0 20 IF(L1 121 921 923 21 EFL=2. GO TO 24 23 EEL=1. 24 XII=PPHI/(2.*PI)*EEL 30 RETURN END 61 SUBROUTINE PPDD(SS,XLAM*XMU*LLMAXIPPIDP) XNU=SS DD= 1 TEYl=O. TEM2=0. ZX=(l.-XMU)/2. IF(ABS(ZX)-OaO00001)35,35931 31 CONTINUE DO 40 L=l,LLFIAX ZL=L TEMl=TEMl+DD*ZX*+(L-l) TEM~=T~M~+(ZL-~O)*DD~ZX**(L-~) DD=DD*(ZL-XLAM -1 1 * (ZL+XLAM)/(ZL+*Z+ZL*XNU) 40 CONTINUE ARTFL=((~.-X~UI/(~O+XMU))**(XNU/~~) PP=ARTFL*TEMl DP=-XNU*PP/(l.-XMU**2)-0o5*ARTFL*TEM2 GO TO 41 35 PP=O. DP=O 41 CONTINUE RETURN END 62 SUBROUTINE LEQ(AIBINEQSINSOLNS~IA~IB~DET) CLEQLINEAR EQUATIONS SOLUTIONS FORTRAN I1 VERSION LEQF0020 C SOLVE A SYSTEMOF LINEAR EQUATIONS OF THE FORMAX=B BY A MODIFIED LEQF0030 C GAUSS ELIMINATION SCHEME LEQF0040 C LEQF0050 C NEQS = NUMBER OFEQUATIONS AND UNKNOWNS LEQFOG60 C NSOLNS = NUMBER OFVECTOR SOLUTIONS DESIRED LEQF0070 C IA = NUMBER OF ROWS OF A AS DEFINED BY DIMENSIONSTATEMENT ENTRY LEQF0080 C r6 = NUMBER OF ROWS OF B AS DEFINED eY DIMENSION STATEMENT ENTRY LEQF0090 C ADET = DETERMINANTOF A9 AFTEREXIT FROM LEi) LEQFOlOO C LEQFOllO DIMENSIONA(IA9IA)98(IB9IB) LEQF0120 NSIZ = NEQS LEQF0130 NBSIZ = NSOLNS LEQF0140 C NORMALIZEEACH ROW BY ITS LARGESTELEMENT. FORM PARTIALDETERNT LEQF0150 DET=1 a0 LEQFOl6O DO 1 I=lrNSIZ LEQF0170 BIC=A(I,I) LEQFOlSO IF(NSIZ-1)50,50951 LEQF0190 51 DO 2 J=Z,NSIZ LEQF0200 IF(ABSF(BIG)-ABSF(A(I,J)))) 39292 LEQF0210 3 BIG=A(I,J) LEQF0220 2 CONT INUE LEQFC230 BC=l*O/BIG LEQFO240 Dfl 4 J=l,NSIZ LEQF0250 4 A(IIJ)=A(I,J)*BG LEQF0260 DO 41 J=l,NBSI7 LEQF0270 41 B(IIJ)=B(IIJ)*BG LEQF0280 DET=DFT*RIG LEQF0290 1 CONT I NlJE LEQF0300 C STARTSYSTEM REDUCTION LEQF0310 NUMSYS=NSIZ-l LEQF0320 DO 14 I=l,NUMSYS LEQF0330 C SCAN FIRST COLUMN OF CURRENTSYSTEM FOR LARGEST ELEMENT LEQF0340 C CALL THE ROW CONTAININGTHIS ELEMENT, ROW NBGRW LEQF0350 NN= I +1 LEQF0360 BIT,=A(III) LEQF0370 NBC,QW=I LEQF0380 DO 5 JzNNINSIZ LEQF0390 IF(ABSF(BIG)-ASSF(A(JIII))) 69595 LEQF0400 6 BIC=A(J,I) LEQFG4lO NBGRW= J LEQF0420 5 CONTINUE LEQFG430 BG=l .O/RIG LEQF0440 C SWAP ROW I WITH ROW NBGRW UNLESSI=NBGRW LEQF0450 IF(NBGFW-I)791017 LEQF0460 C SWAP A-MATRIX ROWS LEQF0470 7 DO 8 J=IrNSIZ LEQF0480 TEYP=A(NBGPW,J) LEQF0490 A(NRGRW,J)=A(I,J) LEQF0500 I3 A( I ,J)=TEMP LEQFO5 10 DET = -DET LEQF0520 C SWAP 6-MATR IX ROWS LEQF0530 DO 9 JZlrNBSIZ LEQF0540 TEMP=B(NBGRWIJ) LEQF0550 B(NBGRW,J)=B(IrJ) LFQF0560 9 R(1,J)=TEMP LEQF0570 C ELIMINATE UNKNOWNS FROM FIRST COLUMN OFCURRENT SYSTEM LEQF 0580 63 10 DO K=NN,NSIf13 LEQF0590 C COMPUTE PIVOTALMULTIPLIER LEQFOhOO C APPLYPMULT TO ALL COLUMNS OF THECURRENT ,4-MATRIX ROW LEQF0620 PMULT=-A(K,I)*BG LEQFC610 DO 11 J=NN,NSIZ LEQF0630 C APPLYPMULT TO ALL COLUMNS OF MATRIX B LEQF0650 11 A.