THE AFCRL-64-684 UNIVERSITY 3648-7-T OF MICHIGAN ITERATIVE SOLUTIONS OF THE HELMHOLTZ EQUATION by R.E. Kleinman August 1964 Report No. 3648-7-T on Contract AF 19(604)-6655 Project 5635 Task 563502 3648-7-T = RL-2107 Prepared for AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSPACE RESEARCH UNITED STATES AIR FORCE BEDFORD, MASSACHUSETTS - -

-- THE UNIVERSITY OF 3648-7-T MICHIGAN Requests for additional copies by Agencies of the Department of Defense, their contractors, and other Government agencies should be directed to: DEFENSE DOCUMENTATION CENTER (DDC) CAMERON STATION ALEXANDRIA, VIRGINIA 22314 Department of Defense contractors must be established for DDC services or have their "need-to-know" certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to: U. S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON, D. C. 20230 ii

THE UNIVERSITY OF MICHIGAN 3648-7-T TABLE OF CONTENTS I. Introduction 1 II. A General Representation Theorem 5 II. Application of the Representation Theorem to a Class of Scattering Problems 12 IV. The Iteration Procedure 17 V. An Example: Scattering of a Plane Wave by a Sphere 25 REFERENCES 44 APPENDIX: Convergence of the Neumann Series 47 -- iii

THE UNIVERSITY OF MICHIGAN 3648-7-T I INTRODUCTION This report presents a new approach to the solution of the problem of scalar scattering by three-dimensional finite bodies. Specifically, a method is found, of general applicability, whereby the solution of the static potential problem for a Dirichlet boundary condition on a particular surface is transformed, by successive operations, into the solution of the scalar Helmholtz equation satisfying the same boundary conditions. Lord Rayleigh (1897) considered the relationship between potential problems (boundary value problems for the Laplace equation), and scattering problems (boundary value problems for the Helmholtz equation). In a typically virtuoso performance, Rayleigh considered two as well as three dimensional problems in the electromagnetic (vector) as well as acoustic (scalar) case. In particular he showed that the potential of an obstacle in a uniform field not only was the near field limit of the solution of the corresponding scattering problem but also could yield the first term of an expansion of the far field. He gave explicit results for a general ellipsoidal scatterer including many limiting cases of interest such as the sphere, spheroid and disc. Since that time, considerable effort has been spent in deriving higher order terms in the expansion of these as well as other shapes. Long sought in this work is the development of a systematic procedure which will generate the solution of the Helmholtz equation, satisfying a particular boundary condition, from the solution of 1

THE UNIVERSITY OF MICHIGAN 3648-7-T Laplace's equation which satisfies the same boundary condition. It is toward the achievement of this goal that the present work is directed. The major drawback in most of the methods proposed heretofore is their intrinsic dependence on a particular geometry. That is, the techniques result from the (often adroit) exploitation of the geometric properties of the surface on which the boundary conditions are specified. Thus, restricting attention to three dimensional scalar problems, we find a variety of methods for obtaining the low frequency expansion for a disc (and an aperture in a plane screen); see Bouwkamp (1954) and Noble (1962) for an extensive bibliography to which we may add Heins (1962), de Hoop (1954 Senior (1960) and Williams (1962a). However, success in generalizing these techniques has been limited to a class of axially symmetric problems, (Collins (1962), Heins (to be published) and Williams (1962b)), and explicit results have been obtained only for a spherical cap (Collins (1962) and Thomas (1963)). For those shapes where the Helmholtz equation is separable, of course, the low frequency expansion may always be obtained from the series solution provided sufficient knowledge of the special functions involved is available. A method for obtaining low frequency expansions for bodies which are intersections of such "separable" shapes has been proposed by Darling (1960) though as yet has been applied only to a spherically capped cone (Darling and Senior, to be published). Most low frequency techniques, however, have as their starting point the formulation of scattering problems as integral equations using the Helmholtz repre sentation of the solution in terms of its properties on the boundary and the free space 2

THE UNIVERSITY OF MICHIGAN 3648-7-T Green's function, e.g. Baker and Copson (1950). This formulation is also vital to the proof of the existence of solutions for a general boundary given by Weyl (1952) and Miiller (1952) as well as that of Werner (1962). Noble (1962) shows how this integral formulation may be used to obtain a representation of the solution of a scattering problem for a general boundary as a perturbation of the solution of the corresponding potential problem. Each term in the low frequency expansion is the solution of an integral equation which differs only in its inhomogeneous part from term to term. However, this formulation does not yield an explicit representation for successive terms in general except as the formal inverse of an operator. The present work describes a method whereby the solution of the general Dirichlet problem for the three dimensional Helmholtz equation is explicitly expressed in terms of the Green's function for the corresponding potential problem. A new integral equation for the scattered field is derived whose kernel is the potential Green's function for the surface instead of the free space Green's function for the Helmholtz equation. Despite the fact that the integral operates over all space, rather than just the scattering surface, and is really an integro-differential operator, it is still possible to solve the equation iteratively in a standard Neumann expansion which has a nonzero radius of convergence and may be interpreted as a partial summation of the low frequency expansion. The results are valid for complex as well as real values of wave number, k, with no restriction on the sign of the imaginary part provided k is sufficiently small in absolute value. The present work also provides a 3

THE UNIVERSITY OF MICHIGAN 3648-7-T constructive proof of the existence and uniqueness of solutions of the Dirichlet problem for the Helmholtz equation based on the existence and uniqueness of the potential Green's function. The results stem from an integral representation of functions which are regular at infinity in the sense of Kellog (1953). This representation, which is a direct consequence of Green's theorem, is derived in Section 2. Wave functions, i. e. solutions of the Helmholtz equation which satisfy a radiation condition, are not regular. However it is possible, using an expansion theorem (Wilcox, 1956b), to modify them so that the representation theorem applies. This is done in Section 3 where a new integral equation for wave functions is derived. In Section 4 this equation is solved iteratively as a Neumann series and the relation between this series and the Rayleigh expansion is given. As an illustration and a check, the method is applied to the classic problem of scattering by a sphere in Section 5. This example serves not only to corroborate the analysis but also provides further insight into the manner in which the truncated Neumann series, i. e. the Nth iterate, approximates the solution. A rigorous proof of the convergence of the Neumann series is given in the Appendix. 4

THE UNIVERSITY OF MICHIGAN - 3648-7-T II A GENERAL REPRESENTATION THEOREM We begin with a statement of Green's theorem or Green's second identity (e.g. Stratton, 1941, p. 165) which states that if u and w are twice differentiable functions of position everywhere in a closed region of space, V, bounded by a regular surface S then u2 _ ] udV \ a - A dS] ' (2.1) V LS where the normal derivative a/an is directed out of the volume V. We remark that the conditions for the validity of (2.1) may be weakened but for present purposes we consider only smooth regular surfaces, deferring consideration of bodies with edges. We choose to consider the surface S as consisting of a small sphere, S1, with center at xl, y1, z1; an arbitrary smooth surface, S2, not necessarily connected, consisting perhaps of a finite number of closed smooth surfaces; and a large sphere, S, containing S1 and S2. Further, we erect a rectangular Cartesian coordinate system with origin in S2, see Fig. 1. 2' The usual procedure in formulating an integral equation for a wave function then involves identifying one of the functions in (2.1) with the free space Green's function, e~ /R, and the other with the field scattered by S. The integral over S is shown to vanish by virtue of the radiation condition and the integral over S evaluates the scattered field. The volume integral vanishes since both functions are The sign ambiguity is removed with a particular choice of harmonic time factor. 5

I THE UNIVERSITY OF MICHIGAN 3648-7-T chosen to be solutions of the homogeneous Helmholtz equation yielding the well known result (XY1) aw a (i kR. n W- an BR dS. (2.2) "3 FIGURE 1 Here, however, we wish to employ, not the Green's function for the Helmholtz equation (though this is our final goal) but the Green's function for the potential 6

