THE UNIVERSITY OF MICHIGAN 7359-1-T ON THE HELMHOLTZ EQUATION FOR AN ACOUSTICALLY RIGID SCATTERER by ErgurAr May 1966 Report No. 7359-1-T on NSF Grant GP-4581 Prepared for MATHEMATICAL SCIENCES SECTION NATIONAL SCIENCES FOUNDATION WASHINGTON 25, D.C.

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THE UNIVERSITY OF MICHIGAN 7359-1-T ACKNOWLEDGEMENTS The author is most grateful to Dr. Ralph E. Kleinman for suggesting the problem. The present work would not have been possible without his original idea. To his chairman, Professor Gerald W. Hedstrom, the author expresses his indebtedness for assistance, encouragement, and for providing some key counterexamples. The author is grateful to Drs. Raymond F. Goodrich, Olov Einarsson, and Vaughan H. Weston for listening patiently and critically to the arguments involved in connection with the various aspects of this Work. Thanks are due to Miss Beverly Ballard for her excellent typing and editing of the manuscript. This work has been supported under the National Science Foundation Grant GP-4581 through the Radiation Laboratory of the Department of Electrical Engineering, The University of Michigan. The author expresses his thanks for this support. iii

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THE UNIVERSITY OF MICHIGAN 7359-1-T TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii INTRODUCTION 1 CHAPTER I THE NEUMANN PROBLEM 4 1.1 A Representation Theorem 4 1.2 A Representation of Wave Functions 10 CHAPTER II THE SOLUTION OF THE INTEGRO-DIFFERENTIAL EQUATION FOR THE NEUMANN PROBLEM 14 2.1 Preliminaries 15 2.2 Construction of the Space 17 2.3 The Solution 19 2. 4 Remarks on the Low Frequency Expansion 39 CHAPTER III AN APPLICATION: SCATTERING OF A PLANE WAVE OF SOUND BY AN ACOUSTICALLY RIGID SPHERE 43 CHAPTER IV CONCLUSION 48 4.1 Functional Analytic Aspects 48 4.2 Non-Separable Surfaces. Bodies with Edges 49 4. 3 The Radius of Convergence of the Low Frequency Expansion 50 4. 4 Extension to Vector (Electromagnetic) Problems 50 4. 5 Two-Dimensional Low Frequency Scattering Problems 50 RE FERENCES 52 v

THE UNIVERSITY OF MICHIGAN 7359-1-T INTRODUCTION The classic three-dimensional scalar scattering problem consists of determining a function 0 exterior to a smooth finite boundary B which is a solution of a scalar Helmholtz equation, satisfies a Dirichlet or Neumann boundary condition on B, and obeys a radiation condition at infinity, i. e., (V2+k2)s = (1) s - or ac - on B (2) On an lim r a 0 -iks ) =0, (3) r ---- c where 0 is the incident field which is known everywhere including the bound - ary B. The study of the relation between this problem and potential problems (boundary value problems for the Laplace's equation, V20=0) goes back to LordRay(33) leigh. The general problem is one of generating solutions of the Helmholtz equation (vector or scalar), which satisfy prescribed conditions on a given boundary in terms of solutions of Laplace's equation. Physically, this amounts to an attempt to infer the manner in which an obstacle perturbs the field due to a source of wave motion from a knowledge of how the same object perturbs a stationary (non-oscillatory) field, e. g., determining an electromagnetic field from an electrostatic field. The advantage of such a procedure derives from the fact that stationary fields are physically simpler than wave phenomena and associated mathematical problems, though often still formidable, are always more easily handled. 1

THE UNIVERSITY OF MICHIGAN 7359-1-T Interest in this problem has gained new momentum in recent years (5) (31) (21) (see Bouwkamp, Noble, Kleinman for an extensive bibliography). The major drawback in most of the methods heretofore proposed is their intrinsic dependence on a particular geometry. That is, the techniques result from the exploitation of the geometric properties of the surface on which the boundary conditions are specified. 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. Most low frequency techniques, however, have as their starting point the formulation of scattering problems as integral equations using the Helmholtz representation of the solution in terms of its properties on the boundary and the free space Green's function; i.e., 1:p-I s a )upp(4) Sp s(B naPBan u(pB,)-a, u( pB) (PB)} dB (4) 4' %8 n'B B an B'f(4) B where ikR(p, kR(p, pB) B the integration is carried out over the entire scattering surface B, the normal here is taken out of B, p is the general field point, and PB a point on B whose coordinates are the integration variables, and R is the distance between them. This formulation is also vital to the proof of existence of solutions for a general boundary by Weyl(5) Muller ), and Leis 3. The investigation of the solutions for the scattering problems with the help of integral equations originated by the works of Rothe(3, Sternberg(41), and Kupradse(22).. Werne4549) also provides different existence proofs.for acoustical as well as electromagnetic scattering problems. More will be said on the existence question in the Conclu(31) sion. Noble shows how the integral formulation (4) may be used to obtain a 2

THE UNIVERSITY OF MICHIGAN 7359-1-T 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. Long sought has been the development of a systematic procedure which will generate the solution of the Helmholtz equation, satisfying a particular boundary condition, from the solution of Laplace's equation which satisfies the same (21) boundary condition. This goal has been achieved in a limited sense by Kleinman for the Dirichlet problem on which the present work is based. In Chapter I an integral equation for the scattered field is derived, whose kernel is the potential Neumann 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 and the surface and it is really an integro-differential operator, it is still possible to solve the equation for the wave numbers k (assumed complex) with sufficiently small modulus. This is done in Chapter II. Also, the relation between the low frequency expansion and the Neumann-Liouville series for the solution in indicated. In Chapter III the procedure is applied to an acoustically hard sphere. Since the exact solution in this case is known, a check (for the first three terms in the low frequency expansion) is provided. 3

THE UNIVERSITY OF MICHIGAN 7359-1-T CHAPTER I THE NEUMANN PROBLEM 1.1 A Representation Theorem Let B denote the boundary of a smooth closed bounded surface in E3 and V is the volume exterior to B. Erect a spherical polar coordinate system with origin interior to B and denote by p a point (r, 0, 0) in V and by PB a point (rB eB, B ) on B. The distance between any two points p, plV = V+B will be denoted by R(p, p ) and is defined as R(p,P) [r +r -2 rr (cs os cos 0+sin0 sin01 cos(0 -o0 l0 (1.1) Furthermore, let c=maxrB so that B is contained in a sphere of radius c and assume that the normal to B is directed inward (out of V). Definition 1. 1 A function f(p) of the coordinates of p is said to satisfy a Holder condi - tion at (or is Holder continuous at) p if there are three positive constants, A, B, and a such that f(p) - f(p) A. Ra(p, p for all points p for which R (p, p ) B. If there is a region G in which f(p) satisfies a Holder condition at every point, with the same A, B, anda, f(p) is said to be uniformly Holder continuous. Definition 1.2 We shall define a surface Bto be smooth (or regular) if (a) it can be described by an equation rB =g(0,0) (1.2) 4

THE UNIVERSITY OF MICHIGAN 7359-1-T where g is a continuously differentiable function of 0 and O, and (b) n rB Ah A rl~~~~B B (where n is the unit normal, rB is the unit radius vector) is uniformly Holder continuous on B. Definition 1. 3 A real valued function f: V-* E is defined to be regular (in the sense of Kellogg) at infinity if 2 df rf = 0(1), r - (1) as r - o (1.3) ar uniformly in 0 and 0. A complex valued function is regular if both real and imaginary parts are regular. Definition 1. 4 The Neumann potential Green's function for the surface B, the existence (2Q) and uniqueness of which is proven by Kellogg, is defined to be a function Go(p, P1) of two points and may be written in the form oG ~- 4rR(pp) +Uo(PI p') PI peV where u (p, 1) has no singularities in V and (a) V u (p,p) =, p, pEV (b) Gn (PB P) [This notation is used repeatedly and has dn o B 1 n t the following meaning. Let Vbe the gradient operating on coordinates of p and n the unit normal on B directed out of V. Then define 0 ^ tsp)A VGo(p, pl) =VG (p, p) and Go(PB ) p)n VGo(p, 0 B' 1 o an o B' 1 o B' 1 peB (c) uo is regular at infinity. (1 5) 5