(K,J)=PMUL T*A( I rJ)+AIK*J) LEQF0640 DO 12L=l,NBSIZ LEQF0660 12 B(K,L)=PMULT+B(I,L)+B(K,L) LEQFO670 13 CONT I NUE LEQF0680 14 CONT I NUE LEQF0690 C DO BACKSUBSTITUTION LEQF0700 C WITHB-MATRIX COLUMN NCOLB LEQF0710 50 DO 15NCOLB=lrNBSIZ LEQF0720 c DO FOR ROW = NROW LEQF0730 DO 19 I=lrNSIZ LEQF0740 NROW=MSIZ+l-I LEQF0750 TEMP=O.O LEQF0760 C NUMBER C OF PREVIOUSLY COMPUTEDUNKNOWNS = NXS LEQF0770 NXS=NSIZ-NROW LEQF0780 C ARE WE DOING THEBOTTOM ROW LEQF0790 IF(NXS)16117916 LEQFOBCO 5 NO LEQF0810 16 DO K=l,NXS18 LEQF0820 KK=NSIZ+l-K LEQF0830 18 TEMP=TEMP+B(KK9NCOLB)*A(NROW~KK) LEQF0840 17 B(NROW,NCOLB)=(B(NRCW~NCOLB)-TEMP)/A(NROW~NROW~ LEQF0850 c HAVE WE FINISHED ALL ROWS FOR 6-MATRIX COLUMN = NCOLB LEQF0860 19 CONT I NUE LEQF0870 c YE 5 LEQF0880 C HAVE WE JUSTFINISHED WITH 6-MATRIX COLUMN NCOLBzNSIZ LEQFO€!9O 16 CONT I NUE LEQF0900 C YES LEQF0910 C NOW FINISHCOVPUTING THE DETERMINANT LEQF0920 DO 20 I=l,NCIZ LEQF0930 C WE ARE ALL DONE NOW LEQF0950 C WHEW. s s LEQF0960 20 DET=DET+A(I,I) LEQF0940 RETURN LEQF0970 END I EQF0980 64 SUBROUTINE COFACT(ADFTIIIMAX,ICORD,JCORD,DETCOE) DIMENSION ADET(50,50),ROET(50,50),CDET(~O950) M= 1 MM= 1 10 N=l MN= 1 IF(M-ICORD)20912920 12 MM=M+1 20 IF(N-JCORD)23,22923 22MN=N+1 23 B~ET(M,N)=ADET(MM,MN) N=N+l MN=MN+l IF(N-IIMAX+l)20920931 31 M=M+1 YM=MM+l IF(M-IIMPX+l) 10~ln935 35 rTERM=IIMAX-1 DO 37 I=l,TTERM DO 36 JzlqITERY CDET( I,J)=O. 36CONTINUE 37 CONTINUE CALL LFO(BDET*CDET,ITERM,0950~5O~DETCOE) RETURN END C SECOND PROGRAM C DETERMINATION OFTHE COEFFICIENTS OF THE EIGENFUNCTION EXPANSION FROM C THE INITIALDATA DIMENSION F(70)~FUP(70,7O),FBT(70,7O),SFL(7O,70) DIMENSION SUMJ(70),SUML(7O)~BMIN(7Ol~DMIN(70) LLMAX=50 IDIM=70 MAXU1=60 MAXTOP=60 MAXU2=60 MAXBOT=60 1000 READ(5,lOOl)II,NP'AX,LMAX 1001 FORMAT(315) IFlENDFILE5) 9999,1002 1002CONTINUE IF(I1 mEQ* 1) 101,103 C C C ODD FUNCTION 191 IIMAX=LMAX+NMAX h'RITE(6,1003) 1003 FORMAT(lHl* ODD FUNCTION*) NCALL=NMAX LCALL=LMAX GO TO 104 C C EVENFUNCTIOF' 103 IIMAX=LMAX+NMAX+Z WRYTE(6r1004: 10C4 FORMAT ( 1H19*EVEN FUNCTION* NCALL=NMAX+l LCALL=LMAX+l 104 CONT I NUE PI=3.1415926 