THE UNIVERSITY OF MICHIGAN 7133-5-T AFCRL-67-0360 LOW FREQUENCY SCATTERING BY SPHEROIDS AND DISCS J. S.\Asvestas and R. E. Kleinman The University of Michigan Radiation Laboratory 201 Catherine Street Ann Arbor, Michigan 48108 Contract No. AF19(628)-4328 Project 5635 Task 563502 Work Unit No. 56350201 Scientific Report No. 4 June 1967 Contract Monitor Robert A. Shore Microwave Physics Laboratory Distribution of this document is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general public. Prepared for Air Force Cambridge Research Laboratories Office of Aerospace Research United States Air Force Bedford, Massachusetts 01730

THE UNIVERSITY OF MICHIGAN 7133-5-T ABSTRACT The problem of scattering of a scalar plane wave by a spheroid of revolution is solved for either Dirichlet or Neumann boundary conditions, arbitrary major to minor axis ratio, and arbitrary incident direction. The solution is obtained by using an iterative method applied to solutions of the corresponding potential problem and is expressed as a series of products of Legendre and trigonometric functions, and ascending powers of wave number. A recursion relation for the coefficients in this series is derived. These results are employed to calculate the scattering cross sections for 2:1, 5:1, and 10:1 prolate spheroids. 11iii

THE UNIVERSITY OF MICHIGAN 7133-5-T TABLE OF CONTENTS ABSTRACT iii I. INTRODUCTION 1 II. GENERAL CONSIDERATIONS 3 III. THE DIRICHLET PROBLEM 11 3.1 The Iteration Scheme 11 3.2 The Recurrence Relations 13 3.3 The Far Field and the Scattering Cross Section 22 3.4 Nose-on Incidence 23 IV. THE NEUMANN PROBLEM 26 4.1 The Iteration Scheme 26 4.2 The Recurrence Relations 29 4.3 Nose-on Incidence 33 V. THE OBLATE SPHEROID AND THE DISC 35 5.1 The Oblate Spheroid 36 5.2 The Disc 40 VI. NUMERICAL CALCULATIONS 44 APPENDIX A: The Regularity of the Function w(p) = e ik( r) uS(p) 64 APPENDIX B: The Surface Integral for the Dirichlet Problem 67 APPENDIX C: Evaluation of the Integral C(, s)Qr)d 72 Ct ( ~'~1 ~s s APPENDIX D: Derivation of Relation (3.21) 76 APPENDIX E: Derivation of Equations (3.24a, b) 78 APPENDIX F: The Surface Integral for the Neumann Problem 80 APPENDIX G: The Integral d Kt(., 1 s)Q() 92 5sP ( ) 9 APPENDIX H: Redefinition of Qm 94 n PPENDIX I: The Far Field for the Disc 97 REFERENCES 102 i- -- ^ -- _ _ _ _ _ V i i -- - _ i* _ _ _

THE UNIVERSITY OF MICHIGAN 7133-5-T I INTRODUCTION This report presents the complete low frequency expansion of the field scattered when a scalar plane wave is incident from an arbitrary direction on a spheroid of revolution (prolate, oblate, or disc), on which either Dirichlet or Neumann boundary conditions are imposed. The expressions for the field are valid everywhere in space and for all values of the ratio of spheroid dimension to wavelength within the radius of convergence of the low frequency expansion. The work began as a demonstration of the efficacy of a recently derived technique for solving boundary value problems for the Helmholtz equation by iterating the Green's function for Laplace's equation. This new method had been applied to the problem of scattering by a sphere both for a Dirichlet boundary condition (Kleinman, 1965) and a Neumann boundary condition (Ar and Kleinman, 1966). The prolat spheroid was selected to provide a more substantial test of these methods, which proved to work even better than anticipated. The problem of scalar scattering by a prolate spheroid for both Dirichlet and Neumann boundary conditions has been extensively treated. F.B. Sleator (1964) pre sents an exhaustive bibliography. Exact solutions are known in terms of spheroidal wave functions and both low and high frequency approximations have been found. The standard methods for obtaining low frequency approximations, either by direct expansion of the terms of the spheroidal function series in powers of wave number or by determining each term in the expansion as the solution of a potential problem (cf. Noble, 1962), are somewhat cumbersome. One may question the purpose of finding low frequency expansions if the exact solution is known. The answer lies in the complexity of the spheroidal functions which make analysis and computation difficult. The present approach, although certainly not a trivial calculation, avoids entirely the use of spheroidal functions on the one hand and, on the other, obviates 1

THE UNIVERSITY OF MICHIGAN 7133-5-T the need for solving more than one potential problem. The solution is found in the form of a series of products of spheroidal potential functions, i.e. Legendre functions, whose coefficients are determinated iteratively. While this in itself might be ample justification for presenting the results, their value is considerably enhanced by the fact that a recurrence relation for the coefficients is found. This means, in effect, that the iteration process may be carried out completely and the complete low frequency expansion obtained. This is carried out explicitly for a plane wave incident from an arbitrary direction on a prolate spheroid for both Dirichlet and Neumann boundary conditions. In addition to expressions for the field valid everywhere in space, the simplifications occurring in the limiting cases of far zone and nose-on incidence are explicitly given as is the expression for scattering cross section. The corresponding results for an oblate spheroid and the important limiting case, the disc, may be obtained by a simple transformation and these results are also presented explicitly. Some numerical calculations of scattering cross sections of prolate spheroids have been carried out. These results are presented and compared, where possible, with existing data. In Section II, the iteration method is adapted to take advantage of the symmetry of prolate spheroid geometry. The method is applied to the Dirichlet problem for the prolate spheroid in Section III and the Neumann problem in Section IV. Section V contains the detailed analytic results for oblate spheroids and discs. The numerical calculations for prolate spheroids are presented in Section VI. Much of the detailed mathematical analysis has been relegated to a series of appendices in the hope of making the method and the results more accessible. This work was supported by the Air Force Cambridge Research Laboratories under contract AF 19(628)-4328 and by the National Science Foundation under Grant No. GP 6140.......; —- 2 i

THE UNIVERSITY OF MICHIGAN 7133-5-T II GENERAL CONSIDERATIONS In this section we present the problem, the method of attack, and some definitions essential to a clear understanding of the procedures followed. The problem we are concerned with is the determination of the scattered field which results when a plane wave of arbitrary incidence impinges upon a prolate spheroid. With respect to a rectangular system of coordinates (x, y, z), the prolate spheroid is oriented with its axis of revolution (major axis 2a) coinciding with the z-axis, and its geometrical center at the origin. The minor axis is 2b. Then the relations between prolate spheroidal coordinates (Q, ar, 0) and rectangular coordinates (x, y, z) are r~ 2 2 x =c ( 1)(1- r) cos (2.1) y = c /(2- 1)(1-r12) sink (2.2) z = ceri (2.3) where c is half the interfocal distance of the spheroid, and 1 < oo, -1 < < +1, 0,<, 27r. The surfaces e = constant represent confocal prolate spheroids. The metric coefficients of the spheroidal variables are given by by 22 222 h =c -; h c; h =c (2 -r2). (2. 4) Having defined the prolate spheroid, we now turn to the definition of the incident plane wave. Without loss of generality, we take the x-z plane as the plane of incidence. The direction of propagation forms an angle 0 with the positive z-axis, 0O For a detailed discussion of the geometry of the prolate spheroid see Sleator (1964). i, i i ii i. ii ii iu ii ii 3 -

THE UNIVERSITY OF MICHIGAN 7133-5-T z 0 p ~( Px,y,z) K'I^ -C FIG. 1-1 (see Fig. 1-1), and, if p is the observation point with coordinates (r, 0, ), we write -ikr cos O ui(p) = e, (2.5) where u (p) denotes the plane wave as observed at p and cos = cos 0cos0 +sin0sin0 cost. (2.6) o o0 When 0 is reduced to zero, the plane wave is seen to propagate along the negative z-axis. The time dependence is harmonic (e-i). We now state the problem: Let S designate the surface of a prolate spheroid with surface coordinate As, and let V be the volume exterior to it. Designate by V the union of S and V: V = SUV. Finally, let u (p) be the resulting scattered field due to the presence of the prolate spheroid. We wish to determine a function u(p) such that 4 -------------------------

