QUARTERLY OF APPLIED MATHEMATICS 51APRIL 1979

RESONANT FREQUENCIES IN AN ELECTROMAGNETIC ECCENTRICSPHERICAL CAVITY*

By

JOHN D. K.ANELLOPOULOS and JOHN G. FIKIORIS

National Technical University of A thens

Abstract. The interior boundary-value electromagnetic (vector) problem in the re-gion between two perfectly conducting spheres of radii /?!, R2 and distance d between theircenters is considered. Surface singular integral equations are used to formulate theproblem. Use of spherical vector wave functions and related addition theorems reducesthe solution of the integral equations to the problem of solving an infinite set of linearequations. Their determinant is evaluated in powers of kd = 2ird/\ to a few terms. It isthen specialized to the axially symmetric case and set equal to zero. This yields closed-formexpressions for the coefficients gns in the resulting relations wns(kd) = w„s(0) [1 + gns{kd)2+ •■■] for the natural frequencies of the cavity. Numerical results, comparisons andpossible generalizations are also included.

Introduction. The interior boundary-value acoustic problem in the region betweentwo spheres of radii , R2 and distance d between their centers (see Fig. 1) for bothDirichlet and Neumann boundary conditions, has been solved elsewhere [1], The presentpaper deals with the corresponding electromagnetic problem between perfectly conductingeccentric spheres and should be read in conjuction with [1], The motivation for consid-ering analytical and exact solutions to such problems, their advantages over numericalones and the possible generalizations to other shapes are discussed in [1] and will not berepeated here. An updated and extensive reference list may also be found in [1]. However,the significant differences between acoustic and electromagnetic cavities that will bebrought to light in this paper require a separate treatment.

As in [1], let small letters , q2) denote points on 5! and S2 and capitals (P, Q) pointsnot on 5, , S2 . A fixed point on Si or S2 is designated by p^ (0! , 00 or p2{62^7 , cj>2), avariable one by q^, <p[) or q^d2, (f>'2), with the primes indicating, in particular, variablesof integration. Under this notation Ri , (or R2, d2, <j)'2) indicate spherical coordi-nates of px (or q2) with respect to centers (or 02) of the surface Si (or S2) to which thepoint belongs. However, pi(Ri2, 9l2, <f>12) or q^R'^ , 621, (p2l) are spherical coordinates ofthe same points with respect to the other center, 02 or , indicated by the secondsubscript. Normal unit vectors npl, iiQ2 on Sj, S2 are directed out of V, as shown in Fig. 1.

The electromagnetic cavity problem in V, with perfectly conducting boundaries and

* Received December 19, 1977. Partially supported by the National Research Foundation of Greece. Theauthors wish to thank Dr. E. Lecatsas for his assistance in the numerical evaluation of the gv, .

52 J. D. KANELLOPOULOS AND J. G. FIKIORIS

S2, can be formulated in terms of homogeneous surface integral equations on , S2 [2, 3,4] for the unknown surface current densities jiiPi)'-

-2ttj\ 00 = npi X [ t/i (<7i) X VG(pi, <70] dSqlJs,

+ "pl X Is I/2 (?2) X7%, <?2)] </S,2 , (1)

-2tt/'2 (j02) = «p2 X f [j\ (qi) X V'<?(/?2, ̂ 0] dSqi

+ np 2 X f fc(?,)XV%,?2)]flt2, (2)J

where

G(/>, 0) = C(/?) = exp (/A:/?)//? (7? = />£) (3)is the free space Green's function and V' operates on the prime coordinates of the point qx(or q2). The integrals involving V'G{pi , q 1) or V'G(p2 ,q2) are convergent singular surfaceintegrals [2, 4, 5].

In the following, analytical solutions of the preceding integral equations are obtained,after properly evaluating the singular integrals by a convenient limiting process. Use alsois made of translational addition theorems for spherical vector wave functions [6]. In thelimit of small kd and in the particular but important case of axially symmetric fields, anexact evaluation of the elements of an infinite determinant and of the determinant itself isachieved that yields the values of the resonant frequencies a>vs(kd) to second order in kd, inexact analogy with the methods of [1].

Solution of the integral equations. The unknown surface current densities j 1 (<71),ji (#2), being tangential to the surfaces Sx and S2 of the spheres, may be convenientlyexpanded in terms of a complete set of complex spherical surface vector functions B, C ofthe angles d, 0 [7], The latter are directly related to the complex spherical eigenvectors m,n, I by the definitions:

rhmn(r, d, <fi) = [n{n + 1 )]l/2zn(kr)Cnm(d, (j>)

= zn{kr) exp (/w0)^ p. (COS (4)

nUr, 6, 0) = n(n + Pmn (0, d>) + [n(n + 1)]"* Z-^ Bmn {d, 0)

exp (im<j>)kr ,n{n + \)zn{kr)Pmn (cos 6)r + Zn(kr)

3? » + K7 (5)

where m = —n, — n + 1, • • • 0, 1, • • • n, zn(x) is the general spherical Bessel function andz%x) = lxzn (■*)]'• In this notation, similar to Cruzan's [6], P'nm(x) (m > 0) is defined by

pnm(x) = (-1 r ~ pmM (6)

RESONANT FREQUENCIES 53

differing by a (— l)m factor from another common definition [7, 8], Anyone of the abovecomplex vectors (m, n, B, C etc.) is related to the corresponding even and odd real vectors,defined in [7, 8] for m > 0, as follows:

cmn = C£n + iC°n , C.mn = (-1 r ~ [C<n - /£*,], (m > 0), (7)

where mmn, nmn , Bmn etc. may be substituted for Cmn . Relations among the surfacevectors P, B, C and their orthogonal properties are as follows:

f X Pmn = 0, r X Bmn = ~Cmn , r X Cmn = Bmn ;