ALPHAzPI 12. XMU= 0 PPHI=~O~~(PI-ALPHA/~.1 XNUl=PI/PPHI HALPHI=PPHI/2* EPS =0.000001 WRIPE(6,502) NMAXILMAXPIIMAXIALPHAYPPHI~XNU~~XV~U 5 02 FORMAT ( * NMkX=*,I5,5X,aLMAX=9,I5,5X,sIIMAX=*,I5/5X, * lALPHA=*,E12.5~5X~*PPHI=*,E12.5,5XI+XNU1~*~E~~~5~5X~*XMU~~~ElZ~5~ h00 READ(5,lOO)XLAM IF(ENDF1LE5) 1000,1111 1111 CONTINUE 100 FORMAT(5F15.O) READ(5,100)(BM!N(Il,I=1,NCALL) READ(5,lGO)(DMIN(L)*L=l,LCALL) UG=O MAXPUS=MAXTOP+l INDEX=l WRITE(6932) XLAM 3 2 FORMAT( 32 * XLAM=*,E15.8) WHITE(6,503) 503FORMAT(//* COEFICIENTS CFEIGENFUNCTION*) WRITE(6r304) (BMIN(I),I=l,NC4LL) WRITE(6,304) (DMIN(L)rL=l,LCALL) 66 304 FORMAT(~E~OO~/(~X,~E~OO~)) DELUl=(lo-UO-2o*EP.S)/MAXU1 MAXUST=MAXUl+l UK=UO+EPS DO 5 I=l,MAXUST TOTARGtOo CALL FFF(UK~INDEX~MAXTOP~FTII~PPHI,IDIM) DO 4 J=l,MAXPUS FUP(I,J)=F(J) 4 CONTINUE IROW= I DO 13 J=l,NCALL IF(1I oEQo 1) 111,113 C C ODDFUNCTION 111 CONTINUE XVAL=J CALL SININT(FUP,XVAL,IROW,MAXTOPIPI,O,IDIM 9TRIINT) GO TO 114 C C EVEN FUNCTION 113 CONT I NUE XVALZJ-1 CALL COSINT(FUP~XVAL~IROWIMAXTOP,PI,OIIDIM 9TRIINT) 114 CONT I NUE SS=XVAL CALL PPDD(SS,XLAM,UK,LLM4X,PPIDP) TOTARG=TOTARG+BMIN(J)*PP*TRIINT 13 CONT I NUE SUMJ(I)=TOTARG UK=UK+DELUl 5 CONT I NUE TOTEMl=SUMJ ( 1) DO 19 1=2,MAXUl COEF=3o+(-lo)*+I TOTEMi=TOTEMl+COEF+SUMJ( I) 19 CONT I NUE TOTEMl=(TOTEMl+SUMJ(MAXUST))*DELU1/3o MAXPUS=MAXPOT+l INDEX=2 DELU2=(1o+UO-20*EPS)/MAXU2 MAXUSM=MAXU2+1 UK=-lo+EPS DO 9 I=l,MAXUSM TOTARG=Oo CALL FFF(UK,INDEX,MAXBOT,F~II~PPHI~IDIM) DO 8 J=l,MAXPUS FBT(I,J)=F(J) 8 CONTINUE IROW= I DO 23 L=l,LCALL IF(II oEQo 1) 1219123 C C ODD FUNCTION 121 CONTINUE XVAL=(2*L-l)*XNUl CALL SININT(FBT,XVAL,IROW,MAXBOT,HALPHI,O,IDIM 9TRIINT) GO TO124 C C EVENFUNCTION 123 CONT I NUE XVAL=2*(L-ll*XNUl CALL COSINT(F5T~XVAL~IROW,MAXBOT~HALPH~~~~IDIM9TRIINTl 124 CONT I NUE SS=XVAL UF=-(JK CALL CPDD(SS,XLAM,UF,LLMAX,PP,DP) TOTARG=TOTARG+DMIN(L)*PP*TRIINT 23 CONT I NUE SUML( I)=TOTARG UK=UK+DELU2 9 CONTINUE TOTEM2=SUML(l) DO 29 I=2,MAXU2 COEF=3-+(-1- )**I TOTEM2=TOTEM2+COEF-~SUML(Il 29 CONTINUE TOTEMZ=(TOTEM2+SUML(MAXUSM))*DELUZ/3. EEEM=TOTEMl+TOTEM2 WRITE(6,ll)EEEM 11 FORMAT( /20X,~K(LA~DA)=*,E15.8) WRITE(69401) 40 1 FORMAT(////) GO TO 400 9999 STOP EN 0 68 SUBROUTINE SININT~FP~XVAL~IROW,M~XST~UPLIM,SOTLIM,I~IM~TRIINT~ C FILONIS METHOD FOR THENUMERICAL EVALUATICN OF TRIGONAMETRICAL C INTEGRALS--(INTEGRAND=FP(P)*SIN(X*P) 1 