THE UNIVERSITY OF MICHIGAN 7359-1-T In terms of this Green's function we may state an integral representation of functions regular at infinity. This is contained in Theorem 1. 1. If uw V -- E is a function which is twice differentiable in V and regular at infinity, then W(p) = Go(P, Pl)V2w(pl)dv- G(pG(P, pB n1 (pB) B (1. 6) vV v B where Go(p, p) is the Neumann potential Green's function, dv is the volume element and V2 is the Laplacian both expressed in coordinates (r1, 1 0) deB is the surface element and a/an the inward normal derivative. (out of V) both expressed in coordinates (rB, 0B, B). Proof. Let r = B+B +B, where B is the surface of the body assumed to be regular in the sense of the Definition 1.2, B1 is the surface of a small sphere, with radius e> 0, with the center at the point p, B is the surface of a large sphere containing B and B1. Further we erect a rectangular Cartesian coordinate system with origin inside B, and let V" denote the volume bounded by B, B1 and B2 (see figure below) z x z B n O6.f A -/ --

THE UNIVERSITY OF MICHIGAN 7359-1-T The functions w, G and the surface r are sufficiently regular so that we can 0 apply the Green's second identity; thus, dG C 2 2 OG 0 (G-V -V Go)dv (Go ^) ) da. (1.7) n 0'dn don v" r Here we note in passing that the usual formulation of an integral equation for a wave function involves the application of the above identity to the free space + ikR Green's function e- /R (the sign ambiguity is removed with a particular choice of harmonic time factor) and the other field w scattered by B. The integral over B2 is then shown to vanish by virtue of the radiation condition and the integral over B1 evaluates the scattered field. The volume integral vanishes since both functions are chosen to be solutions of the homogeneous Helmholtz equation yielding the well-known result + +ikR a +ikIC1 e(xy, z) =o w a (e - ] d. (1.8) B Here, however, we wish to employ, not the Green's function for the Helmholtz equation but the Green's function for the potential (Laplace) equation G, given by (Definition 1.4) G(P P1) = 4RR(p, P1) + Uo(PP) Substituting this into the Eq. (1. 7) we obtain 4rR 4rR o an an 4rR V" _r=B+B +B 7

THE UNIVERSITY OF MICHIGAN 7359-1-T Integration over the surface B1: u Let Y(P*) denote the mean value for some p*eB, of u -w IL B on an on B1. Then C au 2 (uo an )d = Y(p*) 4r2. (1.10) B1 Thus, as the radius e -- 0 this integral vanishes. a o Also, on B - -—, and for the spherical coordinates (R, e, ) with the 1 an oaR the center at (x, y, z), we have lim. \' d o d R s in0[4R a- R (x+R cos0 sin0, y+Rsinp sine, z+R cose) R —,0 J 0 0 v 2n -w(x+RsinO cosp, y+Rsin0sino, z+Rcos)- 1R =lim -. dO S sinO 0 0 * [R- +w(x+Rsinocosp, y+Rsin0sin0, z+Ros = x, y, z). (1.11) Combining (1.10) and (1.11), \ (Godn -w-n ) da = w(x, y, z) (1.12) B1 Let Vt = lim. V". Incorporating (1. 12) into (1.9) and observing that _ 0, (1.9) reduces to

THE UNIVERSITY OF MICHIGAN 7359-1-T 2 va a~ d - (X, y, z)= G V wdv- G do (G -W )an. (. 13) J o an oan n. Vt B B2 Integration over B2: OnB2, an ar B2 B2 lim ( P)nn i(p0) -W(pB)n G p p) d 2 B lim do d< r s pine pG (ppJ).) ) G,.xp (p 2 0 0 2 2 2 2 2-r= lim G ( dPB2 sInS[o(PB2 Ga B2 ] =lim do0 d0 -f rB Go Br aU(P )+ - r (p r ( 2 P2 B 2P B2 2 (PB 2o B2 2 2 9~~~20)(1. 14) since G and d are both regular in the sense of Kellogg (Definition 1. 3).

THE UNIVERSITY OF MICHIGAN 7359-1-T Since V=lim Vt, with (1.14), (1.13) becomes rB- * 0 2 IJ( X, Y. )2 dw (PB ) (x, y, z)= S G(P )V ()dv1- \G(p PB) a do d o 1 1 o(n d~B V B proving the theorem. 1.2 A Representation of Wave Functions A function u: V-) E is a scalar wave function for the volume V if (a) u(p) is twice continuously differentiable in V (with the understanding that if pEB the limit is taken from the exterior, V), (b) (V2+k2)u(p) =0, p, (1.15) (c) r (a -iku) = o(1), as r - oo, uniformly in 0 and. Other statements of the radiation condition are possible [Wilcox( 53 (39) but this form, as given originally by Sommerfeld ) is quite adequate for our purposes; it may be stronger than necessary but it does what we want it to do, namely, characterize radiating solutions of the Helmholtz equation. We wish to employ Theorem 1. 1 to represent scalar wave functions, (20) but they are not regular at infinity in the sense of Kellogg.. In order to modify them we employ a well-known expansion theorem given with varying restriction (2) (4Q) (4) by Atkinson, Sommerfeld, Barrar and Kay and most generally by Wilcox(). Theorem 1.2. If u is a scalar wave function for the volume V, then ikr f (,0) r=-'-a n' (1.16) n=0 r 10

THE UNIVERSITY OF MICHIGAN 7359-1-T where the series converges absolutely and uniformly for r) c+c, E> 0 and c= max rB. Furthermore, the series may be differentiated term by term with respect to r, 6 and 0 any number of times and the resulting series all converge absolutely and uniformly. It follows from this expansion theorem that if u is a scalar wave function, then - ikr (p) =e ikr(p) (1.17) is regular and satisfies the hypothesis of Theorem 1.. Lemma 1. 1. The function u (p) defined by (1. 17) satisfies 2, 2ik a VU = - (r C) (1.18) r ar Proof. Since (V +k )u=0, it follows that (V +k )ei u=0 In spherical coordinates 2 a 22 1 a a1 _2 v2 a+ -- + -- (sin 0 + (1. 19) V 2 rar 2 DO a' 2 Dr r sine r 2sin 0.2 Since a ( ikr ikrf a P e u)=e i ku+ a (1.20) Dr ar and 2 21,) Da ikr, ikr 2. Di u ( e k)=e + 2-ik + a (1.21) r2 ar 2r 3r Ir J clearly (e u)+ (e u)e i+ - u(r u) t (122) Dr2 r rar or rr or r r11 11

THE UNIVERSITY OF MICHIGAN 7359-1-T 2 2 ikr Hence, with (V +k )e u =0 and (1.18), (V2+k2) ikrf ik r 2 ika (+ V e = (1. 23) ( ke e \ (r') + V 0 (1: 23) from which the lemma follows.. Next we note that a *(p B =V. ir - ikrikr au (pB) aUpB)=eV =nVe = - ikrB (pB)+e B (1u24) ann an B B A A where n and r are unit vectors in the normal (inward) and radial directions B A A respectively at p. (If B is a sphere, n -r.) Incorporating these results B B in the representation theorem (Theorem 1.1) establishes the following. Theorem 1.3. If (a) u is a scalar wave function for V, the exterior of a smooth, closed, bounded surface B and (b) G (p, p ) is the Neumann potential Green's function for this surface ( G (p, P )=0) then'u(=e u) may be represented as an o B' Go(P, P1) (p)=-2ik r r [r()]dv+ ik G(p, ^ ) V B -ikrB au(pB) G(p, B)e an (1.25) 12

THE UNIVERSITY OF MICHIGAN 7359-1-T Included in this theorem are representations of the solutions of the two most common exterior Neumann problems for the Helmholtz equation and the surface B. au(pB) a ikR(p.P ) an a n PR(PB' (1.26) then u represents the regular part of the Neumann Green's function for the Helmholtz equation. If au a iki \ I an = a- (e ik ) on B (1.27) then u represents the field scattered when a plane acoustic wave is incident in A the direction a on a rigid surface B. Note that the representation (1. 25) is in ikr terms of'u but u is easily found by multiplying with the phase factor e. 13