Pmn ' Bmini Pmn ' Cm'n' Bmn ' Cmn = 0; (8)

JJ Pmn ■ Pmn dtt = JJ Bmn ■ Bmn dtt = JJ Cmn ' Cmn dQ

= Jj" Bmn • €m'n' dSl = 0; (9)

JJ Pmn ■ P-m'.n'dil = JJ Bmn ■ B_m,,n,d& = JJ Cmn • C.m',n'dQ

4x (- l)m5mm' bnn' , (10)2 n + 1

where dtt = sin 6 dd d4> and the integration is over the ranges 0 < 6 < ir, 0 < <j> < 2tt\ n, n'= 1, 2, • • • ; m = —n, -n + 1, • • • 0, 1, • • • n; m' = —ri + 1, ■ • ■ 0, 1, ■ ■ • 8nn• = 1 for n=ri and 8nn* = 0 for n 4- n'.

In expanding now jx (qx) and j2 (q2) no loss of generality occurs if only one value of theindex m, namely m = M, is considered. As in [1] this is due basically to the fact that withOi 02 along the z axis the azimuthal angles remain the same (<t>[ = <p[2, <j>i = <t>2i) whenreferred to Oi or 02. Therefore:

/i(<7i)= Z W^+l)]1S= |M|

/2(<?2)= D W^+l)]1s = I Af |

lis («•«).Sfz bMs (d'2, 00 + cMs (d'2, <t>'2)

y 8\X2J X2js\X2)

(11)

(12)

where Xi - kRi, x2 = kR2. The more general case would simply involve a superposition(or further summation over M = —s, —s + 1, • ■ ■ • 0, 1, • • • s).

Use will also be made of the well-known expansion [7]:

6(P, Q) = /g£jpi = ik g ^ (_„r [nt-[n(Q)mmi'(P)

+ «(im,n(Q)«mn (P) + fl(« + 1) P»m,n (Q)f^ (/»)], (r = OP > / = OQ) (13)

for the free space Green's dyadic in terms of the complex spherical eigenvectors. Thesuperscript (1) is associated withy„(x), while (3) implies use of h(x) = hn(x). On thebasis of (7), written for the m, n, I vectors, it is easy to show that (13) is equivalent to thealternative formula involving the even and odd vectors [7], In (13) / is the identity dyadic(A-1 = I-A = A) and in case r (= OP) < r1 (= OQ) one simply interchanges the

54 J. D. K.ANELLOPOULOS AND J. G. FIKIORIS

superscripts 1 and 3 of the spherical eigenvectors, or, equivalently, their arguments Pand Q.

In evaluating the surface integrals of (1) and (2) it is convenient to consider initially theintegral

L(P) = V X f j(q)G(P, q)dSQ = V X f j\q) ■ G{P, q)dSQJ s J s

= fj(q) X V'G(/>, q)dSQ, (14)

where S is a spherical surface of radius a and center O, q(a, 6', 0') is a point on S, P(r, 9, 0)is a fixed point not on S (r > a or r < a), R = qP and j{q) is a vector tangential to S, whichcan be expanded in a series like (11) with the subscript 1 deleted (x = ka). Under theseconditions it is well known [2, 5] that L(P) is an analytic vector function of P(r, 0, 0). Thelast expression for L(P) can be deduced from the first and the relation V X [j(q)G(P, q)] =- j(q) XVG= /(<?) X V'G.

Substitution of (13) into (14) and term-by-term integration yields, in case r > a, termsof the form

riimn (P) f j(q) ■ ri±m]n(q)dO.Q, where dSQ = a2di1q.

For r < a the superscripts of the eigenvectors are interchanged. Noticing that V operateson r, d, 0 of P, that V X [ = 0, V X m = kn, V X n = km and that j ■ Pmn = 0 (j beingtransverse and P radial), one finds

03 2n 4- 1 n CL(P) = ix2 £ Z (-1 )m[n^(P)jn(x) J Rd\ 0') • C_m.„(0\ 4>')dn,

+ tfimn (py^ fj\d\ 0') • 0') daq (15)

where rf££n(q) and ri±„ln(q) have been replaced in accordance with (4), (5). Substitution of(11) (with the subscript 1 deleted) for f(q) in (15) and use of the orthogonal properties (9),(10) yield:

L{P) = 4irix £ [FMsn^(P) + AMsm^(P)], r > a. (16)S= \M\

For r < a a similar procedure yields:

L(P) = Atvix £S= I M|

f hs(x) n hs(x) idj (x) yd(v) Ms (17)

On the basis of (17) and (16) it is now possible to evaluate immediately the non-singular integrals in (1) and (2), respectively:

riD, X. c°

/ t/2(?2) X V'G^J, q2)]dSQ2 = 4irixrfpl X £" S2 n = I A# I

HMn T"7 T nMn (^12 5 ^12 > 012)Jn\X 2>

+ GMn ^ WlMn (P12 > $12 > 012)J n\X 2)

(18)

RESONANT FREQUENCIES 55

where (12) was used instead of (11) and the coordinates of px (Rl2, d12, 012) with respect tocenter 02 have been explicitly shown. Also:

n„2 X/* CO

/ [/i(?i) X V'G(p2 , qi)]dSQi — 4iriXiiiP2 X ^ [pMn^Mn (^21 > $21 > 02i)J Q1 n - I Uln=\M\

+ AMnm{Mn (^21 , $21 - 021 )] ' (19)