THE UNIVERSITY OF MICHIGAN 7133-5-T (i) u(p) = ui(p)+uS(p), peV (2.7) (ii) (V2+k)us(p) =0, peV (2.8) us (iii) lim r -- iku =0 (2.9) r r —o co (iv) Either (a) u(p ) =0 p S (2.10a S S a(b) ( (b) u, nS (2 b'an s Equation (2.9) implies a suppressed time harmonic dependence e Moreover, boundary condition (2. 1 Oa) refers to the Dirichlet problem and (2.1 Ob) to the Neumann problem and the two problems will be treated separately. The approach employed in solving the problem is based on a new method of finding iterative solutions of the Helmholtz equation (Kleinman, 1965; Ar and Kleinman, 1966). Inherent to this method is the assumption of long wavelength compared to the dimensions of the scatterer. The original iteration scheme was phrased in spherical coordinates and much of the analysis depended upon expansions in these variables. Here we essentially rederive these results in prolate spheroidal coordinates in which form the iteration becomes more tractable. We start with a representation theorem (Kleinman, 1965; Ar and Kleinman, 1966): Theorem: Any function w(p), defined for all p V, which is twice differentiable in V, and regular at infinity satisfies the integral equation -------- 5 ---— i

THE UNIVERSITY OF MICHIGAN 7133-5-T w(p ) = Go(pl, p (p )dV+ (p)a G(plp)dS, (2.11a) Jv s where G is the normalized static Green's function of the first kind, and the inte0 gral equation w(pl) = GN (pp' v)V 2(pv)dV- GN(PlPs) (nn(Ps)dS (2.11b) V S where G is the normalized static Green's function of the second kind. The normal o ized static Green's function G (P1i p) of either kind is defined as follows: (i) V2G(p, P) = 6Pllp) Pp (2.12) (ii) G (P1, p) regular at infinity (iii) (a) G (Pl ps) = 0 (first kind) (b) a-n Go(P1P) = 0 (second kind). The normal is directed out of the volume V. Moreover, we define a function f(p) to be regular at infinity if it satisfies the Kellog (1929) conditions 2 a&f(p) 0 0< 7T lim Jrf(p)| <oo and lim r - <ao, (2.13) r- 00co r — 0.<.0< 2r. Using expressions (2.4), it can be readily shown that in prolate spheroidal coordinates.... —-------- 6 ---

THE UNIVERSITY OF MICHIGAN 7133-5-T 12(~ 2-) a2 V 222) 2 Lr2 T4 at + n 2 + ar[ ( r2 1 -r2) 2 Q2aj (2.14) dV = c3(2-n 2)ddrid (2.15) 2 2 2 2 dS c2 ( -r2)( 2-1) drld (2.16) 5s s a - 1 __ __ a (2. 17) s The function we wish to substitute in the representation theorem equations is the scattered field u (p). This function, however, is not regular at infinity but, as we have shown in Appendix A, the function e -( u (p) is. For this reason we let -ikc( + r)) s w(p) = e u (p) (2.18) in equations (2.11a) and (2.11b). From (2.14) and the Helmholtz equation (2.9) we have V2(p) =_ 2ik 2-_1) (2-1) (1 ) + (g )w(p) (2.19) c(2_ 2) 2 a Substitution of equations (2.15), (2.16), (2.17) and (2.19) in (2. lla) and (2. llb) gives, -7

THE UNIVERSITY OF MICHIGAN 7133-5-T 00 +1 2r p1) = -2ikc2 d5 d0l dr GD(, 1 p) (2 1) ( (P) +( ++1 227r -1 o s for the Dirichlet case, and dary condition)2 defined by (2.12) and given by+) sm +1 2nm ) +c(m 1) dr) doG pP)- p), U (2.20b) -1 0 for the Neumann case. These are the integrodifferential equations that we have to solve. The first one involves the normalized static Green's function of the first kind (Dirichlet boundary condition) defined by (2.12) and given by OD n <pi p (-lm (2n+l) [((- ]2 -1 m) 0 4,f c m Ln+m)_ com n m!n pm(- ) m )ro n s Qn(s P (17 )p (1) I - )Qn(Q (2.21) n P n ( n n See for example, Morse and Feshbach (1953, p. 1291). The existing differences are due to a different noalization and a different definition of the Legendre functions. i —----------- 3 -------------— i.m

THE UNIVERSITY OF MICHIGAN 7133-5-T The corresponding Green's function of the second kind is of similar form except for involving the ratio of the derivatives of Legendre functions so that the boundary condition (2.12.iiib) is satisfied] Equation (2.20b) involves the normalized static Green's function of the second kind (Neumann boundary condition) defined by (2.12) and given by O0 n G(i P) - (-l)e (2n+1) [ ) cos m(1- ) n=0 m=0 m (n+m) 1 pm(L n s nm n{pm()m( (n i) Qm(5 )Qn( j (2.22) n ni where a prime on a function denotes differentiation with respect to 5. The symbol s e is the Neumann factor defined by, m=0 e = (2.23) m L2, m=1,2,3,... The associated Legendre functions are defined as follows: _mW, 1 r(n+m+l) (1_ 2)m/2 - - n 2m r(n-m+l)r(m+l ) 2F1(m-n, +m+1;m+ ), Iu-11 <2. (2.24) n (n+ 2) 2 n-m m-n+ m-n 1 1 nf2 (n- m+ 1)r(1/2) W 1) 2 n2 2;2 —2; |Iu|>l; |arg(~u+l)| <7. (2.25) ---... --------. ------------

THE UNIVERSITY OF MICHIGAN 7133-5-T Qm(-)= (-1)m r(n+m+l)r(l/2) (2 1) 2 Qn+m+2 n+m+1 3 -n 2n+l 3,n n+m+1 2 1 2' 2 2>n 2 r(n+- ) 2 J| | >1; | arg(p - 1) < i. (2.26) Definitions (2. 25) and (2. 26) agree with those given by Magnus and Oberhettinger (1949, pp 64 and 60, respectively), while (2.24) differs by a factor of (-1)M. To solve the integrodifferential equations (2.20) for w(p) we proceed as follows: We write w(p) as a power series in k of the form 00 w(p) = E (-ikc)M w(p) M=O and we substitute in equations (2.20) to obtain an iteration scheme for WM(p). We subsequently show that these coefficients of k are of a particular form and develop recurrence relations through which wM(p) can be found for arbitrary M. --—.. —.... --—.,. —-.10

THE UNIVERSITY OF MICHIGAN 7133-5-T III THE DIRICHLET PROBLEM 3.1 The Iteration Scheme The appropriate integrodifferential equation for the Dirichlet problem is (2.20a) which we repeat here for convenience co +1 27r -, w(p1) = -2ikc2 d dG dD(p, dG (p l -p) ) +(2-1) +(r) 7w s - c(s- 1) 5d\ ddw(ps) a- Go P) (3.1) The appropriate Green's function is given by (2.21) and the boundary condition satisfied by w(p) is seen to be, from equations (2.7), (2.10a) and (2.14), -ikc( +~n)) Ux(p~ )=-u (p~ )e ~s (3.2) S S The incident plane wave u (p) is given by (2.5) which can be written in prolate spher oidal coordinates as = -i kr = -ikc coso0 1+sin0 \I-1 -jl-2 cos] ui(p) = e cos = e. ] (3.3) Denote the surface integral of (3.1) by I (pl): +1 27r ispl) a D IS(p1) = -c(~2-l)5 dn d pw(p) -a GD(p p). (3.4) J. 1 \cQ \- 1) do S 0 0' ~ 5 In Appendix B we show that I (pl) may be written in the following form 11 1

THE UNIVERSITY OF MICHIGAN 7133-5-T -ikc~ OIS(p e sc S (-i kc)MI(p) (3.5) M=O where IMP) = = AQ' m( e )P ()Qn( )cos m01 (3.6) with -E F7o (2+1) (m) M+ even M m A (5 AM m(s ) = 0, M+Q odd. (3.7) s Note that IM is independent of k. Moreover, let +ickS (p) = e s(pl (3.8) where b(p1) is assumed to have a power series expansion in k of the form co (p) = I (-ikc)M M(p).(3.9) M=0 Substitution of (3.9) in (3.8) and the resulting equation together with (3.5) in (3. 1) gives 12