THE UNIVERSITY OF MICHIGAN 7359-1-T CHAPTER II THE SOLUTION OF THE INTEGRO-DIFFERENTIAL EQUATION FOR THE NEUMANN PROBLEM We write the Eq. (1.25) of Chapter I in the operator form u = L.t+u (2.1) with G (p p) wo- L.w= -2ik dvl ~ - [rl (pP) 1 r1l V +ik (ppB)nB p r(PB) (2.2) and u du-BBGO(p, pnB)e an *(2.3) B An explicit solution for (2. 1) may be given in the form of a NeumannLiouville series. That is, we rewrite (2. 1) in the form o) -1 (0) u'u (I-L d (2.4) and formally expand the inverse, obtaining GO a) u = (2.5) If we denote by u the partial sums (N) Ln.U(o), (2.6) n:O 14

THE UNIVERSITY OF MICHIGAN 7359-1-T it follows immediately that for N > 1, u(N) also satisfies the recursive relation N). Lu (N-1+u. (2.7) Our main task is to define a proper normed linear space and show that u (the solution we seek), u) (the known term), and all the iterates u(, N 1, are elements of this space and that in the sense of the norm of this space (N) ~ lim u u. (2.8) N- cao 2.1 Preliminaries First we mention some well-known properties of spherical harmonics and expansions of the potential Green's function [e.g. Kellogg2 (p. 143), (40) Sommerfeld, (p. 123), which will subsequently be used. We denote by Y (, 0) an nth order spherical harmonic n Y (0, ) = AmnP (cosO)eim0 (2.9) n _ mn n m=-n and by Y (, 0; 0,1' l a symmetric nth order spherical harmonic n Y (, 60;, 0 )- 0) A P(cos (cos )cos m(0- (0), (2.10) n mn n n 1 1 m=0 where P are the associated Legendre functions. n 15

THE UNIVERSITY OF MICHIGAN 7359-1-T These functions have the orthogonality property dnY(m, 1, 1 0)Y(O, 0; 1 01) =, m n =Y( ~, m5n, (2. 11) Y Y(, ), m=n where f is the unit sphere and dQ= sinOeded is the element of solid angle on it. The static Green's function for our surface B may be written nY(O) ~~aG (p, p(l ) R. + 1 n+1 6 r< (2.12) = 1- 4IR(pp) + ~l', rAd, (21) r n=O a in a an n i i p r>6, r < 6 (2.13) a) n 4R(pR(,) n+lp (2. 1) n=O r r =max(r,rJ, r =min(rr). 10 where the series are uniformly and absolutely convergent and may be differentiated or integrated any number of times with respect to r, 8, or 0; 6 = c+E, e> 0; and c the radius of the smallest sphere enclosing B. The source term may also be expanded in spherical harmonics wO n 1R Z pi:r+1n [oscoso 1+sinosino8lcos(0-01 (2.15) R(p, p1) r 1 n n=0 > where r> max(r r), r< =min(r,r1) (2.13), (2.14), provided r /r1 16

THE UNIVERSITY OF MICHIGAN 7359-1-T 2.2 Construction of the Space -ikr We recall from the last chapter that the function u= e u, where u is a scalar wave function, is an analytic function in the complex 1/r-plane having the expansion U En:, nrl >b 6=c+c, (2.16) n=O r where c is the radius of the smallest sphere containing the surface B, > 0. 1 Putting z=-, we have n+l 1 Co at= L...fn(O, ) z Iz < L (2.17) n=0 Clearly we may assume 6 = 1. Next we define the following function space. Definition 2. 1. Let W be the set of functions defined on V such that (a) e C2(V), W C ) (b) c is analytic on the closed unit disc having the expansion CD Z (, 0) zn+l IzI = 1 n=O (c) f (,0) = y (0, ), where Y is an mth order spherical m=n (2.18) 17

THE UNIVERSITY OF MICHIGAN 7359-1-T harmonic, i.e. m Ym(O.) Z Al P (cos )e0 m I mI m I= -m It follows immediately from this definition and the Cauchy's integral formula f (O, d d n 2i n+2 dr C where C is the unit circle around the origin, that fn(,0)| < max 1w(l)l ~EC We define the following norm on W. Definition 2.2 |11i = max I|(p)I + max w(0,0,-)|. (2.19) pEV I zi 1 06o,2r With this norm it is clear that | fe(., 0)| | ull (2.20) We now proceed to solve our operator equation in the space w || | W. 18

THE UNIVERSITY OF MICHIGAN 7359-1-T 2.3 The Solution Lemma 2.1 If a is uniformly Holder continuous on B (Eq. (2. 3)) then (0) u EW. Proof. The definition of u, (2.3), shows that u( consists of the potential -ikrB du of a single layer distribution of density e - plus another term (correan sponding to the regular part of the Green's function) which is at least as well behaved. Therefore the differentiability of u is essentially that of the poten-ikrB du tial. If the density e n is piecewise continuous then according to (20)O Kellogg (p. 122) the potential is infinitely differentiable in V, thus, in partic(o) ular, the potential (hence u ) is twice continuously differentiable in V. Furthermore, if the density is uniformly Holder continuous then, again, according (20) (o) to Kellogg (p. 165) the potential (hence u ) is continuously differentiable in the closure V. Since rB is a continuously differentiable function of 0 and 0 -ikrB it follows e is uniformly Holder continuous, hence so is the product -ikrB du (o) e Gn. Therefore u satisfies (2.18a). an G (p, p ) may be expanded in an absolutely and uniformly convergent series of spherical harmonics of the form [e.g. Kellogg(20)(p. 143) G (p,B)= Z Z Amn(PB) n+P (cos)eim, r>6. (2.21) o B mn n+1 n n=0 m=-n (o) Thus for r> 6 we may rewrite u as co n n\ ~-ikr u(~) _ n+1 Pm(cos )e im A (p )e B U d (2.22) u - n+l n n mn B dn B r " n=0 m=-n B 19

THE UNIVERSITY OF MICHIGAN 7359-1-T which is of the form o0 )1u E... (2.23) n+l n=0 r and satisfies (2.18 b) and (2.18 c), thus, proving the lemma. We next split the operator L defined by (2.2) as follows. Definition 2.3 Let L kL= kO+kO1 (2.24) where k is the wave number. Thus L1, 0, and 01 are all independent of k; they are given by 41' Oow= -2i dvI r [r1o(p)] (2.25) V and w- O w= 1 j du BGo(P, pB)no B (2.26) B Lemma 2.2 The operator 0, defined by Eq. (2.25), maps the space W into itself. Proof. We have Eq. (2.25) G (p, p) O w=-2i dv ~ 1 [rw()] 1 r ar rl (Pl 1 1 V We separate- the volume over which the integration is performed into an infinite volume, Vt' where r1 6 = 1 and the Theorem 1.2 of the last chapter holds, and a finite volume, V., between the sphere and the surface B, where the mt expansion theorem does not hold. We define two functions 20

THE UNIVERSITY OF MICHIGAN 7359-1-T G (P' P ) (O~ P))^e -2i dvl r;- d [rlw(P )] dv o xt =_2i r ext \ G(pP) a rG ((p)] (O.w(p)t=-2if dvl. 1 1 r (2 o (P) int 2 i dvl r ar 1r w(P1) (2.27) V. int If we can demonstrate that the functions defined in (2.27) are in W, then, since the space is linear, it follows that Ow= (Oow) xt+(Ow)int is also in W. Consider (O0w)int first. It is the potential of a volume distribution mt with the density - a [r 1(p1l which is certainly continuous; therefore, r ar L1 1 J [Kellogg )(p. 122)J the potential is infinitely differentiable, and hence (ow)int eC(V). Also, [Kellogg(20)(p. 151, 152)] (Ow)int cC(v). When r> 6 = 1, since r1,6=1, from the formulas (2. 12) and (2. 15) co n co I XI c0 y11 (r, i) 1 n'r G (p, +l n ~ cosecoso +sinesin1e cos(0'-0 + ) n'' o l bwt.....4...n+1 n L 1 11.J.-.. n+1 n=0 r n=0 r (2.28) or, since P is an nth order spherical harmonic, n OD n | 1 m im0 o P n+ Amn(P1) P (cose)e (2.29) n=0 r m=-n The series converges uniformly, (2.12), as does the derived series. We substitute (2.29) into (2.27); and since the integration is carried out over the finite limits, we may change the order of integration, thus (O )inO= 1 mm de im )int n+l n (c )e dv (-2i) pl) a w( n-j 1 r1. L1 p Jrar n0 r m=-n Vint (2. 30) 21