In order to evaluate the singular surface integrals in (1), (2) it is convenient to return to(14) and cross-multiply it by hp = rp , wherep{a, 0, 0) is a point on S, on the same radius asP{R, 0,0), with r > a or r < a. It is then possible to make use of the classical result [2, 4, 5]:

fPxf \j{q) X V'G{p, q)]dSQ = lim [rp X L(P)] T 2trj(p), (20)J S P-p

in which the upper/lower sign of the last term corresponds to P approaching p from theexterior/interior of S, i.e., for r > a/r < a. Before the limit is taken, it is possible, asbefore, to use (16) (for r > a) or (17) (for r < a) to obtain

X f [/(<?) X VG(p, q)]dSQ = -2vj(p) + 4irix lim £ [FmJp X ri% (P)J S P-*p s = I Ml

+ Amsrp X m(Ms (^)]

= —2irj(p) + Airix £ ls(s + 0]1/28= |A#|

FMs ^ CMS (0, 0) + AMshs(x)BMs (0, 0)

hs{x)

(21)

X f \j(q) X V'GO, q)\dSq = +2irf(p) + 4irix lim £J -S P-p s= | Ml

FMsjs(x)

hds(x) 1j%x)

rP X n% (P) + A Ms ™ fp X m'Jf. (/>)

= + 4irix £ [.5(5 + l)]1I Ml

/is(*)/1(jy)ws */.(*)

4>) +AMs js{x)BMs{d, 0)] , (22)

in which (4), (5) and (8) were also used. The two results for the singular surface integralare identical, if the relations

f(p) = Z [4*+ D]1/2s * |M|

~ = j8(x)hds(x) - hs(x]jds(x)

are also invoked.

56 J. D. KANELLOPOULOS AND J. G. FIK.IORIS

It is now possible to substitute back into the integral equations (1) and (2). Recallingfrom Fig. 1 that nPi = —rPx , nPi = rPl, it is convenient to use (22) in (1) and (21) in(2). Together with (18), (19) and (12) instead of (11), wherever j2(q2) is involved, onefinally gets:

£ [*fr + 1 )]12[ FMshiX\KiX:]CMs(d, 00 + AMs-0^1/,(xi)BMs(du 0,)] + x2rPis = I A#| XlJs{Xl) Js\* l)

£ HMn W* (*» -e» ' M (/?.., 0», <!>»)n=)M\ L Ja\X2) Jn(X2)

= 0, (23)

Xi fp2 X ^ [F(^21 ? 021 i 021)"t" A Mn^Mn (R21 > 021 » 021)ra = I Af I

+ £ [*(*+1)]1

[ H Mshds(x2)CMs (02, 02) + GmsX^s (x2)BMs(d2, </>2)] — 0' (24)

The second/first term in (23)/(24) contains spherical eigenvectors with respect to originOJOx. They may be expanded into sums of spherical eigenvectors with respect to 0,/02(like the other term of the equation) using the well-known translation al addition theoremsof Cruzan [6]:

m'tin (Rtt , 0,2, 0.2) = £ £ [A"in> (*> - - 00 + K" KV (*. , , 0i)], (25)S = 1 H= -8

"Mn (RU , 012 , 012) = £ £ "m" (*1 , 01 . 0l) + (*. , 0. - 0l)]i (26)= 1 11= -8

Fig. 1. The geometry of the cavity.

RESONANT FREQUENCIES 57

m^n (*2i , 021 , 0«) = £ £S = 1 H= -s

+ (*2, 02 , 02)], (*» > <0, (27)

«(jL (/?«, 02i, 0«) = £ £ [c*^1 (/?2, e2,02)s = l H= -s

+ D^m^f (R2 , S2, <M], (*» > <0, (28)

with

= (-If Z) a(M, n|-/x, j|/»)a(n, 5,p)jp(kd) PMp" {cos d0) cxp{i{M - m)<M, (29)P

BT» = (_iy Z a(M, n\-fi, s\p + 1 ,p)b(n,s,p + 1 )jp+1 (kd) PMp+>i (cos 0O)P

•exp (i(M - n)<t>0), (30)

C"ns = (-1V X a(M> n\~n, s\p)a{n, s, p]jp(kd) PMp~" (cos 8'0) exp (/(A/ - m )<££), (31)P

DTs = (-If 2 fl(^, «I_M, j|/> + l,p)b(n,s,p + l)/p+i (W) (cos 0£)P

■exp (z'(Af - n)4>'o), (32)

where d, 60 , <t>0 are the coordinates of Ox with respect to 02 , d< 6'0, </>i those of 02 withrespect to Ox and the summation index p varies from | n — j | to n + ^ by steps of 2. Finally,the symbols a(M, n\ ~n, s\p), a(M, n \ ~n,s\p + 1 ,p), a(n, .?,/>) and £(«, .?,/?) are defined inthe Appendix, where certain particular values of them are also given. The expressions forBMJl and D^l , Eqs. (30) and (32), differ by a minus sign from Cruzan's values [6], asdiscussed in the Appendix.