THE UNIVERSITY OF MICHIGAN 7133-5-T OD OD oio )+1 27r E, (-ike)MM(p) = -ikkc2 d (-ike)M d \ d d GoD(p, P) M=0 M=01 s -1 0 *(-D i )~IM(P) n)2-) OD ) 2 M (2- a M +() M(P + (ikc M The interchange in differentiation and summation, and summation and integration was made by assuming (3.9) to converge absolutely and uniformly and to be term by term differentiable with respect to each of the variables and the resulting series to be- uniformly and absolutely convergent. Collecting the coefficients of equal powers of k in the above equation, we arrive at the following iteration scheme: (p =I (P (3. 10a) C00 +1 2 2 2r 2 _(P) QM+l(pi)2c dg drl do GD(pip)KQ2-l) - M +(n2 M +( +n)(p - ) + ( M=0,+,2,... (3.'10b)| 3.2 The Recurrence Relations We shall now solve for the M+ 1st iterate in (3. lOb). In order to do this we need to establish the fact that OM may be written as M M t 0o(P) = IoE D (ps)Q()P (n)cosI (3.11) VM() t= r=O Pa0 r,Mt s r tM(3) for all M (M = 0, 1, 2,...). This is accomplished using the principle of mathematical induction, that is, first we show that (3.11) holds for M = 0 and secondly we show that if it holds for M, it also holds for M+1. -— 13

THE UNIVERSITY OF MICHIGAN 7133-5-T That the representation holds for M =0 is obvious since, with equations (3.10a) and (3.6), ~o(p)= A0,'(~)Qo(?) (3.12) 0 0 S 0 which clearly is of the form (3.11). Next assume that (3.11) holds for M. We wish to show that M+ (p) may then be written as M+1 M+1 t Z7) DM+1, Wb (p) = / (^ ) P (i Zcos I M+1 rt (sr)Q ()tcos t=O r=O P=0 The analysis which establishes this is somewhat tedious; however, in the process we actually arrive at an expression for DM' in terms of DM which in fact is r, t r, t the major goal of this section. First note that the second term in (3.1 Ob) has already been shown to be of the form (3.11) [see (3.6), (3.7)]. Next denote the volume integral of (3. lOb) by I +l(Pl) and substitute in it the Green's function of (2.21). Then, O0 n SCO +12 (n- m) =. (-1)m (2n+1) L (Irl M+(Pl) = - 2nO m=O (rl) m + m 2oo +1 27M dC c( 1 \ m drPcos m(r) — 1 0 4 O -- $ 2 ___ 7 (3.13) here 14

... THE- UNIVERSITY OF MICHIGAN 7133-5-T?( i)Qn( ) pm > 1 C Qm( )Q( —-. (3.14) WQn (Q) Qn:,s) s < g Substitution of (3.11) in (3.13) leads to oD n M M t M1(P ) =- 2-r O, ( -1l)mE (2n+l) (nm) n=O m=O t=O r=O i=o ao +1 2 dQP 2 ) dp..(2 S dcosm(0-0)cos0L (- 1)PIr1) --- - (r2- 1)QI) (d r t r dis + ( + )Q (M)Pdt (rr) r tJ Performing the angular integration and rearranging terms we get co M M t +1(P1) = - Z (-1) (2n+l) [ DMt Q n=O t=O r=O:=0 Go +1 ~p"( 01)10~41Spn 2 d I], *P ^)cos^^IC ) dr.r)2_ 1)s) l-~+ ( rl) -- J-1 * Q?) [ (2 r12 -1)d + ] pf ( )Q^Q ( r) ((3.15) I 15'"15

THE UNIVERSITY OF MICHIGAN 7133-5-T To perform the r1 integration we use the relation (Magnus and Oberhettinger, 1949, pp 61-62) IP(z) P (z)l ^H n )(I) n+l 2 d z (n+l)(n- +1) 1_ dz J-; 2n+l = n n+1. n(n + () 2n+; n, = 0, 1, 2,... (3.16) n- 1 Substituting this relation in (3.15), we get n= t=O r=O 1=0 00 M 2t2t,oo -+1 1 2_ d QIQ) PC (rl) t(t ) (1 s) dg r t 2t+ 1 t-l()Qr( oo M+l M t =-C ~ (-1)~-(2n+l ia-)r p co n=O t=O r=O n=0 Lnr1 nsl 1 oo00 &2 ++1 Et-1 dr C nQ,, t s d?7Pn (77 Pt6- 2t -+T3 r, t+l (3.17) 16 -

THE UNIVERSITY OF MICHIGAN 7133-5-T where in this last expression we have adopted the convention that D' ( ) is idenr, t s tically zero whenever any of the subscripts or the other superscript is greater than M. We now employ the following orthogonality property for the Legendre functions of the first kind (Magnus and Oberhettinger, 1949, p. 54), +1 \ \ \ 2 (n+ )'. dxP (x)P (x) = 6 \ n( m ) 2n+i (n-i)' nm J-1 to obtain t=0 r=O 0' t+ d I(S I t(t - _ _ DM: + ) (t+ 1)(t+i + 1) M g IF (p C =- E E -1) D ) Dt, 1 | t sl's + 2t- 1 r,t-ls - 2t+3 r, t+l sQr() ^s + Dr, t s) - 1) d+ Qr( Employing once more the relation (3.16) in the equation above, we write M+1 M t oo IM+v (p =-2 ( -1) (t+ —' Pt(11)C~S01 d Ct('M 1 s t=0 r=O 1=0 (r+ 1)(r-I + 1) M I t(t-l) M, I 2r+1 D't ts)Q r 2t-1 1 s + (t+l)(t+ +l) DM, ]. Q I r(r+) DM, )Q - 2t+3 r, t+1 r 2r+1 r, t (s r-1 (cont'd) 17...

THE UNIVERSITY OF MICHIGAN 7133-5-T M+1 M+1 t =-2 Pt.-1) (t-) l)cos1- i D, 2ajZ r(r-i) DM, I t=0 r=0 t=0 t(t-_) M, I + (t+ 1)(t+I + 1) M,i +2t 1 Dr'() Dr, r, + 2t-1 rt-1s - 2t+3 r,t+ (r+ 1)(r++) DM, I )Q I( 2r+3 r+l, t t (3 19) M, gs' In arriving at this last expression, one must bear in mind that D' ( ) is idenr, t s tically zero whenever r, t or I is greater than M. As shown in Appendix C, 00 r r t. (3.20) Furthermore, whenever r=t in(3.19), the bracketed coefficientis equal tozero. This follows from the fact, established in Appendix D, that the relationship DM' r) = ( +)tD) (3 21 ) r,t ( t,r s ( holds among these coefficients. Thus we need not evaluate terms in (3.19) when r = t. Substituting, then, (3.20) in (3.19) we get......... —---- 18

THE UNIVERSITY OF MICHIGAN 7133-5-T M+1 M+1, t v r 1 7 ^7yM 1 j D' ( v (p -2 r MQ ) iM+t1 1 = r(r+'l)-t(t+l) 2r-1 r-1, t( s t=O r=0 1= L t(t - D) M, I + (t+ 1)(t+Q + 1) DM,I ( + 2t - 1 r,t-1(s- 2t+3 r, t+1 s) (r+ 1)(r+i + 1) DM,'L- Q<()-Q P (1)cos 33 Qtl) D_1 LQQ( ) 6I (3.22) where the prime on the summation for r indicates that the term t = r must be deleted. This may be rewritten as M+i M+1 t IM+i(Pl) r=' E. r, t (s' q r(l)Pt( 7ll)C~S01' -(3.23) where M+1, I 2 r(r-e) M,k I t(t-) M,, ( E (QD I ___ r,t s r(r +1)- t(t+ 1) 2r-1 r-, ts) + 2t- 1 rt-1 s +(t+l)(t+i+l) DM1, I (r+l)(r++l1) DM1 0 ( 1 2t+3 r,t+1 s 2r+3 r+l1, t s (3.24a M+1, t M+ Q E IQM+ r ( s) E3M+ ( 3.24b t,t s rt (s) t S and the prime on the summation indicates that the term r =t must be deleted (see Appendix E). Thus (3. lOb) can be written as -..... —-- 19

THE UNIVERSITY OF MICHIGAN 7133-5-T M+1 M+1 t I'' t')+I, ) Z M+i(Pl)= IM+(P)M+i(P ) = 1 t (S ) ( +)PI (t )r t=0 r=O 1=0 (3.25) where M+1i, M+1, D -E (c) r t r,t r,t s M+1 I )+AM+i =Ett'()+A^, r=t. t, t (S Equation (3.25) is clearly of the form (3.11) which is what we wished to establish. Not only have we completed this inductive proof but, in the process, we have derived M, S recurrence relations for the coefficients D' (l ): r, t s M+D, ( 2 r(r-O DM, t D Q D Q ( r, t (s r(r+l)-t(t+l) L2r-1 r-l, t s 2t-1 r,t-1 s + (t+l)(t+ +l) DM, )_ (r+ )(r++l1) DMU. r t 2t+3 r,t+l s 2r+3 r+1,tsJ' M =0,1,2,.. (3.26a) M+1, ) DMt'()= Q D)A' ); M r = 0,M+, 2,... M+D r(+s)A t, t rt s t s t s (3.26b) with 20

