acoustic resonant frequencies in an eccentric spherical cavity

Acoustic resonant frequencies in an eccentric spherical cavity

John D. Kanellopoulos and John G. Fikioris

Department of Electrical Engineering, National Technical University of Athens, Greece (Received 2• January 1977; revised • December 1977)

The interior boundary-value scalar (acoustic) problem in the region between two spheres of radii R•, R• and distance d between their centers is considered for both Dirichlet and Neumann boundary conditions.

Surface singular integral equations are used to formulate the problem. Their solution is obtained in terms of spherical wave functions in combination with related addition theorems. It is then specialized to the case of small values for kd = 2•r d/X to yield exact, closed-form expressions for the coefficients g•s in the resulting relations OOn, (kd) = OOn, (0) [1 + gn, (kd): +'"] for the resonant (natural) frequencies of the cavity. Numerical results, comparisons, and possible generalizations are also included.

PACS numbers: 43.20.Ks, 43.20.Bi

INTRODUCTION

Analytical solutions of interior or exterior boundary- value problems in potential theory, acoustics, .electromagnetism and related fields are severely limited bythe shape of the boundaries. The separation of variables as a method of solving three-dimensional problems, in particular, is limited to cylinders, spheres, ellipsoids, and certain conical structures. For more general

shapes, both interior and exterior problems can be formulated in terms of surface (two-dimensional) singular integral equations, an approach that, although quite old, has come into practical use in the last 15 years or so and is often referred to as the boundary-integral- equation method. The scalar equations of classical

3

potential theory can be found in Refs. 1-4. Jaswon, in particular, treats two-dimensional potential problems, and so does Symm, 4 who proceeds to solve numerically the related integral equations. The method has also been applied to elastostatics s'6 and elastodynamics, 7,8 to transient heat conduction, 9 and to water waves. 10,11

More related to this paper are applications of the method to acoustics. Thus, Banaugh and Goldsmith •' formulate and proceed to solve numerically the integral equations of two-dimensional acoustic scattering by cylinders of arbitrary cross section. Numerical tech- niques are also applied in Refs. 13-15. In particular, Schenck •4 (whose paper contains many numerical results and a good list of references) and Chertock •5 are also con- cerned with questions of the uniqueness and existence of solutions of the singular integral equations and with the connection between the solutions of the interior and ex-

terior problem of the same geometry. In particular, they analyze the apparent difficulties of the exterior solution at the critical wavenumbers of the interior problem. Such problems are also considered in Refs. 16 and 17. Schenck 14 but mainly Chertock ls'17 suggest ways of avoiding such complications.

The method has also been applied to transient and dynamic acoustic and/or elastic problems, interior and

•)Partially supported by the National Research Foundation of Greece.

exterior, by Shaw in a series of papers. 18,-22 Finally, the vector equations of electromagnetism are considered in Refs. 2, 23, and 24.

In this paper the interior problem in the region V between two eccentric spherical surfaces S• and S•. with centers O• and O•. and radii R• and R•. respectively, and eccentricity O•O 2 =d, as shown in Fig. 1, is solved analytically for the scalar (acoustic) Helmholtz equation. In particular, the characteristic equation for the natural frequencies w,s of the acoustic microwave cavity is obtained for both Dirichlet and Neumann boundary conditions. More importantly, when the condition kd=2•Td/ ), << 1 is imposed, one is able to determine in exact, closed form the coefficients g,s in the resulting expressions w,s(kd)=w,s(0)[1 + g,s(kd) •' +... ], where the w,s(0) are the resonant frequencies for concentric spheres, for which d =0. The corresponding electromagnetic cavity problem has also been solved, but for reasons discussed later, the vector case is treated in a separate paper. •'s Finally, the exterior (scattering) problem of an eccentrically coated sphere is, at present, also under consideration.

FiG. 1. Geometry oœ the cavity.

286 J. Acoust. Soc. Am. 64(1), July 1978 0001-4966/78/6401-0286500.80 ¸ 1978 Acoustical Society of America 286

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.114.34.22 On: Wed, 26 Nov 2014 01:05:51

287 J.D. Kanellopoulos and J. G. Fikioris: Resonant frequencies in a spherical cavity 287

Beyond the obvious success in solving analytically and exactly one heretofore unsolved problem, it is hoped that the approach can be generalized to other shapes. In particular, what is shown is that certain singular surface integrals can be evaluated analytically by a proper limiting process, and that this result can be generalized to cases where a convenient expansion for the Green's function G [here G =exp(ikR)/R] can be found, even though boundaries do not fit coordinate surfaces of the system in which G has been expanded. In this connection, mention of Ref. 26 should be made, where, in a rather different context, the same general idea is employed. Specific shapes for which generali- zation of the method seems possible will be discussed at the end.

Let now small letters (p•, q•.) denote points on S• and S 2 and capitals (P, Q) points not on S• and S2. A fixed point on S• or S 2 is designated by p•(O•, ½•) or p2(0•., ½2), a variable one by q•(O•, ½[) or qz(O•, ½•), with the primes indicating, in particular, variables of integration. Un- der this notation R•, 0•, • (or R2, •, ½•.) indicate spherical coordinates of p• (or q•.) with respect to centers O• (or 0 2) of the surface S• (or S•.) 'to which the point be- longs. However, Pl(Rl•., Ol•., ½12) or q•.(R•, O•l , ½•1) are spherical coordinates of the same points with respect to the other center, O•. or O•, indicated by the second subscript. Normal unit vectors •, •q•. on S•, S•. are directed out of V, as shown in Fig. 1.

The acoustic microwave cavity problem in V with Dirichlet or Neumann boundary conditions on $• and S 2 can be formulated in terms of a scalar function u(P) (PGV) satisfying

(a) Dirichlet problem:

vu(P) + k'u(P) =0; P v;

u(p0=0, u(p•.)=0; p•S•, p•.•&.; (1)

(b) Neumann problem:

vu(P) + k'u(P) =0; P v;

au(p0 -0, au(p•) =0; p• s•, Pz• sz. (9.) Applying Green's second identity to u(P) and the free

space Green's function

G(P; Q) =G(R) =e•kR/R (R =PQ) (3)

in region V, then letting P approach p• on S• and P2 on S•. and using the corresponding boundary conditions in conjunction with well-known limiting values of certain integrals •'2a leads to the following set of homogeneous surface integral equations on S• and

Dirichlet problem:

fs•D(qt)G(pt, qt)dSqt + ;szD(q2)G(pt, q2)dS=2 =0, (4) fs D(q•)G(p•., q•)dS• + fs D(q•.)G(p•., q2)dS•2 =0, (5)

au(q) D(q)= an• (q=qx or q2), (6)

Neumann problem-

-2;ru(px) = fs u(qx) aG(p,,qx) dSqx 1

+ fs u(q•.) aG(px• q•) dS•.o (7)

- 2mr(P2)= fs u(qx) 1 aG(p2• q!) dS•

•ql

+ fs u(q2) aG(p•,q2) dS•2 ø (8) All surface integrals involving G(pi, qi), G(p2, q•.), or their normal derivatives are convergent singular surface integrals. •,•a

In the following pages, analytical solutions of the preceding integral equations are obtained after properly evaluating the singular integrals by a convenient limiting process. Use also is made of translational addition theorems for spherical wave functions. •'?'•'ø What also proves crucial is the solution of a set of in- equalities involving indices of summation. This allows proper isolation of one index and eventually leads, par- ticularly for small kd, to the exact evaluation of second- order terms (in kd) for the elements of an infinite determinant and, finally, of the determinant itself. It is then possible to obtain the resonant frequencies w,s(kd ) to second order in kd.