With C?! 02 along the z axis one has 60 = ir, 6'0 = 0. Therefore, P"~" (cos 7r) = PMp~» (cos0) = 0 for n f M and Pp(cos ir) = (— Iy = (-l)"+s, Pp(cos 0) = l. As a result theexpressions (29)—(32) simplify to

ATs = (-l)n+sCK = (-l)M + n+s £ a(M,n\-M,s\p)a(n, s, p)jp (kd), (33)P= \n-s\

BTs = (-l)"+s+1 DTs = (-\)M+n+s+1

• £ a(M, n\-M, s\p + \, p)b(n, s, p + \)jp+1(kd), (34)P- Ira — sI

while in (25)—(28) n takes only the value n = M, forcing 5 to start from s > \M\.Substituting in (23), (24), interchanging the summations over s and n and using therelations

r X mMs = [j(.s + l )]I/2 zs(kr)BMs , r X nMs = - [5(5 + l )]1/2 CMs ,

58 J. D. KANELLOPOULOS AND J. G. FIK.IORIS

one finally obtains:

*1 Z [4* + 1)]L 1 J8K-* 1

hn(x2)

— F hS(X\\) s(X\) ^ ,n , \ , A M*i) : (y \ D (Q ^ \Ms V / /v- \ 1 01) ' A Ms V Js\Xl) t>Ms \Vl » <Pl)

XlJs\Xl) J s\x l)

x2 X [^+1)]1/2 £ \Hs=\M\ n = | MI

' Mn

+ BmUs^Xl) B Ms (di , 4> i)

- BWlJ^- CMs (0i , <MXi

jn{x2)

hn(x 2)

rsi M sMs ^MsCl > <Pl)*1

^iWn -a, \M**) ^M?ys(^i) 5Ms (i?! , 00

*1 X t^+D]1/2 X V7

= 0, (35)

Cm" " " Cms ($21 <t>2) DMnshs{x2)B Ms{62, 02)]x2

+ A Mn\CMnhs (x2)Bms(62 , (j)2) — DmI ~~ " Cms ($2 > 4>i)x2

+ Z [^+1)]1/2S-IMI

[ WjijAj(x2) Cms ($2 > <t>2) "F GMsx2hs{x2) BMs (^21 02)] — 0. (36)

Invoking the orthogonal properties of the BUs and CMs vectors over the sphericalsurfaces Si and S2, one finally gets four sets of homogeneous linear equations for theexpansion coefficients AMs, FMs , GMs, HMs ofj\ (Pi),j2 ip2)• The last two, originating from(36), express GMs, HMs in terms of AUs , Fms ■ When substituted back into the first twoequations, originating from (35), they produce two sets of homogeneous linear equationsfor the determination of AMs, FMs • The final formulas are:

CMs=-- i (F„nD&» + AMnC%1),HMs= - ^ £ (FMnC%1 +AMnD%1), (37)x2 n=\M\ x2 n = |Af|

h"s(.Xi) _ y ^id(\■ \ ^ ^Js\Xl) 1/ = I Af | n=|A#|

• tn = I Af I

P hs(x 1) _ y> j V1S I <Y "l £-> M" £->v=\M\ n = \M\

hn(x2) nMnnMv i hn(x2) -Mn/^Mu• / \ D M Mn ' -d/ \ f* M s*- Mn

Jn\x 2) Jn\x 2)

hn(X2) RMnCMv+ WX2)LUx2) BmsCm» + fn(x2)

DMn/^M v _i_ nv- 2' A Mn r\M v&Ms^Mn ' ;d( >. AMsuMn

y . fMvv =\M\

= 0, (38)

hn{x2) jMnnM" 4- hn{x2) nMnrMv• / \ /*Ms^'Mn ' -d/ \ *^Ms^Mn

Jn\x2J Jn\x2)- Z fm

v=\M\

n = \M\= 0- (39)hn(x2) . Mn/^M u i hn(x2) nMnr\Mv

• / \ ^Ms^Mn ' •(// \ ^Ms^MnJn\X 2) Jn\X 2)

Setting the determinant of the coefficients equal to 0 provides the equation from whichthe resonant values kns = w„s/c are determined. For general values of kd one can procedefrom here by numerical methods only. The complications and uncertainties of such anapproach are discussed fully in [1],

For small values of kd, however, an analytical solution is possible. In particular, for kd= 0 it is obvious from (33), (34) and the fact thatyn(0) = 0, n £ 0,j0 (0) = 1 that B%1 = D%ns= 0 and Am1 = Cm" = &ns, where 8ns = 0 for n ± s and <5SS = 1. The last result follows from(25)—(28) if it is noticed that for d = 0 there is no translation and that the eigenvectors m, n

RESONANT FREQUENCIES 59

form a complete and orthogonal set. It follows from (37)—(39) that

GMs = - (xl/x2)AMs, HMs = ~(xJx2)FMs (.s = \M\, \M\ + 1, ), (40)

K(xi) h"s(x2) j N hs(x0 hs(x2) n,AM ~ "0 (clectnc modes); iw " ZST <ma8n,!"c mode>)' <41'

The resonant frequencies ojs„(0) are found from the solution of Eqs. (41) (j = \M\, \M\ +1, • • •; n = 1,2, • • •) and are independent of M (apart from the restriction s > \M\).Moreover, there is no coupling between electric and magnetic modes. All these results arewell known from separation of variables.

For kd ^ 0, but small, it is observed from (33), (34) and the relations

2"n' (kdVjn{kd) ̂ (2n + *i)! W[1 + °(*2rf2)J for » + Mkd) 1 - + 0(kW)