THE UNIVERSITY OF MICHIGAN 3648-7-T (Laplace) equation. Thus we identify u in equation (2.1) with a function of the following form u = - R + U 2 2 2 2 where R = \x-xl) +(y-y1) +(z-zl) and V u = 0 even at R = 0, i.e. u1 is regular everywhere in V in addition to the interior of S1. Equation (2. 1) then becomes -+u V2 -+ [(wa ( + u)] dS v s 23 (2.3) If w is assumed regular in the interior of S1, then the integral over S1 may be evaluated in the usual way; first integrating then letting the radius of S approach zero. Clearly the regular part, au-w dS-7 [1 an a-1 and, since on S an- =- R ' 1 On OR'I 27r R O0 de O 1 R o(x s sin0, y +Rcos sin si, nOz +Rcos 0 0( - W(x +Rsin cos O, y +Rsinesin(, z1+RcosG0)a 4' = o(xl,yl,z ) (2.4) 7

THE UNIVERSITY OF MICHIGAN 3648-7-T Hence (2.3) becomes - WV - ul Vd a tw(Xl'Yl' Zl) = 4 R 47rR +u an 47rR J 4+R - S2+S3 a 1. dS (2 5) an 47rR Now we further specify ul so that ( + u p =0, that is + u 1 47rR Vs 47rR 1 2 is the static Green's function of the first kind (Dirichlet condition) for the surface S We shall denote this function hereafter as G (p,1) = - 4R +u (2.6) where the dependence on two points is indicated. The notation is a shorthand for Go(p,P1) G (x,y, z, x1,y1 z) = G e[r,,,rl01, (2.7) The volume or surface symbols V and S will be used as subscripts to indicate the variables of integration. With this in mind, equation (2.5) may now be written W(pl) = Go(pV, Pl) V (pV) dV + W(PS) (PS 'n G(Ps P) d V S2 [Go(P'P) ( - W(P) an G(ps' Pl dS S3 (2.8) 8

THE UNIVERSITY OF MICHIGAN 3648-7-T Recall that thus far, w(p) is arbitrary except for differentiability requirements. On the other hand, G (p, p1) is completely specified and one of its more i portant properties is the fact that it is "regular at infinity" in the sense of Kellog (1953, p. 217), i.e. I r, I I lim lim 2 Co r-o rG < o an r < a (2 9) where r is the radial polar coordinate of the point p (see (2.7)). Let us now consider the integral over S3. It is our aim to let the radius of S3 increase so that the volume under consideration is all of 3-space exterior to the surface S2. If G were a Green's function for the wave equation and w were a wave o function then the radiation condition would imply that the integral over S vanish. However G is not a wave function and since we would still like the integral over S3 to vanish, we must determine the requirements on w which will accomplish this. Since S is a sphere of radius r with center at the origin, n = r and we wish S3 3 to determine conditions on w so that lirm 1 [(PS, a W(p -(p) G(p pj dS -- \ o o(PS' PI an (PS) - (PS an 0pS' dS S3 27r r lim 2 a a -= lr dp dO r sinO GO(P, p) r w(p) - @(p) r G P'P = (2.10l 0 0 9

THE UNIVERSITY OF MICHIGAN - 3648-7-T r-lim d0 dO r2sinO [G(P, pl ) -r p) ( a G o Pj r J ---: o [o y27r or o lim dO f\d -r G (p, p1) ar w(p) +| r (p) ar Go(p P) (2.11) and conditions sufficient to force this limit to vanish are lim |2 r -m Ir2Go(ppl) ar O(p) =0 and (2.12) lim 2 rlim Ir(p) ar G(p'pl) = 0 r —>- oDar =0 Rewriting these equations slightly as follows, lim 1 lim r li 1 I 2 G r-=-oo r ar r — oo r ar (2.13) clearly indicates that requiring w(p) to satisfy the same regularity conditions at infinity as does Go(p, P1) (i.e. eq. (2. 9)) is certainly sufficient to guarantee the validity of (2.10). Thus the contribution of the integral over the large sphere S3 is nugatory provided w is regular at infinity. We note in passing that this condition on w(p) may be weakened without invalidating the result but since the functions with which we shall eventually be concerned satisfy the stronger condition, we defer consideration of this refinement. 10

THE UNIVERSITY OF MICHIGAN 3648-7-T We may summarize the results obtained thus far in the following theorem: Let V be the volume exterior to S, the union of a finite number of smooth, closed, bounded, disjoint surfaces, and let G (p, P1) be the static Green's function of the first kind for this surface (i. e.: a) V2G(p,1) = 6(r r-l) b) Go(p, p1) = 0 c) Go(p, p) regular at infinity); then any function w(p) which is twice differentiable everywhere in V and regular at infinity satisfies the integral relation w(P) G(P. p V2w(pV))dV + W(P) Go(P, pS)dS (2.14) v JS where the normal is directed out of V. Having evaluated the integrals over S and S we hereafter will denote the surface as S rather than S2. 2 -HThe 6 function is normalized so that the free space static Green's function is 1 4Jr|-t| *l 11

THE UNIVERSITY OF: MICHIGAN 3648-7-T ifI APPLICATION OF THE REPRESENTATION THEOREM TO A CLASS OF SCATTERING PROBLEMS We now consider the time harmonic scattering problem for the surface S with Dirichlet boundary conditions. This may be considered as determining the per5 i turbation, u, of an incident field, u, due to the presence of the surface S. Specifically, for a given u (plane wave, point source, or superposition of such sources) we seek a function of position u(p) such that i S a) u(p) = u(p) + u (p) b) (V2+k2)u(p) = 0, EV = Ext S c) u(p) + uS(p) = 0 p~S d) uS(p) satisfies a radiation condition, lim r(- iku = 0. r --- oo ar In this formulation a time dependence e is assumed which gives rise to the radiation condition given in (d) and implies that the free space Green's function is ik|rri | e +iwt 4 & -- l. The comparable expressions for a time dependence e are found by replacing the k by -k throughout. It is our intent to represent the scattered field with the integral relation derived in the previous section; however, we cannot identify u (p) with the function O(p) since u (p) is not regular at infinity in the required sense although it does satisikr S e fy the radiation condition. To be more specific, even though u -- f(e, P) for large r which implies that Iru I will be bounded, r - u will not be bounded, l9 arg rwihipistat Jr! 12

I THE UNIVERSITY OF MICHIGAN 3648-7-T hence u will not be regular at infinity. It is possible, however, to find a function closely related to the scattered field which does satisfy the regularity requirement. This is evident from the expansion theorem for "scalar radiation functions" (see Wilcox, 1956a, b; Atkinson, 1949; Sommerfeld, 1949) which may be stated for present purposes in the following form: If S is a surface satisfying the requirements for the representation theorem, and u is a function satisfying (a) - (d), then the unique expansion ikr u (O, u ik — -— i r>c (3.1) run n=O r is uniformly and absolutely convergent for all r, 0, p provided r > c. It is not difficult to show that c may be chosen as the radius of the smallest sphere entirely enclosing the surface S. The question of whether this is the smallest c for which (3.1) remains valid is not without interest but is not our present concern. Now we are interested in the behavior of u at infinity and a glance at (3.1) reveals that while u is not regular in the sense of Kellog (eq. (2. 9)) the function -ikr s 1 ) (3.2) e u = - (3.2) r - n n=O r is regular in this sense. Thus we may identify the function w(p) of the representation theorem, not with the scattered field, but with this related function, viz., W(p) - e u (p). (3.3) I 13

THE UNIVERSITY OF MICHIGAN 3648-7-T I I II With w defined in this way it is easily seen that (V2 +k2 )u = 0 =~(V2+ k2) eikr w(p) = 0 or., explicitly, ~a2 2rar 12 a2 cotOe a 1 a2 2 32 r r r~a r2 e r2sinOe ao2 Since arr and a2 rikrl 2ikr ikraco 2 ~ W(PU -k e (o(p) + 2 ik e + 2 r eikrCO = (3.4) ikr a 2 e 2 w 0p) it follows that -ir I2a ar I 3 r L. Making use of (3. 5) i 2 ikr ikr ao ikr a 2 2ik ikr = -ke w +2ike - +e -w(p) + ar r2 r 2eikr a r ar 2 ikr ikr 1 a 12aw \2ik eikr ___ = -ke (0+e r- +ar) 2 ar \ar/r ar r,n (3.4) leads to ikr 2 2ik ei a e V+ () r ar (3. 5) (3.6) 14