I. SOLUTION OF THE DIRICHLET PROBLEM

The unknown surface function D(q) in (4)-(6) is expanded in a complete set of spherical surface harmonics on each sphere:

(9)

Despite the circular symmetry around the • axis (Fig. 1) the ½ dependence cannot be ruled out, because, in general, it may be due to the (not necessarily axisym- metric) initially impressed field that generates the oscillations inside the cavity. However, with O•O•. along the z axis it is obvious that ½i =½I•., ½I =•l and, with no loss of generality, it suffices to consider only one value of rn(= M):

D(q•) = Y• s=l MI

D(q•.) = Y• Bus, Ps•(cosOi)e 'u*•-. s '--I MI

(10)

The more general expansion (9) can be realized from (10) by superposition.

Use will also be made of the well-known expansion •'9'aø

J. Acoust. Soc. Am., Vol. 64, No. 1, July 1978


288 J.D. Kanellopoulos and J. G. Fikioris' Resonant frequencies in a spherical cavity 288

G(P, Q)=ei•n/R =ik • (2n + 1) •.• (n- m)! p•(cosO)Py(cosO,)eim(•_•, , ,=0 m=-, (n+m)! [h.{•)(kr,)j,(kr)j, r<r,(r=OP, r'=OQ),

where Jn and h{, •) are the SPherical Bessel and Hankel functions. Deleting from here on the superscript (1) in h,, substituting (10) and (11) in (4) and (5), using the orthogonal properties of the surface harmonics, and noticing that Ozp• =R•z<R•. and O•pz =Rzl>R•, it is possible to evaluate immediately the nonsingular integrals

fs D(q•.)G(p•, qz)dSq•. =4•rikR[ L B•s'hs'(xz)js'(kR•.)PsU'(cøsO•z) e•= , (12) •. s'=l MI

fs D(qOG(pz' qt)dSql =4•ikR• • A•sja(xt)hs(kR•.t)PsU(cosOzt)e'•zt , (13) 1 s I

where x• = kR•, and x•. = kR•..

For the singular integral over St, it is convenient to consider a point P(R, O, •)•i St and evaluate it on the basis of the classical result •'•"•'s

;s D(q•)C(P•, qx)dS• 1 1

= lim Is D(qOG(P'ql)dS•' (14) P-}Pl ' 1

where P can approach Pt from either side, i.e., with O•P=R>R• or R<R•. However, before the limitP--p• is taken, it is possible to use (11) for G(P, q•) to obtain

fs D(qOG(p•,qOdS•= lim 4•rikR• 1

x • A•sP•(cosO)e '• (15) L(n)J '

where the upper (lower) form corresponds to R from higher/lower values. In the limit both values coincide and one gets

;s D(qOC(p•, q•)dSq• =4•ikR• 1

M iMq) x (cosOe

Similarly:

fs •.D(qz)C (p•., q•)dSq•. = 47rikR• x • B•s,js,(xz)hs,(x•.)p•(cosO•.)e•*z. (17)

s'=l MI

Substituting now the results (12), (13), (16), and (17) in (4) a• (5) and, as before, noticing t•t ½• =½• and =½•, there results the following set of homogeneous linear equations for the e•ansion coefficients A•s a•

R• • A•sjs(x•)hs(xOP•(cosO 0 s=l MI

+ R• • B•s,hs,(xz)js,(kRiz)P•(cosOiz ) =0,

s'=l MI

The second [first] term in (18) [(19)] contains a spherical wave function with respect to origin Oz[O•]. It may be e•anded into a sum of spherical wave functions with respect to O•[Oz] (like the other term of the equation) us•g the well-knom translatioml addition theorem ,zv, z0

(18)

(19)

(20)

where zs is a general spherical Bessel function; and (r, 0, ½), (r', 0', ½'), and (d, 00, ½0) are, respectively, the coordinates of some point P with respect to origin O, of P again with respect to another origin O', and of O' with respect to origin O. For r '> - d the variables r', 0', ½' are simply interchanged with d, 00, ½0 in the right- hand side of (20). In particular, for zs = js either form is applicable without restriction on the relative size of r' and d. The. summation index p varies from Is-nl to s +n by steps of 2. Finally, the coefficients a(m, s l - •, nip) are defined in the Appendix, where certain particular values of them are also evaluated.

Returning to (18), it is observed that 0•.01 =d, O0=•r , P•(cos,r) =(-1) • for t• =0 and 0 for t• • O, r=Ri•., 0 =01•, ½ =½•.=½•, r' =R•, 0' =0•, ½' =½•, and therefore

Js'(kRla)Ps•'(cøsO•') = n--• i"+•'s' p= I

x (2n + s'l O, 1)"j•(xOP•(eosO•),

where, of the two possible forms of the expansion, we choose [with d, 00, ½0 and r', 0', •' interchanged in (20)] the first term of (19), which admits only one




form. It is then seen that OxOn. =d, 00=0 , P,• (cos0) = 1 for t_t = 0 and 0 for t_t =• 0, r =R21, 8 = 02• , • = •2• = •2, r' :Ra>d, •': •a, •': •a, and therefore

hs(kRal)P•(cosOal) =• i n=O P nl

• now (21) [(22)] is substituted into (18) [(19)], it is possible to isolate the total coefficient of P•(cos0•) [Psg(cos01)] in the e•anded term and to use the orthog- oal properties of P•(cos0) to decouple the equations for s (or s')=[M[, IM[ +1, .... Thus in (21) [or (22)] and for s' (or s)=1MI, I MI + 1, .... all possible values of n should be found for the fixed value p: s (or p: s'). Under the restrictions

[n-s'[•s•n+s' and s+n+s' is even, (23) it is shown in the Appendix tMt the totality of such values of n are such tMt

I s-s'•n•s+s' by steps of 2. (24) This result can also be directly deduced from the general properties of the symbol a(m, sly,nip), which s•te that the symbol is 0 unless the triangular condition Is- n l•p • s +n is satisfied cyclically by all indices s, n, p and unless s +n +p is even. This is a direct consequence of the corresponding properties of the Wignet 3-j symbol

0

as discussed in the Appends. However, because of the reader's presumed unfamiliari• with such symbols a short, independent derivation of (24) from (23) Ms been included in the Appendix.