(42)that for n =)=.? and n = s

AM? , CK ~ (kdy-'i [1 + O(k'cF)]- BTs , DM? ~ (W)'"-S' + I[l + 0(*W)]. (43)These results, as in [1], simplify Eqs. (37)—(39) a great deal, but not to the extent achievedin [1]. The reason is that for M ^ 0 the coefficients AMs remain coupled to the FMs (an m orn mode is expanded under translation into both m and n modes, unlike the scalar case,Eqs. (20) or (21), (22) in [1]). Omitting details, it is found that in the determinant D{ans) ofthe coefficients of AMs, FMs terms of order kd that must be retained are found not onlyalong the diagonal (ass) and the ones next to it (as,s+i ; as+i,s), as in [1], but, in addition,along the next two diagonals (as,s+2 ; as+2,s)- The evaluation of D(ans) in such a case canstill be carried out, as will be shown in a forthcoming publication on the exterior(scattering) problem. However, here these difficulties can be avoided if one restricts theexamination to the case M = 0. It is to be remembered here that the axial symmetry of theconfiguration makes this the most important case, since azimuthal dependence (M ^ 0)may be introduced only by an initial non-axisymmetric impressed field that generates theoscillations inside the cavity, not by the configuration. For M = 0 the m and n modesbecome completely decoupled. Indeed, from (34) and (A.5) of the Appendix (and forunrestricted values of kd) one has:

a(0, n\0, s\p + l,p)= 0; = Z?„°s" = 0. (44)

These values imply that in (25)-(28) the m/n modes are expanded under translation intom/h modes only, while (37)—(39) simplify to:

Gos = - ^ E Co- Aon ; H0s=-^± Fon (^ = 1,2,--), (45)•*2 n = l %2 n = 1

A hj(Xl) _ y y K(X2) 0n ̂ _ „A°s jd(Y \ L Aov :d(v \ Aos C0n - U, (46)

J s\* 2) i/=i n = 1 Jn\X 2)

p hs{X i) _ y y hn(X2) 0n _0s ; (Y \ : /v \ A0s U. (47)

Js\-X l) v = 1 n = 1 Jn\X2)

In these equations the indices 5, n start from n, s = 1 (not 0), since m00 = 'ho = 0, as seenfrom (4), (5). Both (46) and (47) are similar to (29) or (67) in [1] and can be put into the

60 J. D. K.ANELLOPOULOS AND J. G. FIK.IORIS

forms

X as,Ao„ = o, jtfs-For = 0: f,s= 1,2, (48)V=1 V=\

_ hs(xi) _ fln(x2) 0n^Qs _ _ y hn(x2) J Qn£0V / / \ss ,-d/v \ /d/v \ ' "s" ;d/v \ "^Os^On V* T V

ysV-^U n = l Jn\^2) n = l Jn\X2)

r _ /?s(*l) _ hn(X2) .0n ^0s r _ __ hn(X2) . on/^Ou /„ J. \ fSfhJss • / x ^ ■ / \ ^Os ^On 5 Jsv ~ ^ • / \ ^Os *-0n Wv/

Js\Xl) h=l JnyXi) n=l Jn\X2)

similar to (30)-(32) in [1], From here on it is obviously possible to treat only the secondcase for the coefficients fsv (magnetic modes), and obtain results for the asi, (electric modes)by mere substitution of zn(xi), zn(x2) by Zn(*i), z^(x2), respectively, zn being eithery„ or hn.

For small kd, reference to (33), (42), (43) shows thatAS? C& ~ (W)'s-n| + '"-nl[ 1 + 0(W], (51)

implying that

/„ = Css + C"s(kd)2 + 0(k4d4)- U = CUkdy*-">[ 1 + 0(k2d2)], {v * s), (52)

in which the C's are independent of kd. From this point on the procedure follows stepsidentical to [1], leading to an explicit evaluation of the determinant D(fsv) to order (kd)2(Eq. (41) in [1]) and to a closed-form expression for the coefficients gvs in the relationw„(rf) = w„s(0)-[l + gvs(kdf] for the resonant frequencies. The latter (Eq. (55) in [1]) isrepeated here for convenience:

gvsdCU-Xi)

-Xl dx i -1c c c cI ^v,v-l,l> s~ilf -l / i \ /f 1\L~y, + —7 CmJ (f,s= 1,2, ••■). (53)

where, in differentiating dCvXxi, x2)/dxi, one writes x2 = rxi with r = xjx^ = RJRi =constant.

There remain the explicit expressions for the constants C. They are obtained from (33),(42), (51) and (52):

{ — c A- C" llrl\2 4- DtL-*/1*\ = _ hnjxz) .0n(^0n _ hn-i(Xi) AO.n-lronJ nn ^ nn ' ^ nn\KU) i U ) • / \ • / \ ^On^On • / \ ^*0 n *-0,n-l

Jn\Xi) Jn\X2) Jn-l\X2)

- tHtY ASnn+1CTn+1 + O(kW), (54)Jn + 1\X2)

r _ _ hn(X2) a onfO,n+l _ ^n + ljXj) 0,n+l /"0,n+l i n>( b3J3\Jn,n +1 • / \ ■^0n*-0n • / \ n ^0,n+l ' vy^/v u )

Jn\X 2) Jn + l\X2)

= C',n+1(kd) [1 + 0{k2d2)], (55)

r _ _ hn(Xj) 0n „Qn _ hn+ i(X2) ,0n+1 ^0n , f-.,, i»iJ n + l,n ■ , \ ^ o,n+l*- On • / \ ^O.n + 1 o,n + l ~ " )

J n\X2) Jn+l\X 2)

= c;+l,n(kd)[ 1 + Oik'd2)]. (56)

Using from the Appendix formulas for a(0, n|0, s\p)a(n, s, p) one gets:

A°0nn CZ = [a(0, n |0, n\0)a(n, n, 0) (1 - k2d2/6)

RESONANT FREQUENCIES 61

+ a(0, «10, n\ 1 )a(n, n, 2) (kd)2/15 + 0(k4d4)]2

= tM£^ \2 I + 2("2 + n - 3)+ 0(k4d4), (57)(2n + 3) (2« — 1) J

A°oWn~l CT,n-i = [-fl(0, n - 110, n\\)a(n - 1, n, l)(kd/3)+ 0(k3d3)][a(0, n\0, n - 111)