THE UNIVERSITY OF MICHIGAN 3648-7-T or finally, 2 -2ik 8. V2 2k (r). (3.7) r Or Now we make use of the representation theorem of the previous section which with (3.3) and (3.7) enables us to write pikr G0 r(wpp') dV+ -ikrl s o V,Pi) rvxOv) (p1) e u (pl) = -2ik r dV + VG (p v v ar[ (P) an Go (PS P) dS. (3.8) Furthermore the boundary condition for the scattering problem states that s i u (p) = -u (p), p S which implies that on S, ((p) = -e-ikrui(p). (3.9) Making use of this in (3.8) yields the representation f -ikr oG (p1) = -2ik \k' W(P = -2ik 5 V i- [rvw(Pv) dV - \ e Sui(ps) Go(PS ) dS rV arv Js no V (3.10) We may summarize the results of the present section inthe following theorem: If (1) V is the volume exterior to S, the union of a finite number of smooth, closed, bounded disjoint surfaces, (2) G(pP1) (= 4 1- + u(P P1)) is the 1 15 -~

THE UNIVERSITY OF MICHIGAN3648-7-T static Green's function of the first kind for this surface (Go(p, 1 ) = 0, pe S) and (3) u(p) = u (p) + u (p) is the solution of the time harmonic (e -i) Dirichlet scattering problem for this same surface; then the scattered field satisfies the following integral equation ikrl Go(PVI) krV -e e an Go(Ps, Pl) dS (3.11) where the normal is directed out of V. 16 -

THE UNIVERSITY OF, MICHIGAN 3648-7-T IV THE ITERATION PROCEDURE With the theorem of the previous section established, an iteration scheme is clearly indicated. Here again it is convenient to work, not with u, but the related function -ikr s(.,n) = e u (p) in terms of which the integral representation is given by equation (3.10). If we rewrite (3.10) in operator form w = kO o + f (4.1) where 0 denotes the volume integral and f the known surface integral, the form of (4.1) suggests that a solution may be found using the Liouville-Neumann series of Fredholm theory. That is, we rewrite (4.1) as w = - ko] v f (4.2) and formally expand the inverse, obtaining o o = kn on of. (4.3) n=0 Denoting by w(N) the partial sums N )(N)= Z knOno f (4.4) n=0 it follows that 17

THE UNIVERSITY OF MICHIGAN 3648-7-T w(0)= f (N1= kOwN) +f, N 0. (4.5) If, as is the present case, f is not independent of k but has a power series representation o f= a k ^-J m m=0 then, substituting in (4.3) and formally employing Cauchy's form for the product of two series yields o m=O Co kn+m On a m=O m m kno a n=O m-n (4.6) If we denote by w the sum m m = 0 o a m n=O m-n n=0 (4.7) It follows that w = a 0 0 m+l m+l = O am+l-n n=0 m = am+ + _0n+l n= = a +Oow M m+l m m-n m >0 (4.8) 18

THE UNIVERSITY OF MICHIGAN 3648-7-T We have thus formally produced two representations for the function o, namely lim (N) ( - N-+co (4.9) (N) where (N) are defined in (4.4) and (4.5) and ( Zkm (4.10) I-0 m where o are defined in (4.8). m There is of course a relation between the two. Clearly the first N+1 terms in the low frequency expansion of w are given by N >7w km m=0 m whereas the Nth iterate, eq. (4.4), includes these as well as terms of all order in k and may be considered as a partial summation of the low frequency series. Ex(N) plicitly, (N) may be written, (4.4) and (4.6), N oD ((N) Zkn+m o a =k o a n=0 m=0 or, adding and subtracting the same quantity (N) \ Zkn+mon n+m n = - O ko a m n=0 m=0 n=N+l m=0 19

THE UNIVERSITY OF MICHIGAN 3648-7-T Using Cauchy's form of the product of two series to rewrite the first sum and adjusting the index of the second enables us to write oo m co oo (N) =o a - a a X 1 k n+m+N+1 On+N+l (/ - / oa - 0o a. -m-n 7 nm=0 n= n= m=0 Splitting the first sum and again adjusting the index of the second yield three terms N m oo m (N) 7 Wmnn V 7'mn = o a + o a -o n mm-n I m=0 n=Oi m=N+l n=O oo m - 7 '_ km+N+1 0n+N+l a m=O n=O The first term is seen to be, with (4.7), the sum of the first N terms of the low frequency expansion while the remaining terms may be combined by further reordering to yield N oo m (N) 7 m 7mn (N = _. k + 7 k 0 o a ' —m n m-n m=0 m=N+l n=0 oD m-N-1 - Z kmOn+N+l m=NI n=O rnm-N-l-n m=N+l n=0O ' w + k O o a m=0 m m=l nm nm - - kion o a] n=N+l and finally 20

THE UNIVERSITY OF MICHIGAN 3648-7-T N oo N k(N) Mw k + k o a (4.11) m m-n m m=N+1 n=O In what follows we shall assume that w may be expanded in a power series in k satisfying sufficient convergence conditions to enable us to make explicit the formal results obtained above. However, in the Appendix we show that no assumptions on the existence of a convergent low frequency expansion are necessary and in fact these formal results are not only valid but constitute a constructive proof of the existence of the low frequency expansion. For the present, however, let us assume a convergent expansion of w(p) in the form o0 w(p) = m(p) k (4.12) m=0 and in addition assume that the related series Or [rW ) km (4.13) m=0 r converges absolutely and uniformly both for all points p in V and 0 < k < k where O k is some finite radius of convergence. Then, since the spatial differentiability of w(p) has been assumed, it follows that the order of summation and r-differentiation may be interchanged (Whittaker and Watson, 1952, p. 79) yielding a- [rm(p) km= ar r (p)] km (4.14) m=O m=O I 21

THE UNIVERSITY OF MICHIGAN 3648-7-T Substituting (4.12) and (4.14) in (3.10) yields -0(Pl)km = -2ik G rv m- P) - e u(p) a Go(PS' P) dS. S (4.15) The incident field, whether plane wave or point source, is an entire function of k as is e u (p) which may therefore be expanded in a Taylor series co e ui(p) = A (p)km (4.16) m=0 where 1 dmn r-ikr il dkm k=O Hence (4.15) may be written (Pm)k = -2i G v aV [rvwm(pv)]km+l dV - A (P)km an Go(pS p ) dS (4.17) The convergence of (4.16) is uniform and absolute for any finite k and in par ticular for k < k; hence,the order of summation and integration in the second (suro face) integral on the right hand side of (4.17) may be interchanged. Similarly, since j _ - - 22 -- - -- -2

THE UNIVERSITY O0F MICHIGAN 3648-7-T the sum appearing in the volume integral in (4.17) has been assumed absolutely and uniformly convergent, the order of summation and integration may again be interchanged. The justification of this interchange is slightly more involved since the volume integral extends over an infinite range whereas the surface integral is over a finite range; however, in either case the result is contained in or a minor extension of the theorems of classical analysis (e.g. Apostol, 1957, p. 451). With this interchange of order of operation in (4.17), we equate coefficients of like powers of k obtaining o(p1) = Ao() Go(Ps, P) dS S (P)Go(Pv P) L rvm(PV) dV "m,(Pl) = -2i m rv ar V (P) a Go(S P1) dS, m = 0, 1, 2... (4.18) where A I dm r-ikr ip ) m(P m! dkm e k=0 pES Equation (4.18) is the explicit form of (4.8), the low frequency expansion. The explicit form (4.5), the partial summation of the low frequency expansion is given by 23

THE UNIVERSITY OF 3648-7-T M IC HI GA N (0) w ( 5'S -ikrs i a e u (pSan 001(p5, p 1 )dS Lk G (pvP Pi) a [w(N)(pV dV 1 e S ~~s uI(p)a G G(pS, pl) dS (4.19) N = 0, 1, 2,...9 24

THE UNIVERSITY OF MICHIGAN 3648-7-T V AN EXAMPLE: SCATTERING OF A PLANE WAVE BY A SPHERE We now apply the methods derived in the previous section to the specific problem of scattering of a plane wave by a sphere and compare the results of the iteration process with the known exact result. We fix the origin of our coordinate system at the center of a sphere of radius a and consider a plane wave of unit amplitude propagating down the z-axis (see Fig. 2). Thus o ui = e- = e - (-i)n(2n+l) j (kr)P (cos0) (5.1) n=O n n is and the known expression for the scattered field such that ul = (u + u ) 0 r=a r=a is (e.g' Morse and Feshbach, 1953, p. 1483) o h (kr) u = - (-i)n(2n+1) h (ka) jn(ka) Pn(cos) (5.2) n=0 n where P are Legendre polynomials and j and h are spherical Bessel functions n n n and spherical Hankel functions of the first kind respectively. The static Green's function for this problem is also well known (Stratton, 1941, p. 201) and is given by G(p,pl) -, + r1 (5.3) 0 1 47r V+r2+r- 2rr cost 4 2 1 2 1 + cos1 4 V -- - 2rr cosy where cos y = cosOcos 01 + sinOsin cos(p- - ). I 25