The end result for (18) and the corresponding one for (19) are

s' = I• I n s'l

• i"*'"' (2•+ t)(- t)"a(•, s'10, •• s)y.(•)= 0,

s: I MI, I MI + X,...,

s=l MI n= ' I

= I MI, I MI + X, '".

•bstitution of R[ B•s, from (26) into (25) gives

A•shs(xi )_ • (_ 1)s. s, hs,(Xa) •,• h,(xa)

v=l MI

where

(27)

= •. i""'"(2n+l)a(m, ulO, n[s)j,(kd). (28) n=lv-sl

Interchanging the order of summations in (27) yields the final set of homogeneous linear equations for the expansion coefficients Aus:

Aushs(xx)- L Auvj,,(Xl) L (-1) s*s' v=lMi s'=l MI

O(MI , x , Ia)(MI s, =o or

•'. a,,,Au,,=O for ,s=lMI, v=IMI

where

ass=hs(Xl)-is(X1) Z (- 1)

h•(xz) b(M[ s, m I ka)b(M I m, s] ka) (81) x 5•(xz) ,

a•. = - 5.(x•) • (- 1)•'• •(xz)

x(ml,mlkd)(mlm, slkd), Setting the determight D(as,) of the coefficients as,

equal to 0 provides the transcendental equation for the evaluation of the resonant values kns =Wns/C. For general values of kd one can proceed, from here, by numerical methods only. The difficulties of this approach are discussed later. However, for small values of kd an analytical solution is possible. M particular, for kd =0 it is obvious from (28) tMt 5,(0):0 for n• 0 and 50(0)=1. Therefore, •(m]u,s]0):0 for u* s and b(MI s, sl0)=l, as shown in the Appends, gq. (A3). This implies tMt all as,(O)=0 for s• u and that

a•(0) = a•(x•)- 5•(x•)a•(xa)/Y•(xa). (aa)

The determimnt becomes diagonal and the resonant frequencies w,•(0) are found from the equations as•(O) =0(s= IMI, IMI+I,...; n:l,2,... ), a result independent of M (apart from the restriction s•lml) and well knom from separation of variables.

For kd• 0, but small, it is obse•ed from (28) and the relations

2"n ! (kd)"[ 1 + O (kad z) ] for n • 0,

5o(•) • • - (ka)Vo + O(k4ah

tMt for u • s and u = s,

This implies tMt

C" (kd) a +O(k4d4), a• =C•+ s• (30) a•,:C•,(kd)•*'"•[1 +O(kada)](u* s),

in which the C's are independent of kd, and allows a closed-form evaluation of the determimnt D(a•,) to order kad z. To this end the following development of an n Xn determinant is employed' One writes the pr•uct




a•a22 .. ß a,, of the diagonal elements and then similar products of n elements by carrying out all permutations of the first subscripts keeping the second subscripts fixed, or vice versa. As a result n! terms are obtained and their sum, with the proper sign, equals D,(as,, ). The sign convention assigns a +/- sign to a term all of whose permutations (in the aforementioned process) are formed by an even/odd number of inver- sions of the permuted subscriptSo With this and (36) in mind, it is obvious that (a) the diagonal product alullulalul+z, lul+z... in D(a,.s) is of order 1 +O(kd)Z+.'..; (b) products all of whose elements excepi one are diagonal do not occur (i.e., the only possibility in a• ß .. arn.•,m.•a•mam+•,rn+• ... is thatj =m, since no other value remains for the subscript j in the aforementioned permutation process); (c) products with three or more nondiagonal elements are at most of order (kd) s and can be neglected; (d) products with only two nondiagonal elements are at most of order (kd) •' and, according to (36), the only ones that do not exceed (kd) •' in order include the elements nearest to the diagonal a.,.+•a.+•,. (n = I MI, I M{ + 1, ß ..). This means that only products of the form alullul ß an.l,n. lan+l, na m•+la•+•.,•+•. ß ß ß should be retained in this class, all others yielding order higher than (kd) •'. Moreover, to such remaining products the sign convention assigns a - sign. Therefore, up to order k•'d •',

D(as") =P(ass) ( 1- ,, ••ul a..•,.a,,..q] = an, n, an+l..+l / '

P(ass) = al ,vl I MI al MI +1, I MI +1 ø ø ø ass ø ø ø ß

(37)

(38)

Furthermore, using the notation of (36), it is obvious that up to order k2d 2

# 2 P(ass) =P[Css +Css(kd) ]=

a.+•..a...+• _ C.+•..C...+• (kd)•- -- .

a.. 1, .l C,,.C.I, .1

Substituting in (37) yields

(39)

(40)

C•+l'nC•'"*l• O(k3d3)] -- Cn.C.+l,.+l ! + D(%,,) =P(Css ) [1 + (kd) •' *( c.,; n=l MI Cnn

(41) Exact expressions for the coefficients C.,, C,'%,

C•,,,l (• = I/V/I, IMI + 1,... ) appearing in this final form can be found by referring to equations (28), (31), (32), (34), and (36)'

h.(x•) .... Fn.(x,) n.(x•.) l C..:h.(xO- j.(xO j.(xe)=•].txl•kj-•- j.---•)j=a..(C),

(42) where n.(x) is the spherical Neumann function and, as before, the result a(M, n10, 0In) = 1 has been used. AI-

c.",, : j.(x j.(x,.) [« + x a(M, n l O, 2 l n) ]

_ h.•.(x,) a(M, nlO, l{n-1)a(M,n-l}O l{n) '

SO,

h.+•(x•.) , 1)a(M, 1{0, 1In)}, (43) - nlo 1In+ n+

[h.+•(x•.) C•,+•,.=j.(xOa(M,n{O, 1{, + 1) LJ.+---•'•) - j.(x•.)J' (44)

C/,,.+, = j.+,(x,)a(M, n + 1{0 1In) ' - ]' (45) Substituting from the Appendix [Eqs. (A4) and (A6)] the values

a(rnn{o, 1ln- 1)- n+r. 2n+1'

a(m,n- 1lo, 1In)- n- m 2n- 1'

a(rn, n ]0 2In) = 3[(n + 1)•' - rn•'] 3(n •' - rn 2) ' 2(2n + 1)(2n +3) + 2(4n 2 - 1)

(46)

1

(47)

one obtains the remarkable result that, like C.., the re- ' C' and in particular, maining coefficients

C,"• turn out to have imaginary values, too'

C.• =ij.(xl)jn.x•.) • (n+l) [ j.(x•.) (•n + 1)(2n + 3) + 4n •' - 1

n..•(x•.) n •' - M •' n.+•(x•.) (n + 1)•' - M •' } (48) - j..•(x•.) 4n•"-•'• - - j.+•(x•.) (2n + 1)(2n •õ) '

C•+•, =/j.(xO n + 1 - M[nf+•(x•.) n.(x•.)'{ " 2n + 1 L./.+•(x•.)- j.(x•.)]' (49)

C' n + l + M[n.._2•.•(x•.) n.(x•.)] (50) •,•,•=ij•.•(x•) 2n+3 [y•q(x•)-jn(x•) ' This, according to (36), means t•t all elements of the determinant D(asv ) are imagi•ry, at least to order (kd) •, and, therefore, that the resonant frequencies w•(kd) resulting from the equation D(asv ) =0 are all real, as predicted by theory for all cavities and made e•licitly evident by the latter result (55). The authors consider this outcome as a very convincing check on all relations and results ob•ined in this paper.