THE UNIVERSITY OF MICHIGAN 7359-1-T which is of the form x (e, 0) (~ ~)int E n+1 * r>6 = (2.31) n=O r and satisfies (2.18b) and (2.18c). Therefore, (O )it EW. (2.32) Now we consider (O )ext). If Vext is replaced by a large but finite volume then again it follows [Kellogg 0)(p. 122, 152)] that, in the volume considered, w is twice continuously differentiable, and it is once continuously differentiable in its closure. Next we study the exterior integral as volume extends to infinity. Explicitly, p 2v 2 G (p, p)' 2 oi (Pl ar (0o) t=-2i lim dr df r1 sin1 r - [r (pr)], (2.33) ext I-1 roo _r1 1 1 0 0 100 and it is sufficient to show that the integrand is of 0(- -) for large r1. Since 2r w(p1) eW, it follows that rl O f(81 I1 W(pl =/, 1 rl 6 = 1 n+1 n-0 rl and, therefore, that 00nf (Oi*Oi) W [r p])]:-i n+1 (2.34) n=1 r1 r\ - a 1 Thus for larger1, r1 wr(pl)]= O ) ). Furthermore, the expansions of 1 dar 1 1 2 ri 1 c) r1 rl 1i G (p, p1) o'1 G (P, p) given in (2.13) and (2.15) show that for large r1. = 0 1 rr Thus, as rl - oo the integrand is of O(-~) and (O)ext exists. r1 22

THE UNIVERSITY OF MICHIGAN 7359-1-T Now we pursue this calculation carefully to show that (O00 xt) satisfies the expansion properties required of elements of W. Thus, we rewrite (2.33) for r, rl>6 l (see expansion(2.14)for G) as oD 2 fe, eO T ( f 1 0), ^ n 1m=0 (rrl n=O rl 1 0 0 (2.35) where we have absorbed the factor 2ikn in the functions f (01, 0). Now conn 1 1 sider separately the integrals involving the regular and singular parts of the reg static Green's function, treating the regular part, (O0e), first. In this case ext' both series are uniformly convergent, and the integral has been shown to exist; thus we may interchange the order of integration and summation and perform the integration using the orthogonality properties of spherical harmonics (2. 11) and the definition of f (0, 0) (2. 18 c) to obtain OD 2 (0 w)reg dr ( 1d dOsinOl m 11) n(l'l ext 1 1d01 1 1 2... m+1 m+n+l 1 \, u m=O n=l r r 1 0 0 n=1- m~n (mEEnrm+l *(2.36) (m+n)r n=l m=n 23

THE UNIVERSITY OF MICHIGAN 7359-1-T Absorbing the constant factors in the spherical harmonics and shifting index, we have co y +n+co (o 0) (O')reg -m+n+ (2. 37) ext Z m+n+2 (2. n=O m=O r Using the identity oo co co n Z Z Q(m,n) Z Q(m, n-m) n=O m=O n=O m=O and since the series involved are absolutely convergent, we obtain reg 0n 0)I (00 )reg =2 Y n+1( 38 ext n+2 (2.382 n=O m=0 r The coefficients in Yn+1(0, 0) may depend on m, but the summation over m is still a spherical harmonic of order n+1, hence (2. 38) is of the form 00 (o_ reg n+(00) e(O w xt n+2 (2. 39 n=0 r and satisfies (2. 18b) and (2. 18c). We now pursue the analysis involving the singular part of the Green's function. The expansion of 1/R, (2. 15), is not convergent at r= r. From (2. 35) we see that ( sing 1 sin01, fn (0 ext= - dr1 dl 1 R(p, pl) n(2.40) 1 0 0 n=l 1 24

THE UNIVERSITY OF MICHIGAN 7359-1-T Since the series in (2. 40) is uniformly convergent and the infinite integral has been shown to exist, we may interchange the order of summation and integration, and absorb the factor (-~) into f, obtaining 4w n 00 2r Ir sing d d R(p sie f(r0) ( ext d R(p p n (2 1 0 0 We now employ the expansion for 1/R, (2. 15), to obtain ~sing x sine r + dr \ d0l O r n si 1 = E m+1 Y( P 1 01 ext =n n - r n= 0 0 1 m=0 sin 01 oo rn J ^ ^ ~rl m=O r (2.42) Although the inner summation is singular at r=r1, 0=01, 0=01 it is a straightforward matter to exclude a small neighborhood of p = (r, 0, 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 arbitrarily small by taking the neighborhood sufficiently small [e.g. Kellogg 0)(p. 148)] Thus we find, again using the orthogonality properties OD OD m-n+l1 0n 00 Y (O,0)(r mn 1) Y (0) }(2. 43) sing= r -1)+ ext m+1 n er (m-n+l) (n+m)r nsl m=n 25

THE UNIVERSITY OF MICHIGAN 7359-1-T We note that in obtaining (2.43) the condition (2. 18 c) is necessary, since without this property, terms involving logr would occur. Again absorbing the constants in the spherical harmonics from (2. 43) we obtain O.osing= r1 ~0 0 C0 Y~ Y (O, 0) (Ow) = n Y Y(0O,0) + Z (2. 44) ext n n m+l1 r r n=l m=n n=1 m=n Using the identity following Eq. (2,37) on the second series of (2. 44), and using the similar argument, we have 00 OD OD (O~ 0):~ xt ext n m n+2 (O.w)sing ii 1 ] Y (O,)+Z Yn+(0~) +) * (2.45) 2 r n= 1 m=n n= 0 Now we combine the Eqs. (2.39) and (2 45) to obtain, for r > 6 = 1, 00 Co OD 0 reg sing 1 n+l( ) (O ow) =(Oaw) +(0 ) ext sg= r n n+2 Ym( 0)+ n n=1 m=n n=O r (2. 46) or, by shifting index, ~~00 00 co ~~Y (06 0) (Oow)ext = -+ Ym(0 0)) n+ (2. 47) r r n=O m=n+l n=l This is of the form (O~w)ext n+ r 6 1 (248) ^^ext- 2-. -iTT - ^6=1 -(2 <48) n=O r satisfying (2.18b) and (2.18c), 26

THE UNIVERSITY OF MICHIGAN 7359-1-T with 00 o m o Z Y(v,0) (2.49) m=l 0O f Y (0,0), n>l. (2.50) n Z.- m m=n Hence, (O ) extW. (2,51) ext Finally, the Lemma 2.2 follows when we combine (2. 51) with (2. 32). Lemma 2.3 The operator 0, defined by Eq. (2.26), maps the space W into itself. Proof. We have A A 0 O c = i da G (p, B ) n r u (PB) 1 B o 0 J B B B dB[_~~B +uP lrgo(PB),, (2:26) = i dB 4-R(p, pB) +UO(P' PB nrB(pB) 2 B where u is a regular potential function at all points pEV. The Eq. (2.26) 0 shows that 0 1e is the potential of a single surface layer distribution of density in.rBw(p ) plus another term (corresponding to the regular part u) which is at least as well behaved. If the density in-rBw(pB) is piecewise continuous then [Kellogg (p. 122l the potential is infinitely differentiable in V, thus, in particular, the potential OO wu is twice continuously differerntiable in V. 27

THE UNIVERSITY OF MICHIGAN 7359-1-T Furthermore, if density is uniformly Holder continuous then [Kellogg (p. 165) the potential is continuously differentiable in the closure V(1 eC (V)). But 1 1 c~W ensures that weC (V). With the fact that the surface is closed and finite, it follows that upB) is uniformly Holder continuous. Thus the density inr B(pB) will be uniformly Holder continuous if no r is. This however, is one of the B smoothness requirements on B. (See the definition of smoothness for the surface B given in Chapter I. Indeed, the above need promoted us to give this particular smoothness definition. ) For r > 6 = 1, using the previous similar argument, we have oD n Substituting into (2.26), we obtain Go n ZI 1 Pm(cos0O)eim jA n(P) in rB (P B (2.53) 1 n+1 mnB B BB n=0 m=-n which is of the form Oe 0 - r A 6=, rB (2.54) n=O r and satisfies (2. 18 b) and (2. 18 c). With this, we conclude that 0~ n W proving the lemma. From the Lemmas 2.2 2 and 2. 3 and from the Definition 2. 3 (2. 4) it immediately follows that Corollary 2. 1 The operators L1 and L=kL =k O+kO1 map the space W into itself. 28