THE UNIVERSITY OF MICHIGAN 3648-7-T 1 z / /(r, o0,) (r1, 01, 01) / / x FIGURE 2 The expansion of the Green's function in spherical harmonics, which will prove useful in the subsequent analysis, is 1 < 1 a G(pp) = - Z(~) p(cos"y) + P (cosy) G(PP).= r A=r r 47n+ r> n=O > (rlr)n+l (5.4) where r>= r if r > r and r< = r if r < r1 =r if r>r and= r1 if rl<r. First we shall calculate the first few terms in the low frequency expansion of the function w = e u using equation (4.18). In the present case, the surface a a is the sphere r = a on which = -- and an ar 26

THE UNIVERSITY 3648-7-T OF MICHIGAN A (P) = 1 d r-ikr ui( ~ dk m k=O pES = () am( 1+cos O). m: Utilizing these facts in (4.18) allows us to write W (P1) = do dO a sinO a G (p, pl) 0 r=a 0 (5.5) and 0o o 27T 2 p, p w m+(p ) = -2i \ dr d dOr sin0 r m+1 1 r a 0 o C27T ir m+m \da sinO (m+il) (1 +cos) ar ar a [rtp)] ar m Go(P, Pl) r=a I (5.6) m = 0,1,2,... where G (p, p1) is given in (5. 3) and (5. 4). Explicitly ar o(P'Pl) r=a OD 1 4-w n=O n-1 a__ (2n+ 1)P (cos'y) n+l n 1 (5.7) hence wo(pl = - - 0 1 47r n+l rJ 27r rr (2n+l) \ d dsinOP (cosy). o 0 (5.8) 27

THE UNIVERSITY OF MICHIGAN 3648-7-T In this section we shall assume that all interchanges of summation and integration are justifiable. Recalling the definition of cosy (eq. (5.3)) and making use of the addition theorem for Legendre polynomials (Magnus and Oberhettinger, 1949, p. 55), namely P (cos ) = Pn [cos cose1 + sinO sinO1 cos(-pl) (5.9) OD = P (coso)P (cosl)+2 (n m) P(cos)P (cosl) cosm(P-), n n 1 =1 (n+m;m) n n 1 ml1 it is clear that 2 2 7r ir \ do d desineP (cosy)P (cosd) = 2 Pn(cosl) desineP (cose)P (cos0) 0 0 0 42+ Pm(cosl) n=m 2m+ m 1 (5.10) 0 n m Since P (cose) = 1 it follows that 0 (P) = -a/rl. (5.11) Proceeding now to the next term, since r (p =r (-a) = 0 (5.12) we have () = a ) (2n+1) d d sine(l+cos0)P (cos y). (5.13) 1 1 -r =0 rn 1 n 10 I 28

THE UNIVERSITY OF MICHIGAN 3648-7-T Since 1+ cose = P (cose) + P (cos0), this may be evaluated using the orthogonality o 1 of Pn, eq. (5.10), yielding (5.14) 2 ia ( = P) rI 2 +ia( -a) P (cos01) 1 1 Proceeding to the next term, since -r a r2 ia+ ar a-r P1(co).3 ia 2 P1(cos0), r (5.15) we have 00dr o2(P1) = -2i dr a 7r dO 0 27r 3 do r2sin Go(p, p) Ei(a )Pl(cos0) 4 a 2 2ir 7r dp dO sinO(l + cos0)2 a G(p, P) 0 J0 r=a (5.16) Substituting the explicit forms for G (p, P), (5.4) and (5. 7), leads to r(l 2 1 o2(P1) = - dr dp a 0 7r 3 dO r sing (a )P1(cos0) Or 0 n=O n + =(cos) n=0 r l oo + 21 dr rl 27r d 0 rr 0 d r2sinQ(r Jo 3 ) P 1(cosO) oo n Z r1 n+1 p(cos7) n=O r (cont'd) 29

I ~~THE UNIVERSITY OF MICHIGAN 3648-7-T 2 5 dr 2r 7ir 3 do der sine a~ P1(cose) 0 JO 2n+l a P (cos'y) (rr )I+ 1 n 1 2 6 n+1 27 + a (2n+ 1) do ~de silne(1+cos0)2 P (Cos 'Y) 8ir n=O 1 (5. 17) The angular integration is trivial usin the orthogonality relation (5. 10) and the fact that (1 +CosO) 2= 2P2(cosO) +2P1(Cos e)+-4P (Cos90). (5.18) This results in w2(lpl) 5r1 2 Sdr( 3 a 2 a P1(CosO1 ) + -9 1 1 3 3 3 -~P(cosO ) 21 1 "NO 3 a r d 3 1P(cosOs) r r1 + P22(cos ) (a5 + 1P(cose 0) a 1 ~1 r 2 4 a 3 r 1 (5.19) The r integration is simply performed yieldin 30

-- THE UNIVERSITY OF MICHIGAN 3648-7-T 2( w2(Pl) = 3 a 2 3 -2 P1(cosl)(r -a)+ 3a r1 P1(cos l) rl 2r r=r 2 3 6 a Pl(cos1) r1 a0 r=a + a _ Pa(Cos0 ) 3 r1 P2(cos1) +2 (a) Pl(cOsO) + 1 2 2 a a - 3r (5.20) or w2(P) = a2 P1(Cs1) - P~~l(~l)r 2 Pl(cosel) (a 31 1 + r1 P (cos1 ) 3 r 1 1 - P2(cos )3 P(s1) 3 1 2 Pl(C 1 +2 3 a rr I (5.21) This may be further simplified as follows 2 w2(pl) = a P(Cos)( ) 2 1 r1 + r P (Cos ) + 1 1 2 a 3 r 0 (5.22) Collecting our results we have, with (5.11), (5.14) and (5.22) 00 (p1) = k n(P1) n=O = to +kw 1+kw2 +O(k3) 0 o a rl +ika rD-,[ 1 + (r) Pl(coso) + (ka)2 L3 3 P2(~osl ) () 1r +r Pl(cosl) + 3 } +O(k3). r 1 1 3 n11 (5.23) 31

THE UNIVERSITY OF MICHIGAN 3648-7-T We wish to compare this with the exact result, calculated from eq. (5.2) to the same order in k. To calculate these terms we first observe that since (Magnus and Oberhettinger, 1949, p. 22) 1 ip.-n-1 nI (n+, m) h (p) = e 1 () P (2P m=O (2ip)m (5.24) where 1 (n+,o) = 1 2' and 1 (n +, m) = 2' E2n+1)2- ) 2n+ 1)22... R2n+12- (2m- )2] 22m, 2 ml r(n+m+ 1) r(n- m+ )m! (-1)m(-n) (n+ 1) m; m m =M! m = 1,2,3,... r(z+l) (z)= r(z); h (kr1) the ratio h (ka)may be written h (ka) n n Z-lim 1 (n+, m) (2ikrl)m 1 - h (kr ) ik(rl-a) n 1 ( a e h (ka) r or, reversing the order of summation and or, reversing the order of summation and n (n+ 1, m) (-1)m -- 2, -m 2 m=0 (2ika)m simplifying, 32

THE UNIVERSITY OF MICHIGAN 3648-7-T n 1 m 1 e (5.25) n ((n+ 2' n-m)(2ika) m=O With equations (5. 2) and (5. 25), the exact form of w(p1) may be written -ikr (p ) = e u (1) n + n+1 (-1)m(n+ n-m)(2ikr1)m = -e -i)n (2n+1) (ar) j (ka)P (cosO n (-l)m(n+ 1, n-m)(2ika)T m=0 (5.26) Note that the quotient of two polynomials in k with non-vanishing constant term is expressible as a power series in k with non-vanishing constant term (see footnote at the end of this section). Also, from the definition of j (p), namely n WOD 2m jn(P) n+ (-l)m (p2) (5.27) 2" 1 m=0 m.r(m+n+ 3 2 we see that jn(ka) may be written as a power series in ka whose lowest order term -ika 2 and, since e 1-ka- ) + (k3), (5.28) 2 we may write the exact result, (5.26), as 33