THE UNIVERSITY OF MICHIGAN 7133-5-T D () = Ao, ( s) (3. 26c) (see equation 3.12). We are now ready to write the expression for the scattered field u (p1). By equations (2.18), (3.8), (3.9) and (3.11) we have u(p) - +ikc( 1t() u (p) = e (pl) -ikce +ikc(.i+y1) = e e (p) + 0o -ikc5 +1ikc(E1 1) (i ) = e e (-ike (p) M=O 1/ ) oD M M t -ikc(g i +l ) If we expand e in a power series of k and employ the Cauchy formula for the product of two infinite series, the above expression becomes ik1 o n _>.n- M M M t e e k (-i1 M ~). DM ( ) 1n=O MM=O t=O r=O0 =0 r, 2 1 ~Qr(9l)Pt(rl)cos01. (3.28).... 21 IQ? ) ^ ) o ^ mI~ If(3.28)'%er%1

THE UNIVERSITY OF MICHIGAN 7133-5-T M S where, in both of the above equations, D' ( ) is given by equations (3. 26a, b, c) r, t s and, in turn, At (5 ) is given by (3.7). 3.3 The Far Field and the Scattering Cross Section From the definition of Q (e) in (2.26) and (3.28) the far field is given by ikce O n _ n-M M t sf )= e 1T - i_,kc)n E Z ZI(- 1) Y! DMeIt ) 1 n=0 M=0 (n- M)' t=_ o,=0 Pt(nl)cos l. (3.29) Since the incident wave is of unit amplitude and r -- c in the far field, the scattering cross section is given by 2r2 sfn2 2 u o= lim 4rc2 1 2 u (p) 1= 47rc2 (l-ikc)n () (3.30) 1 —- co n=0 where n ( )n-M M t (^ = 1 -- (n-M)' E (-1)! D' (D Q)Pt(r )cos0 IA M =0 M) t=0 1=0 O (3.31) Assuming k real, we can rewrite (3.30) as follows: 22

THE UNIVERSITY OF MICHIGAN 7133-5-T 2 n sf 2u 2 sf a = 47rc2 (-ikc) un (p) = 47rc2 (-ikc)u (p1) (-ikc) u (p n00 n n=0 pn=0 42 c kc)n sf )n usf =47rc (-ik(p ) (ikc) u (P1) 00 n n=O n=O n=O n = 47c i (-ikc) n (-1) u (pl)u (p) n=0 m=0 m0 2n = 47rc 2(kc) 2n (-1)n+m usf (P )U sp (3.32) 3.4 Nose-on Incidence In the case of nose-on incidence (0 = 0) quite a few simplifications occur. o If we set 0 = O0 in (3.7), it becomes obvious from the definition of the Legendre function Pm(), |,- 1 < 2, equation (2.24), that AM'm( s) becomes zero n' s unless m = 0. We then conclude that in the case in which the incident plane wave propagates along the z-axis there is no dependence on the azimuthal angle i. This simplifies the results as follows: Equation (3.6) can be written M IM(p) = AM (S )Pt()Qt(333) t=0 with 23

THE UNIVERSITY OF MICHIGAN 7133-5-T (+ 1)M(2t+ 1) M+t even r 2M+1 (M-t) ( 2 2) Qt( s At (s) = t S 0, M+t odd.(3.34) Equations (3.26a, b, c) become M+1 2 r2 M t2 M - 2t+3 r,t+1 s 2r+3 r+l,ts s] M=0,1,2,... (3.35a) M+1 f M+, ( \. r M+1 M+1 DM+1(- ) =_ - r(gs 2 r Dr + l (') M=0,1,2,... tt s -r=O Q( ) r,t s t s (3.35b) with D ()= A( (3. 35c) The scattered field, u (p ), becomes ikc ao n ( _ n-M M M tkts Q rt s s us(p) = e (n-(ikc)n )! r) tZs)QD(j)Pt ) 1n=O M=O r0 t=o (3.36) and the far field, ikc^ oo n n )n-M M sf p,= e (-t s)Pi(r ). (3.37) 1- n=O M=0 t'

THE UNIVERSITY OF MICHIGAN 7133-5-T The expression for the scattering cross section remains the same except for sf un (P): n n ( _ )n-M M f) E ( n-MD) )Pt )(r (3.38) M=O- t=O25 - 25

THE UNIVERSITY OF MICHIGAN 7133-5-T IV THE NEUMANN PROBLEM 4.1 The Iteration Scheme The appropriate integrodifferential equation for the Neumann problem is (2.20b) which we repeat here, oo +1 2v7 w(pl) =-2ikc2 d dr1 d GN(pl ) 21) a(p) (r2 -1) s +(e+r)U(l+ c(2s- 1) dr1\ d G (pl, PS) as (ps), (4.1) -1 0 s with G (p1, p) given by (2.22). The appropriate boundary condition is given by (2. 10b), which through equations (2.17) and (2.18), may be written, 2 + c 2 2 a P s a-r) s Excluding the case in which the prolate spheroid degenerates to a wire of finite length (Q = 1), we can write 5~~s L (iiks + )+) (pj = 0 (4.3) s s from which 8u(p ) -ikc( tr) au(p ) S) S a U'(ps) - =-ikcw(ps)-e s (4.4) 5 R~~~~~(4.4) Substitution of (4.4) in (4. 1) leads to 26 -

THE UNIVERSITY OF MICHIGAN7133-5-T 001 27r (p) = -2ikc2 d dn S dG N(pl p ) l 2- 1) +(- ) — 1 J -ikc +1 2 i (ps) 2 7ikdcs'2S d +ikn GN(p, au P - c( -d (4.5) S l \ 0 S -1 Denote by IS(p1) the second surface integral in (4.5) 2 -ikcgCl 2 i N aui(p) -s, 2 ) s \ik cr7lN \Nusi\ I(pl) = -c(s -1)e dr] d e+ i G (pl s)'s (4.6) -1 SI \0 In Appendix F we show that I (p1) may be written in the following form 00 -ikc 0 IS() = e s (- ikc)M IMs 1) (4.7) M=i where M I IM(P1) = Z A)P (rl)Q (l)cos m, (4. 8) i=0 m=0 with I —--- - 27

THE UNIVERSITY OF MICHIGAN 7133-5-T (e +co M-1 2 (M2 2 Q(s) r coseO 1 - r - cose 1ti * \t s o ) + l \s g0 — ) O Me~ 0~+ Cos s 0+ 1Cos2 (0 cosO ti osjl - U+ m)( +cos0 )m 1K ~cos0, for M+ I even, AM (_m )=0, for M+ odd. (4.10) t S From now on, the procedure for developing an iteration scheme parallels that of the Dirichlet problem. After writing ikc? (p) = e w(pl) (4.11) and assuming a low frequency expansion in powers of k for ((p1), (P) = (-imkc) M(P), (4.12) M=0 l we substitute (4. 11) in (4.12) and the resulting expression together with (4. 7) in (4.5). Equating coefficients of equal powers of k, we obtain the following iteration scheme: ip(p1) = 0 (4.13a)....I- --.... —. —. - - - 28

THE UNIVERSITY OF MICHIGAN -. 7133-5-T M+l(Pl)=2c d d2 0_G (pi - M+1 1: a(r s +1 2 27t +( 1 s ) dr 0 d0GoN(Pl ps) M(Ps) -1 O +IM+1(P1); M = i0 1, 2,... (4.13b) 4.2 The Recurrence Relations The procedure we shall follow here is practically identical to that for the Dirichlet case. We assume,M(Pl) to be of the form M M t DM(Pl)= _DM' 1 M (p =,t ( s)Q r( )Pt (r)cosl M = 0, 1, 2,... (4.14) which we'substitute in (4.13b) and solve for iM+l(P1). If IM+l(P ) turns out to be of the form (4.14), then because of 0 (P1) being zero we can conclude that (4.14) is true. The volume integral of (4.13b) is practically identical to that of (3.10b) except for the Green's function. From equations (2.21) and (2.22) we see that these two functions are identical except for their dependence on the surface coordinate.s If we denote the volume integral of (4.13b) by IVM+1(), we can use the result of (3.19) and write 29 ___________