It is now evident from (41) that setting C,, =a,•(0) =0(v=lMI, IMl+l,...)yields

D(asn) =(kd)•. P(Css) C•+•.,. ,,,,,+• "'"'•C"'h" Cv v .v Cv+l, v+l Cv-1, v-1

In other words, the fixed roots of D[as.0) ] =0 do not, in general, satisfy the equation D[%.(kd)] =0. Instead, this latter equation requires that

i ** (C,•+•..C•,,+• (kd) 2 - .:•'•1 x CnC.+•.,.• - C.". ) (51) Cr/r /

With d as small as one may wish it is obvious that this equation can be satisfied by values of the only varying parameter k =•o/½ that make the denominators Cn• and C•+•,•+• as small as required by the large values of 1/ (kd) 2. In other words, the resonant frequencies w,,s(kd ) correspond one-to-one and have values very near the w,,s(O) of the cocentric cavity (v >_ {MI, s = 1, 2,... ). Setting k,,s(d ) = k,,s(O ) + • k,,s, or more compactly k•as = k•øs + 5k,,s, one has

J. Acoust. Sec. Am., Vol. 64, No. 1, July 1978


291 J.D. Kanellopoulos and J. G. Fikioris: Resonant frequencies. in a spherical cavity 291

+

Substitutions in (51) and retention of large terms only fi•lly yields

t 0 t 0 C t 0 t 0

For v = I MI, in particular, the middle term should dis- appear, since the index n never arMins the value I MI - in (51). This, however, is automatically satisfied ff one observes from (40) and (50) that C{•,•.xC{•.t,• • is 0. Therefore (53) holds for all values of v • I MI and s= 1, 2, .... With •k-(kd) • it is more convenient to write

kvs(d) = k,=(0)[ 1 + gvs(kd) z] or (54)

= + g.,(kg) =]

and obMin the altersrive final e•ression

= = , J

Cvq'l, vCv, v+l Cv, v-lC v-l,v -. C t!

where all C's inside the second set of brackets are

evaluated at k = k,s(0) = k,øs. Also, in differentiating dC,,(x•,x•.)/dx•, one writes x•.=rx• and dz(x•.)/dx• = rz ' (x •.) , where r = x z/x • = R•./R • =constant.

(55)

The fact that the 6k,s turn out to be proportional to dZ--in other words, that d =0 is an extremum of the k,s(d) as functions of d--could be anticipated from geo- metrical symmetry. Positive or negative d (placing O• to the left or right of O a in Fig. 1) would not alter the geometry of the cavity and the values of k,,(d). The re- versal of the z axis that this implies and the consequent sign change in the • dependence of the (unaltered) initial excitation do not affect these remarks. In fact, only • is involved in (53) or (55), as seen from (42) and (48)- (50). In particular, C•+•..C•'..+• involves M only in the factor (n + 1) z- M •. Such observations constitute a further check on the results of this paper. In any case, unlike the w,,(0), the w,,(d) do depend on M.

II. SOLUTION OF THE NEUMANN PROBLEM

In analogy to the Dirichlet problem, one expands the unknown function u(q) in (?) and (8) in a manner analo- gous to (10):

u(q•) = • A,,P• (cos0•) exp(iM½l),

u(q•.)= • B,s,P,•,(eosO•.)exp(iM•b';.); (56)

then expands the related Green's function from (11), noticing (Fig. 1) that

O/On•a =o/oRa and

oG(P, qi) = • ;o =;ik • •(2n+l)

x • (n- m)! •:-n (n +

X [h•(kRa)jn(kr) , r< (r=OiP , Ri=Oaqa, i=1 or 2), where the upper (lower) sign is used when i = 1(i = 2).

As previously, the nonsingular integrals in (7) and (8) may be immediately evaluated:

• • G(p•,ps)dSqs =4wik•R[ s u(qs) s' =1 MI

fs a G(pa ' ql)dS• 1 _ 4•ikaR • ,u(ql) an• = X • .t M tM•g 1 A•,j,(xx)h,(kRzx)P, (cos0zx)e . (59)

•=1 •1

For the singular integral over Sz, again a point P(R, O, •)• Sx is considered and the integral evaluated on the basis of the classical resultZ'z'z::

fs a G(p•,q•)dSq• •u(qO anq-•

fs a G(P q•)dS•+ 2;ru(p•) (60) = lim u(q•) an•---• ' . ' P•/•I 1

in which the upper (lower) sign of the last term corresponds to P approaching P l from the exterior (interior) of S1, i.e., for 01P=R>R•(R<R•). Again the relation a/a•l =-a/aR1 should be taken into account. Before the limit P--P• is taken, it is possible, as before, to use {57) to obtain




where the upper (lower) form corresponds to R-R• from higher (lower)values. That the two forms are equivalent is immediately verified if one makes use of the Wronskian js(x)h;(x)- hs(x)j;(x ) =i/x •' and the first Eq. of (56), for u(PO. Similarly (with O/On•2 =o/aR2) ,

fs a G(p•.,q•.)dS•. •.u(q•.) an•-•

fs a G(P, q•.)dS•. ß 2rru(p•.) = lim u(q•.) an•--•

=:• 2rru(p•.) + 4•rix•. Ms,Ps, s' =1 MI

Is'

Xe i•ee (62)

where both forms are equivalent for the same reasons. Finally, substituting (58), (59), the lower form of (61), and the upper of (62) into (7) and (8), and noticing that ½• =½• and ½• =½•, one gets:

-x• • A•,P,(cosO,)h}(xOj,(xO

+x• • B•s,h•,(xa)js,(kRz•)P•(cosO,•)=O , (63) s' =1 MI

- s=l MI

+x[ • B•,,P,•(cosO•)j•,(x•)h,,(x•)=O. (64) s•=lMI

Comparison with (18) and (19) shows tMt these •o sets of homogeneous linear equations are essentially the same, apart from a sign change in A•s and the replace- ment of certain Bessel functions of xz or x• by their derivatives. However, the terms that are being reex- panded by means of the addition theorem remain the same, and the procedure following (18) and (19) similarly applied in the present case leads to the set

