-THE AFCRL-64-684 UNIVERSITY OF MICHIGAN 3648-7-T 1 ITERATIVE SOLUTIONS OF THE HELMHOLTZ EQUATION by R.E. Kleimnan August 1964 Report No. 3648-7-T on Contract AF 19(604)-6655 Project 5635 Task 563502 Prepared for AIR FORCE CAMBRIDG(E RESEARCH LABTOATORIES OFFICE OF:AEROSPA'CERESEARCOH UNITED STATES AIR.:FO:RCE BEDFORD, -MASSACHUSE TTS

I m ~~ THE UNIVER SITY.OF MICHIGANAIEJ 3648-7-T 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 I ii

I THE UNIVERSITY OF MICHIGAN ----- 3648-7-T TABLE OF CONTENTS I. Introduction 1 II. A General Representation Theorem 5 I. 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 s;olution of I a 1

W - Y V o r - n InrV T s Wir V F w SV d A YT - a'II- AL U N V f J b A X UL x r x l l Aoi A IN 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 L I 3

I I 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 [uV o-wV u] dV= i [u - ]an dS, (2.1) V S where the normal derivative 8/8n 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 x1, yl, Zl; an arbitrary smooth surface, S2, not necessarily connected, consisting perhaps of a finite number of closed smooth surfaces; and a large sphere, S, containing S and S2. Further, we erect a rectangular Cartesian coordinate system with origin in S2, see Fig. 1. 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 ~ikR + function, e i/R, and the other with the field scattered by S2. The integral over S is shown to vanish by virtue of the radiation condition and the integral over S 3 1 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. I I~~~~~~ 5 5

THE UNIVERSITY OF MICHIGAN 3648-7-T chosen to be solutions of the homogeneous Helmholtz equation yielding the well known result "1 w(xl' Y1 z1) =-1 47r aw a )e dS. an - w an R (2.2) s3 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 -47rR 1 r1 2 2 2 2 where R = \(x-x ) +(y-y) +(z-z ) and V ul = 0 even at R = 0, i.e. ul is regular everywhere in V in addition to the interior of S1. Equation (2.1) then becomes ( + u wdV = 4 u ) + u) dS 4irR R 1 d an an 47r R 1 2+3 (2.3) If w is assumed regular in the interior of S, then the integral over S may be evaluated in the usual way; first integrating then letting the radius of S approach zero. Clearly the regular part, u a au, dS-U 0 1 an - w an'. 1 on- W T~. j dS — l o S1 a and, since on S n = - 1 an aR' liraa R-im de0\ d R sinO 4-R aR w(xl+R cosP sin0, y1+Rsin sine, z+Rcos0) RO7' U i |_47rR O R 0 J - w(xl+ R sin0 cos, yl+ R sine sin ), z + R cos ) 1-a 4 1 1 1+c ) \.47rR/ = w(xl' Y' l) (2.4) I ~ I I ~I 7 I I II

I THE UNIVERSITY OF MICHIGAN 3648-7-T Hence (2.3) becomes 1 2 (xl'Yl' Z1) = 4R r+ ul V2dV - + R) an S2+S3 an 4rR+u] dS (2.5) Now we further specify ul so that + up =, that is + u ~1 \4ir R Is 4irR 1 is the static Green's function of the first kind (Dirichlet condition) for the surface S2. S2 We shall denote this function hereafter as G (p, pl) = - 4?R + Ul (2.6) where the dependence on two points is indicated. The notation is a shorthand for Go(p, p) Go(x, y, z, xl1zl) GO [r,,0,rl101 (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 a <(pl)- = Go(PV, Pl) V 2W(PdV+ w ) 1) V S2 - [G(Ps' Pl)'n (p) - (p) Pn Go(Ps' Pl dS 3v~S3~~~~ ~(2.8 (2.8) I 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 important properties is the fact that it is "regular at infinity" in the sense of Kellog (1953, p. 217), i.e. lim lim 2 oG rGo < oo and r - < co (2.9) r —> oI oI r -- co ar 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 o S3 to vanish, we must determine the requirements on w which will accomplish this. 3 Since S3 is a sphere of radius r with center at the origin, n = and we wish 3 an ar to determine conditions on w so that lim ( a w r 2r 7r lirm d a a oo L \Od r2sin (P P r (p) - w(p) r ( = (2G10: o o0 9

I THE UNIVERSITY OF MICHIGAN 3648-7-T 02r pr lim rn d r sinO Gp a (P) -(P)a- Go r -— >co 0 o' 1 3r J or e27r re limr 2 a ro n d (Pr P1 ) jr (p)p r w(p) a G (P., P (2.11) 0o Jo and conditions sufficient to force this limit to vanish are lim 2 a r o r Go(PP WP) and (2.12) lim 2 li Ir 2 (p) arr Go(P Pl) I 0 r -4- oar^ =0 Rewriting these equations slightly as follows, lim 1 2 3 lim 1 P) 2 o0 i I- rG r - I(p) = 0 and -r r —poo rd r — or Or (2.13) clearly indicates that requiring w(p) to satisfy the same regularity conditions at infinity as does G(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 S is 3 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. 1 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.P) 6(=-1lr- )+ O b) G (p,p) = 0 c) G (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, PV) V2 w(p)dV + \ (Ps) Cn G (p, P)d (2.14) S where the normal is directed out of V. Having evaluated the integrals over S and S3 we hereafter will denote the surface as S rather than S2. 2~ The 6 function is normalized so that the free space static Green's function is 1 4 — 41r rI —|-*' *- 11 -