DIMENSIONFP(IDIM9ICIM) HH=(~PLIl~-BOTLIM)/~AXST S~S=OO~*FP(IROWI~)*SIN(XVAL*BOTLIM) DO 14 J=3,MAXST,2 P=BOTLIM+(J-l)*HH S2S=SZS+FP(IROW,J)*SIN(XVAL*P) 14 CONT I NUE J=MAXST+l S2S=S2S+0.5*FP(IROW~MAXST+1)*SIN(XVAL*UPLIM) 52 SM=O DO 16 J=2 ,WAXST92 P=BOTLIM+(J-l)*HH S~SM=S~SM+FP(IROWIJ)*SIN(XVAL*PI 16 CONT I NUE THE=XVAL*HH IFITHE-0.2)25,21921 21 ALPHA=(THE9*2+THE*~IN(THE)9COS(THE)-2."SIN(THE)**Zl/TtiE~*3 BETA=;.*(THE*(1o+COS(TH~)**2)-~o*SIN(THE)*COS(Tti~))/THE*~3 GARM=4.*(SIN(THE)-THE*C@S(THE) )/THE**3 GO TO 31 25 ALPHA=2.*l~Ea~3/45.-2.*THE~'~5/315.+2o*THE.~*7/472~. BETA=2./3.+2.+Ttit**2/1~.-4o*THE~*4/lO5.+2o~THE~*6/5670 GARM=~./~.-~.*THE**~/~~~+THE**~/~~OO-THE**~/~~~~O. 31 FA=FP(IROW,lI FR=FP(IROW,MAXSTtl) TRIINT=HH*(-ALPHA*(FRaCOS(XVAL*UPLIM)-FA*COS(XVAL~BOTLIM))+B~TA~ lS2S+GARM*SZSMI RETURN END SUBROUTINE COSINT(FP,XVAL,IROW,MAXSTIUPLIMIBOTLIM,IDIM,TRII~T) C FILONtS METHOD FOR THE NUMERICALEVALUATION OF TRIGoNAMETRICAL C INTEGRALS--(INTEGRAND=FP(P)*COS(X*P)9 DIMENSIONFPIIDIM,IDIM) HH=(UPLIM-BOTLIM)/MAXST S2S=Oo5*FP(IROW9l)*COS(XVAL*BOTLIM) DO 14 J=3rMAXST,2 P=BOTLIM+(J-l)*HH S~S=S~S+FP(IROWIJ)*COS(XVAL*P) 14 CONT I NUE J=MAXST+l S2S=S2S+0~5*FP(IROWeMAXST+l~*COS~XVAL*UPLIM~ S2SM=0. DO 16 J-2rMAXST92 P=BOTLIM+(J-l)*HH S2SM=S2SM+FP(IROW*J)*COS(XVAL*P) 16 CONTINUE THE=XVAL*HH IF(THE-0.2)25,21921 21 ALPHA=(THE*+~+THE+SIN(THE)*COS(THE)-ZO*SIN(THE)**~)/THE**~ ~ETA=~~~(THE*(~~+COS(THE)**~)-~O*SIN(THE)*COS(THE))/THE**~ CARM=4.*(SIN(THE)-THE*COS(THE) )/THE**3 GO TO 31 25 ALPHA=~O*T~E**~/~~~-ZO~THE%*~/~~~~+Z**THE**~/~~~~~ BETA~2~/3o+20*THE*~2/15o-4o*THEw+4/105.+2o*~HE**6/567o GARM=4./30-2.*THE**2/15o+THE**4/2lO*-THE**6/1134Oo 31 FA=FP(IROWql) FB=FP(IROW,MAXST+l) TRIINT=HH*( ALPHA*(FB*SIN(XVAL*UPLIM)-FA*SIN(XVAL*BOTLIM))+BETA* l.%?S+GARM+SZSM) RETURN END I' SUBROUTINE FFF(UU~INDEXIMAX*F*II,PPHI*IDIM) DIMENSION F(IDIM1 PI=3.1415926 IFtINDEX oEQ. 1) 4r6 4 DELPSI=PI/FLOATIMAX) GO TO 7 6 DELPSI=PPHI/FLCAT(2*MAXl 7 CONTINUE PSI =O. MAXPUS=MAX+l DO 19 N=lrMAXPUS BPSI'PSI-PPHI/2o CALL FSF(BPSI9UU9FS rPPHI) FA=FS BPSI=2.*PI-PSI-PPH1/2. CALL FSF(EPSI9UU9FS *PPHI) FB=FS IF(I1 WEQO1) 14916 14 F(N)=(FA-FB)/2. GO TO 17 16 F(N)=(FA+FB)/2. 17 CONTINUE PSI=PSI+DELPSI 19 CONTINUE RETURN END SUBROUTINE FsF(BPsI9uu9Fs9PPHI) PI=3.1415926 PI23=2.*PI/3. P143=4.