- s•=l MI

S+S •

n=[ s-s • [

s=l MI n=l s-s' I

x a(M, s]O,n ]s')j,kd) +x• B,,, j•, (x2) = O, (66)

equivalent to (25) and (26). Substituting x[B•s from (66) into' (65) leads to an equation similar to (27), and interchanging of the summation order in this equation yields the final homogeneous set of equations for the AMs:

A•sh}(x O- •, A•,j;(x•) Y• (- 1) s*s' •1 MI s'=l MI

x h•,(x2) 0(MI., [ a)0(MI Ia) =0. (07)

This is similar to (29) or (30) except that the Bessel functions of x• and x•. are replaced by their derivatives The procedure following (30) remains unaltered. The case d =0 yields the well known result

'! ! a**(0) =h;(x0- ?,(x•)h,(x•.)/?s(x•.)=0, (68)

whereas the case of small kd leads again to relations (54) and (55), in which kv•(0) = k,ø• [or wv•(0)] are now the roots of (68) and the C's are defined by (42)-(45) [or (48)-(50)] if the Bessel functions of x• and x•. appearing in them are replaced by their derivatives.

III. NUMERICAL RESULTS AND DISCUSSION

The roots(x0•,s(v=0,1,2,...; s=l,2,...)of (33) or (68) were obtained by a modified Newton-Raphson method, called the Regula-Falsi method, and in the case of (33), were checked against tabulated values. 3• Four values of the ratio r=x•./x• =R•./R• were used' r =1.2, 1.35, 1.5 and 2.0. The roots (x•),s of (33) (Dirichlet problem) and the corresponding valuesg•s from (55) are given in Tables I-IV, those of (68) (Neu- mann problem) and the corresponding g,s in Tables V-VIII. M=0 throughout.

The case v =0 for the Dirichlet problem requires special treatment and serves to point to the complications that are inherent in any direct numerical deter- mination of the roots of D(avs)=0. Equation (33) for v=s=0 reduces to tanx•=tanx•. or tanx•=tan(rxO. Its roots are (x00, = s•r/(r- 1)(s = 1, 2,... ) and in a number of very common cases, in particular when s/(r- 1)= m

TABLE I. Dirichlet problem (M=0), r=x2/xi =1.2.

Ocl )vs

gi•$

1 2 3 4 5 6

0 5•r 10•r 15•r 20•r 25•r 30•r 1 15. 76064 31. 44237 47.14150 62. 84508 78. 55033 94. 25650 2 15. 86550 31. 49526 47.17682 62. 87154 78. 57154 94. 27419 3 16. 02151 31. 57444 47. 22976 62. 91128 78. 60335 94.30070 4 16. 22717 31. 67971 47. 30025 62. 98423 78. ,64574 94. 33604 5 16. 48061 31. 81082 47. 38822 63. 03036 78. 69869 94.38019

0 8. 333333 8. 333333 8. 333333 8. 333333 8. 333333 8. 333333 i 6. 044379 6. 027234 6. 045676 6. 075658 6.108474 6.151726 2 1. 457082 1. 439718 1. 440276 1. 444396 1. 451305 1. 458969 3 O. 6911515 O. 6753853 O. 6738132 O. 6759440 O. 6790528 O. 6838808 4 O. 4114470 O. 3969645 O. 3946462 O. 3952597 O. 3967629 O. 3992045 5 O. 2758927 O. 2629380 O. 2604798 O. 2604439 O. 2617776 O. 2633834



293 J.D. Kanellopoulos and J. G. Fikioris' Resonant frequencies in a spherical cavity

TABLE II. Dirichlet problem (3//= 0), r=x•./x! = 1.35.

1 2 3 4 5 6

(xt)vs

0 8.975979 17.95196 26.92794 35.90392 44.87990 53.85587 1 9.057011 17.99305 26.95539 35.92452 44.89639 53.86962 2 9.216926 18.07494 27.01022 35.96571 44.92936 53.89711 3 9.451597 18.19702 27.09224 36.02739 44.97876 53.93831 4 9.755591 16.35861 27.20123 36.10947 45.04456 53.99319 5 10.12253 16.55865 27.33688 36.21183 45.12667 54.06172

0 - 3.67320 - 3.67280 - 3.67254 - 3.67238 - 3.67232 - 3.67232 1 -1.69979 -1.50011 -1.47109 -1.45986 -1.45024 -1.43245 2 0.5499949 0.5308630 0.5276770 0.5268086 0.5263193 0.5252350 3 0.2650623 0.2514990 0.2480633 0.2467400 0.2460978 0.2458565 4 0.1590243 0.1491328 0.1480499 0.1448228 0.1443263 0.1438982 5 0.1062775 0.0997481 0.0969786 0.0958526 0.0953720 0.•949660

293

(or • +«) (m being an integer), they satisfy tanx• =tanx•. =0 (or •), implying sinx• =sinx•.=0 (or cosx• =0) and leading to indeterminate forms 0/0 (or •/•) in the expression forg0s, which in terms of sin and cos functions reads

The indeterminate forms result in large computer errors. In particular, the values r= 1.2, 1.5, and 2.0 (and many other common ones) make tanx• = tanx•. =0. This was the reason for also considering the value r = 1.35. In such cases a direct evaluation of (69) yields

for tanx•=tanx•.=0' go$=[3(r - 1)•'] '•, (70)

for tanx•=tanxa =øø' g0s=-r[3(r- 1)a] '•. (71)

It is worth observing that in these cases g0s is independent of s. This result persists, to a very good approxi- mation, in all other cases, v =0 and v• 0, even for the Neumann problem, as a glance over the results in Ta- bles I-VIII reveals. In other words, the coefficients gvs are almost independent of the order s of the resonant frequency.

The question arises whether the case • =0 for the

Neumann problem causes similar troubles. Equation (68) in this case (similar to the Dirichlet equation (33) for • = 1) reduces to (tanx•-x•)/(1 +x• tanx•) = (tanx•. -x•.)/(1 +x•.tanx•.). It is not satisfied by tanx• and tanx•. both equal either to 0 or to •. In very special cases (particular values of r), either tanx• or tanx•. may turn out to be 0 or •, but this, in general, will not lead to indeterminate forms and computer errors. Similar remarks hold for all other cases, y• 0.

In any event, the numerical difficulties of the Dirich- let case v =0 are a first indication of the very severe

problems that may crop up in a direct numerical search for the roots of D(av$)=0. Another obvious difficulty in such an approach is the truncation of D(a•,) to a finite order, an unavoidable necessity in direct numerical computations. Truncation of D(a•,) to, say, a 10 x 10 determinant may be good for the evaluation of roots of order v up to 7 or 8, but becomes irrelevant to the de- termination of roots of higher order. For small kd the significant elements of the determinant that affect the value of ½ov$(d) are the diagonal ones av.•,•.•, a•v, as well as the off-diagonal ones nearest to them a•,•.•, a,,.•,•, a•+•,.•, av,•+•. Properly retained, as in (55), they yield the correct value of •o•,(d). The other elements of D(a•,) are insignificant in this case, causing confusion in the direct numerical computation of its roots and complicating their classification in terms of the indices

TABLE III. Dirichlet problem (M = 0), r=xx/x! = 1.5.