THE UNIVERSITY OF MICHIGAN 3648-7-T m 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 pers 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) (V +k )u ( p) =0, pEV = ExtS c) ui(p) + u(p) = peS s lim fa S d) u (p) satisfies a radiation condition, ro r ik-u = 0. -iwt 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 -; -i 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 w(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 uS r- f(e, ) for large r which implies that Irus will be bounded, r -r u! will not be bounded, I 12 --.. ~~-~~

THE UNIVERSITY OF MICHIGAN 3648-7-T I S 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 (eO) s e 7 n u =, r>c (3.1) r n' n=O r is uniformly and absolutely convergent for all r, 0, ~ 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 ODu (0,) -ikr s 1 \.n e u = - (3.2) r -- n n=0 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) ikr P (33) w(p) e u (p). (3.3) I. 13 I L I~ I -L ~

I T~ TH UNIVERSITY S T OF MICHIGAN 3648-7-T With w defined in this way it is easily seen that (V +k2 )u = 0 (72+k2 ) eikr(p) 0 or, explicitly, a2 aLr2 +2 a 1 a2 rr r2 02 r 09 cotO a 2 30 r + 1 r sin 2 a2 +k2 eik p) = 0 (3.4) Since ar [el )(p)] = ikee kr(p) + eir o<p ft Tr and a2 ar r ikr(pl= _k2ik ikrikr aw ikr Lew (pU = -k e ikr(p) + 2ike r + e a2 2 o(p) 8r it follows that 1 8 2 ar r 2 a2(e krW ] 2k ik r ar r O -k e r + 2ike ar J ar + kr 82 r 2 ~r 2ik eikr W(P) + r- r ikr r ar -k2 ikr 1 =- -ke w+e -2 2 r r (r 3r) ikr 2ike a(rw) r ar (3.5) Making use of (3.5) in (3.4) leads to ikr 2 2ik ekr e V + r a( Or (r) = 0 ar (3.6) I 1.4- ~ 14

THE UNIVERSITY OF MICHIGAN 3648-7-T or finally, 2 -2ik 8 ( (37) V t = (w) (3.7) r ar Now we make use of the representation theorem of the previous section which with (3.3) and (3.7) enables us to write -ik(p) e u(1) = -2ik a rv (pv) u(pj~ e uS(p1 ) -2ik dV + 1 1 \ ~ ~r v Q (Ps) n Go (Ps Pl) dS (3.8) S Furthermore the boundary condition for the scattering problem states that s i uS(p) = -ui(p), p S which implies that on S, W(p) = -eikui(p). (3.9) Making use of this in (3.8) yields the representation \ (PV'Pl) a -ikr )s, a W(pI) = -2ik \ Ga [rV(P) p dV - [ e u Ps an Go(PS'Pl)dS (3.10) 1 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 (p, P) (=- - + u(p, P1)) is the 15 I I

I THE UNIVERSITY OF MICHIGAN 3648-7-T static Green's function of the first kind for this surface (Go(p p) = O, pe S) and (3) u(p) = u (p) + u (p) is the solution of the time harmonic (e- ) Dirichlet scattering problem for this same surface; then the scattered field satisfies the following integral equation us(p) = -2ike G(P P) e irvu(pV) dV rV v ikr -ikr - ke ue (ps)a Go(Ps l) dS JS (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 us, but the related function v -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 ow+f (4.1) where O 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 = 1-ko] - f (4.2) and formally expand the inverse, obtaining Go w0 = knOn f. (4.3) n=0 Denoting by (N) the partial sums N (N) = knno f (4.4) n=0 it follows that 17.. I II

a THE UNIVERSITY OF MICHIGAN - 3648-7-T (0)= f (N)= kO N) +f, N>0. (4.5) If, as is the present case, f is not independent of k but has a power series representation oo f= a km f-Jm m=0 then, substituting in (4.3) and formally employing Cauchy's form for the product of two series yields nFm n o =A) k Ona (4.6) n=0 m=O oo m = Z Zkmono a m= n=-n If we denote by w the sum m m w= > o amn (4.7) m m-n n=0 It follows that ( =a o o m+1 m o+ 0 = a + Z n o a (m+1 n= Onam+-n m+lln m+ - m-n n= am+ + 0 n=0(4 am+ 0+o (, m>O (4.8) m+1 m~I ] 18

THE UNIVERSITY OF MICHIGAN 3648-7-T We have thus formally produced two representations for the function o, namely lim (N) ( 0) =-+ W (4.9) (N) where w are defined in (4.4) and (4.5) and o ( = i 0 k (4.10) =0 m where 0 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 I'1km M=O 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) Z Ez n+mn m n=0 nm=0 or, adding and subtracting the same quantity - ao - oo oo nN m==0 Em n=N+1 m0 n=0 m=0 n=N+l m=0 19 M _,I

I I 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 co m Co Co W(N) VW 7m on 7 - > kn+m+N+1 On+N+1 (N = z k oa - O 0 a. m=0-^ ^m m-n m=0 n=0 n=0 m=O Splitting the first sum and again adjusting the index of the second yield three terms N m o m (N) 7 WkmOn + mn m =., kO o a + / kO0oa im-n nm-n m=O n=0 m=N+l n=0 O m -m=I Zkm+N+l n+N+l a mO k n=oa m=O n=0 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 I N oo m (N) \7m m n 0) =.km m+ + kmon o am m=0 m=N+l n=0O - I Z kmO n+N+ 1 -" 1 0 amN-l-n m=N+l n=-n N co N m kmm + Z kmOn a - kmOn aj mm ~- I^-m-n N1mm=0 m=N+l n=0 n=N+l and finally 20