THE UNIVERSITY OF MICHIGAN 7133-5-T M+1 M+1 t IM+l(P) = -2 (-)1 (t-C:)! (t-1 )[ Pt(r71)e~osl t=0 r=0 k=0 r(r —!) M, T t(t-, ) D M (t+l)(t+ +1) M, 2r- 1 Dr-1 t(~ ) Tt-) D D 2 r-1 lt(~s) 2t-1 rt-(1s 2t+3 r, t+l(s) (r+1)(r+ +1) M, f I 2r +3 Drtl tD 11 tdQKt( 1Q, )QQ (4.15) - 2r+3 r+l rt0 5s where t s( ) t (~ )Q() (4.16) It is understood in (4.15) that D't ( ) is identically zero whenever r, t or i is r. t s greater than M. From Appendix G, J-)_ ( t+ Qs' g ODKt(S''' ~i~ Q t( l)QI Q < sd~KtQ: ts )Q () r(r+l)-t(t+l) (t- )! [ Q Q s r t. (4.17) Furthermore, through an inductive argument identical to that given in Appendix D for the Dirichlet case, we can show that M: I Q +'r+-t M. DrM ( ) =( 1) D r (s). (4.18) r,t s t,r Employing (4.17) in (4.15), we obtain. —------. —-------- 30 ---

THE UNIVERSITY OF MICHIGAN 7133-5-T M+1 M+1 t V (p) = -2 D r- 1-Fr ) rDM,. D ) M+i(1) = = E E r(r+l)-t(t+l) 2r-1 r-l t(s - t(t-e) DM, I (t+l)(t+ +1) MD, 2t- 1 r,t-l(s) 2t+3 r, t+1s (r+ 1)(r+1 +1) DM I _ _ I ( t P o' (4.19) Having evaluated the volume integral of equation (4.13b) we now turn to the surface integral of the same equation and we denote it by I(p1), I(p1) = -) dn\ doG(p Pi)bM(P) * (4.20) -1 J0 1 2 (nQ-rm)! M Substituting equations (2.22) and (4.14) in (4.20), we obtain I(p1) = c (- (1)e (2n+l1) ( 20 X1 2 n=0 m (Omn L(n+ m)j n M M s m ), *2ZZZZ'eDMKs m m s=O r=O r=0 t r s s Q n J n s +1 27r -1 J Using equation (B. 4) for the Wronskian and at the same time performing first the integration with respect to L and then the integration with respect to r) according to (3.18), we obtain 31........

THE UNIVERSITY OF MICHIGAN 7133-5-T M M t I( = D ) Q )t Qt( 6))P)(rl)cos?1. (4.21) t=0 r=O 0 rt s Q ( From equations (4.19) and (4.21), it is clear that ~M+l(P1) of (4.13b) is of the form given by (4.14). At this point, then, we not only have concluded the inductive argument that the representation (4.14) of M(pP1) is correct, but in exactly the same fashion as in the Dirichlet problem we end up with the following recurrence relationships: M+1 ) 2 r(r-1) M,J ) - t(t-) DM, i r,t (s) r(r+l)-t(t+l) 2r-1 r-l,t(s 2t-1 r.t-l(s +(t+l)(t+ +l) M, (r+)(r++) 2t+3 Dr, t+(s) 2r+3 r+l3 t, I M = 0,1,2,... r t (4.22a) M+1, I (K7 M+1: I Q r s M., I M+ 1. IQ Dtt ) )=- Dr t ( )+ - r.t ( )+A () t, t (s r, t s s, s' s r=O Q(0) r=0 Qt(rs)t Qt ts M =0,1,2,... (4.22b) with D0'O() =0, (4.22c) where AtM ( ) is given by (4.9a, b). The prime on the first summation in (4.22b) denotes that the term r = t must be deleted. 32

THE UNIVERSITY OF MICHIGAN 7133-5-T The scattered field u (pl) for the Neumann problem is given by the same expressions as for the Dirichlet problem (equations (3.27) and (3.28)) with the under M1 standing of course that the coefficients D' (g ) are this time given by equations r,t s (3.22). The same is true for the far field and the scattering cross section expressions (see Section 3.3). 4.3 Nose-on Incidence When 0 =, (4.9a) becomes zero unless m = 0. This is so because of the o definition of the Legendre function P (m) for Iu- 1| < 2, equation (2.24). Consen quently, when the incoming plane wave propagates along the z-axis, there is no dependence of the scattered field on the azimuthal angle 0, a result we should expect since the z-axis is the axis of symmetry of the prolate spheroid. Due to the substantial amount of simplification, we redefine our results for the Neumann problem as follows: Equation (4.8) becomes M I^(sp) =p A^sM tS^^ ()P t' r~l, M - 1, 2,3,... (4.23) t=0 where (2t+1)( +-l)M-l - 1Y7 M +_ -_ 3, M +t even AM(~~ ) 2 (M>Q r(f_+t_ 2 ) At () 2 (4.24), M+t odd. The scattered field u (P1) is given by (3.36) with DM (S ) given by *I-*33 rt s ------------------------- 33....

THE UNIVERSITY OF MICHIGAN 7133-5-T 2 2 DM+1 2 r _Mi t Mi r, t () = r(r+l)-t(t+l) L2rI r-l,t( s)+ 2t-1Dr,t-l( s) +_(t+l)2 M (r+1)2 M M=0,l,2,... 2t+3 r,t+l(s) 2r+3 r+lt Qs' r t (4. a) Di t ( - - Q ). Dt ^ -^ Q ). Qt~s)^ (gs) e M =0,1,2,... (4.25b) with Do o()=. (4.25c) The expression for the far field is the same as the one for the Dirichlet problem (equation (3.37)) with D t( s) as above. The same is true for the coefficient of the ot s scattering cross section, u (P), which is given by (3.38). 34

THE UNIVERSITY OF MICHIGAN 7133-5-T V THE OBLATE SPHEROID AND THE DISC The method employed in the preceding sections to determine the field scattered by a prolate spheroid can be employed in a straightforward manner to determine the field scattered by an oblate spheroid. This is not necessary, however, since we can transform the prolate spheroid into an oblate spheroid and utilize the results already obtained to determine the field scattered by the oblate spheroid. Specifically, if we let e - iS and c -— ic, the prolate spheroid is transformed into an oblate one with the axis of revolution (minor axis 2b) coincident with the z-axis of a rectangular coordinate systems (cf. Morse and Feshbach, 1953, p. 1502). The ranges of the variables now are 0, < oo, -1,r< < +1, and 0 < 27r. Moreover, ax, yz)'t.,yc+FIG. 5-1. if we let s - 0, the oblate spheroid degenerates into a disc of infinitesimal thickness, radius c (the semifocal distance), and coplanar with the x-y plane. In the remainder of this section we shall treat each body separately. ------------- 35 -------------

THE UNIVERSITY OF MICHIGAN 7133-5-T 5.1 The Oblate Spheroid With the incident field given by (see equation (3. 3)), u) -iikr cose - 2ikc cos 0 +s in 2 coss] ui(p) = e = o the scattered field is given by (3.28) with g = iS and c = -ic and can be written as follows * o n /n-M M M t u(p) eikc1 (-ikc) n 1(-i) -MM QI n=O M=O n t=0 r=0 = M I the coefficients D'(i? ) with r, t s( +s )M Pt(S 0 -i~gs'+cos0'.o ( 2t+ 1)) A' (i4 ) =2a Qbt 3is 3t s M+t even 0, M+t odd (5.2) For the Neumann problem the coefficients D' (i^ ) are relatedthrough equations r, t s (4.22a, b, c) with 36

THE UNIVERSITY OF MICHIGAN 7133-5-T M- 1 (Mt )W ) + [(i t~cose )M1 Mt (2t + 1) f ip cos~ i (iS cosO to TeLgnrfuci Q P( i ingo ( 5.1) t nowboredfine s tee a -(t+ I)( tcose )p+ I os S 0 -1 i Cos QL M+t even, (5.3) and A i ()i= 0, M+t odd (5.4) The prime on Qt(is ) in (5.3) implies differentiation with respect to i. TepieoQt S S The Legendre function Q (i ) in (5.1) must now be redefined since can now assume values between 0 and 1 as well as values greater than 1. This has been done in Appendix H where we show that 1 (-1)m r(n+m+l)r(1/2) (j2+1) n ^1.n+l2m 3 + 2/ )n-m+1 1 3 1 F -m+l,'-m; n+-;- - 2,?>0. 2 1 2 2 + +) (5.5) The far field is given by (5.1) by letting 1 — > oo,........ —-- 37