1 2 3 4 5 6

0 2x 4x 6x 8x l(hr 12x 1 6.38580 12.61895 18.88476 25.15918 31.43708 37.71673 2 6.58613 12.72351 18.95504 25.21203 31.47942 37.75204 3 6.87535 12.87882 19.05999 25.29113 31.54283 37.80495 4 7.24246 13.08311 19.19908 25.39622 31.62720 37.87539 5 7.67580 13.33421 19.37160 25.52701 31.73236 37.96326

0 1.333333 1.333333 1.333333 1.333333 1.333333 1.333333 1 1.238751 1.211458 1.206728 1.205421 1.206037 1.207849 2 0.3084221 0.2931413 0.2894204 0.2883314 0.2878781 0.2879462 3 0.1493206 0.1396848 0.1365644 0.1353658 0.1349980 0.1348218 4 0.0885560 0.0835308 0.0808307 0.0797393 0.0792489 0.0790213 5 0.0576548 0.0562019 0.0539658 0.0529585 0.0524692 0.0522467




TABLE IV. Dirichlet problem (M=0), •'=x2/xl=2.0.

i 2 3 4 5 6

(xl)vs

gvs

0 • 2• 3• 4• 5• 6•

I 3.28600 6.36067 9.47718 12.60587 15.73963 18.87597 2 3.55578 6.51306 9.58126 12.68459 15.80286 18.92877 3 3.92252 6.73556 9.73553 12.80189 15.89729 19.00774 4 4.35839 7.02183 9.93792 12.95687 16.02247 19o11260 5 4.84099 7.36467 10.18591 13.14838 16.17778 19.24300

0 0.3333333 0.3333333 0.3333333 0.3333333 0.3333333 0.3333333 I 0.4350069 0.4114638 0.4056064 0.4034784 0.4026381 0.4023395 2 0.1100840 0.1024208 0.0988994 0.0974523 0.0967263 0.0963975 3 0.0504863 0.0500031 0.0475824 0.0463832 0.0457631 0.0454110 4 0.0268988 0.0302005 0.0287286 0.0277343 0.0271788 0.0268539 5 0.0152122 0.0201402 0.0194881 0.0187097 0.0182259 0.0179155

v and s. In this connection the close proximity of the roots (x•)•s for the same s and variable v should be noticed. For large kd, where the present theory is no longer applicable, one loses track of the significant terms in the evaluation of a specific •v,s(d). At the same time, the aforementioned difficulties in a direct numerical search for the roots of D(a,s)=0 are not ameliorated and, as matters stand so far, the problem appears in- tractable.

In order to substantiate these remarks, the Dirichlet case r = 2.0 was solved by direct numerical evaluation of D(a,s) truncated to a 5 x 5 determinant. Two cases kd =0 and kd =0.001, were considered. The range x• =6.32-7.47 was investigated (corresponding to s =2, • = 1 - 5) by varying x• by steps of 0.01 and evaluating Dsx 5 for each such value of x• in the above range. For d=0 the exact roots are found from Table IV' (x•)• =6.36067, (x•)22=6.51306, (x•)32=6.73556, (x•)4•. = 7.02183, and (x•)5• = 7.36467. Without going into de- tails, the direct numerical evaluation yielded roots in the ranges 6.36-6.37, 6.73-6.74, and 7.36-7.37 (corresponding to (x•)•, (x•)3•, and (x•)s•., respectively). A root was detected by the simultaneous passing through 0 of both the real and imaginary parts of Dsxs. The other two roots, x• =6.51306 and 7.02183, were missed, because of the proximity of a pole of Dsx5 near these valuesø Poles of D•xs were detected by large values

and simulatenous sign changes in both the real and imaginary parts of D. Such poles were also observed at other values of x•, away from the above roots. It is obvious that the numerical approach fails in detecting roots in such cases, unless one greatly refines the step in the x• values near such poles. Even that will fail if the pole coincides with the root. It must also be re- marked in this connection that the troublesome pole may be due to an unrelated element of D. In any event, the above computations require long execution times. Finally, similar results were observed in the case kd =0.001. Poles and roots appeared near the same values of x• as before.. The simultaneous passing through 0 of both the real and imaginary parts of Dsxs (implying real roots •v,) was not as well satisfied as in the case d=0, the discrepancy becoming worse (but never severe) with increasing order • of the root. Moreover, the root (x•)• was found near 7.41 rather than the correct 7.36 due to the truncation of D at • = 5.

As mentioned in the Introduction, the corresponding electromagnetic cavity between conducting eccentric spheres is treated elsewhere. 2s Two sets of expansion coefficients on each sphere are required, which, when M s0, remain coupled to one another. This is due to the fact that under translation an m eigenvector is expanded into both m and • eigenvectors, unlike in the scalar equations (20) or (21), (22) of this paper. In the de-

TABLE V. Neumann problem (M=0), r=x2/x! = 1.2.

I 2 3 4 5 6

0 15. 76064 31 . 44237 47.14150 62. 84503 78. 55033 94. 25651

1 15. 81404 31. 46894 47.15920 62. 85830 78. 56094 94. 26535

2 15. 92032 31. 52202 47.19458 62. 88483 78. 58217 94. 28304

3 16. 07848 31. 60147 47. 24759 62. 92461 78. 61399 94. 30956

4 16. 28705 31. 70711 47. 31819 62. 97760 78. 65640 94. 34491 5 16. 54418 31. 83867 47. 40629 63. 04379 78. 70939 94. 38908

0 - 9. 76468 - 9. 92679 - 9. 93969 - 9. 92502 - 9. 90297 - 9. 87075

1 - 3. 85651 - 3. 95959 - 3. 97441 - 3. 97368 - 3. 96774 - 3. 95889

2 1. 425339 1. 431182 1. 436598 1. 445069 1. 452458 1. 463524 3 0. 6768214 0. 6711512 0. 6720519 0. 6742397 0. 6783057 0. 6829348 4 0. 4034726 0. 3944625 0. 3937332 0. 3949337 0. 3971019 0. 3997259

5 0.2710316 0.2614214 0.2599339 0.2599410 0.2612095 0.2631526




TABLE VI. Neumann problem (/14 = 0), r=x2/x! = 1.35.