THE UNIVERSITY OF MICHIGAN 3648-7-T N oo N (N) = E _km + E km 2_ on oa (4.11) m=O m m=N+1 n=O -n 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 co p(p) = E m(p) (4.12) m= and in addition assume that the related series ii ar [rw(p) km (4.13) converges absolutely and uniformly both for all points p in V and 0 < k < k where 0 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 r =oo [tmpk=eoD ar2 [rw -(p k = ar I rw (p)] km. (4.14) m=r m=O I 21 IL I I ---

THE UNIVERSITY OF MICHIGAN 3648-7-T Substituting (4.12) and (4.14) in (3.10) yields m (p )km -2ik (PVPi) L ( kmdV m=0 rv m=O rv ~~~~Ov ~mpkd -ikr - e i () Go(PS P1) dS. JS (4.15) The incident field, whether plane wave or point source, is an entire function -ikri of k as is e u (p) which may therefore be expanded in a Taylor series 00 e-ikru(p) = A (p)km (4.16) m=0 where A / 1 dm F-ikr i, I Am(P) u( m MK mLeu ^Pl Hence (4.15) may be written OD\,) 00 I wm(pl)k =-2i 1 [rg Vm(PV)km+l dV m=0 rV m=O V mm m E A (PS)k 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 (surface) integral on the right hand side of (4.17) may be interchanged. Similarly, since 22

I THE UNIVERSITY OF 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 W-P\ =- Ao(Ps) an (Ps ) dS S nm+l(P1) = -2i G(P P) r (p)dV m+P 1 rv arV rvm(pv V a A m+(P) an Go(Ps Pl) dS, = 0, 1, 2... (4.18) where Ap) - l d"m r-ikr i(p Am(p) - 1! dkm u *dk k=O 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 I _,1 I I I 23

I THE UNIVERSITY 3648-7-T OF MICHIGAN w((p ). - Si a e u (PSan Go(Pis,1)dS (N+l)(p w (P1) = -2ik G(PVP) [ r(N) dV V rv \ -ikr - Se sui(Ps) anGo(PS, l)ds sN = 0, 1, 2, N=O, 1, 2,... (4.19) I -- W 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 oo u = ek = eikr co = (-i)(2n+ 1) j(kr)P (cos0) (5.1) n=O and the known expression for the scattered field such that u = (u + u ) = 0 r=a r=a is (e.g. Morse and Feshbach, 1953, p. 1483) 0D a)z h (kr) (-n(2n+l) h (ka) jn(ka) P (cosO) (5.2) n=O 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 1 1 G(PP) 1, + 1 (5.3) 47rT Jr +r - 2rr1cos'y 2 1)2rrlco where cosy= cosOcosO + sinsin cos(2rrcos). where cos7 cosecos 0 + sinesinlcos(A- pi). 25

L THE UNIVERSITY OF MICHIGAN 3648-7-T 1 / (r, 0, ) z / / (r, 01 4 1) / I x FIGURE 2 The expansion of the Green's function in spherical harmonics, which will prove useful in the subsequent analysis, is (PP) 4 rcos ) 1. 4 where r>= r if r >r and r< =r if r < r = r if r > r and = r if r < 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 is the sphere r = a on which - - and an 8r. 26 26

I THE3[ UNIVERSITY OF -MICHIGAN I 3648-7-T A (P) = m M! d" -ikr ui(p m e u(p) dk k=O pES (-i) am(l + os )m Utilizing these facts in (4.18) allows us to write wo(p1) = d0s dOasin0 - Go(p, P ) I0 r=a (5.5) and 00(p) =r G (p, p) P) = -2i dr d dOr2sinO ar Lrm 1p eO2ir om+ 2T d 7 am+l (_ia)m+l + d d a2 sineO -ia) 0 0 (1+ cos0) 1 ar (Pl) r=a (5.6) m = 0,1,2,... where G (p,p1) is given in (5.3) and (5.4). Explicitly ar Go(p, p) ar o 1 r=a 1 n a= — 4 - na (2n+1)P (cosy) n=0 r (5.7) hence wo(pl) = 1 n+1 4r n=Z r 2r(2n+(cos y) (2n+l) \ d 0o dO sinO Pn(cos y). J0 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 cosBy (eq. (5.3)) and making use of the addition theorem for Legendre polynomials (Magnus and Oberhettinger, 1949, p. 55), namely Pn(cos7) = P [cosecose1 + sine sine1 cos(-01) (5. 9) oD = P (cos0)P (coseo)+2 > (n-m)! P m(cos0)Pm(cos0 )cosm(-0), n n 1 (n+m) n n 1 1 nm=l it is clear that 2ir 7r Ir \ ds desineP (cosy)P (cosO) = 2r Pn(cosO) dOsin0P(cos)P (cosO) \ \ n m nn1 m 0Jo 0 0 4._.PE (Cos_.. n m 2m+ P (cos) n=m (5.10) 0 n m Since P (cose) = 1 it follows that 0 Wo(P) = -a/r. (5.11) Proceeding now to the next term, since ar ro(pP = -r (-a) = 0 (5.12) we have O a n+ l 27 \ r ia W1^ (P ^ Z-iC'^J (2n+ 1) do d sin0(l+cose)P (cos'y). (5.13) ~=0 1 0 I — 28 -