THE UNIVERSITY OF MICHIGAN 7359-1-T Our next task is to show that operator L (2.24) is bounded. However, to do this we shall need the following estimate. Lemma 2.4 For every WEW the estimate I - 01 < M Wl[ pEV z J. I r 2 rir', r is valid. M is a constant independent of ow f, the so-called "radiation pattern", is the first coefficient in the expansion of w. Proof Recall we may assume 6 = c + e = 1 For r< 1, since If ||II|I (Eq. (2.20)) "- < 4~ L + -. l l + 2. (2.55) r <| r r2'I'_ r r r 00 _of For r >, 6= 1, since - is analytic its maximum is achieved on n=1 r the circle z | == 1; therefore, | w-l - r- |= 2 1 I E lZf | (2. 56) r, - 2 r r r n=l n.l On the other hand f I f |f | IJrn+1 = w- r' >.6 = 1. (2.57) n+ 1 r r n=1 Since f | |11wI and 1/r. 1,, it follows that n=1 29

In particular, at r = 6 = 1 OD fn 2 Nwll. (2.58) n=l Substituting this in (2. 56) yields fo 2 -- <-| j ll,Iiw r>6=1. (2.59) r From (2. 59) and (2. 55) the lemma follows Lemma 2.5 Operator L (2.24) is bounded. Proof. We want to show that there exists a constant M < oo such that if teW then IL; Wl < M ||w|. Since L1- + 0 (2. 60) and IIL1 Wo||= IloW0 + o W||l |I Ioow11 + loo11 l||, (2.61) it is sufficient to show that lloIwIl <: M1 IIwll, M < o (2.62) and 1ol wll |< M2 |uII, M2 <oo. (2.63) Consider (2. 62) first. We integrate (2.25) by parts with respect to r once to obtain G (p, pl1) V V | 2 1dB G(P, ) [r, PB r + 2 1dl V r a[r1 G(p, [(1> j v v B V (2.64) 30

THE UNIVERSITY OF MICHIGAN 7359-1-T where f (0, ) is the "radiation pattern" for. o The integrated term vanishes at the upper limit, r - oo, since (see Eqs. (2.13), (2.15) and (2.16)) rG (p, P)[rl(p)- f(1)] = O () as r- oo, 1 1 1 1 and gives rise to the surface integral at the lower limit. Making use of the estimate "-0 ~ M: 1 W- - - 11 2| v pev |z= | = 1 1 2 Ir 1 of Lemma 2.4 in Eq. (2.64) we see that for peV 1 (2M2 G PB. 6 Id a 2 r G)| dvl B V (2.b r65)r The surface integral may be rewritten, separating out the singular part of the Green's function, as IG~p~p)I1 BJ1oP'PBIr B G (PPB) IrdB B 4 rBR(PP B) B B B The first term on the right is the potential of a single layer distribution of density 1/4rB. Since rB 0 (the origin was taken within B) and the surface is smooth, closed and finite, this density is uniformly Hb'lder continuous which means [Kellogg(20)(p. 165)] t that the potential is continuously differentiable for all points p~V. The second term on the right hand side of (2. 66) is the integral of a bounded function over a finite surface and hence is also bounded. Thus for some N < oo, 31

THE UNIVERSITY OF MICHIGAN 7359-1-T S B B G(ppB) dc < N, pcV. (2. 66) B The volume integral in (2. 65) is also bounded since the integrand is sufficiently well behaved. At the singularity of G, 0 2__ 0~1(rG) = O(1/R 2 1i2 ) =al (~ o 0(1/Rl) as R -- 0, rl 1 1 and is therefore integrable over any finite volume containing the singularity [Kellogg(2 (p. 148)]. Furthermore,.(r G) 1G 0(r|/r) as r1 - oo 2r l 1 thus for some N < co \ r1 2 [rlGO(ppl jldvl < N r, pEV (2.67) r 1 With (2. 65), (2. 66) and (2. 67) we have for some constant N3 |o0*w N3* ||lw|l, peV. (2.68) In particular this is true for the maximum value of lO w |, therefore, renaming the constants, max |0owL| M1 IwiU (2. 69) peV 32

THE UNIVERSITY OF MICHIGAN 7359-1-T Also, since G (p, p) has its singularity only for real values of r at P=P1 (integration is carriedout over the real range 1 4 rl ao), it follows from (2. 65) and the expansion of G that for a positive constant M independent of w, we have 0 2 max lo l< M2 IIll. (2. 70) 2 Hence with (2. 70) and (2. 69), 110|wl|| M 11l11 (2.71) for an appropriate constant M. Next we establish (2.63), thus proving the lemma. With the definition of 01 (2.26) we see that 1 o WI < | GO(P, PB)I In r Ipg (2. 72) B By definition, I ) (P) f 1l. (2.73) Also n and r are unit vectors, B n rBI 1. (2.74) Thus, Io WIIU 5IGO( PBp)| dB (2. 75) B By arguments similar to those above, we have 33

THE UNIVERSITY OF MICHIGAN 7359-1-T max 10lOW < N IIWJ|, (2.76) peV and max 0o1 wJ1< N2 j1WI |z l 2 for some constants N1, N. Hence for an appropriate constant N > 0 Hlbowl1,I N Jlw(. (2.77) With this the lemma is proven. Since L = kL1, and LI has been proven to be bounded, we immediately have the following Corollary 2.2 IILI| < 1 for sufficiently small I kJ Lemma 2.6 If u is a scalar wave function (see (1. 15) then u=e u is an element of W. Proof u is a scalar wave function, and as such (see Chapter I, Section 1.2) 2 - ikr u C (V). Since e is analytic in r and r is continuously differentiable - ikr on B (see Chapter I, Definition 1. 2), ~u = e u satisfies the requirements 2 that ueC (V) and uEC (V) Furthermore, scalar wave functions may be expanded in spherical harmonics [e.g. Sommerfeld(40)(p. 143)] in the following well-known manner 0D u (p)= h (kr)Yn(0,), r = (2. 78) n=O where h (kr) are spherical Hankel functions of the first kind, 34

THE UNIVERSITY OF MICHIGAN 7359-1-T ikr.-n-1 h (kr)=e (n+ (2. 79) m-0 Equations (2 78) and (2. 79) yield - i' Y D(mn)O,' D(m, n +m))' n=0 m=0 n=0-ikr m=0 and absorbing the constants into spherical harmonics, (2. 80) reduces to Y( O e 0) (2.80) r (n-m)! m 0 (2 ik)m n=0 m=0 r or E n(' f ) (2. 82) n=0 r where 00 n o 00o Go > fn(' D(m) n): 0 D(mZ n+Y 0m) =0 mm=n proving the lemma. co OD f (0, 0) y (, 0) Y n(me' n=O m=n We are finally in a position to state and prove our main result. 35

THE UNIVERSITY OF MICHIGAN 7359-1-T Theorem 2. 1. 3 Let B be a closed, finite surface imbedded in E and let it be described by the equation rB g(0, 0) where g is continuously differentiable for 0< 0 < Ir, 0O0 2r. Let n-rB be uniformly Holder continuous. If u(p) is a scalar wave function for V, the exterior of B, then there exists a disc around the origin in the complex wave number plane I kl < k | such that for k inside this disc u(p) is given explicitly by the convergent expansion 0o u(p)= ekr Ln u(o), (2.83) n=0 where (o) G(p, A )(p (2 84) Lou( =-2ik dv1 U (ik G(p pB)n ), 1 1i V B -ikr uO G(p) - p,p)e B ) d(2.85 0 B D'n Bu(P) dB 285 B G (p, B) is the static Neumann Green's function - G(p, ), and the normal is taken out of V. Proof. -ikr Multiplying both sides of (2. 83) by e we obtain o u=Z Lnu. (2. 86) n=O We proceed to prove (2. 86), since this would be equivalent to proving (2. 83). u, u ), and the partial sums u) are all in W (Lemmas 2.1, 2. 6, Corollary 2.1). Thus - u(N) | is meaningful for any N 0. We shall prove the theorem by showing that for any E> 0, there exists a positive integer N (c) such that 36