THE UNIVERSITYI 3648-7-T OF MICHIGAN (ka) 2-a /a2 w(p1) = - -ik — -ji(ka) -' 4'-' 1 2 - r 1 o j1 (ka) P1(cos61) - 5 (a i)3j(ka) P (cose0 ) 1 2 2 1 r 2 2' I(-, 2) L2 5 - (-2 1) (2 ikr ) 2' 1 - (-5 1)(2ika)2' + (- 0) (2ikr )2H 2' 1 + (- 0) (2 ika)2I 2'j + 0(k3). (5.29) Making use of the notational definition in (5.24) we have (-3, 0) = 1 2 (-,P 0) = 1 2 (-, 1) = 2 (-5 1) = 6 2' (5.30) (-5 2) = 12 2' and from the definition of jn (5. 27), we see that j (ka) = 1- + (ka )+0(4 o 6 (ka 3 ) j1(ka) = T: +0(k) 2 12 Substitutin (5.30) and (5.31) in (5.29) we obtain (5.31) 34

I m THE UNIVERSITY 4 3648-7-T OF MICHIGAN w(p1) = -[1-ika- (ka)] a r1 [1-] + 3i [ - ika] 2 (a) r 1I ka 2- 2ikr -P (cos ) 2 - 2ika 3 1*" 2-2ika 3 12 12-12ikr -4(kr )2 + 5 ( ka) 1 5 (C21 r1 15 2 1 12- 12ika-4(ka)2 + O(k3) (5.32) This may be further simplified noting that, to the required order, 1 - ikr - (kr ) - 1 3 2- - 1 + O(k). 1- ika- - (ka)2 Hence (5.32) becomes ((P1) a [ - ika - 2 (ka)2 6 + i (Ar ) ka Pl(cos 01)(1 - ikrl) + (ka)2 3 (r~1) P2(cos 01) + O(k ). (5.33) I Collecting terms in like powers of k we find 2 ((p ) = - a + ika a-a + (-) p (cos0 1) 1 r1 1 1 + (ka)2 [2 a 3> r^1 + a P (cos0 ) r 1 1 + P2(co~sO1 + O(k3) (5.34) 35

THE UNIVERSITY OF MICHIGAN 3648-7-T Comparison of (5.23) with (5.34) verifies the fact that to this order in k the exact result and the low frequency iteration method are in complete agreement. Next we consider the perhaps more important iteration method given in (4.19) which represents, as was pointed out, a partial summation of the low frequency series. Specializing the surface S to be the sphere of radius a and making use of the incident field and Green's function representations given in (5.1), (5.4) and (5.7) we rewrite equation (4.19) as follows: 27Tr 7T s o (0) 1= o - ika- ika cos e (p) 4 d sin (2n + 1) P (cosr) e I -ika e 4ir n a (2n+ 1)(2m+ 1)(-i)mj (ka) n=0 m=0 1 d \ dd0sin0P (cosy)P (cos0) 0 JO0 (5.35) (N+l) k) I r 27r 7T CO ik 2 n-1 ar na) P1) = + 2 dr d dOr2sinO n+l pn(Cos)r rw (p o n=O r1 co 27r r Co n + 2 d r r ine n+2 rr a ( + d \ r \ d \ dO r2sinO. n+2 P (cosy) - Lrw Np)1 r n=O r 1 Co 2\r 2 2n+o dr dr2sin a Pr (cosy)- Lrw (p) 2ir n=O r(rr n+l ar a n 0 (0) +w (p) 36

THE UNIVERSITY OF MICHIGAN 3648-7-T With the orthogonality relation (5.10) we find that (O)(p) 1 (P) 00 -ika (2n+ 1)()n n=0 - -e n+l 1r n(ka) Pn(cos 1) I1n (5.36) Now calculating w(l)(p ) using (5.35) and (5.36) we have 1 a r w() (p -ika Or Lr PJ = e 00 /(2n+l)n(-i)n n=l r n+1 ) Jn(ka) P (cos0) nrn and j(1)(pl) OD 00 = ka (2m+ 1) m(i)mj (ka) ~ n=0 m=l dr adr d27 dOr 2sinO r an-o1 + 0 dersn r1 0o 0o r P (cosy) P (cos ) 1(c (cos P (cos7) P (cos 0) n m Oo 27r nr + rl r d do de r sin0 - r 0 0 1l (r OD - \ dr a S2r d0 Jo 2n+l 2 an dOr sinO a +2 n+l n+2 or1 r +1 P(cP (cos)P +(cos ((p,). (5.37) (5.37) Using the orthogonality relation (5.10) this becomes 37

(1) THE UNIVERSITY 3648-7-T OD (p ) = 2ik e1k ~Zm(- i) mj m(ka co m m+1 O + dr ra - adr r1 )P (cosO1 OF MICHIGAN I which., on carrying out the r-integration becomes w 1 p)= 2ik e1ik ~m(-j)mj (ka) P (cosO0 ) m=1m 1 +a - a 2mrm 2mr I 1 This may also be written = -e -k -) m4j m+1 (a L jm (ka)Pm (cosO1)(2m+ 1)(-ik)(r 1- a) -e -ika (2m +1) (-i)m m+1 ) cs m=O 1 (5 have 1~ m m r Continuing the iteration weI wrr (p] = - e1 38

THE UNIVERSITY OF 3648-7-T MICHIGANI and w(2) ( w(1) r1 ~5> (ik)2 -ika - V OD 4~J(-i)mj (ka)(2m+1)a l n= e2-r 7Tn1 dO r 2sinOr P (cos'y)P (cosO) S ~ n+1 n m F-m+i1 + ma7 -rm rM~ +5d jdr 2ir 0 2T n dO r sinO 1 p (cos'y)P (cosO)m+ n+2 n m 0r r + ma~ rm 7F 2n+l dOr 2sine n+a + P (cO5y) P(CO.C. s O) [-m+1 - m (1) + (5.41) Using the orthogonality relation (5. 10) this becomes w(2) ( ) =2jk2 e-ika (- i)mj (ka) am+l P (CosO0) m1 m m 1 t dr a (-m+ 1)r Fm+1 + ma7 r1 [0(-m+ 1)ri + da ma rr2m+1.-j 000 -\dr 3a 2m+l a m+1 [(-m+i) + L-rm (1) (5.42) 39

THE UNIVERSITY OF MICHIGAN 3648-7-T which, on carrying out the r-integration becomes OD 2 (2) 2 -ika mm+1 J (-m+1) r U (2) )( 2(ik) e-a (-i)j (ka)a P (cos1) - rm+1 2 m=ma(r a) 2 2 1 (-m+1) ma (-m+ )a2 a2 (1) m+1 i + i - m ma (1 rl (2m- 1)rm- 2mr (2m- 1)rm+ 2mr +l 1 1 1 1(5.43) This may also be written (2) 2 -mika +1 W (p) = 2(ik) e (i) j(ka) ) (2m+l)P (cosO) m \r m 1 m=l 1 r(-m+l) (r2 2a (r- + (1) L2(2 - 1) 1r (p) (5.44) With (5.39) we rewrite this as (2) -ika a__(Pl r (p) =- ie (i)m j(ka) (2m+1)P (cos1) 2m 1 r -(ka)2 + ka(ka ) - ikr + ika + - e-a a j (ka) (5.45) m-1 1 m r We could of course continue iterating but it will be seen that this is not necessary in order to see the sense in which successive iterates approach the exact result. If we re-examine the known exact result for w(p1) given in (5.26) we see that, separating the n = 0 term, this may be written as follows, 40 l

m I THE UNIVERSITY OF MICHIGAN 3648-7-T -ika a w(p,) = -e r1 (ka) - ika jo(ka) - e 0 00,(-i)n(2n+ 1) n=l n+1 (-ar- jn(ka)P (cos 1)S (5.46) (5.46) where n II-1)m(n + m=0 - n-m)(2ikr1) 2'p S n II m ll I n O (-1)r(n+ 1i 1m, n-m)(2ika) 2 and the notation (n + - n-m) is defined in (5.24). 2' Since n > 0 in (5.46), the quotient of the two polynomials, S, may be written n (- 1)m(n+, n-m)(2ikr1) m=0O n n (n + n)+ (-)m(n + 2 m=l m-1 1 rn-m i, n-m)(2ika)m 2 p 1 or, dividing numerator and denominator by (n + - n) 2' S n n (n+ 1 n-m) n +, 2 zL(-1) (n+ n) (n+- n-r) 1+ (-) (2' (2ika)m m=l (n+ 2 n) 2 (5.47) We may always choose ka small enough so that n I(-l1)m m=l (2ika)m < 1 (5.48) 41