THE UN~IVE1RSITY OF MvlICHHIGAN. m 3648-7-T Since l+cose = P (cose)+Pl(cosO), this may be evaluated using the orthogonality of Pn, eq. (5. 10), yielding.2 ) ia Wi(P = ~~1 r 2 + ia(a P(cos) 1 r1 (5.14) Proceeding to the next term, since a [r l (P)] = a 2 ar 1'l(~ ar -~" a 3 r P (cos0)] r 1.3 - ia 2 Pl(cos0), r (5.15) we have = -2i adr a r r2r - 3 S dO do r2sinO Go(p, p) [i ( )P(cos0) 0 JO 4 r2r r - a d2 dOsinm(l+cos0)2 arG (p, P) 2 ar o 1 JO 0 r=a (5.16) Substituting the explicit forms for G (p, P), (5.4) and (5. 7), leads to r1 w (p ) = dr a 2z2~ 7r 0deTr~si ne (a 3 de r2sin0( ) P1(cos0) 0 00 n n+ P (cosy) n=O rl 1 + SOdr rl 2-r1 S IT rr 0 drsine( l: 3 (cos ) P (cos0) n r1 ~1 P (cosy) r (cont'd) 29

F THE UNIVERSITY OF MICHIGAN 3648-7-T Co 1 d -2- dr a S27r r \ do\ de r sinO (- )3 P1(cos0) n=o 0d0 2n+l a( P n(cos 7) (rr1) 2 a 8r n= n=0 n+l (a > (2n+ 1) d 5 dOsin0(l+cos802 (coS ).) 0 (5.17) I The angular integration is trivial using the orthogonality relation (5.10) and the fact that 2 2 4 (1 +Cos0) = -P(COSe) + 2P (Cos)+ 4 P (Cose). 3 2 1 3 o (5.18) This results in (rl (p ) = 2\ dr a 3 a P (cose) + rl oD 3 2 a r1 3 dr 3 P1(Cso r Jr - ra 3 dr ( Ia 3 a P1(cosol) + rl P2(cos1) (r 2 1 3 +2P (cosOl)(- )+ 1 1 rI 4 a 1 3 r 1 (5.19) The r integration is simply performed yielding 30

I 2 w2(P) 3= 2 3 I THE UNIVERSITY OF MICHIGAN 3648-7-T 2Pl(cosO)(r - a)+3-a rPl(cos01) ( r 2r r=r 2 22 + a K} P1(cosO ) + a 3r 1 1 {3Plcsa)a 1 P1 (cosV1(r1a ) + 3 P2( + (cos1) + 2r-+ 3 "2<rPl' r - ~~~~a 2( ) 3 2 a 2 2_} wap) = a P(co(COs )+ — Pl(c2LS31) ro1 \r1'r11' 13rJ 1~~~, 31 or w2(1) = a 1 3 This may b Collecting i a (5.20) 1 - (5.21) (5.22) k). 23 3). (5.23) our results we have, with (5.11), (5.14) and (5.22) 00 (P1) = wk ~n(Pl) n=O -= + k + k 22 + O(k3) 0 1 )(cosO1) - + 0( a rl +ika a -Yl 1)2 1r Pl(cos 01 + (ka) 2 EI.31

T.HE 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) ip.-n-1 (n+,m) h (p) = e 1 j )(-l)m m=O (2ip)m (5.24) where 1 (n+ o0) = 1 2' and 1 (n +, m) 2' (2n+1)2 - 1 2n+1)2 - 32].. 2n+ 1)2 -(2m- 1)2 22m m 2 m! r(n+m+ 1) = (n- m+ 1)ml (-1)m(-n) (n+ 1) ml m = 1,2,3,..., z r(z+,) i) ~ h (kr ) r n 1 the ratioh (ka) h (ka) n may be written (n+ -, m) h (kr) ik(r -a) ( (1), m ) hn 1 ae 1- 1m(2ikrm n 1k) a e h (ka) r n 1 n 1 - s — (n+ 1, m) " (-1)m - m=0 (2ika)m or, reversing the order of summation and simplifying, 32

! THE UNIVERSITY O-F MICHIGAN 3648-7-T n (kw /a r1'^r8' ^-1 m2r h (kr ) n+l ik(r1- a) -2'. n-m)(2ikrl)2 n 1. (-21- ) In m=0 (5.25) h (ka) / r n n (-1)m(n+, n-m)(2ika)M 2' m=O With equations (5.2) and (5.25), the exact form of w(p1) may be written -ikr o(pl) = e u (p1) n oo. y (-l)'(n+ -n-m)(2ikr )m = -e ( —i)(2n+1) (- (ka)P (cos&O) n=O 1 x n| (- l)m(n+, n-m)(2ika)m 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 -o ji(P) = m =7. (_1)m 2n+l m=O (p/2)2m (5.27) m'r(m+n+ ) 2 we see that j (ka) may be written as a power series in ka whose lowest order term is (ka). Thus all terms in the series (5.26) for n > 2 are of order k or greater, and, since -ika (ka)2 3 e = 1 - ka- - + O(k ), 2 (5.28) we may write the exact result, (5.26), as, I ~ I i, I I" 33 _ --

I THE (p1) = - 1- ika UNIVERSITY OF MICHIGAN 3648-7-T 2 aJ r jo(ka) - 3i ( ) jl(ka) P1(cos) l5) kP~1 3, O) 2ikr 3 2- 5 a J2(ka)P2(cos1). -.O) 2ika 25 5r) 0)(2i 5( 1)(2ikr ) + (-, O)(2ikrl] 2' ~1 2 + O(k3). 5 5 (- 1)(2ika) + (2' 2 (5.29) Making use of the notational definition in (5.24) we have 5 (, 0) = 1 (2' 0). 1 ( 1) = 2 2' (5.30) (- 1) = 6 2' (, 2) = 12 2' and from the definition of jn, (5.27), we see that n (ka)2 4 jo(ka) = 1 - (k + O(k) j (ka) = + 0(k3) 1 3 2 j (ka) = (+ O(k4). Substi g 15 Substituting (5.30) and (5.31) in (5.29) we obtain (5.31) 34