1 2 3 4 5 6

(Xl)vs

0 9. 057011 17. 99305 26. 95539 35. 92452 44. 89639 53. 86962 1 9.140954 18. 03449 26. 98296 35. 94518 44. 91291 53. 88338 2 9. 306749 18.11714 27. 03801 35. 98646 44. 94592 53. 91090 3 9. 550490 18. 24044 27.12039 36. 04829 44. 99541 53. 95214 4 9. 886902 18. 40362 27.22986 36.13058 45. 08130 54. 00709 5 10.24988 18. 60569 27. 36610 36. 23318 45.14354 54. 07569

0 - 3. 44889 - 3. 61499 - 3. 64554 - 3. 65788 - 3. 66213 - 3. 66646 1 - 1. 29936 - 1. 38030 - 1. 34699 - 1. 63611 - 1. 49274 - 1. 47432 2 0. 5210806 0. 5239118 0. 5243346 0. 5250741 0. 5248595 0. 5251067 3 0. 2528332 0. 2476472 0. 2462032 0. 2456530 0. 2455917 0. 2452324 4 0.1529289 0.1470467 0.1450920 0.1442791 0.1438942 0.1437191 5 0.1030884 0. 0984443 0. 0963269 0. 0964987 0. 0950396 0. 0947616

terminant D(a•) terms of order kd, which must be retained, are found not only along the three diagonals ass, as.•,s, and as,s.•, but along the diagonals as.2, s and as, s.2 next to them as well. Only when M= 0 does this additional complication disappears. However, even then the w•(d) for the so-called magnetic modes are no longer the same as those of the Dirichlet scalar case, as happens to be true for cocentric spheres (d = 0).

The analytical approach developed in this paper could be extended to exterior problems as well as to other shapes. Beyond the obvious extension to cylindrical geometries, it is believed that, for instance, regions between cocentric spheres and spheroids (and corresponding generalizations) can also be treated in case of small spheroid eccentricities. All that is required is an expansion (similar to the addition theorems among spherical wavefunctions) of a spherical wavefunction into a sum of cocentric spheroidal ones, and vice-versa. Recently, cylindrical geometries were considered, in particular, eccentric waveguides and eccentrically coated waveguides. Our results were found in excellent agreement with those of Ref. 33, in which the cutoff frequencies of the lower TM and TE modes were obtained numerically and experimentally for eccentric, perfectly conducting waveguides. For small kd agreement to four decimals was obtained. Even for the largest kd considered in Ref. 33 the agreement extended to the first

two decimals, an indication that the restriction kd << 1 of the present method is not really as severe as it may appear. This is further corroborated by the fact that the g•s get smaller fast with increasing ratio r, as seen from the tables. Besides, ,when r is small and both g• and x• become relatively large, kd necessarily gets small due to the physical restriction d < - R•.- R• or kd <- (r- 1)x•. This application to waveguides will be the subject of a forthcoming paper.

APPENDIX

The symbol a(rn, nl!•,vlp) in (20) appears also in the expansionU.?, •.8

P,P• = • a ,

(p=n+ v,n+v-2,..., In-v[), (A1) and has the value zo'az

a(rn, n]•,v[p)=(- 1)m+"(2p + 1)

x + m)' + - m - '/" - rn) !(v -/•)!(p + rn +/•)! J

rn•g -m- ' 0 ' where

(A2)

m 1 •/t•. •/t 3

TABLE ,VII. Neumann problem (M=0), r=xx/xl=l.5.

2 3 4 5 6

(xt)us

g¾$

0 6. 38580 12. 61895 18. 88476 25.15918 31. 43708 37. 71673

1 0. 49506 12. 67248 18. 92028 25.18577 31. 45833 37. 73443 2 6. 70893 12. 77893 18. 99112 25. 23886 31. 50080 37. 76982

3 7. 01910 12. 93712 19.09694 25. 31832 31. 56440 37. 82283 4 7. 41515 13.14536 19. 23719 25. 42390 31 . 64901 37. 89342 5 7. 88594 13. 40157 19. 41119 25. 55531 31. 75449 37. 98143

0 - 1. 78442 - 1. 93965 - 1. 97137 - 1. 98245 - 1. 98662 - 1. 98768

1 - 0. 695226 - 0. 770642 - 0. 786446 - 0. 791939 - 0. 794139 - 0. 795051 2 0. 2824335 0. 2850451 0. 2857384 0.2 862614 0. 2865309 0. 2869954 3 0.1391835 0.1361287 0.1347892 0.1343883 0.1342457 0.1343751 4 0. 0839762 0. 0816423 0. 0798521 0. 0791852 0. 0789036 0. 0787410 5 0. 0554573 0. 0551164 0. 0533492 0. 0525855 0. 0522077 0. 0520564

J. Acoust. Soc. Am., Vol. 64, No. 1, July 1978 ß



TABLE VIII. Neumann problem (M = 0), r=x2/x! = 2.0.

i 2 3 4 5 6

0 3.28600 6.36067 9.47718 12.60587 15.73963 18.87597 I 3.45575 6.44261 9.53101 12.64600 15.77164 18.90260 2 3.77681 6.60434 9.63799 12.72596 15.83550 18.95577 3 4.21943 6.84200 9.79684 12.84518 15.93092 19.03530 4 4.75049 7.15069 10.05760 13.00283 16.05744 19.14093 5 5.33882 7.52517 10o26264 13.19787 16.21449 19.27232

0 - 0.484320 - 0.606347 - 0.637989 - 0.649959 - 0.655542 - 0.658488 i - 0.225467 - 0.246795 - 0.257032 - 0.261054 - 0.262956 - 0.263963 2 0.0904794 0.0954562 0.0952398 0.0952550 0.0953214 0.0953891 3 0.0417336 0.0476865 0.0460822 0.0454208 0.0450970 0.0449557 4 0.0209622 0.0297034 0.0280587 0.0272368 0.0268192 0.0265983 5 0.0102417 0.0205632 0.0192720 0.0184409 0.0180105 0.0177506

is the Wigner 3-j symbol. A useful property of the latter is that

•1 TF/2 TF/3 '

unless rn• + m• + ms =0, unless the triangular condition [j•-j,.I- < ja- < j•+j,. is satisfied cyclically by all indices

j•, ja, ja and unless Ira, I • j,. Moreover,

0 0 =0 unless j•+jl+ja =even.

Applying now (A1) for • =p =0 one gets P,•Po=P• =a(m, nl O, 0ln)P•, from which

a(m,n[O, Oln) =1. (A3) For r = 1 and • =0 Eq. (A1) yields P• =xP•(x)=a(m, •0, 11•- 1)Pn• • +a(m,•lO, 11• + 1)Pn• •. However, since xP• • (n + m)/(2n + 1)P•.• + (n m + 1)/(2• + )P•+,, equating of the ri•t-hand sides and using the orthogonal property

1

Pn•u•=O for •r, 1

yields

a(m,•l 0, 11• - 1) =(• + m)/(2• + 1), (A4) a(m,•gO, 11• + 1) =(•- m + 1)/(2• + 1).