THE UNIVERSITY OF MICHIGAN 3648-7-T and, with this restriction, then expand the denominator in (5.47) obtaining n1 1 00 n t 1 \ (n+ n-rm) m (n+,n-m) S = (-1) - (2ikr m) (-1). (-1) -- -- (2ika) m=0 (n+v n) SE=0 m=l (n+ n) 2' 2' (5.49) We now calculate the first few terms of S expanded in powers of k. It is a convenience, in this calculation, to note that (n+1, n-m) (-n) 1 (-2n) m (5.50) (n + n) m (-n)m + m 1 m A gross estimate of how small ka must be is found by noting that ( 2n) m Using this fact we find, after substituting (5. 50) in (5. 58), that (n+, 'n-m) _-___(-n) (1)m 2 (2ika)m =. (-)m (2ika)m m=1 (n+ 2 n) m=l m, kal-m IEakal m< --- m' _ m.= m=l m=l hence requiring Ikal < log 2 is sufficient to guarantee that n (n+ 1 n-m) ( nl)+m n-r- (2ika)m < 1. m=l (n,+ n) 42

THE UNIVERSITY OF MICHIGAN 3648-7-T With (5.50) we fihd that Sn= 19 - ikrl - (-2n)(-2n+ l)2! 4(kr ) 2+ O(k 3 + + ( -2n)(-2n+ 1)! 4(ka) 2 - l ' (-2n)(-2n+ 1)2 1 + (-2n)(-2n +1)2 -(ka)2 + O(k3) = 1 - ikr1 +ika+ 2n- kr2 - (ka)] + ka(kr1 - ka) + O(k3). (5.51) Substituting (5. 51) in (5.46) we find that provided ka is small enough the exact result may be written o0 c(p1) = - eka a j (ka)-e - ka (i)n(2n+1) ) j(ka)P (cosO -ikr r1 on=1 1 ika + i ka(kr -ka (ka)2 + 0(kjn. (5.52) Comparing this form of the exact result with the expression obtained by our iteration process, (5.45) shows that the two are in complete agreement to order k3 in an expansion of the quotient S. n 43

THE UNIVERSITY OF MICHIGAN 3648-7-T REFERENCES Apostol, T.M. (1957) Mathematical Analysis (Addison Wesley, Reading, Mass.). Atkinson, V.A. (1949) "On Sommerfeld's Radiation Condition", Philos. Mag. XL, Series 7, pp. 645-651. Baker, B.B. and E.T. Copson (1950) The Mathematical Theory of Huygen's Principle (Clarendon Press, Oxford). Bouwkamp, C.J. (1954) "Diffraction Theory", Rep. Progr. Phys. 17, pp. 35-100. Collins, W. D. (1962) "Some Scalar Diffraction Problems for a Spherical Cap", Arch. Rational Mech. Anal. 10, pp. 249-266. Darling, D. A. (1960) "Some Relations Between Potential Theory and the Wave Equation", The University of Michigan Radiation Laboratory Rep. No. 2871-5-T. Darling, D. A. and T. B. A. Senior (To be published) "Low Frequency Expansions for Scattering by Separable and Non-Separable Bodies." Heins, A. E. (1962) "Function-Theoretic Aspects of Diffraction Theory", appearing in Electromagnetic Waves, ed. R.E. Langer (University of Wisconsin Press, Madison). Heins, A. E. (To be published) "Axially-Symmetric Boundary Value Problems", Bull. Amer. Math. Soc. deHoop, A. T. (1954) "On the Scalar Diffraction by a Circular Aperture in an Infinite Plane Screen", Appl. Sci. Res. B4, pp. 151-160. Kellog, O.D. (1953) Foundations of Potential Theory (Dover Publications, New York] Magnus, W.(1942) "Uber Eindeutigkeifsfragen bei einer Randwertaufgabe", Jahresber. Deut. Math.-Verein 52, pp. 177-188. Magnus, W. (1949) "Fragen der Eindentigkeit und des Verhaltens in Unendlichen fuir Losungen von Au + k u = 0", Math. Seminar der Univ. Hamburg 16, p. 77. Magnus, W. and F. Oberhettinger (1949) Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea Publishing Co., New York). I 44

THE UNIVERSITY OF MICHIGAN 3648-7-T Morse, P.M. and H. Feshbach (1953) Methods of Mathematical Physics (McGrawHill Book Co., New York). Miiller, C. (1952) "Zur Methode der Strahlungskapazitat von H. Weyl", Math. Z. 56, pp. 80-83. Noble, B. (1962) "Integral Equation Perturbation Methods in Low Frequency Diffraction", appearing in Electromagnetic Waves, ed. R.E. Langer (University of Wisconsin Press, Madison). Lord Rayleigh (1897) "On the Incidence of Aerial and Electric Waves upon Small Obstacles in the Form of Ellipsoids or Elliptic Cylinders and on the Passage of Electric Waves through a Circular Aperture in a Conducting Screen" Philos. Mag. XLIV, pp. 28-52. Rellich, F. (1943) "Uber das Asymptotische Verhalten der Losungen von Au+Xu = 0 in Unendlichen Gebieten", Jahresber. Deut. Math.-Verein 53, pp. 57-64. Senior, T.B.A. (1960) "Scalar Diffraction by a Prolate Spheroid at Low Frequencies' Can. J. Phys. 38, pp. 1632-1641. Sommerfeld, A. (1912) "Der Grensche Funktion der Schwingungsgleichung", Jahresber. Deut. Math.-Verein 21, pp. 309-353. Sommerfeld, A. (1949) Partial Differential Equations in Physics (Academic Press, New York). Stratton, J.A. (1941) Electromagnetic Theory (McGraw-Hill Book Co., New York). Thomas, D.P. (1963) "Diffraction by a Spherical Cap", Proc. Cambridge Philos. Soc. 59, pp. 197-209. Werner, P. (1962) "Randwertprobleme der Mathematischen Akustik", Arch. Rational Mech. Anal. 10, pp. 29-66. Weyl, H. (1952) "Kapazitat von Strahlungsfeldern", Math. Z. 55, pp. 187-198. Whittaker, E.T. and G.N. Watson (1952) A Course of Modern Analysis (Cambridge University Press, Cambridge). Wilcox, C.H. (1956a) "An Expansion Theorem for Electromagnetic Fields", Comm. on Pure and Appl. Math. IX, pp. 115-134. I 45

THE UNIVERSITY OF MICHIGAN 3648-7-T Wilcox, C.H. (1956b) "A Generalization of Theorems of Rellich and Atkinson", Proc. Amer. Math. Soc. 7, pp. 271-276. Wilcox, C. H. (1959) "Spherical Means and Radiation Conditions", Arch. Rational Mech. Anal. 3, pp. 133-148. Williams, W.E. (1962a) "Diffraction by a Disk", Proc. Roy. Soc. London, Ser. A 267, pp. 77-87. Williams, W.E. (1962b) "The Reduction of Boundary Value Problems to Fredholm Integral Equations of the Second Kind", Z. Angew. Math. Phys. 13, pp. 133-152. 46

THE UNIVERSITY OF MICHIGAN 3648-7-T APPENDIX: CONVERGENCE OF THE NEUMANN SERIES This appendix is devoted to the task of -showing that the unique solution of the operator equation w = kO o +f (A.1) where G(Pv' P1) a (A. 2) 0 o = -2i dV oVi a Lr o (A.2) arv V V V and -ikr f=- e u(p) n G(Ps Pl) dS (A. 3) S is given by the Neumann series 00 w = /k0o f. (A.4) n=0 Specifically we shall show that the series (A. 4) converges, establish the sense in which it converges, and demonstrate that it converges to the solution of (A. 1). This will be accomplished by proving that f, w, and all the iterates (N), i.e. first N+1 terms of (A. 4) are elements of a normed vector space which is mapped into itself by the operator kO. Further we shall show that for |k sufficiently small, this operator has norm less than unity. The convergence of the Neumann series, in this norm, then follows as does the uniqueness of the solution. 1 47