I I I I THE UNIVERSITY-S I~ OF MICHIGAN 3648-7-T = [(] (ka)2] a = - l-ika+ 2 r1 + 3i [- ika a 2 k 2-2ikr1 (_) 3 P1(COS01) 2 -2ika 3 (ka2 12-12ikr -4(kr1) +5 ( ) (ka) P2(cosel) 1 1 1 l 15 12-12ika-4(ka)2 + O(k3) (5.32) This may be further simplified noting that, to the required order, 1 - ikr - (kr 1- ika- (ka)2 3 = 1+0(k). Hence (5.32) becomes a r1 J-ika 2 a rl..1 (ka 6 2 + i (-a-) ka P1(cos0 1)(1 - ikr1) 1 (ka)2 + 3 3)P2(cos1) + o(k3) -a- P2(cosO) + O(k ). 1 (5.33) Collecting terms in like powers of k we find 2 (p ) = -A- + ika -a- + Pl(Cose) rl r1 \r1 1 + (ka)2 2a 3 r + -r P1(cosOl) r 1 + (-) P2(cosOj1 + O(k3) (5.34) 35 I 35l - -~-.p~~e —-— ~-l l~L~-r- - ~- I plk-9 I —--s - — e-c

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: 27r 7T 00 (0) 1 -ika - ikacos6 (p1) d d O sin (2n+1)P (cosy)e mn=O m=0 1 I 27 d \ dOsinOP (cos ) P (cos0) \ 3n m 0 0 (5.35) a rl r27r 7 0 (N+i) ) = + id [ n(N) p poo p~ir a 2-r n=o Jo n~n= O r1 1+ \dr do dOr sinO n P) ) rp) 2r J J0d J n=O r n ar 00 27r 70 0 2r(rrn+l ik a n=0 a a + w(0)(pl) I I' ~ 36

T.HE UNIVERSITY OF MICHIGAN 3648-7-T With the orthogonality relation (5.10) we find that (0),) (0 (p) = -e 1 00o,-ika (2n 1)(-i)n n=O n+l 1r in(ka) P (cos 1) n n 1 (5.36) I Now calculating (1)(p1) using (5.35) and (5.36) we have e c 6 I a r (0) (pn -ika 3r Lru W = e 0^~-.0,n+1 (2n+ 1) n(-i)n () j (ka)P (cos0) n=l and L(1)(pl) 0 p1) 0O 00 ik -ika 77^n. m = - e %Z (2m+l)m(-i) j (ka) n=O m=1 dr r La 52r 2 n-1 dp5 dOr sinO r m+1 (a) P (cosy) P (cos0) r n m OD 27T r n r + dr do\ d r sin n+2 0 m+1 (a)'r/ P( s)P ( cos y (os0) n m o00 - \ dr a do 0?r 2nSnl r 2 a211-l dOr sinO n+l nt+2 0 r1 r a +1 )(0) )(f )P (cos )P (cos) +& )(P ). (5.37) Using the orthogonality relation (5.10) this becomes 37 ~I I - ~. - - - I-I I I r - -

I THE UNIVERS'ITY OF MICHIGAN 3648-7-T I I (1) -ika ( (p ) = 2ike Sa) + \ dr rl dr m(-i)mj (ka)P (cos01) \ — j m m 1 m=l / am+l (a) r 1 m m+l r a 2m+1 r o00 - dr > a + (O0)(p) +w(1 which, on carrying out the r-integration becomes ()(p1) = 2ike-ika C) p1) 00'~Zm(-i)mj (ka) P (cosO1) m m m 1 E1 m+l a 2mrI 1 m+2 a I 2mr1 +~ (pl). (5.38) This may also be written (1) (p1) 00 -ika \..m = -e /, (-i) m=l m+1 (-) j (ka)P (cos0 )(2m+1)(-ik)(r -a) r /m m 1 1 00 -eika (2m+1)(-i)m m=O (a r1 m+1 j (ka)P (cosO1). m m 1 (5.39) Continuing the iteration we have 00 a r (i) ~ -ika'Tarr )(p)J = -e m m=l m m m () (ka)P (cos)(2m+ l)(-ik) a r + ma m+l r + a r (O)(P -arL PJ (5.40) 38,I -

I THE UNIVERSITY OF MICHIGAN 3648-7-T I and (2) (p1) = 2( -ika I 2 e (-i)m j m=l (ka)(2m+ 1) a idr dr a 2ir \ d0 Jo n-1 dOr2sinO n+ 0 rl P(cosy)P (cose) [-m m r + ma m+l r + dr rl 2i7r do 0 S2T ^n dOr sinO 2 P (cosT)P (cosO) e-m n+2 n m m 0 r r 0n 2n + ma r r 00 - \ dr a 27r 0 JO C 2 a2n+l 2 a \dOr sinO n n2 P (cos7)P (cos0) Jo rl n+2 n m 0 rI r -m+l [m r + + w(1) (p ) - (5.41) Using the orthogonality relation (5.10) this becomes (2)p) = 2( -ike a -- 00 (-i)mj (ka)a P (cos ) m=1 m m m=l 4Idr dr a (-m+ 1)r m+L L r _mam+l r1 r O + da r1 (-m+ )rm 1 2m r m marl 2m+1l r \dr - dr a 2m+l a m+l rl (-m+1) 2m r + w( )(p ) ( 1) +w P (5.42) - 39