■a(n, n - 1, 1) (*<//3) + 0(k3d3)] = (W)2 + 0(k4d4), (58)

^oV+1C°0%+1 = [-a(0, n + 110, n\\)a(n + 1, n, \)(kd/3) + 0(k3cP)] [a(0, «|0, n + 11 1)

■a(n, n+ 1, 1) (kd/3) + 0(k3d3)] = — 3) (W)2 + 0(k4d4)-

(59)Therefore:

. ̂ b ( y„\ n (n (= /„„(0), (60)

r = hnjXi) _ hn(x2) = .jn(.X 1) 7„(X2) '

M*i) _ nn(x2).jn{x 1) y„(x2) J

c„ _ (2n + 3) (In - 1) + 2n2 + 2n - 6 h„(x2) _ n2 - 1 t(;t2)3(2/7 + 3)(2n - 1) 7^2) 4/j2 - 1 jn-i(x2)

_ n(n + 2) hn+l(x2)(2n + 1) (In + 3) jn+i(x2)

n2 — 1 n(n + 2)An2 — 1 (2n + 1) (2n + 3) _

_ n(n + 2) nn+1(x2)(2n + 1) (2/7 + 3) jn+i(x2)

Also, following similar steps:

nn(x2) _ n2 - 1 nn-i(.x2)jn(x2) 4/72 - 1 jn-l(x2) (61)

= n + 2n'n+1 2« + 3

r, _ «n+1'n 2/7 + 1

hn+i(x2) _ hn(x2)-jn+ l(x2) jn(x2) J

6n-uCx2) _ M*2)-jn + l(X2) jn(x2) .

. n + 2" ' 2h + 3

2/7 + 1

-jn+l(x2) jn(x2) .

nn+i(x2) _ wn(JCa)

jn + l(X2) jn(x2) .

(62)

(63)

As in [1], one obtains the remarkable result that all the C's turn out to be imaginary. Thismeans that all elements /„, of the determinant D(jvs), at least to order (kd)2, are imaginary,and therefore that the cvns(d) are real, as predicted by theory for all cavities. This is a veryconvincing check on all relations and results obtained in this paper.

For the electric modes one simply replaces jn(x) and nn(x) in (60)-(63) by jdn(x) =[xjn(x)]' and ndn(x) = [jc/7„(jc)]'. Comparing with the results of [1] for M = 0 one observesthat the uns(d) for the magnetic modes are no longer the same with those of the Dirichletscalar case, as happens to be true for concentric spheres (d = 0).

Numerical results. The roots (xi)„s (v = 1,2, ■■•;s= 1,2, •••) of (41) were obtainedby a modified Newton-Raphson method, called the Regula-Falsi method. In the case ofthe magnetic modes these roots are the same as those of the Dirichlet scalar problem andhave been checked against tabulated values, as explained in [1], For M = 0 the roots (*i)„s

62 J. D. KANELLOPOULOS AND J. G. FIKIORIS

for the magnetic modes and the corresponding g„s from (53) are given in Tables I-IV forfour values of the ratio r = xjxx = RJR u namely, r = 1.2-1.35 - 1.5-2.0. Similar valuesfor the electric modes and for M = 0 are found in Tables V-VIII.

One general observation is the fact that the g„s are almost independent of the order ^ ofthe resonant frequency; in other words, the percentage change of u^d) from «„s(0) isalmost the same for all 5 (same v), particularly for s > 2. As far as the complications of adirect numerical evaluation of the roots of the equation £>(/„„) = 0 are concerned, thereader is referred to the detailed discussion of this aspect in [1],

Reference to [1] should also be made concerning the possible generalization of thepresent approach to exterior problems (scattering by an eccentrically coated sphere is atpresent under investigation), as well as to other geometrical configurations.

Recently, cylindrical geometries were considered, in particular eccentric waveguidesand eccentrically coated waveguides. Our results were found in excellent agreement withthose of [9], in which the cutoff frequencies of the lower TM and TE modes were obtainednumerically and experimentally for eccentric, perfectly conducting waveguides. For smallkd agreement to four decimals was obtained. Even for the largest kd considered in [9] theagreement extended to the first two decimals, an indication that the restriction kd « 1 ofthe present method is not really as severe as it may appear. This is further corroborated bythe fact that the gvs get smaller quickly with increasing ratio r, as seen from the tables.Besides, when r is small and both g,,s and Xi become relatively large, kd gets necessarilysmall due to the physical restriction d < R2 — Ri or kd < (r - I)*! . This application towaveguides will be the subject of a forthcoming paper.

Appendix. The symbols a(M, n\ - n, j|/j), a(n, s, p) etc. appearing in (29)—(32) aredefined as follows [6]:

a(m, s\p) = (—1 )m+^(2p + 1) (n + m)l(s + nV(P ~ m — n)l_ (n — m)\(s - m)!(/> + m + n)\_

n s pL0 0 0.

n s pm n —m — n_ (A. 1)

a(m, n\n, s\p + 1, 9) = ( — 1 )m+"(2p + 3) (n + m)l(s + n)\(p + 1 — m — n)\__ (n — m)\(s - n)\(p + 1 + m + n)\_

n s qL 0 0 0.

n s p + 1m fx — m — n (A.2)

a{n, s, p) = ;'p+s_"[2i'(i + 1) (2s + 1) + (s + 1) (n - s + p + 1) {n + s - p)