I THE UNIVERSITY OF MICHIGAN 3648-7-T First we record some properties of the spherical harmonics and known expansions of the static Green's function which will prove useful. Denote by Y (0, p) an nth order spherical harmonic n Y (0) = E PP (cosO)ei (A.5) n m-n mn n m=-n and by Yn (, 0;01, 01) a symmetric nth order spherical harmonic n Y(0,;01 1) = A P (cosO)P (cosO )cosm(-o1). (A.6) n m=O mnn n These functions enjoy the orthogonality property IrN 2ir \ d6O \ dsin0Ym(0O)Yn(0=;01 ) 0, m n J0 JO = Yn(01,p1) m=n. (A.7) Here it must be kept in mind that Y (0, 6) and Y n(0 01) occurring in (A. 7) are not necessarily the same function but are elements of the same equivalence class. That is, they both may be written in the form (A. 5) but the constant coefficients A may differ. In what follows, it is often unnecessary to distinguish between spherical harmonics of the same order thus we denote them all with the same symbol. This should not be overlooked in any specific calculation of the coefficients where a more precise specification is required. 48

THE UNIVERSITY OF MICHIGAN 3648-7-T It is well known (e.g. Kellog, 1953, p. 143) that potential functions may be expanded in spherical harmonics. In particular the static Green's function for the surface S may be written 1 + Y (0O,) = - 4R(pp) n+ r >a (A.8) 1 n=O r + r > a (A. 8) =- 47R(p,p1) +n=O rn+11 A.) +1 I ea E a( n=O (rr n+1 =_4iP P ) + nr1> a (A.10) where the series are uniformly and absolutely convergent and may be differentiated or integrated any number of times with respect to r, 0, or (; a = c+e, e > 0; and c is the radius of the sphere enclosing S. The reciprocity relation is explicitly exhibited. It is useful to note that the source term may also be expanded in spherical harmonics, OD n 1 - _<- P [cosocoso + sin0sin0 cos(0-Q)i (A. 11) R(p, p) n=0 r+ n where r>= max(r, rl), r<= min(r, rl). Note that the expansion has the same convergence properties as the series in (A. 8) - (A. 10) provided r j r1. In addition to the orthogonality of spherical harmonics, it will be useful to define a related property. 49

THE UNIVERSITY OF MICHIGAN 3648-7-T Definition. A function f (, 0) will be called a "pseudo-spherical harmonic of order n if order n" if dO 0 d sinefn(0e, )Ym(0,; 01, 01) = m<n 0 J0 =Y(0 ( 01) m >n (A.12) With the understanding that zero may be considered a spherical harmonic of any order (all coefficients in (A. 5) are zero) if follows that any spherical harmonic of order n is also a pseudo-spherical harmonic of order n. Now we are in a position to define a particular function space in which we will establish the convergence of the iterations. Recalling that V is the volume exterior to the surface S and a is the radius of a sphere entirely containing S in its interior we define W as follows: a) wec (V) 1 fn b) w = n- n, r >,a and the series is uniformly r n=O r and absolutely convergent, term by term differentiable, W = w with respect to r, 0, or 6 and the resulting series are uniformly and absolutely convergent. c) f (e0, ) are pseudo-spherical harmonics, i.e., satisfy (A. 12) (A. 13) 50

THE UNIVERSITY OF MICHIGAN 3648-7-T Further we specify the following norm, implied both by the pointwise convergence of the series, (A. 13b), and the fact that elements of W are twice differentiable everywhere in V, IIWli = maV w(p)I (A. 14) pEV It is clear that much more could be said of W than that it is a linear normed vector space; however, rather than investigate this space in general, we confine our attention to those properties necessary for our present purpose. These are established in the following lemmas which are then used to prove the main result of the report. Lemma 1: fe Proof: We complete the definition of f given in (A. 3) by restricting u to be either a plane wave or a point source not on the surface S. Then, since the surface is finite and G (ps, p1) is infinitely differentiable with respect to coordinates of the point pi, as long as this point does not lie on S, it follows that the order of integration and differentiation may be interchanged and fe c (V). In fact fe c (V). Actually Kellog (1953, p. 172) established that the potential due to a double layer, with twice differentiable moment, of which f is an example, is also continuously differentiable for p on 1 - S, i.e. fe (V). Furthermore when r >a we utilize (A. 8) and (A. 11) to obtain no | (p,p - 4 n+l Pn [Cosecos + sinOsinescos(-p S) + Yn(e ) n= rS n=O r (A. 15) I 51

THE UNIVERSITY OF MICHIGAN 3648-7-T or since Pn is an nth order spherical harmonic 00 n G(P PS n+ A)(p ) P (cos0) eim. (A. 16) n=O r m=-n This series converges uniformly as does the derived series; therefore, we may rewrite (A.3) as f = - n+1 Pn(cosO) eim e S ui(ps)anA (PS), dS n=O r m=-n (A. 17) which is again of the form OD Y (op) I f = n rl (A. 18) n=O r Hence conditions (A. 13b, c) are satisfied as well as (A. 13a) and the lemma is proven. Lemma2: If w W then O o wW. Proof: With the definition of 0, eq. (A. 2) we write O ow = -2i \ dV G(PP a r w(p ) (A.19) rv 'V v We separate the volume over which the integration is performed into an infinite volume, Ve, where rV a and the expansion theorem (3.1) holds, and a finite volume, V. where it does not. V. thus is the volume interior to the sphere of a radius a and 1 1 exterior to the surface S. Thus we define two functions 52 --

THE UNIVERSITY OF MICHIGAN 3648-7-T w(p) = -2i dV G0(P, [rvw(Pv)]. (A.20) e rV arv i V e i Clearly w (p) + wi(p) = O o w and if we can demonstrate that w and wi are elee 1e i ments of W then, since the space is linear, it follows that 0 o w is also in W. Consider first the finite volume. w.(p) is the potential of a volume distribution which Kellog (1953) has shown to be twice differentiable, for finite volume, provided the density is piecewise continuous (p. 156). This is certainly satisfied in the present case since we W which implies that the density ~ [r w(p is con2 r -V v ontinuously differentiable. Therefore, w.(p)ec (V). When r >a the expansion of the Green's function (A. 16) is valid, with PV replacing PS. The uniform convergence of the expansion and the fact that the integration is carried out over finite limits permits interchange of order yielding 11 (pm) P (cos) eim dV(-2i) mn i r,a (A.21) which is of the form w(p) = r1 r>a (A.22) n=0 r hence ence (p) W. (A. 23) 1 I 53

THE UNIVERSITY OF MICHIGAN 3648-7-T Turning now to w (p) we see that if V is replaced by any large but finite e e volume then the fact that w ec (V) again follows from Kellog's work. It is only e necessary to show that w remains well defined when V becomes infinite. Explie e citly w () limr 2i d d T d 2i 0 o a r w( e p -> OCD 1inaV rv -v r ar vW(Pv (A. 24) and it is sufficient to show that the integrand is O(1/r2) for large rV. Since w(pV) E W it follows that w(pE) = nVV r >a a(A. 25) n=0 rV and therefore that n n +1 (A. 26) V n=- r1 V Thus for large rV, r- vw(pv] = o(1/rv). Furthermore, the expansions of G (p, p1) given in (A. 9) and (A. 10) show that for rV sufficiently large, Go(pp P) 2 r V = 0(1/r ). Hence, despite the factor r2 in the volume element, the 2 integrand is indeed O(1/rV) and it makes sense to let the rV integration extend to. This calculation may be pursued more carefully to show that in addition, w (p) e satisfies the expansion properties required of elements of W. Thus we rewrite (A. 24) for r,rV >a as I 54

THE UNIVERSITY OF MICHIGAN 3648-7-T Go y2 m=O (r) - (A.27) where we have absorbed the factor 2in in the functions f (9V 8 ). Now consider separately the integrals involving the regular and singular parts of the static Green's function, treating the regular part, w,re first. In this case both series are uniorder of integration and summation and perform the integration using the pseudo-orthogonality condition (A. 12) to obtain wreg(p) = drX d dvsinv E m + 1 m+n+1 Jo Jom=O n=l r rV Y (0,0) m(A. 28) = xn- +ll n=1 m=n (m+n)a r Absorbing the constant factors in the spherical harmonics and renaming the second summation index yields reg0 Y (O,9i) () m+nl Wreg() m+n (A.29) n=1 m=O r 55