I THE UNIVERSITY OF MICHIGAN 3648-7-T which, on carrying out the r-integration becomes oo 2 (2) ) 2 2 — ika \i)mj ml(-m+1 (1 a2 ma1 -(a) (a (-m+a m+l)a2 ma ) This may also be written + m +l + +.....m r() 2 -1r il) () 2(ik) e (-i)m (ka) ( ) (2m+l)P (cos m=l 1 F(-m+1) r2 2 + a, ~ a (I). 2(2m-1) 1 r 1. I -(rl-a)+-r(rl + l (5.44) With (5.39) we rewrite this as W(2(p) = -e- ik (ka (2m+lPP (cos Q) 1)(kr1)2 111 1 m ) = 2 1L(i ) ( -) rk -(ka)2] +ka(k"-r) ka - ka + ika + i - eika a j(ka). (5.45) 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(p ) given in (5.26) we see that, separating the n = 0 term, this may be written as follows, I 40

THE UNIVERSITY OF MICHIGAN 3648-7-T 00 w(pl) = -e - j (ka) - e (-i)n(2n+l) (- j(ka)P (cos01)S w() = e a jo(ka) j n n + 1 n=l 1 (5.46) where n -l)m(n + 1, n-m)(2ikrl) S m=O n n (-l)m(n+ -,n-m)(2ika) 1 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 (-l)m(n+ 2, n-m)(2ikrl)m m=0O S n n (n+ - n) +,(-)m(n+, n-m)(2ika)m 2 - 2 m=l or, dividing numerator and denominator by (n + -, n) n (n+, n-m) 2 z (-1)Tm 1 (n+, n) S 2 (5.47) n n ( 1 ( n'- m 2(n+ 2 n-m) 1+ I _(-I)M 1- 2- " (2ika)m m=l (n+, n) We may always choose ka small enough so that n 21 (ia <1(54 (n+ 1n-m) (-1)m 12' (2ika)m < 1 (5.48) m=l (n+-, n) i 41 L I I I

THE UNIVERSITY OF MICHIGAN 3648-7-T and, with this restriction, then expand the denominator in (5.47) obtaining n 1 F 71n + 1 (n+ n-m) S= (-1)M 2 (2ikr)m (-1) -)m — ( —2 (2ika)' (-1)m0 2'k1 (n+ 1,n-m) m=O (n+,n) =0 m=l (n+,n) (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 1 (n+, n-m) (-n) _2 — = 7 m.(5.50) + 1 - (-2n) m! (n+-,n) m (-n) m im A gross estimate of how small ka must be is found by noting that ( ) (( 2 m Using this fact we find, after substituting (5.50) in (5. 58), that | 1 -)m 2- - (2ika) = (-1) (-) (2ika) n 1 D,(n+ i ka, < kaiM e Ia ) m=l~ (n+,mn) m(n =l mn) In2 (-D — (2ika)' < 1 m=l (n +,n) 0 I 0 42 I- -I

I THE UNIVERSITY OF MICHIGAN 3648-7-T With (5.50) we fihd that Sn =1 - kr- (-n)(-n+1)2 4(krl) O(k 3 [1 + a (-n)(-n+ ) 2' 4(ka)2) n [_ l1r! (-2n)(-2n+1)21. 1 (-2n)(-2n+1)2. - (ka)2 + O(k3 = 1 ikr1+ ika+ (2 (1) Fkr) - (ka)2 + a(kr - ka) + (k (5.51) Substituting (5. 51) in (5.46) we find that provided ka is small enough the exact result may be written oo, ^Fn+l () =-e -ka a j (ka)eika (-i)n(2n+l)(a-) j (ka)P (cos0 1 -ikr +ika+ka(kr1 -nka)l+2 1 3 + ika + ka(krl -ka) (-n+ 1) ( ~^ kr )2 - (ka)3 (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 I I I -I - -— 43 _ I, I,

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 fur 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 I d I a8 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 Courseof 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 _ Il I I

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 w+f (A.1) where G (Pv, Pl) _ ~ r [r o o = -2i \dV ) i-.. [r o (A.2) -ikr f=- e S l(p) G(p, ) dS (A. 3) is given by the Neumann series 00 w = knOno 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 w(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. 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,) = P Pn (cos0)eim (A.5) n b mn n m=-n and by Y (0, 0;0, ) a symmetric nth order spherical harmonic n Yn(0,;O1,10) = iAmnnn )(cos)m()-1) (A.6) These functions enjoy the orthogonality property 5 dO = d0sin0Ym (0,)Y n(O,;0 P ) - 0, m n JO JO0 = Yn(0ll) m=n. (A.7) Here it must be kept in mind that Y (0, 6) and Y (01 1) 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 -1-7 1 -- I

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 Go(p,pl) 4srR(ppl) n=O rn+l 1___ _ Y (Oi, 1) G1 + n(,.n+i1 rl >a (A.9) o _ 47rR(p, p) + n 1 1 n=O rr) 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, 10 n 1 r P [cosocoso1+ sinOsinO, cos(p- )) (A. 11) R( = 47rR(p p) + 1^r n= O r where r>= max(r, r ) r= min(r, r ). Note that the expansion has the same convergence properties as the seres in (A. 8) - (A. 10) provided r b rl. In addition to the orthogonality of spherical harmonics, it will be useful to define a related property. 0 49