THE UNIVERSITY OF MICHIGAN 7133-5-T ikcC 1 OD n vM sf = e I( ikc ()n -,n-1 s) u (p) (icnM M=t I n=0 M0 -O M t / ^ )(-1)' DM (iSP.' (5.6). 0 -1) Q'Do, t (iS)Pt (rll)cO 1 (5.6) The scattering cross section is given by (3.30) and, in the present case, can be written in the form oo 2n 2 2n X +m sf sf a = 47rc (kc) (-1) u 2n- (pl) )' (5.7) n=0 m=0 where we have taken k to be real, and n n-M u (pl) = (-i) n ( 1) -Q Dt (i)PM(r1 )CO n M= 0' t=0 1=0 -o, t s 1 1 (5.8) Nose-on Incidence: When 0 = 0, we can rewrite (5. 1) as follows: 0 i k o (ik n )! M M ikc - 4 W +uS(pl) =e (-ikc)n -i) r D (i)Q(i )Pt(n1)' n=0 M=0 (n DM r= t=0rt, (5.9) where, for the Dirichlet problem,.........._____ ~38.... —--------

THE UNIVERSITY OF MICHIGAN 7133-5-T 2 2 r, t (is = r(r+1)-t(t+l) L Dr-l1, + 2t- r, t-( s r+l (ig)- M t(i ~. rD t 2t+13 rt+1( s 2r+3 Dr+l, t s' M = 01,2,... (5.10a) M+1 Q DM+1i(i,) )=_ s DM+ii M+A Dt t ( =- (i) Dr t (i )+At (is); M = 0,1, 2,... (5. 10b) D~ (i A~(i ), (5.10c) O,0 S 0 S with (2t+1)(i + 1)M M5^Ml/-VyMt^ ^:, M+teven F~fM+1' P M+t + 3 M: 2 ) A (i ) ='i 2 M ( t s t s 0, M+t odd (5.11) The corresponding expressions for the Neumann problem are, 2 2 M+1 2 r2 M.. t M r,t s r(r+l)-t(t+l) 2r 1 r-l,t(is 2t-1 r,t-l( s 2t+3 2r+3 DM+_ M D (i)+AM (ii); M+l.,. V ^sL-M~l.r1 r-li 2t-1 r, s M+l M = 0,1,2,... (5.12b) D (i) D (i ()+At5.12c) trO Qt s Q'r= Q t(i s' r, s s'3 0,0 S 39

- THE UNIVERSITY OF MICHIGAN 7133-5-T with 1 t2t+l)(i -+ 1)M-1 F"-i'fM Si"(.)/+'2 M+t even AMi 2~M+1 rMt(iM ) = At (i) = 2 Qt(is 0, M+t odd (5.13) In both the Dirichlet and Neumann problems the far field is given by u (p = e l ikc)n (( I En-M) Do t(is)Pt(rl) 1 n=O M=(nM t=O O., (5.14) and the scattering cross section be equation (5.7) where in the present case, n i. n-M M U sf( i)n -1 M E (n-M)s Do t(i s)pt(l) ) (5.15) 5.2 The Disc As we mentioned earlier, when e = 0 the oblate spheroid degenerates to a disc of radius c in the x, y plane, with center at the origin (Morse and Feshbach, 1953, p. 1292). It is easy to verify from the corresponding formulas for the oblate spheroid that the scattered field due to the presence of the disc is given by OD n n-M M M t s iku~1 n u (p ) = e 1 (-ikc)n (-.) ( M M (0) (Pt) = e1 n: ~ (n- M t)(0. ) Q=(il)P^t(r1)cosr1=, (5.16) 40

THE UNIVERSITY OF: MICHIGAN 7133-5-T where, for the Dirichlet problem, the recurrence relations (3.26a,,b, c) hold among the coefficients D' (0) with r, t (+cos0 )M Pit Gos -E c\I' (2t+l) (t- +): \c ~ Ie^ 2 M+1 (t t r(M+t+ 3 I A' (0) =2 2 r2 + t M+t even 0, M+t odd (5.17) For the Neumann problem, the coefficients D' (0) are related through equations r,t (4.22a, b, c) with (+ os )M-1 AM (O)=-c/Tcosoo 2M+1 (M-t +) LM - t)(t- +) Pt+l s +(M+t+l1)(t+t)Pt-1(' M+t even, (5.18) A' (0) =0, M+t odd. (5.19). From (5.16) the far field is ikc1 oo0 n M _ -M M3 t n= (0-ikc) (-i) (n-M)' (-1)( 0 Do)t (~) Pt (r )cos. (5. 20) 41

THE UNIVERSITY OF MICHIGAN 7133-5-T The scattering cross section is given by (5.7) with n n-M M t sf n- 1 -_:( - M)'-17' DM' > (p) ==0 ( i) Do (O)Pt (r1)cos l (5.21) Normal Incidence: When e = 0, we can write, as we did in section 5.1, 0 ci < oo n ( )n-M M M s ikCel n n -, u5(p1) = e (n-ikc) ( M)! DM )Q(ig Pt(l)t uspl= L1(n-M) r,n-; ~n=uM~=0 r=u t=r (5.22) M where, for the Dirichlet problem, the coefficients D (0) are related through r, t equations (5. lOa, b, c), with 2t+ 1 -V'~(+1) 0 M+t even A (0) ( - + 2 Qt(= 0, M+t odd (5.23) For the Neumann problem the coefficients DM (0) in (5.23) are related through r,t (5.12a, b, c), with 2t+ 1 T -' (+-)1M _M+lM-.t\ - M+t even M 2M+ M-t +, A (0)= 2 -2 K 2 Qt() t 0, M+t odd. (5.24) _______________________ 42

THE UNIVERSITY OF MICHIGAN 7133-5-T Equations (5. 23) and (5.24) were derived from (5. 17) and (5. 18), (5.19), respectively, by letting 0 = 0 and I = 0, and they are in agreement with the corresponding equations (5.11) and (5.13) for the oblate spheroid. For both the Dirichlet and Neumann problems the far field is given by (p1) ikcc)n (i)n1 (n)M M M O (5.25) while the scattering cross section is given by (5. 7) with n )n-M M sf (.)nl 7' un(p) ( I)-1 I) g DM(0)Pt(). (5.26) n ( = (n- M)' o. In Appendix I, we give the first six terms of the far field expansion for both the Dirichlet and Neumann problems. 43 -

THE UNIVERSITY OF MICHIGAN 7133-5-T VI NUMERICAL CALCULATIONS As a demonstration of their usefulness, the theoretical results have been employed to calculate the scattering cross section of some representative prolate spheroids for both Dirichlet and Neumann boundary conditions. The prolate spheroids considered had major to minor axis ratios of 10:1, 5:1, and 2:1. Back scattered and forward scattered cross sections were determined as functions of wavelength, and complete polar diagrams of bistatic cross section were obtained for a few special values of kc. All calculations were carried out for a plane wave incident along the axis of symmetry of the spheroid. The expressions employed for this calculation, which we repeat here for convenience, were equation (3.32) o0 2n o = 47rc2, (kc)2n E (-1)n+m u (sf 1) 6) ~.c /jU2n-m _r1 (6m 11) n=0 m=0 and equation (3.38) u (r1) = J D )P.(ri ) (6.2) m j=0 i= (mj), s I where DJ. in (6.2) is given by (3.35) for the Dirichlet problem and (4.25) for the o, i Neumann problem. The series in, (6. 1) was terminated at n =10 for the 2:1 and 5:1 spheroids and at n=9 for the 10:1 spheroid. Thus the cross section values included terms up to and including (kc)2 and (kc)8 respectively. The back and forward scattering results were also obtained for smaller values of n so as to reveal the manner in which the inclusion of higher order terms improves the Rayleigh approximation. 44 -

THE UNIVERSITY OF MICHIGAN 7133-5-T Figures 6-1 and 6-2 present the back scattering (r1 = 1) cross sections of sof and hard spheroids respectively. The cross section values are normalized with respect to the geometric optics value r4 2 (21)2 7f 2 s c.o = - = 7r 2 (6.3) g.o. 2 2 a The number associated with each curve indicates the value of n at which equation (6.1) was terminated. The Rayleigh curve (the curve obtained by terminating (6.1) at the first nonvanishing power of kc) is denoted by n= 0 for the soft spheroid and by n =2 for the hard spheroid. The exact result shown in Figures 6-1 and 6-2 was computed from the prolate spheroidal function series (Senior, 1966). Also included in each figure is the maximum value of ka (= kce ) for which the series in (6. 1) converges, i.e. the radius of convergence, as estimated by Darling and Senior (1965). The present low frequency calculations have no precedent except in the case of the 10:1 hard. spheroid where similar calculations (though not as extensive) were reported by Sleator (1964). Figures 6-3 and 6-4 present the forward scattering (ri =-1) cross sections o the same spheroids. The cross section values are normalized with respect to the limiting form of the bistatic geometric optics value 2 22 a =ra = c2. (6.4) g.o. s As before, the number associated with each curve designates the value of n at which the series in equation (6.1) was terminated. No exact results were available for comparison in this case. Figures 6-5 through 6-9 present polar diagrams of the bistatic cross sections of the same spheroids. Since the polar diagram is symmetric for nose-on incidence, 45