THE UNIVERSITY OF MICHIGAN 7359-1-T 1|-u -(N < E provided N> N. (2 87) -# A (o) From our operator equation u = Lu + u and from the definition of partial Neumann-Liouville sums (^\^N-! Tn (o) (N) T (N-I) (o) u(N)_ Lj *u which implies u = Lu +u n=O we have, by induction, Pi (N) N+1 u -u L ~u, (2.88) and hence li'- uU(N) I LIlIN1 ||q| (2.89) Since I|k < I|k, we have IILII < 1 (Corollary 2.2), and since I||1l is bounded, for a given E > 0 IILII N+ 1tf11 < e (2. 90) provided log ijII N+1 > log IILII (2.91) This proves the theorem. Representations of wave functions in two important special cases follow immediately. Corollary 2. 3 The Green's function of the second kind for the Helmholtz aGk equation and surface B (-n =0 on B) is an ikR(p, po) (P' Po)- 4rR(p,p) + uP ) (2. where u(p, p ) is given explicitly by (2. 86) with 37

THE UNIVERSITY OF MICHIGAN 7359-1-T (o -ikr ik p u()(P,' G(PB) e B [ eP]) ] d B. (2.93) 4 LrB 0 k R (PBIPo I B This, of course, follows immediately by taking the normal derivatives of both sides of (2.92) and observing that 0 O then by substituting the result a a B a' ikR(pB, P) an u (PB Po) = n 4R L (PBPO) (2 94) into the Eq. (2.85) Corollary 2. 4 The velocity potential u when a plane acoustic wave A a)(ut is incident in a direction a on a rigid surface B (- = 0 on B) is dn t ikr a Q u =e +u(p),(2.95) where u (p) is defined explicitly in (2. 83) with (O)(p) G(p, p)e-ikr B \ e ik B () -ikrB a ( eB' ). (2. 96) B Similarly this follows from the fact that normal derivative of the velocity potential vanishes on the surface. Theorem 2.2 (Uniqueness) If u1 and u2 are scalar wave functions for V and au au 1 2 =,- on B dn an then uu2, peV Proof. Suppose u1*u2. Then subtracting the equations 38

THE UNIVERSITY OF MICHIGAN 7359-1-T (0) U1= L~ u1 + u (o) (2.97) u2= Lo u2 + u (o) (where u is the same in both equations since the normal derivatives are equal) we obtain u1- u2= L (u -U2) and IIul-u211 IILII Iul1-u211 (2.98) Since 11|u-u211 f O0 we may divide obtaining IILII > 1. This violates Corollary 2.2, the boundedness of the operator, and theorem is proved. 2.4 Remarks on the Low Frequency Expansion We have shown that the solution to the equation 1, *. (o) u-kL1 u =u is given by 00 -ikr x,' kn n (o) u =e u = kL ou (2. 99) n=0 where the operator L is independent of k. If the boundary data is analytic in (o) k, as is the case in Corollaries 2. 3 and 2.4, then u has the expansion D00 (o) n u)(p) = a(p)kn (2. 100) n~-0 n=O Substituting (2. 100) into (2. 99) and observing that both series are absolutely convergent, we obtain ao m u = k L1 a (P), (2. 101) m~O n=0 39

THE UNIVERSITY OF MICHIGAN 7359-1-T or oo AO k m mu k (2. 102) m mX0 where m urn = Lna (p). (2.103) m'a 1 m-n n=0 From (2.103) it follows that u =a o o (2. 104) u =a +L u m m 1 m-1 Equations (2. 102) and (2.103) represent a low frequency expansion of u. If we had assumed the expansion (2.102), substituted it, together with the expansion (2. 100) for u into the equation u= kL *u +u( and equated the coefficients of k, then we would have obtained the u exactly as given by (2. 103). For the scalar wave function u(p) we have the corresponding expansion oo oo oo n up eikr^ Y k Er) n km ukn (ir) u(p)/ e\U n! \ m k, k (n-m)i um n=0 m=0 n=0 m=0 ~~~~~~~~~~ooD ~ n m (2. 105) n (ir) n m' k - L a (p) E (an-m)! n n=0 m=0 v=0 For these expansions the radius of convergence in the k-plane is |k | > 0 (Theorem 2. 1). Next we note the relation between the low frequency expansion and the (N) Neumann-Liouville expansion. Specifically, partial sums u in the Neumann-Liouville series are (wvith (2. 100)) 40

THE UNIVERSITY OF MICHIGAN 7359-1-T N N oD u(N) Ln U (o) knLn L Z ak. (2.106) n=0 n=0 m=0 Adding and substracting the same quantity, we have o (N)o a n u(N)= k +mLn - Zkn+mLl *a. (2.107) t m, 1 m n=0 m=0 n=N+l m=0 Using Cauchy's form of the product of two series to rewrite the first sum and shifting the index in the second enable us to write oo m oo oo (N) > E mL n >a' ~ kn+m+N+lL n+N+1 u k L *a -k L a / 1 m-n 1 m m=0 n=0 n=0 m=0 Splitting the first sum and again adjusting the index of the second yields three blocks of terms N m o m (N) n u(N)= k nak L a -n+ km L na m=0 n=0 m=-N+1 n=0 oo m -^r"_"m L n +N+l L+N+ 1 m-n m=0 n=0 The first block is seen to be, with (2. 102) and (2. 103), the sum of the first N terms of the low frequency expansion while the remaining blocks may be combined by further reordering to yield 41

THE UNIVERSITY OF MICHIGAN 7359-1-T N oo m oo m-N-1 (N) n n Tm m n - T C T rm n+N+1 u = k u + Z 2 k L m oa mn- k L k -N a 1-n m=0 m=N+1 n=O m=N+1 n=O N o m m kmu + Z kmL oa kmLoa /, m ^_I l 1 m-n 1 m-n m=0 m=N+1 n=0 n=N+l m=0 m=N+1 n=0 With (2. 103) we see that the first sum on the right represents the first N terms of the series (2. 102). Thus the Nth term of the Neumann-Liouville series is seen to contain terms of all order in k, the first N of which corre - spond exactly to the first N terms of the low frequency expansion. 42 42

THE UNIVERSITY OF MICHIGAN 7359-1-T CHAPTER III AN APPLICATION: SCATTERING OF A PLANE WAVE OF SOUND BY AN ACOUSTICALLY RIGID SPHERE Both as a check and an illustration we apply the techniques described in the previous chapters to a specific problem —one which is indicated by the title of this chapter. In this case, the exact result is known and we are able to show, not only that the iteration produces the correct result, but how the Nth iterate approximates the exact result. The surface B is now a sphere of radius a whose center is taken as the origin of the coordinate system. The static Green's function of the second kind for this sphere is (e.g. Morse and Feshbach ) 1 r n 2n+l 1 1 (< n a o 1 4 i n+1 n+1 (r n+1 n n=O r> where r = min (r, r) r max (r, r) <' 1> A A and cos y = r * r = cos 0 cos+ sin sin cos (0- The incident field is a plane wave which, without loss of generality, is chosen as propagating down the z-axis, i. e., ^A i z- -ikz -ikr cos 8 u =e = e = e (3.2) The boundary values of interest are (with the well-known plane wave expansion) 00 i = =-k (-ikr os ( n+j(ka ) P (cos e) (3.3) an ar |._j n n r=a = n=O 43

THE UNIVERSITY OF MICHIGAN 7359-1-T where j is the spherical Bessel function and the prime denotes differentiation n with respect to ka, i.e., jn(kaa ) ka)[( ) J / (ka) (3 4) n d(ka) 2ka +(a The scattered field, u (where au/dn = -du /an on B), is given by the methods described previously as ikr (N) u = lim ei u( (3.5) N- oo where N (N)'n (o) u )= DL U ou C (3.6) n=0 (o) ik < n a 1 3 ( 1 d Lu (P)=2 + n+l n+1 n raru( r )d n=o (r>rr ZP a (cosu (p )d (3. 7 47r n+1 n+1 n B B n=O r m= (3.8) (cos 44always involves coordinates of p and the integration variables) and -ik 2OnD l aO u O(p)= ke 2n a p (cos -y) (-i)m(2m~il)j( (ka)Pc 4ir n+1 n+1 n (u dm(co (3.7B n=O r=O ~~~~~~B ~(3.(8)