THE UNIVERSITY OF MICHIGAN 3648-7-T Definition. A function f (0, )) will be called a "pseudo-spherical harmonic of n order n" if S7T 2 ir 0 ~0 s dO rO dp sinef (0 = 0mm(el,; 1,1 m <n =Y(01, 1), 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) b) w =- E n-, r r>a and the series is uniformly r n=O rn and absolutely convergent, term by term differentiable, W = 8 w with respect to r, 0, or and the resulting series are uniformly and absolutely convergent. c) f (0, ) are pseudo-spherical harmonics, i.e., satisfy (A. 12) (A. 13) I,I, - 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, IHIW = pV I W(P) I * (A. 14) 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: feW 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, pt) 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. f c (V). Furthermore when r >a we utilize (A. 8) and (A. 11) to obtain 1 r1 rn G( = P 47n+l Pn cosOcoso8S+ inOsi os(P- S)+ Yn() n=0 r n=O r (A. 15) I - -— l r. ~ s = I W 51 ~II I

THE UNIVERSITY OF MICHIGAN 3648-7-T or since P is an nth order spherical harmonic n 00 n Go(PPS) = n+1 (PP(os) em (A.16) n=O r m=-n This series converges uniformly as does the derived series; therefore, we may rewrite (A. 3) as oD n C f = - P (cos0) e e u A (pS)dS -- n+ n n nm n=0 r m=-n (A. 17) which is again of the form Y(0.) Proof: With the definition of 0, eq. (A. 2) we write n=O r v 0 ow = -2i dV V r w(p ) (A.9 ulx eV 8V v v We separate the volume over which the integration is performed into an infinite volume, Ve, where r >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 exterior to the surface S. Thus we define two functions exterior to the surface S. Thus we define two functions I _ IL -II, 52

THE UNIVERSITY OF MICHIGAN 3648-7-T w(p) = -2i V dV o aV [WV. (A. 20) V )] e i Clearly w (p) + wi(p) = O o w and if we can demonstrate that w and w. are elee 1 e 1 ments of W then, since the space is linear, it follows that O 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 - a [rLw(p)1 is conrv V vw(P tinuously differentiable. Therefore, w.(p) c (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 oD n im ( P 11n ar w^p) = E ^ n Zl E Pm (~~0)e X dV(-2i) r -- [rvw(pv)J n=0 r m=-n V V 1 r >a (A.21) which is of the form wi(P) = n+ r>a (A.22) n=O r hence w.(p)~e W. (A. 23) 1 I r- _ ----- - I - 1 53

I 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 2 volume then the fact that w e c (V) again follows from Kellog's work. It is only necessary to show that w remains well defined when V becomes infinite. Explie e citly Pj 2 (2 G7T (p, pV) w (p) = -2i d d d r sinv rV arv v E ~e - 1 ODa V Vrv, vLrw and it is sufficient to show that the integrand is 0(l/rV) for large rV. Since w(PV) ~ W it follows that w(P) E n+ + rv a (A.25) n=O rV and therefore that WaarPv)vv = J -L L nl V (A.26) V n=l rV Thus for large r, rvw(Pv = (/r. Furthermore, the expansions of V Go(p, p) given in (A. 9) and (A. 10) show that for r sufficiently large, ( = (l/r). Hence, despite the factor r in the volume element, the r V V integrand is indeed O(1/rV) and it makes sense to let the rV integration extend to a 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,r >a as I! 54 I

THE UNIVERSITY OF MICHIGAN 3648-7-T we(p) = ldrv d V dOVrVsin OV 47rR(p, p) a 0 0 v + EI Ym('~ V ) 1 n V'.'.m+f1' n+l m=O (rrV) n=l r (A. 27) where we have absorbed the factor 2in in the functions f (0' v ). Now consider n VV separately the integrals involving the regular and singular parts of the static Green's function, treating the regular part, w, first. In this case both series are unie formly convergent and the integral has been shown to exist, thus we may interchange order of integration and summation and perform the integration using the pseudo-orthogonality condition (A. 12) to obtain wreg(p) = drV \dV \dvsn Y V ninv. ea OV 0o V V m=O n=l r rV m+n m+l - Y-(A. 28) n=l m=n (m+n)a r Absorbing the constant factors in the spherical harmonics and renaming the second summation index yields co _ Y (0,j) reg (P) Ym+n.. (A. 29) ~~e s- ^^ m+n+1l n=l m=0 r 55 r - -- I —a ------- I ~r II I~_- - --

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) = ) (A. 30) e' n+2 n=O m=0 r While the coefficients in Y+1(0 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 wreg(p) C=, E n+2' (A.31) e +2 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 = rl. From (A. 27) we see that pOD /^27r 7r i nv /_ f wsing(p) 1 (7 d A. 32) 4r dr dv (A.32) g(p) _ 1 R(p, pV) 1 l 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 100 20i sinc f (Oe,,.^ we n=mgV V R(p,pv) n Ne drw e y R(pt n s (A. oti33) Now we employ the expansion (a 11) to obtain Now we employ the expansion (A. 11) to obtain I 56

- THE UNIVERSITY OF MICHIGAN 3648-7-T co Sir 2 r oD m ei r i sm0e r sing.sin g Vp) w gyp) = { adrv dov deV -f fn(OV, V) dr y. 1 r0V'n V) n=1 a o O " V m=O r a 0 sm0 rv = is ing nei suicient s s f in, ain uin e pseudo-orthogonality property (A. 12)O 0V V m-n+V m -n m+1 m rY71 J o J1 tv m s(A.34) Although the inner summation is singular t r = ra 0 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) we sing ) y (, ) n+) y ( (A. 35) n=l m=n r (m - n+ 1) (n+m) r Again absorbing the constants in the spherical harmonics we obtain we (p) = PE- 1Y (0 )+ ZL E m+Y ( A. 36) n=1 r rm=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. I____ ____________57