*PI/3. CONS2 3=2 /3 HALPI=DI/Zo HALPHI=PPHI/Z. TESTl=Zo*PI-HALPHI TEST2=2o*PI-PPHI IF(UU) 119 191 1 FO=lo GO TO 31 11 IF(BPS1 .LE. TEST1 .AND. BPS1 oCEo HALPI .OR. BPSI0LTo -HALPI 1 .AND. BPS1 oGEo(-HALPHI-0o0000001~) 1914 12 FO=Oo GO TO 31 14 IF(BPS1 OLEO 0 .AND. BPSI 0GTo -HALPI)12915 15IF(BPL1 oLTo HALPI .AND. BPSI 0GTo 01 16918 16 WRITE(6917)UU9BPSI 17 FORMAT(* PRORAMSTOP IN CASE 1 U=*9E12.5,5X,*BPSI=*,E12.5) STOP 18IF(ABS(BPSI+HALPI) -3.00001) 1991992C 19 FO=Or5 GO TO 31 20 WRITE(6r21) UU9BPST 21FORMAT(* PRORAMSTOP IN CASE 2 U=*9€12.595X9*BPSI=*~~l2.5~ STOP 31 ABSPSI=ABS(BPSI) IF(ABSPS1 .GTo HALPI)32933 32F1=00 GO TO 51 33 RHO=SQRT(UU**2+(1.-UU**2)*SIN(BPS1)**2) TAUO=AS IN (ABS ( UU 1 /RHO 1 IF(UUOGEO 0) 34937 34IF(BPS1 OGEO 0) 35.36 35 TAU=TAUO GO TO 42 36TAUzPI-TAU0 GO TO 42 37IF(BPS1 OLEO 0) 38939 38TAU=PI+TAUO GO TO 42 39WRITE(6940) UU9BPSI 40 FORMAT(* PRORAMSTOP IN CASE 3 U=*~E~~.~~~X~*BPSI=*TE~ZO~) STOP 42CALL CONZD(RHO9TAUrPI239PI43~CONS23~FIF) IF(ABSPS1-HALPI)45943943 43Fl=FIF/Z. GO TO 51 45Fl=FIF 51IF(UU 1 56954953 53 F2=00 GO TO 71 54IF(BPS1 oLTo HALPI .AND. BPSI 0GTo 0) 55956 55F2=00 GO TO 71 56RHO=SLRT(l.-UU**2) IF(BPS1OLEO TFSTl .AND. BPSI .CEO HALPI)57958 57 TAU=BPSI-HALPI GO TO 62 58 IF(BPS1 .LE. 0 *AND*BPS1 *GE*-HALPHI) 59960 59 TAU=BPSI+3e*PI/2* GO TO 62 STOP 62 CALL CON2D(RHO~TAU,PI23~PT43,CONS23~FIF) IF tUU 669'64955IFtUU 64 F2=0*5*FIF GO TO 71 66F2=FIF 71 FS=FO+Fl+F2 RETURN END 73 ...... _. 74 SURROUTINE PPDO(SS,XLAM*XMU9LLMAX,PP,DP) XNU=SS OD-1 fEMl=O. TEM2=0 ZX=(lo-XMU)/2o ?F(ABS(ZX)-Oo000001) 35935931 31 CONTINUE DO 40 L=l,LLMAX ZL=L TEV~=TFM~+DD*ZX**(L-~) TEM2=TEM2+(ZL-loI*DO*ZX**(L-2) DD=DD*(ZL-XLAM -l*)*(ZL+XLAM 1 /( ZL**2+ZL*XNU) 40 CONT INIJE ARTFL=((l.-XMU)/(lotX~U))**(XNU/2o) PP=ARTFL*TEMl DP=-XNU*PP/(l.-XMU**2)-0.5*ARTFL*TEM2 GO TO 41 35 PP=O. DP=O 41 CONTINUE RETURN END 75 c THIRD PROGRAM C DETERMINATION OF THEPRESURE DISTRIBUTION ON SURFACES DIYENSION EDIM~20),P~(~O~~O)~P2(2~~2~~~P3(2~~2~~~Gl~2~,2~~~ 1G2(20~2o)~G3(20~20~,B~IN(20),DMIN(20) LLMAX=F?O DI=3.1415026 ALPHA=PI/2. PPHI=~O*(PI-PLPHP/~O) XNUl=PI/PPHI YALPHI=PPHI /2a DO 106 M=lv10 DO 105b!