- s(s - n + p + 1) (n + s + p + 2)]/[2s{s + 1)], (A.3)

b(n, s, q) = /s+?""[<72 — (5 — «)2]1/2[(^ + n + I)2 - ^2]1/2(2j + l)/[2^ + 1)], (A.4)

where

ji jz jz_ w, m2 a«3_

is the Wigner 3 — j symbol. Some useful properties of the latter are: the symbol is 0 unlessm, + m2 + m3 = 0, unless the triangular condition \jl — j2\ < j3 < j^ + j2 is satisfied

RESONANT FREQUENCIES 63

Magnetic modes (M = 0)Table I. r = x2/x, = 1.2.

s=l s = 2 s=3 s=4 s=5 s=6

v = 1 15.76063 31.44237 47.14150 62.84508 78.55033 94.25652 15.86550 31.49526 47.17682 62.87154 78.57154 94.27419

(x,)„ 3 16.02151 31.57444 47.22976 62.91128 78.60335 94.30074 16.22717 31.67971 47.30025 62.96423 78.64574 94.336045 16.48061 31.81082 47.38822 63.03036 78.69869 94.38019

v=\ -2.94460 -2.98393 -2.98868 -2.98745 -2.98846 -2.979932 .7531856 .7275187 .7258693 .7285967 .7340752 .7403467

6, 3 .5301020 .5099873 .5074118 .5089340 .5115610 .51583214 .3564039 .3393728 .3365035 .3368490 .3381674 .34042835 .2524817 .2378851 .2350669 .2348713 .2360829 .2375913

Table II. r = x2/x, = 1.35.

s=l s=2 s=3 s=4 s=5 s = 6

v = 1 9.057011 17.99305 26.95539 35.92452 44.89639 53.869622 9.216926 18.07494 27.01022 35.96571 44.92936 53.89711

(x.X, 3 9.451597 18.19702 27.09224 36.02739 44.97876 53.938314 9.755591 18.35861 27.20123 36.10947 45.04456 53.993195 10.12253 18.55865 27.33688 36.21183 45.12667 54.06172

s=l s=2 s=3 s=4 s=5 s=6

v — I -1.04854 -1.08817 -1.09551 -1.09844 -1.09953 -1.100422 .2976899 .2714140 .2665779 .2650669 .2642967 .2632253

g™ 3 .2090797 .1917095 .1874811 .1858838 .1851030 .18478044 .1406013 .1285457 .1249803 .1235807 .1229980 .12253095 .0987714 .0909096 .0878274 .0865854 .0860460 .0856145

Table III. r = x2/x! = 1.5.

s=l s=2 s=3 s=4 s=5 s=6

f= 1 6.38580 12.61895 18.88476 25.15918 31.43708 37.716732 6.58613 12.72351 18.95504 25.21203 31.47942 37.75204

(x, i,B 3 6.87535 12.87882 19.05999 25.29113 31.54283 37.804954 7.24246 13.08311 19.19908 25.39622 31.62720 37.875395 7.67580 13.33421 19.37160 25.52701 31.73236 37.96326

v=\ -.548217 -.585778 -.593381 -.596055 -.597151 -.5975662 .1751128 .1529077 .1477128 .1460435 .1453082 .14517013 .1207802 .1077643 .1038657 .1023786 .1018547 .10159024 .0795457 .0726922 .0695374 .0682722 .0676965 .06741945 .0541408 .0516278 .0491016 .0479806 .0474355 .0471801

64 J. D. K.ANELLOPOULOS AND J. G. FIKIORIS

Table IV. r = = 2.0.

s=l s = 2 s=3 s=4 s=5 s=6

v = I 3.28600 6.36067 9.47718 12.60587 15.73963 18.875972 3.55578 6.51306 9.58126 12.68459 15.80286 18.92877

(x.X. 3 3.92252 6.73556 9.73553 12.80189 15.89729 19.007744 4.35839 7.02183 9.93792 12.95687 16.02247 19.112605 4.84099 7.36467 10.18591 13.14838 16.17778 19.24300

e=l -.154770 -.185607 -.193237 -.196091 -.197428 -.1981442 .0701890 .0573415 .0524788 .0505131 .0495400 .0490610

g,,8 3 .0429004 .0402039 .0371142 .0356451 .0348920 .03446514 .0247884 .0270546 .0252153 .0240656 .0234315 .02306185 .0144915 .0188940 .0180254 .0171499 .0166155 .0162764

Electric Modes (M = 0)Table V. r = x2/x, = 1.2.

s=l s=2 s=3 s = 4 s=5 s=6

v = 1 15.76117 31.44230 47.14152 62.84504 78.55033 94.256512 15.86712 31.49547 47.17688 62.87157 78.57155 94.27419

(x.V, 3 16.02478 31.57485 47.22988 62.91133 78.60337 94.300714 16.23269 31.68040 47.30045 62.96431 78.64578 94.336065 16.48902 31.81185 47.38852 63.03048 78.69876 94.38023

s=l s=2 s = 3 s=4 s=5 s=6

v = I -2.91148 -2.97116 -2.97599 -2.96962 -2.95778 -2.944462 .7471076 .7255597 .7258737 .7286365 .7364067 .74481913 .5260365 .5085578 .5066550 .5082459 .51 13348 .51536374 .3540509 .3384358 .3363047 .3370900 .3385343 .34114055 .2511180 .2373111 .2348577 .2349251 .2356447 .2372174

Table VI. r = x2/x, = 1.35.

s=l s=2 s=3 s=4 s=5 s=6

v = I 9.059273 17.99332 26.95547 35.92456 44.89641 53.869632 9.223809 18.07577 27.01047 35.96581 44.92941 53.89714