THE UNIVERSITY OF MICHIGAN 7359-1-T The orthogonality of the Legendre functions enables us to evaluate u( and the first three iterates. Omitting the details, these are found to be ODka ~ n+1l u(~)(p) = ka e- a (n (-i)n a (ka)Pn(cos ) (3.9) (n+l) n+1 n n n=O r (1) -ika a u (p) = ka e j' (ka)(1+ ika) r o wik 2n+n+l +kae ikS 2n+ -)n(a) j (ka)P (cos0)[l+ika-ikr] (3.10) n-1 co (2) -ika i-ika 2n+1. n a n+ u (p)= ka e aj o(ka) l+ika-(ka)2 +kae 2n (i)a n+ r o r n+l r nzl 2 22 x (ka)P (cos ) [+ik(a-r)+kra - - (3.11) n n 2n-1 (n+1) (2n-1) The exact expression for u is (e.g. Morse and Feshbach()29) 00 h (kr) u(p) = - in(2n+l)j (ka) Pcos 0) (ka) (3.12) n=O where the prime again denotes differentiation with respect to ka (see (3. 4)) and h is a spherical Hankel function of the first kind. Expliticly n h (z) e i-n-l _ (n+m) 1 (3. 13) n z ZLi(n-m)' m'i (3.13) m —O m45 0 45

THE UNIVERSITY OF MICHIGAN 7359-1-T With this definition we find that n yj (2n-m)' (n-m)I m ~ (-2ikr)m h (kr). n+1 -n)! m! h = (kr) = ik(r-a)(a m= (3. 14) h' (ka) -ka n h'kan (r > (2n-m)!,) (-2ika)m(n+l -m-ika), (n-m)' m! m=0 The ratio of the two polynomials, of degree n in kr in the numerator and n+1 in ka in the denominator, may be re-expanded, for ka sufficiently small, in ascending powers of k. Thus h (kr) ik(r-a) n+1 Go n e a7kn e a n >~ 0 (3.15) h' (ka) ka n+1 ) (3.15) n Q =0 where the coefficients a are functions of r, a, and n. The first three are found to be a = o a=-ia, n=O i(a - r), n> 0 2 2 = -a, n = 0 (3.16) n-1 2 n +n-1 2 =ar - a n > O 2n- 1 (n+ 1)(2n - 1) In terms of these expansions the exact result for u, Eq. (3. 12), may be rewritten as n+ 1 ik(r-a) ( iP 2n+1 )a j (ka) P (cos0) ak. (3.17) n=0:=0 46

THE UNIVERSITY OF MICHIGAN 7359-1-T If we denote by uN(p) the expression resulting from taking only the first N terms in the expansions in k in (3.17), that is, eo N n+1 ik(r-a) T _i 2n+ an+ uN=kae ( -) n+1 () jn(ka) P (cos ) a k (3. 18) n=O:=0 then we see that, for the values of N computed, ikr (N) uN e u (3.19) (N) where the first three iterates, u, are given in (3. 9) through (3.11). 47

THE UNIVERSITY OF MICHIGAN 7359-1-T CHAPTER IV CONCLUSION The main result of this work consists of (a) the derivation of an integrodifferential equation for the exterior Neumann problem whose kernel is the potential Green's function of the second kind, and (b) an effective approximation method for it. We shall now indicate some of the areas which are the natural extensions of this (and the Dirichlet) problem, and which are not included in the present work. 4.1 Functional Analytic Aspects It is immediately evident from its definition that the space in which the perturbation is performed is not complete. For this reason, it was necessary to show that the solution to the problem was an element of this space and that in the sense of our norm iterates converge to this function. However, let the norm be given as follows: I|u||= maxlu(p) +maxlDu (p)l+max Iu(0,0, 1)|, 0 0<r, 0,,<4<2r, peV peV I z.l where D denotes the first derivatives with respect to any one of the variables 0, 0, or r, and zl= r-. 1 is the unit disc in the complex z plane defined in Chapter II. Then it should be possible to show (the work on this is being completed) that this new space is a Banach space, and that the operator L=kL1, defined in Chapter II, is compact. With these facts, we may appeal to the Banach fixed point theorem to conclude the existence (and uniqueness) of the solution to the equation u+L - = u() 48

THE UNIVERSITY OF MICHIGAN 7359-1-T Another norm with which the present problem may, with somewhat more tediousness, be solved is oo oo y n IIull = max ax u(p)l, max pIV 0 <,2 n=O m=n 0E2r ri b where b some constant greater then 1, and Y are the spherical harmonics m for u, specifically, O f(0, 0) U = " r >.i Z J. n+1 r' 1 n=O r oo f,~(O., ) > n(t,,) = ZYm (oe~) n x i m m=n m (n) An P (cos 0) e m Im m I = -m Again, we note, with this norm the space W is not complete. 4.2 Non-Separable Surfaces. Bodies with Edges. It is now possible to'solve' the Neumann (and Dirichlet) problem for the scalar Helmholtz equation in the regions exterior to a non-separable body, provided k, the complex wave number, is sufficiently small in modulus, and the solution of the Laplace's equation can be obtained for the body in question. This is done for one such body, an ogive (see Ar( )). It should be noted in this connection that the explicit representation of the solution of the Neumann (and Dirichlet) problem has been proven only for smooth bodies (see Chapter I for the definition of smoothness). However, preliminary calculations for the circular disc support the hypothesis that the representation remains valid 49

THE UNIVERSITY OF MICHIGAN 7359-1-T for bodies with edges (e.g. an ogive). To prove this, however, will require a different definition of norm, since it is known (e.g. Bouwkamp 5), Meixner ) that wave functions associated with bodies with edges have singular derivatives. Thus, convergence of the iterates in these cases will have to be established in some other norm, Another non-separable body is the torus. The solution for this problem is also presently under consideration. Since, in this case, the smoothness requirements are satisfied, the present norm is sufficient to justify the iteration. 4. 3 The Radius of Convergence of the Low Frequency Expansion While we have proven that the series convergences for Ikl sufficiently small, that is, there exists some number Ikol > 0 such that the series converges for IkI< Ik 1, no indication was given as to how large Ik i may be. If the boundary data are analytic in k, this problem of estimation is equivalent to finding the radius of convergence of the low frequency expansion. Such estimates are available only for special surfaces (e. g. Darling and Senior () and the general problem remains unsolved. If the exact radius of convergence is found, the analytic continuation into the complex k-plane is then possible. Restriction of this continuation to the real line would in effect "solve" the persistent problem of the "resonance region". 4. 4 Extension to Vector (Electromagnetic) Problems Here the goal is an explicit iterative solution, as opposed to Stevenson's (42) technique in which each successive term in the series solution can be found only by solving a new problem. 4. 5 Two-Dimensional Low Frequency Scattering Problems. The success in three-dimensional problems is due in part to the existence of the expansion (see Chapter I) for the wave functions. While the comparable expansion for two-dimensional wave functions is more complicated (Karp )), its very existence offers some hope that an iterative method analogous to that 50

THE UNIVERSITY OF MICHIGAN 7359-1-T for three-dimensional problems could be found. In one respect, success in this area would be more far reaching than in the three-dimensional case since, in contrast to that case, two-dimensional potential problems are all essentially solvable using conformal mapping; thus, all two-dimensional scattering problems would also be solvable. 51