THE UNIVERSITY OF MICHIGAN 3648-7-T 00 00 00 w (e E) - n Ym(0, )+ n+2 (A. 37) n=1 r m=n n=O r and, with (A. 31), we find that w (p) is also of this form, i. e., w (P) =re( wg( p) + wY(p)+ j Y(,e e e n - n+2 n=1 r m=n n=O r (A. 38) or by a trivial change of notation 1 n w (p) = n+1 Y (O, ).+ 3 9(A. 39) e n= n+1 m n+1 n=0 r nm=n+l n=l r This is precisely the form required for w (p) to be in W, i.e., e w0 f (O, U_ ^f~e^) w(p) = _j n+,' r 3a (A.40) n=O r where 0 0 m=l (A.41) 00 l ZY (0,0), n>1 I n'~-m'I m=n Having shown that w (p) satisfies (A. 13b), it remains only to demonstrate that e f (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 0 JO = 0 dO p E ZY (, )Y1(0,; 0l,0)sinO = o, L<n Jo m=n = Y(0' l), > n Therefore we may conclude that We(p) e W (A.43) which, with (A. 23) proves the lemma. Lemma3: 0 isbounded, i.e., ]Mrea < 3 ||Ioll<M. Proof: If we W then Oow e W by lemma 2. Therefore 0 o w is twice differentiable everywhere in V which guarantees that O o w has no singularities in V. Thus ]M 3 lo w| M w, peV (A.44) and, recalling the definition of the norm (A. 14), I|0 o w|l M. (A.45) sup M Let M = wEW, lwll 1M. This must exist or else for some we W, o owj would be unbounded which would contradict lemma 2, the fact that Oowe W. Thus e p, l110owll < M. (A.46)'The lemma then follows since, by definition, I 59 _ I ~I_ I

I 4 THE UNIVERSITY OF MICHIGAN 3648-7-T o1 = w ll0 = lo - (A.47) weW, iW11 10 Lemma 4: 3k ||kOII <1 if |k < |k |. Proof: Since O is linear it follows from the definition of norm that IlkOil= Iki loll0. (A.48) With lemma 3 we obtain IlkOl.< |k|M (A.49) therefore by choosing Ik < 1/M or equivalently, letting |k | = 1/M, the lemma follows. Lemma 5: wE W. s -ikrs Proof: The definition of U in terms of u, = e u, together with the fact that se u together with the fact that u ec (V) imply that e c2(V). Furthermore, the expansion theorem (3.1) guarantees that we may write p) = _ >r n+1 r a. (A.50) n=0 r It remains only to demonstrate that these f (0, 0) are "pseudo spherical harmonics". To accomplish this we employ the well known expansion of wave functions in spherical harmonics, e.g., Sommerfeld (1949, p. 143), 00 u(p) = h(kr)Y(,) ra (A.51) n n n=0 where h (kr) are spherical Hankel functions of the first kind, n i 60

THE UNIVERSITY OF MICHIGAN 3648-7-T n ikr.-n-1 v-', i a e 1i (n+m)' -1 h (k) (nr)= r (A.52) n r (n - m) 1. m! 2ikr With this expression together with (A. 51) we find that co n -n-i ikr s n E y n(0, )i (n+ m)! (_l)m w(p) = e u (p) = -A.53) n=0 m=O r (n- m)! m (2ik)m or, upon rearranging terms and absorbing the multiplicative factors in the spherical harmonics, co Yco (0) W(p) m r E n+ml' (A.54) n=0 m=0 r This can be rewritten, with the obvious changes in notation so as to correspond to (A. 50) OD f (O, ) w(p) = n+.L=0 r where 0o f (8,) = (8, ). (A. 55) n +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) are finite sums of spherical harmonics is in error. I 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, No 3 W -W(N)I <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 W - w(N) for any N. With (A. 1) and the definition of the iterates it follows that )-f = kO o w (1) 2 2 w-w w) = w-kOof-f =kO ( —f) = k O aw -W(2) = w-kO o -( f = kO. (w-w)) = k303 (N) N+1 N+I1 w-w = k 0 W (A. 56) hence w-WJ < kOH^|W |. (A.57) But lemma 4 states that l|kOl <1 if Ik| < Ik and ||Wij is bounded since wec W (lemma 5) hence it is always possible to find N large enough so that IkoLkO IIW1I <e (A. 58) Specifically, since log I|kOl < 0, we find that (A. 58) is satisfied if 1 j 62

I I - THE UNIVERSITY OF MICHIGAN 3648-7-T log T N+1 > log kO (A. 59) We have thus established that for any > 0, Iw-w( N) <e if N>N and Ikl <1ko log exists by lemma 4. where No is the greatest integer in lo - I and k exists by lemma 4. To prove uniqueness we assume that existence of two solutions of (A. 1), w and w2 such that 2 L1 = kO wo1+f (A.60) 2 2 and 111-211l ~ 0. (A.61) Then w1-o2 = kO o 1- kO o 2 (A.62) and taking norms we obtain w l -"| C rII liw'l (A. 63) y assumption I W 21 - w 0, hence we may divide obtaining 1 l ||kOl (A. 64) hich violates lemma 4. 63

UNIVERSITY OF MICHIGAN 3 9015 03023 8367