Finally, for r =2, • =0,

•• = W(a• • - •)/2 =•(•, •l o, 21 •- +a(•, • o, 2 • •)•f+a(•, • • o, 2 • + 2)P•,

or

a•f = 2•(•, • •0, 21• - 2 )•

+[• +2a(•, • o, 2 •)]•f +2a(•, • • o, 2 • • + 2)•f+•. On the other hand,

2•+1 P•+• +2•+1 •-m+l

=3 (2•+1)(2•+3)[(•- m+2)P•+ (•+1 +m)P•]

+a (2•+•)(2•+a) [(•- •)W+ (•- • +•)•f•];

so that

a(m, n10,21n- 2) = 3(n + m)(n - 1 + m) 2(4n • - 1)

a(m n10,2[n+2)= 3(n- m+l)(n- m+2) (AS) ' 2(2n + 1)(2n + 3) '

(n+l)•._ m •- n'._m •. (2n + 1)(2n +3) +3 4n•._ 1 ' (A6)

A proof of (24) from (23), independent of the properties of the Wigner 3-j symbols, proceeds as follows: Forn-•s ' Eq. (23) yields s'-n-•s-•n+s ' or s'-s-•n, s- s '-• n. Therefore, possible values of n, if any, in this case are found in the interval [ s - s' [-• n •- s'. For

n> s' equation (23) yields n - s' -< s-< n + s' or s - s' -< n -<s + s'. Therefore, further possible values of n in this case are found in the interval s' < n- < s + s', as long as s' TM Is - s' I, or in I s - s' I < n < s + s' as long as s' < I s - s'l. In all cases, therefore, the totality of values of n are those given in (24). In addition, steps of 2 for n are implied by the restriction that s + n + s' is even.

ACKNOWLEDGMENT

The authors wish to thank Dr. E. Lecatsas for his

assistance in the numerical evaluations of the gus.

10. D. Kellogg, Foundations of Potential Theory (Dover, New York, 1953).

2j. Van Bladel, Electromagnetic Fields (McGraw-Hill, New York, 1964).

aM. A. Jaswon, "Integral Equation Methods in Potential Theory. I," Proc. R. Soc. London Ser. A 275, 23-32 (1963).

4G. T. Symm, "Integral Equation Methods in Potential Theory. II," Proc. R. Soc. London Ser. A 275, 33-46 (1963).

5F. J. Rizzo, "An Integral Equation Approach to Boundary Val- ue Problems of Classical Elastostatics," Q. Appl. Math. 25, 83-95 (1967).

•M. A. Jaswon and A. R. Ponter, "An Integral Equation Ap- proach of the Torsion Problem," Proc. R. Soc. London Set. A 273, 237-246 (1963).

VT. A. Cruse and F. J. Rizzo, "A Direct Formulation and Numerical Solution of the General Transient Elastodynamic Problem. I," J. Math. Anal. Appl. 22, 244-259 (1968).

8R. P. Banaugh, "Application of Integral Representations of




Displacement Potentials in Elastodynamics," Bull. Seis. Soc. Am. 54, 1073-1086 (1964).

9F. J. Rizzo and D. J. Shippy, "A Method of Solution for Cer- tain Problems of Transient Heat Conduction," AIAA J. 8, 11, 2004-2009 (1970).

10j. j. Lee, "Wave-Induced Oscillations in Harbors of Arbi- trary Geometry," J. Fluid Mech. 45, 375 (1971).

llF. Raichlen and J. J. Lee, "Response of Narrow-Mouthed Harbors in a Straight Coastline to Periodic Incident Waves," J. Appl. Mech. 38, 572 (1971).

12R. P. Banaugh and W. Goldsmith, "Diffraction of steady acoustic waves by surfaces of arbitrary shape," J. Acoust. Soc. Am. 35, 1590-1601 (1963).

13L. H. Chen and D. G. Schweikert, "Sound radiation from an arbitrary body," J. Acoust. Soc. Am. 35, 1626-1632 (1963).

14H. A. Schenck, "Improved integral formulation for acoustic radiation problems," J. Acoust. Soc. Am. 44, 41-58 (1968).

15G. Chertock, "Sound radiation from vibrating surfaces," J. Acoust. Soc. Am. 35, 1305-1313 (1964).

16L. G. Copley, "Fundamental results concerning integral representations in acoustic radiation," J. Acoust. Soc. Am. 44, 28-32 (1968).

l?G. Chertock, "Solutions for sound-radiation problems by integral equations at the c•itical wavenumbers," J. Acoust. Soc. Am. 47, 387-388 (1970).

18R. P. Shaw, "Retarded potential approach to the scattering of elastic pulses by rigid obstacles of arbitrary shape," J. Acoust. Soc. Am. 44, 745-748 (1968).

19R. P. Shaw, "Diffraction of plane acoustic pulses by obstacles of arbitrary cross section with an impedance boundary condition," J. Acoust. Soc. Am. 44, 1062-1068 (1968).

20R. P. Shaw, "An Integral Equation Approach to Acoustic Radiation and Scattering," in Topics in Ocean Engineering. I! edited by C. Bretschneider (Gult, Houston, 1970), pp. 143-

163.

21R. P. Shaw, "Integral equation formulation of dynamic acoustic fluid-elastic-solid interaction problems," J. Acoust. Soc. Am. 53, 514-520 (1973).

22G. R. C. Tai and R. P. Shaw, "Helmholtz-equation eigenval- ues and eigenmodes for arbitrary domains," J. Acoust. Soc. Am. 56, 796-804 (1974).

2•C. M•ller, Foundations of the Mathematical Theory of Electro- magnetic Waves, English translation (Springer-Verlag, Ber- lin, 1969).

24H. H'dnl, A. W. Maue, and K. Westpfahl, Theory of Diffrac- tion, in Handbuch der Physik 25/J. (Springer-Verlag, Berlin, 1961), pp. 218-573.

25j. D. Kanellopoulos and J. G. Fikioris, "Resonant Frequen- cies in an Electromagnetic Eccentric Spherical Cavity," Q. Appl. Math. (in press).

26j. G. Fikioris, "On the Boundary Value Problem of a Spher- oid," Q. Appl. Math. 31, 143•-146 (1973).

27S. Stein, "Addition Theorems for Spherical Wave Functions," Q. Appl. Math. 19, 15-24 (1961).

280. R. Cruzan, "Translational Addition Theorems for Spheri- cal Vector Wave Functions," Q. Appl. Math. 20, 33-40 (1962). -

29j. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

•0p. N. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

•lE. Jahnke and F. Emde, Tables of Functions (Dover, New York, 1945).

a2A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. P., Princeton, NJ, 1957).

•H. Y. Yee and N. F. Audeh, "Cutoff Frequencies of Eccen- tric Waveguides," IEEE Trans. Microwave Theory Tech. J.4, 487-493 (1966).



acoustic resonant frequencies in an eccentric spherical cavity

Documents