THE UNIVERSITY OF MICHIGAN 7359-1-T REFERENCES (1) Ar, Ergun, "Low Frequency Scattering from an Ogive," The University of Michigan, Radiation Laboratory Report No. 7030-3-T; 1965. (Has been submitted to Quart. Appl. Math. for publication. ) (2) Atkinson, F. V., "On Sommerfeld's Radiation Condition," Philos. Mag., XL, Series 7, No. 305, pp. 645-651; 1949. (3) Baker, B.B. and E. T. Copson, The Mathematical Theory of Huygens' Principle, Oxford: Clarendon Press; 1950. (4) Barrar, R. B. and A. F. Kay, "A Series Development of a Solution of the Wave Equation in Powers of 1/r, " Internal Memorandum, Tech. Res. Group, Inc., NYC. (5) Bouwkamp, C. J., "Diffraction Theory, " Rep. Progr. Phys., 17, pp. 35-100; 1954. (6) Collins, W.D., "Some Scalar Diffraction Problems for a Spherical Cap," Arch. Rat. Mech. Anal., 10, pp. 249-266; 1962. (7) Darling, D.A., "Some Relations Between Potential Theory and the Wave Equation, " The University of Michigan, Radiation Laboratory Report No. 2871-5-T; 1960. (8) Dolph, C. L. and R. B. Barrar, "On a Three-Dimensional Transmission Problem of Electromagnetic Theory," Arch. Rat. Mech. Anal., 3, No. 6, pp. 725-743; 1954. (9) Ehrmann, H., "Iterationsverfahren mit verAnderlichen Operatoren," Arch. Rat. Mech. Anal., 4, No. 1, pp. 45-46; 1959-60. (10) Ehrman, H., "On Implicit Function Theorems and the Existence of Solutions of Non-linear Equations, " University of Wisconsin Technical Report No. 343; 1962. (11) Garabedian, P. R., "An Integral Equation Governing Electromagnetic Waves," Quart. Appl. Math., XII, No. 4, pp. 428-433; 1955. (12) Hartman, P. and C. Wilcox, "On Solutions of the Helmholtz Equation in Exterior Domains," Math. Zeitschr, 75, pp. 228-255; 1961. 52

THE UNIVERSITY OF MICHIGAN 7359-1-T (13) Heins, A. E., "Function-Theoretic Aspects of Diffraction Theory," Electromagnetic Waves (R. Langer, ed.). Madison: The University of Wisconsin Press; 1962. (14) Heins, A. E., "Axially-Symmetric Boundary Value Problems, " Bull. Am. Math. Soc., 71 No. 6; 1965. (15) deHoop, A. T., "On the Scalar Diffraction by a Circular Aperture in an Infinite Plane Screen, " Appl. Sci. Res., B4, pp. 151-160; 1954. (16) Hormander, L., Linear Differential Operators. Academic Press, Inc.; 1963. (17) Ikebe, T., "Eigenfunction Expansions Associated with Schroedinger Operators and Their Applications to Scattering Theory, " Arch. Rat. Mech. Anal., 5, No. 1, pp. 1-34; 1960. (18) Karp, S. N., "A Convergent'Far Field' Expansion for Two-Dimensional Radiation Functions," Comm. Pure Appl. Math., XIV, pp. 427-434; 1961. (19) Kato, T., "Growth Properties of Solutions of the Reduced Wave Equation with a Variable Coefficient, " Comm. Pure Appl. Math., XII, pp. 403-425; 1959. (20) Kellogg, O. D., Foundations of Potential Theory. New York: Dover Publications; 1953. (21) Kleinman, R. E., "The Dirichlet Problem for the Helmholtz Equation." Arch. Rat. Mech. Anal., 18, pp. 205-229; 1965. (22) Kupradse, W. D., Randwertaufgaben der Schwingungstheorie und Integralgleichungen. Veb Deutscher Verlag Der Wissenschaften: Berlin; (23) Leis, R., "Uber das Neumannsche Randwertproblem fuir die Helmholtzsche Schwingungsgleichung," Arch. Rat. Mech. Anal., 2, pp. 101-113; 1958. (24) Levine, L. M., "A Uniqueness Theorem for the Reduced Wave Equation," Comm. Pure Appl. Math., XVII, No. 2; 1964. (25) Magnus, W., "Uber Eindeutigkeitsfragen bei einer Randwertaufgabe," Jahresber. Deut. Math., Verein 52, pp. 177-188; 1942. 53

THE UNIVERSITY OF MICHIGAN 7359-1-T (26) Magnus, W., "Fragen der Eindeutigkeit und des Verhaltens in Unendlichen fur Losungen von Au+k2 u = 0, " Math. Seminar der Univ. Hamburg, 16, p. 77; 1949. (27) Magnus, W. and F. Oberhettinger, Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea Publ. Co.; 1949. (28) Meixner, J., "Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen," Ann. Phys., 6, pp. 2-9; 1949. (29) Morse, P.M. and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill; 1953. (30) Muller, C., "ZurMethoden der Strahlungskapazitat von H. Weyl," Math. Z., 56, pp. 80-83; 1952. (31) Noble, B., "Integral Equation Perturbation Methods in Low Frequency Diffraction," Electromagnetic Waves (R. Langer, ed.). Madison: The University of Wisconson Press; 1962. (32) Odeh, F. and C. Zemach, "Uniqueness of Radiative Solutions to the Schroedinger Wave Equation, " Arch. Rat. Mech. Anal., 5, No. 3, pp. 226-237; 1960. (33) Lord Rayleigh, "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; 1897. (34) Rellich, F., "Uber das asymptotische Verhalten der Losungen von Au+Xu =0 in unendlichen Gebieten, " Jahresber. Deut. Math., Verein 53, pp. 57-64; 1943. IG (35) Rothe, E., "Uber die Integralgleichungen des Skineffekts," J. fur Reine und Angew. Math., 170, No. 4, pp. 218-230; 1934. (36) Saunders, W.K., "On Solutions of Maxwell's Equations in an Exterior Region," Proc. National Academy of Sciences, 38, No. 4, pp. 342-348; 1952. (37) Senior, T.B.A., "Scalar Diffraction by a Prolate Spheroid at Low Frequencies," Can. J. Phys., 38, pp. 1632-1641; 1960. 54

THE UNIVERSITY OF MICHIGAN 7359-1-T (38) Senior, T.B.A. and D.A. Darling, "Low Frequency Expansions for Scattering by Separable and Non-Separable Bodies, " J. Acoust. Soc. Amer., 37, No. 2, pp. 228-234; 1965. (39) Sommerfeld, A., "Der Greensche Funktion der Schwingungsgleiching," Jahresber. Deut. Math., Verein. 21, pp. 309-353; 1912. (40) Sommerfeld, A., Partial Differential Equations in Physics. New York: Academic Press; 1949. (41) Sternberg, W., "Anwendung der Integralgleichungen in der elektromagnetichen Lichttheory, " Zeit. fur Physik, 64, pp. 638-649; 1930. (42) Stevenson, A. F., "Solution of Electromagnetic Scattering Problems as Power Series in the Ratio (Dimension of Scatterer/Wavelength)," J. Appl. Phys., 24, pp. 1134-1142; 1953). (43) Stratton, J.A., Electromagnetic Theory. New York: McGraw-Hill; 1941. (44) Thomas, D. P., "Diffraction by a Spherical Cap, " Proc. Cambridge Philos. Soc., 59, pp. 197-209; 1963. (45) Werner, P., "Zur mathematischen Theorie akustischer Wellenfelder," Arch. Rat. Mech. Anal., 6-, pp. 231-260; 1960-61. (46) Werner, P., "Boundary Value Problems in the Theory of Acoustic Wave Fields," (Two Parts) University of Wisconsin Technical Report No. 233 and 234; 1961. (47) Werner, P., "On the Exterior Boundary Value Problem of Perfect Reflection for Stationary Electromagnetic Wave Fields, " University of Wisconsin Technical Report No. 286; 1962. (48) Werner, P., "Randwertprobleme der mathematischen Akustik," Arch. Rat. Mech. Anal,, 10 pp. 29-66; 1962. (49) Werner, P., "Beugungsprobleme der mathematischen Akustik," Arch. Rat. Mech. Anal., 12, pp. 155-184; 1963. (50) Weyl, H., "Kapazitat von Strahlungsfeldern," Math. Z., 55, pp. 187-198; 1952. (51) Wilcox, C.H., "A Generalization of Theorems of Rellich and Atkinson," Proc. Amer. Math. Soc., 7, pp. 271-276; 1956. 55

THE UNIVERSITY OF MICHIGAN 7359-1-T (52) Wilcox, C. H., "An Expansion Theorem for Electromagnetic Fields," Comm. Pure Appl. Math., IX, pp. 115-134; 1956. (53) Wilcox, C.H., "Spherical Means and Radiation Conditions," Arch. Rat. Mech. Anal.,, pp. 133-148; 1959. (54) Williams, W. E., "Diffraction by a Disk, " Proc. Roy. Soc. London. Ser. A 267, pp. 77-87; 1962. (55) Williams, W. E., "The Reduction of Boundary Value Problems to Fredholm Integral Equations of the Second Kind," Z. Angew. Math. Phys., 13, pp. 133-152; 1962. 56

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