=ls5 P1(b49N)=Oo PZ(M,N)=O. P3(M,N)=O. 105CONTINUE 106CONTINUE 1000 RE4D(591001)II,NMAX,LMAX 1001 FORMAT(315) IF(ENDF1LE 5) 999991002 1002 CONT I NUE IF(I1 eEQo 1) 101,103 C C ODD FUNCTION 101 IIMAX=LMAX+NMAX WRITE(692) 2 FOSMAI(lHl,//* ODD FUNCTION*) NCALL=WMAX LCALL=LMAX GO TO 104 c C EVENFUYCT ION 103 II MAX=LMPX+NMAX+Z WRITE(693) 3 FORMAT(lH19//* EVENFUNCTION *) NCALL=NMAX+l LCALL=LMAX+l 104 CONTINUE XMU=0 o WRITE(69502) NMAXILMAX,IIMAX,ALPHA~PPHI~X~JU~~XMU 502 FORMAT( * NMAX=*,I5,5X9*LMAX=*,I5,5X,*II~AX=*,I5/5X, * lALPHA=*~E12o5~5X~~PPHI=*,El2o5~5X~*XNUl=*~~lZo5~5X~~XMU=*,E~2~5) 110 READ(59100)XLAM*E€FV 100 FORMAT(5F15.0) IF(EYDFJLE 5) 1000,1111 1111 CONTINUE READ(5,lOO)(EMIN(I),I=l,NCALL) READ(59100)(DMIN(L),L=l,LCALL) XYU= 0 0 CALL T102~XLAM,le~EDIM~BBBBB8) AAAAAA=RBBFBB WRITE(6,32) XLAM,AAAAA&,EEEM 32 FORMAT( 32 * XLAM=*~F~4~?~5X~~AAO=*~Fl4o9~5X~*K~~AI~~A)~~~Fl4~9~ 'rriRITE(6,503) 503FORMAT(// * COEFFICIENTSOF EIGEN-FUNCTIONS*) WRITE(63304) (BMIN(I),I=l,NCALL) WRITE(69304) (DMIN(L) ,L=l,LCALL) 304 FORMAT(5€20.8/(5X,5EZOo~)) WRITE(694011 401 FORMAT(////) DELPHI=(PI-HALPHI)/Zo IF(I1 .EQo 1) 2009203 C C ODD FUNCTION 200 DO 202I=ls5 PHI=PI+(I-3)*DELPHI PDUS=O. DO 201J=l,NCALL SS=J CALL PPDD(SS,XLAM*XMU9LLMAX,PP,Dp) SINFU=SIN(J*PHI) PPUS=PPUS+BMIN(J)*PP*SINFU 201CONTINUE Gl(I)=pPUS 202CONTINUE GO TO 209 C C EVEN FUYCTION 2 03 CONT INUE 2 04 DO 207 1~135 PHI=PI+(I-3)*DELPHI PPUS=O. DO 206J=l,NCALL SS= J-1 CALL PPDD(SS,XLA~*X~U,LLMAX,PP,DP) COSFU=COS(SS*PHI) PPUS=PPUS+BMIN(J)*PP*COSFU 2 06 CONT I NUE Gl(I)=PPUS 2 07 CGNT I NUE 209 CALL T~~~(XLAM,AAAAAA.*EDIM,BBBBBBB) DO 228M=l,lO DO 227 N=1,5 Pl(M,N)=PlIM,N)+EDIM(M)*Gl(N)*EEEM 227 CONT INUE 2 28 CONT IWE MYP= 1 PHI=HALPHI DELTH=PI/8. -0oO005 3 00 IF(I1 oEQe 1) 3019400 C C ODD FUNCTION 301 DO 3091~195 THF=PI/2.+(1-l)*DELTH XMU=COS (THE 1 PYIN=O o DO 302LzlPLCALL SS=(2*L-lI*XNUl CALL PPDD(SS,XLAM,-XMU*LLMAX,PP,DPI SINFU=SIN(SS*PHI) PMIN=PMIN+DMIN(L)*PP*SINFU 302 CONTINUE 303 G2(1)=PMIN 305 G3(1)=PMIN*(-la) 3 09 CONTINUE GO TO 500 77 C C EVEN FUNCTION 400 DO 409 1~1~5 THE=P1/2e+(I-l)*DELTH XMU=COS(THEI PM I N=O e DO 402 L=l,LCALL SS=2*(L-l)*XNUl CALL PPDD[SS,XLAM,-XMU,LLMAX,PP,DP) COSFU=COS(SS*PHI) PMIN=PMIN+DMIN(L)*PP*COSFU 402 CONT IrdJE 403 G2(I)=PMIN 404 G3(I)=PMIN 409 CONTINUE 500 CONTINUE 501 DO 505 M=1,10 DO 504 N=1,5 P~(M,NI=P~(MIN)+EDIM(M~*G~(N)*EEEM P3(M,N)=P3(M,N)+EDIM(M)*G3(N)*EEEM 504 CONTINUE 505 CONTINUE GO TO 110 9999 CONTINUE WRITE(699998) 9998 FORMAT(//8X,*PHI*,14X,*Pl~,28X,+XMU*,l4X,*P2*,14X,*p3*) 601 DO 605 N=1,5 PHT=PI+(N-31*DELPHI THE=PI/2e+(N-l)*DELTH XMU=COS ( THE.1 DO 604 M=1,10 WRTTE(6,607) PHI*PI(MIN)~XMUI P2 (MIN 11P3 LMIN 1 607 FORMAT(2E16e5915X,3E16.51 604 CONTINUE 605 