THE UNIVERSITY OF MICHIGAN 3648-7-T Using Cauchy's formula to rearrange terms, which is allowed since the convergence is absolute, we obtain wreg(p) n+,( ) (A. 30) re "_n+2 n=0 m=0 r While the coefficients in Yn+ ( 0) may depend on m, the summation over m is still a spherical harmonic of order n+1 hence (A. 30) is of the form reg(p) = weg - n+2 (A.31) n=0 r The analysis involving the singular part of the Green's function is slightly more involved since the expansion of 1/R, (A. 11), is not uniformly convergent at r = r1. From (A. 27) we see that sing 2 sin n Sdving(P) \r de R(p, pv)n e 4> V V V R(pPv) n 1 S | Since the series occurring in (A. 32) is uniformly convergent and the infinite integral has been shown to exist, we may interchange order of summation and integration, an absorb the factor (-1/47r) into f, obtaining s2r inev f (0ev e plo th expasiorV d(A. 33) n(p) - dry dV R(p, p) n n=1 a O rV Now we employ the expansion (A. 11) to obtain. 56

THE UNIVERSITY OF MICHIGAN 3648-7-T I __ r 2: m e e I ing() r m 7, + \d v dO V n f (0V1 V) j V V m+l n(o V "=O mV V V. 0(A. 34) Although the inner summation is singular at r = rm, 0 = O, p.= 0V, it is a straightforward matter to exclude a small neighborhood of (r, 0, ) from the integral (in which case the interchange of summation and integration is legitimate)and then show that the integral over the excluded neighborhood may be made as small as we wish by making the neighborhood sufficiently small. Thus we find, again using the pseudo-orthogonality property (A. 12) sing ry ( ^)( mn+1 -amn+l Y (, ) W n= m=n r m+(m- n+ 1) (n+m) r Again absorbing the constants in the spherical harmonics we obtain wsing(p) m- 1 + m( (A. 36) e In m+lA 5 n=l r m=n n=1 m=n r The second sum in (A. 36) is of precisely the same form as (A. 29) hence the same argument allows us to write The justification for requiring this apparently artificial restriction on the space W is found here since without this property, terms involving log r would occur. 57 ) I

THE UNIVERSITY OF MICHIGAN 3648-7-T w jL j Y (O, )+ j=1 (A.37) n=l r m=n n=O r and, with (A. 31), we find that w (p) is also of this form, i.e., e w (p) = wreg(p) + wSing(p) = - (O ) + n+ e e n+2 n=l r m=n n=0 r (A. 38) or by a trivial change of notation w(p) = 1 Z (O.,)+ L" (A. 39) n=O r m=n+l n=l r This is precisely the form required for w (p) to be in W, i.e., e (p ) f (,) We() = n r a (A.40) n=0 r where OD f Y (O,p) m=l (A.41) OD ZY (o0) n>1 n E-m ' n m=n Having shown that w (p) satisfies (A. 13b), it remains only to demonstrate that f (0, 0) defined in (A. 41) are pseudo-spherical harmonics. This follows immediately from the orthogonality of spherical harmonics, (A. 7), since 58

THE UNIVERSITY OF MICHIGAN 3648-7-T 7r p27r 5 do \ dfn(O, 0)Y,,(0, 0, ) sine (A. 42) - d \ dd Y (O0,0)YL(0,0; l,0)sine = 0, L<n 0 Jo m=n = YL(e'll) >n Therefore we may conclude that w (p) e W (A.43) which, with (A. 23) proves the lemma. Lemma3: 0 is bounded, i.e., Mreal< o 3 ||IlolM. Proof: If we W then Oow e W by lemma 2. Therefore 0 o w is twice differentiable everywhere in V which guarantees that 0 o w has no singularities in V. Thus 3M w lo Dw.<M, peV (A.44) and, recalling the definition of the norm (A. 14), Il o wll M. (A.45) Let M = W Iw U l 1Mw This must exist or else for some we W, Jo owl would be unbounded which would contradict lemma 2, the fact that OowE W. Thus sup 110. fowll < M. (A. 46) weW, =(A. 46) 'The lemma then follows since, by definition, I 59

THE UNIVERSITY OF MICHIGAN 3648-7-T 11io weW 1 =1 Ilo w 1. (A.47) Lemma 4: 3k 3 |koI| < 1 if Ikl < Ik l | Proof: Since 0 is linear it follows from the definition of norm that l|kOl= I|ki 1101. (A.48) With lemma 3 we obtain Ikollj < I|kM (A.49) therefore by choosing |k I< 1/M or equivalently, letting |k [ = 1/M, the lemma follows. Lemma 5: E W. s -ikr s Proof: The definition of w in terms of u, U = e u, together with the fact that s 2 2 u e c (V) imply that we c (V). Furthermore, the expansion theorem (3.1) guarantees that we may write P)=. f(Op) wp= r>a. (A. 50) n=0 r It remains only to demonstrate that these f (0, r ) are "pseudo- spherical harmonics". To accomplish this we employ the well known expansion of wave functions in spherica harmonics, e.g., Sommerfeld (1949, p. 143), ao u (p) = h (kr)Y (0,0) r>a (A.51) n n n=0 where h (kr) are spherical Hankel functions of the first kind, n 60

THE UNIVERSITY OF MICHIGAN 3648-7-T ikr -n-1 n-, v / m h (kr) = e (A. 52) n r _ f n r m=0 (n-m)! m, 2ikr With this expression together with (A. 51) we find that p) = e r p) = (A. 53) u(p)=m+ (-) n=O m=O r (n-m)! m (2ik)m or, upon rearranging terms and absorbing the multiplicative factors in the spherical harmonics, (E(p) = n m rml (A. 54) n=O m=O r This can be rewritten, with the obvious changes in notation so as to correspond to (A. 50) (0(p) = f (ne, ( ) n+1 1 =0 r where oO fn( ) = Y+, (0 1) * (A. 55) n =0 t=0 The functions f (0, 0) thus obviously satisfy the pseudo-orthogonality condition, (A. 12 and the lemma is proven. Note that this proof essentially duplicates Sommerfeld's derivation of the expansion theorem but that, as is clear from the above, his statement (Sommerfeld, 1949, p. 191) that the f (0, 0) are finite sums of spherical harmonics is in error. 61

THE UNIVERSITY OF MICHIGAN 3648-7-T We now at last are in a position to prove that the Neumann series (A. 4) converges to the solution we seek, and that this solution is unique. First we show that the series converges to the solution, that is, for any E>0, 3N I W-W (N)' <e if N>N. (N) Lemmas 1, 2, and 5 establish that w, f, and all the iterates w are in the space W, hence it makes sense to write |1w - (N) for any N. With (A.1) and the definition of the iterates it follows that w-f = kO o w (1) 2 2 w- w = - kOof-f = kO o (- f) = k O o w (-2 = -k -f = kO o (W-W(1) k 30 (N) kN+10N+1 ()- k N+ N+l (A. 56) hence w-w (Y < IkOM |W|. (A. 57) But lemma 4 states that lkOll < 1 if Ik| < Ik | and ||iil is bounded since we W (lemma 5) hence it is always possible to find N large enough so that IkOUN+l 1wil <. (A.58) Specifically, since log IkOll < 0, we find that (A. 58) is satisfied if 62

THE UNIVERSITY OF MICHIGAN 3648-7-T N+1 > log I (A.59) We have thus established that for any e > 0, |w - (N) <e if N > N and IkI < k | where N0 is the greatest integer in Iog 'k - 1 and k~ exists by lemma 4. To prove uniqueness we assume that existence of two solutions of (A. 1), 1 and j2 such that 0 = kO ow +f (A.60) 2 2 and wit-w2l # 0. (A.61) Then W1 -2 = kO o 1 -kOO ot (A.62) and taking norms we obtain w l-2| kO lW 1l w2| (A.63) y assumption W1 -W2j| j 0, hence we may divide obtaining 1 a ||ekO|| (A. 64) hich violates lemma 4. 63