THE UNIVERSITY OF MICHIGAN 7133-5-T which is the only case considered, each figure includes data for both hard and soft spheroids. The back and forward scattered cross sections lie on the heavy vertical line bisecting the figure with the back scattering (01 =0) value on the upper part and the forward scattering (01 = r) value on the lower. The values of the cross section are normalized with respect to the geometric optics cross section, viz., ag = 4rb4a2 a2(1 +r1)+b2(1- 2) 2 2 2 (6.5) 2 s 1) s = 4rc 2 2 (22 - 1+r1) with rl = cos0. As noted previously, the values presented for the 2:1 and 5:1 spheroids were obtained after terminating the series in (6.1) at n = 10 while for the 10:1 spheroid the series was terminated at n = 9. Similar calculations have been carried out by Spence and Granger (1951) for hard spheroids though the values of s and kc were different from those employed here. In the few cases where comparison was possible (kc = 1, a/b = 5, 10), good agreement was obtained. 46

THE UNIVERSITY OF MICHIGAN 7133-5-T 30 302 8 20 - 4 0 rb4/a2 10 9 maxka =1.06 8 - =1.1547005 7 5sexac exact 4 0.2.4.6.8 1.0 1.2 ka FIG. 6-la: BACK SCATTERING CROSS SECTION OF SOFT, 2:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 47

THE UNIVERSITY OF MICHIGAN 7133-5-T 500 400 300. 10 200 100 90 80 70 60 50 maxka = 1.57 40 - = 1.02062027 30 exact 20 20 Ii I I I 0.2.4.6.8 1.0 1.2 1.4 1.6 ka FIG. 6-lb: BACK SCATTERING CROSS SECTION OF SOFT, 5:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 48

THE UNIVERSITY OF MICHIGAN 7133-5-T 5000 4000 2 3000 2000 r b4/a2 1000 900 800 700 maxka = 1.74 600 = 1.0050378 s 6 500 \ 8 400 - exact 300 0.2.4.6.8 1.0 1.2 ka FIG. 6-lc: BACK SCATTERING CROSS SECTION OF SOFT, 10:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 49

THE UNIVERSITY OF MICHIGAN 7133-5-T 3 2 4 6 8 10.9 - exact.8.7.6.5.4 4 2, 4.3.2 7rb /a maxka = 1.59 = S 1.1547005.01 _.005 I I I I 0.2.4.6.8 1.0 1.2 1.4 1.6 ka FIG. 6-2a: BACK SCATTERING CROSS SECTION OF HARD, 2:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 50

THE UNIVERSITY OF MICHIGAN 7133-5-T.2 4 6 8 10.9.8.7.6.5.4.01 maxkka a2.49 =1.02062027 FIG. 6-2b: BACK SCATTERING CROSS SECTION OF HARD, 5:1.005 PROLATE SPHEROID FOR NOSE-ON INCIDENCE..005 ~ —-— 51

THE UNIVERSITY OF MICHIGAN 7133-5-T 22.9.8.7.6.5.4.3.2 exac.1 4 2 rb /a maxka = 3.28 E0 = 1.0050378.01 _.02 I I I.05 - | o.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 ka FIG. 6-2c: BACK SCATTERING CROSS SECTION OF HARD, 10:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE..... —------ ~9~52

THE UNIVERSITY OF MICHIGAN 7133-5-T 10 9 8 maxka = 1.06 = 1.1547005 s 6 5 2 4 1.5 l l 1 0.2.4.6.8 1.0 ka FIG. 6-3a: FORWARD SCATTERING CROSS SECTION OF SOFT, 2:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. ________________________ 53 ______..

THE UNIVERSITY OF MICHIGAN 7133-5-T 5 maxka = 1.57 4 1.s = 1.02062027 3 _ a 7ra 0.2.4.6.8 1.0 1.2 1.4 1.6 ka FIG. 6-3b: FORWARD SCATTERING CROSS SECTION OF SOFT, 5:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE........ —---- 54

THE UNIVERSITY OF MICHIGAN 7133-5-T 2 maxka = 1.74 s = 1.0050378 2 7ra.5.4 0.2.4.6,8 1.0 1.2 1.4 1.6 1.8 ka FIG. 6-3c: FORWARD SCATTERING CROSS SECTION OF SOFT, 10:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 55

THE UNIVERSITY OF MICHIGAN 7133-5-T 10-1 - B 9 3 10 2 7ra maxka = 1.59 = 1.1547005 10-50.2.4.6.8 1.0 1.2 1.4 1.6 1.8 ka FIG. 6.4a: FORWARD SCATTERING CROSS SECTION OF HARD, 2:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 56

THE UNIVERSITY OF MICHI.,GAN -4 7133-5-T 10 5 3 7 9 10-5 10 10 -6 2.wa 10-7 maxka =2.49 = 1.02062027 s -8 10 - 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 ka FIG. 6-4b: FORWARD SCATTERING CROSS SECTION OF HARD, 5:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 57

THE UNIVERSITY OF MICHIGAN 7 0-8 1 2 7ra 10-9 10 1 o10 maxka = 3.28 Ws = 1.0050378 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 ka FIG. 6-4c: FORWARD SCATTERING CROSS SECTION OF HARD, 10:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 58

THE UNIVERSITY OF MICHIGAN 7133-5-T kc=.2 0O /X / O \kc=.5 kc =.8 \ \. *<- - " ~~Hard -- --- Soft FIG. 6-5: BISTATIC CROSS SECTION OF 2:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 59

THE UNIVERSITY OF MICHIGAN 7133-5-T 01 0 / FIG. 6-6: BISTATIC CROSS SECTION OF 5:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 60 kc = 1.i kc = 1. Hard — Soft NOSE-ON INCIDENCE. 60

THE UNIVERSITY OF MICHIGAN 7133-5-T //0.45 c=.5,06 70 0.0 IA o.04t 90 Hard ------ Soft FIG. 6-7: BISTATIC CROSS SECTION OF 5:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 61

THE UNIVERSITY OF MICHIGAN 7133-5-T \ < — Hard - — Soft / FIG. 6-8: BISTATIC CROSS SECTION OF 10:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 62

THE UNIVERSITY OF MICHIGAN 7133-5-T 0.30.0 ~2 7500 \.0. 4 --- X1800 \'m 900 Hard ~ Soft ip FIG. 6-9: BISTATIC CROSS SECTION OF 10:1 PROLATE SPHEROID FOR NOSE-ON INCIDENCE. 63

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX A -ikc( + THE REGULARITY OF THE FUNCTION w(p) = e i ) uS(p) In this appendix we offer a proof that the function w(p) in (2.14) is regular at infinity in the sense of Kellogg, that is lim | r(p) < oo and lim r2 aw) < ar 0. r->oo r -*oo r (A.1) The proof is based on an expansion theorem (Wilcox, 1956) which guarantees that the field scattered by the prolate spheroid may be written in the form u ) - p) - n r > a (A.2) n=O r where the series is uniformly and absolutely convergent for all r, 0, 0 provided r > a, a being the radius of the smallest sphere completely enclosing the prolate spheroid. From (A. 2) it is clear that u (p) satisfies the first of conditions (A. 1) but not the second and, consequently, is not regular at infinity. The function w(p), however, which by (2.14) and (A. 2) may be written -ik(cS - rt+ cn)C) 1 n w(p) = e - 1 n (A.3) n=O r can be shown to satisfy the Kellogg conditions. The proof is as follows: The variables e and rl are related to the spheroidal coordinates by the equations 1 2 22 ~~ equa s 1 2 +2crcos0+c + r -2crcos0+c 7r [- 2\Ir +2crcos0+c - -2crcos0+c The factor c(tr]) appearing in the exponential of (A. 3) can now be written 64