CONTINUE STOP END SUBROUTINE T~O~(XLA~~AAAAAAIEDIM,BBBBBB) DIMENSIONEDIM(20) COMMON XXXYL XXXXL=XLAM XMAX=Om?989 XO=0~001 XOXL=F3(XLAMrXO) XOXLP=F3(XLAM+lo*Xn) XOXLM=F~(XLAV-~*JXR) XOXLM2=F3(XLAM+2m,XO) 10 FO=(XOXL+XLAM*(XLAM+~~)*XOXLM~/(~~*XLAM+~O))*AAAAAA GO=(XLAM*XOXLM+(XLAM+~~)*~~LAM+~~)*XLAM*XOXLP/~~~*XLA~+~~~~*AAAAAA DELX=O.OOI. x=xo F=FO G=GO I =1 XTEST=OmlOl 20 XKl=Fl(X,F,GI*DELX XL~=F~(X~FIG)*DELX XK2=Fl(X+DELX/Zm,F+XK1/2m9 G+XL1/2o)*DELX XLZ=F2(X+DELX/2mrF+XK1/2o, G+XL1/2o)*DELX XK3=Fl(X+DELX/ZorF+XK2/209 G+XLZ/Zo)*DELX XL3=FL(X+DELX/2o,F+XK7/2e, G+XL2/2o)*DELX XK&=F1(X+DELX, F+XK~~G+XL?)*DELX XL&=F2(X+CELX9F+XK39 G+XL?)*DELX DELF=1 /60* (XK1+2 .+XK2+2o+XK3+XK4 1 DELG=lm/6m*(XL1+2o*XL2+2o*XL4) X=X+DELX F=F+DFLF G=C+DELG IF(X mGEm XTEST)2nlr203 201 FDIM(I)=F XTFST=XTFST+Oml I=T+1 2 03 CONT I NIJE IF(X-0.01)20920931 31 IF(X-0.99)3293293' 32 DELX=Oo 0 1 GO TO 20 33 IF(X-0.9989) 3494094.0 34 DELX=OmOOl GO TO 20 40ZETA=lm-X Cl=-XLAM*(XLAM+1-)/4. C2=Cl**2 C~=((XLAM*(XLAM+I.~)-~~)*C~**Z+~O*C~)/~* GEND=F/(Im+CI*ZETP+C7+ZEPA+*2+C3*Z~TA**~.*4LOG(ZETA)) 41 BBSRBR=l./G€ND EDIM( I )=GEND RETURN END 79 FUNCTION Fl(A,B*C) COMMON XLAM Fl=C RETURN END ._. I FUNCTION F~(DIE) F3=EXP(D*ALOG(EI) RETURN END 82 II - SURROUTINE PPDD(SS,XLAMWXMU,LLMAX,PP,DP) XNU=SS DO=l TEPl=O. TEP2=0. ZX=(l.-XMU)/2. IF(ABS(ZX)-Om0000011 35t35971 31CONTINUE DO 40 L=l,LLMAX ZL=L TEPl=TEMl+OD*ZX*+(L-l) TEM2=TEM2+(ZL-lm)*DD*2X**(L-2) DD=DD*(ZL-XLAP -1. )+(ZL+XLAM)/(ZL**Z+ZL*XNU) 40 CONTINUE ARTFL=((l.-XMU)/(1m+XMU))**(XNU/2m) PP=ARTFL*TEMl DP=-XNU*PP/(1.-XMU**2)-0o5*ARTFL*TEM2 GO TO 41 35 PP=O. DP=O. 41 CONTINUE RETUQN END TABLE I Comparison of numerical results with the exact eigenvalues €or a two dimensional corner. ~~~~ Exact Solution Numerical Solution M# J M= J “ even odd even odd Mc=8 Jc=4- 1,=9 Js=5 Mc=6 Jc=6- -Ms=7 Js=7 0.0 0.0 0.0 2/3 -667394 .669715 1.0 1.0 1.0 1+1/3 1,333342 1.333518 1+2/3 1.668060 1.672477 2. o* 2. o* 2.0 ;‘r 24-1 13 2.333356 2.333793 2.666667 2.666669 2+2/3* 2.669734 2.679591 3. o* 3,o““ 3.0;: .- -;C Double root TABLE ””I1 Some exact eigensolutions for the three dimensional corner with $ = n/2, a = n/4. 84 TABLE I11 Tabulations of theeigenvalues and the coefficients in the series representations of the eigenfunctions (2a = n/2 /3, = n/2). even odd Def. Am Coef. Cj oef. Bm Coef. Dj .426487 .426487 1.135600 -1.44460( 0.0 0.0 .073230 -.097647 x = 0.0 0.0 0.0 x=. 