(x,V. 3 9.465746 18.19879 27.09276 36.02761 44.97888 53.938374 9.779913 18.36159 27.20211 36.10984 45.04475 53.993305 10.16033 18.56318 27.33821 36.21238 45.12695 54.06189

v = I -1.01467 -1.07988 -1.09213 -1.09625 -1.09828 -1.099502 .2935301 .2707237 .2663895 .2646275 .2638187 .2637762

g„ 3 .2067111 .1905825 .1868240 .1856129 .1849674 .18448134 .1398599 .1278972 .1246793 .1233498 .1228392 .12261165 .0989798 .0905222 .0876058 .0864837 .0858591 .0855934

RESONANT FREQUENCIES 65

Table VII. r = xa/x, = 1.5.

s = 1 s = 2 s = 3 s = 4 s = 5 s = 6k= 1 6.3911 1 12.69165 18.88497 25.15926 31.43712 37.71676

2 6.60254 12.72563 18.95567 25.21230 31.47956 37.75212(x,U 3 6.90955 12.88312 19.06126 25.29166 31.54311 37.8051 1

4 7.30225 13.09045 19.20122 25.39712 31.62766 37.875655 7.77013 13.34553 19.37485 25.52837 31.73305 37.96366

v = I -.511572 -.577275 -.589400 -.593255 -.594819 -.5955242 .1730606 .1514145 .1469667 .1455730 .1450011 .14503663 .1203757 .1068492 .1033312 .1020863 .1015466 .10138124 .0805281 .0722090 .0692174 .0680568 .0675526 .06737075 .0557589 .0514180 .0489064 .0478255 .0473364 .0470939

Table VIII. r = x2/x, = 2.0.

s=l s=2 s=3 s=4 s=5 s=6

v = I 3.30922 6.36402 9.47820 12.60631 15.73986 18.876102 3.63013 6.52362 9.58440 12.68592 15.80353 18.92917

(x,)« 3 4.08033 6.75823 9.74207 12.80460 15.89867 19.008534 2.95404 7.06326 9.94941 12.96154 16.02481 19.113945 3.53114 7.43401 10.2043 13.15567 16.18139 19.24505

v=\ -.109733 -.175373 -.188952 -.193740 -.195908 -.1969862 .0739110 .0569686 .0519261 .0500943 .0492965 .0488182

g. 3 .0441847 .0405678 .0368313 .0353987 .0346946 .03432934 -.013230 .0281633 .0251838 .0239423 .0233305 .02297995 -.009227 .0205987 .0182036 .0171116 .0165553 .0162232

cyclically by all indices ju j2,j3 and unless \mn\ < jn. Moreover,

= 0J1 Ji J 30 0 0

unless j\ + j2 + y'3 = even.It is this last property that makes

a(0, n\ 0, s\p + 1, p) = 0, (A.5)

since it contains, according to (A.2), the factors

n s p0 0 0

n s p + 10 0 0

The symbols a(w, n\n, s\p) appear also in the expansion

P™ P* = £ a(m, n\n, s\p)P™+" (p = n + s, n + s - 2, • • • \n - j|). (A.6)p

In particular, for m = n = 0:

66 J. D. K.ANELLOPOULOS AND J. G. FIK.IORIS

»(o, „i o, ,i„) - (?, +1)

[(« + i + p)/2]![(n + s- p)/2]\[(n + p- j)/2]![(j + p - n)/2]\ (A.7)

(A.9)

an equation that has been used extensively in the derivation of (57)—(59).Finally, an independent check of the minus sign correction in Cruzan's formulas for

B^sn, D^n (Eqs. (30) and (32) in this paper) can be provided by evaluating the rectangularcomponents of the rhmn vector and comparing them with those of the me/omn given in [7],Indeed, the rectangular components of mmn are easily obtained if one starts by applying(25) for Ri = 0 and R12 = d. Then:

mllKdJ„,0o)= Z Bfftd, B0, ./.oKV (0), (A.8)(X = -1,0,1

a result based on the fact that

na-i,i(0) = ~(x - iy)/3; n%(0) = 2z/3; «'V,(0) = 2(x + iy)/3

(0) = 0 (all ij., 5); nJ,V(0) = 0 (unless s = 1, n = — 1, 0, 1).

These results follow easily from (4), (5) and the well known relations among spherical (r, #,0) and rectangular (x, y, z) unit vectors. Substitution of (A.9) into (A.8) yields therectangular components of m^] in terms of the symbol (p. = —1,0, 1). This, as ex-plained above, verifies the necessity of the sign correction in (30) and (32).

References

[1] J. D. Kanellopoulos and J. G. Fikioris, Acoustic resonant frequencies in an eccentric spherical cavity, J.Acoust. Soc. Am. 64, 286-297 (1978)

[2] C. Muller, Foundations of the mathematical theory of electromagnetic waves, Springer-Verlag. Berlin (Englishtranslation). 1969

[3] H. Honl, A. W. Maue and K. Westpfahl, Theory of diffraction, Handbuch der Physik 25/1, pp. 218-573.Springer-Verlag, Berlin, 1961

[4] J. Van Bladel, Electromagnetic fields, McGraw-Hill Book Co., New York, N. Y., 1964[5] O. D. Kellogg, Foundations of potential theory, Dover Publ., New York, N. Y., 1953[6] O. R. Cruzan, Translational addition theorems for spherical vector wave functions, Quart. Appl. Math. 20,

33-40 (1962)[7] P. M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill Book Co.. New York. N. Y.,

1953[8] J. A. Stratton, Electromagnetic theory, McGraw-Hill Book Co., New York, N. Y., 1941[9] H. Y. Yee and N. F. Audeh, Cutoff frequencies of eccentric waveguides, IEEE Trans, on MTT 14, 487-493

(1966)