839948 .004226 .031486 0.0 0.0 -.021135 - .016 324 0.0 0.0 .017872 .010371 0.0 -.007779 -.007452 0.0 0.0 -.001831 .005917 0.0 0.0 .007199 -.005446 0.0 .007525 .004085 -496 980 -.662630 2.6 76 300 - 2.6 9270( -. 85376 -.394590 x= -.141180 .188240 x=. 840341 .19722 .070879 1.807053 .224060 .027981 - .086 956 -.032794 -. 131070 -.019274 .024596 .020205 .059485 .013975 -009953 -.014725 -.010481 -.011068 -.022866 .012976 -.016294 .009643 .020424 .024045 -.010008 -.009732 - . 01 8772 .007506 ...... ~ even odd Coef. Am Coef. Cj Coef. Em Coef. Dj .379050 - .505400 0.0 0.0 1.656200 1.188100 3.303556 3.303556 X =l.206 357 .143140 - .038965 x =2.0 0.0 0.0 -.010898 .011436 0.0 0.0 -.008577 -.005016 0.0 0.0 .008833 ,002579 0.0 0.0 -.004721 -.001293 0.0 0.0 .000934 0.0 0.0 .000970 0.0 0.0 ~~ -.172240 .2296 50 -4.072100 -2.750300 -1.263800 2.980100 A= 3.765900 5.021200 1=1.805246 2.193400 .529750 2.814159 1.818200 .6 98380 - .359600 -.135050 -.358120 -.103210 ,101280 .06 5806 .135310 .(I43530 .009582 -,041720 -.032647 -.025606 -.050099 ,032932 -.016271 .018233 .048334 .031678 -.015960 -.0246 99 -.026615 .011970 86 even odd :oef. Am Coef. Cj Coef. Bm Coef. Dj 953654 .953654 2.584400 3.142500 0.0 0.0 1. = -4.047200 5. 396300 A.=2.0 0.0 0.0 2.86 8927 3.291700 1.576600 0.0 0.0 - .6 57200 -. 249830 0.0 0.0 .283730 ~ 1086 90 0.0 0.0 - .081520 - .06426 5 0.0 0.0 -.027367 .045417 0.0 .06 76 22 -038976 0.0 -.060812 .029232 -.208840 .278460 2. 725500 -3. 976900 4.206 800 1. 747500 .358770 -.103810 .077859 .032182 - .06 5595 -.014013 .030583 .006305 -.005977 -.004729 even Coef. Am Coef. Cj .530570 -. 707430 -2.247100 -.855920 1=2.813928 1.003000 5.289000 4.135700 .342700 -.257020 -.116920 -.039995 .064339 .097445 -. 047471 -. 079136 .035604 -. 507640 .676780 2.214800 5.0 76 500 1=2.958670 -3.983000 5.162800 3.733900 .323730 -.242800 -.065636 .lo1760 .024029 - .0357 58 -.010174 .003411 .0076 31 TABLE IV Comparison of the eigenvaluesfor a three dimensional cornerwith -1 those for a,circular cone of the same solid angle@= cos (3/4). Three Dimensional Corner Circular Cone - -1 even odd @= cos (3/4) -n .840341 .839948 .863382 1 1.206357 1.210120 0 1.805246 1.807053 1.893798 2 2.0 2.0 1.961940 1 2.446688 2.477387 0 2.813928 2.814159 2.865298 2 2.958670 2.868927 2.936459 3 3.205219 3.205133 3.190234 1 1 -