THE UNIVERSITY OF MICHIGAN 7133-5-T 2 2 2 c(t r ) = r+2crcos+c r 1/i 2cos(c/r)+(c/r), (A. 4) and, if r is large, c(+tr) = r~ccos0+O(l/r), r — o. (A.5) We can then write for the first Kellogg condition lirm | rn(p)= li e ick os + 0 (r) fn0 r-coo r —co n=O r T ikc cos 0 = e f <oo. (A.6) To show that the second condition is satisfied we need the derivative of w(p) with respect to r =p) -ik[c(+r7) - rk 1k 1~ cos (c/r) fn(O 0) 3r 2 J n+l _r L - + 2cos0(c/r)+(c/ r) n=O r oD (n+l)f (0,0) n=O r -n+2' * (A.7) n=O r For r large 12 1,, 2= 1+ cosO(c/r)+O(l/r2) r-eco (A.8) 1 + 2 cos O0(c/r) + (c/r) so that for the bracketed expression in (A. 7) we can write 1- lc-cos0(c/r) = 1- [lcos0(c/r) r+cos0(c/r)+O(l/r2)] 1 +2 cos (c/r)+(c/r)2 =O(1/r2). (A. 9) 65 ---

THE UNIVERSITY OF MICHIGAN 7133-5-T Employing (A. 5) and (A. 9) in (A. 7) we have that lim 2r aw=) li- m e ikccos0+O(lr) O(1)I fn ) r-'co r->oo n=O r - r|n j _ e ikcos f <o (A 10) nn=din h which shows that the second Kellogg condition holds also. 66

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX B THE SURFACE INTEGRAL FOR THE DIRICHLET PROBLEM In this appendix we evaluate the surface integral of (3.4). Repeating the expression, +1 27r IS(p 1) -c(s2_ 1) d0 doups)as GD(pps (B.s1) -1 1 0 From equations (3.2) and (3.3), -ikc(g t+r) -ikc(cos0 rT+sinO V -1 jl - cos0) w(p )=-e s e (B.2) s and from (2.21), with s <' G (P =- (-1) e (2n+ 1) cos m(- 0 oo n = — 2 n (2n+)11 (2n+ ))n^ cosmos-( 0) n s W[Qn Pn ),Qn m(1 n n s m(n 1 = 2 n m) * oo n ~1 2 7 ~c (2n+l) (nm os - m) 47rc( - 1) n=O m=O s DPm(7 )mPn() (B.3) n n 1 m n s where, above, we used the Wronskian relation w-m(~) pm(~) =pm(~),Q m(5)_Q()pm(~)_ (-1)m (n+m)W^),?^n~ n 2^ ^ = (n-m)' (B.4) 67

THE UNIVERSITY OF MICHIGAN 7133-5-T Substituting (B. 2) and (B. 3) in the integral (B. 1), we have -ikc oo n m s e (n - m)l m n 1 (p1) = - 4r e (2n+1) (n+m)! P' ) m n=O m=O Q () n s +1o- 12 2 f -ikc( cos ~1)r2 -ikc sin 1 - cos drl de ms n) e s cos m(- 0). JSr-1 P d j ~ (B.5) The functions involved in the integrands are continuous in the intervals of integration and the only assumption we made in interchanging integration and summation is the uniformity of convergence of the series (cf. Whittaker and Watson, 1952, p. 78). We now use the expansion (Magnus and Oberhettinger, 1949, p. 155) ikp cos= i J (kp)cosm. (B.6) m= mm m=0 Utilizing (B.6) in (B.5), with kp = -kc sin0 Y 1 ir, results in -ikc oo oo n Qm,s - - __eW e^ (i (n-m)t. ni 1(p E)=-_-e A —/, ~(-i) e4(2n+1)P 1 p47r) o om (n+m)' n Qm() dne 0 m +1 dr e) J ( sinP 0 1 T1 d0 cos m(0- 0)cos 0 | n oVs 1 s+1-1 l~rl n0 (B.7) The integration with respect to 0 can be simply performed, while to integrate with respect to r we use the relation (Morse and Feshbach, 1953, p. 1325), 68

THE UNIVERSITY OF MICHIGAN —-- 7133-5-T rT iz oscos m cosn-m2 m dpe P (cosp/)J (zsinvsin/u)sin= i -- p (cosv)J+ /(z) o (B.8) In this expression we let 7r = coslP z cosv =-kc(C cos0 +1) s 0 r~2 z sinv = kc sinO -1 0 s so that o r- -,2 O 2 s2 cos0 I Z kC( Cs- cos - s 0 We can then write +1 -ikc(Q cosS +~ 1)r dnr e P ()Jm(csin (O 1-l)(l-n )) %-1 = n-m 27r sm cosO B +1 kc( t cosO n ~cos Jn+ 1/2 Lc(s 0 - J 5 0 s 0 Performing the 0 integration in (B. 7) and using (B. 9), we obtain -ikc oo n m 5s I —~ — ~ Q( ) I (P) = - e 2 ~e (-i)n(2n+l) (n-m) P n )osm ~~~1 2 ^^O1~ m(n+m)' P ( m cosmn1 1Bl 2 ~n=O m=O m 1 (r n s 69 -'.... 69 -- 7

THE UNIVERSITY OF MICHIGAN 7133-5-T where, above, we used the relation (Magnus and Oberhettinger, 1949, p. 63) P (-x)= (1) Pl (x) ( -1 <x< 1 (B.l1) n n. B.' We now expand the Bessel function in (B. 10) according to (Magnus and Oberhettinger, 1949, p. 16), CD 21 ~'~z) (c(iz/2) J (z) = (z/2)m ((m+i+1) l argz < (B.12) to get co OD n n+21 s -ikc~ n () ( ~Cos n2 - (p -e (-1)(2n+1) (n-m)! L 1p,= -e 2' m (n+m)' - -~ 2 /r cosO t1 1 Qm( 1 ) n 2-cosO k!r(n++3) n 1Q ( ) 2 1 2 n s oD Fn 2 n-2I -s ei2 1 mL on lI (n-2 - m)! e 2 Em (n- 2~( n ) - 2) + m)' n=O =rn= ikc(Wst+ cos1Pm ( cos 1 2 i n- 21 p +os0 n-2 21 c+ mo s3 \s- o/'r(n-2 + Cs n _ -2m 1 -7-2- k' -— s70

THE UNIVERSITY OF MICHIGAN 7133-5-T cos +1i? S(-COSs a +os )M 0) __' M (M+Q/ 3 P (771 M cosm= 0, (B.13) 2_ 2 2, S where in the above series in k the only nonzero contributions to the coefficients are made by terms for which M+Q is an even integer. We have then written the surface integral (B. 1) as a power series in kc of the form -ikc OD SI (P =e s (-ikc)M IS(P) (B. 14) M=O where I,(P1) is given by IM(P1) = Am( Q) pQ (e )QQ (1 )2os m 1 (B. 15) with ( cos0 t 1 O p -S, M + Q ( ~cos ) M (m iK ~cos0 71 n M+1 v2l)(j+m)t M- /M+iQ 3 ms m bes M+i even 0, M+ odd (B.16) 71~~~~~o _

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX C EVALUATION OF THE INTEGRAL Ct (, t,)Q (S)d S d~ Ct MI 1 s)Q (Q)= ((+ dQ(+ \ deP (1)Q( )Q( ) t sr t it r tit r s s 1 PI(^) 00 -t S | d Qt(~ )Q (I)Q (I) (C.1) Q )t 1 t rs t ~s From Legendre's associated equation we have that d ([)2 r + Lr(r+ 1)& - 2 QS = 0 d r 2 dt r a22] d 1-2) d- + (t+l) - Q2 t ()=o d dr ] i 1 _22 t Multiplying the first of these equations by Q (0) and the second by Q (~) and subtracting the second from the first we obtain the following: dQ1r(_) - d Qt( ) QI() d 2 )r Q d1 2 d~ d~ -r d1 -L dj + [r(r+ )-t(t+ 1)] I Q()Q() = 0 Integrating this expression we have that 72

THE UNIVERSITY OF MICHIGAN 7133-5-T - i d 2dQ ()Q () r(r + 1) - t(t + 1) ) d dQr(Q)) 1' -) o - d2 r ( 2 Qt) r(r+ 1) - t(t+ 1) - Q r()) - Is d 1 (~)(1 -1 fir+ 2^ hw1 t i t-m+l)g +1 r(r+l)-t(t+l) 2r+ 1 -r+ )- (r+l)(r+ rQPr(E) L 2t+ [(tt-+l)Qt+1 ()-(t+l)(t+g)Qt (; rft4O. (C.2) Similarly, 1 rFt r - r d Substituting ( 2) and ( 3) in (C. 1), we obtain — 1 - Pt(-) 3 I - QPr(r ) 2 r (t-f+1)p1 M -(t + 1)(t 1; rftS0. (C.3) 2t++ 1 t+l )-(t+l)(t+ )Pt_l(...... ~~~~~~~73

THE UNIVERSITY OF MICHIGAN 7133-5-T O de t' 1 esQre) {r+g >__________ dQ ( sQ00 i 1 2 r Q1 l )dg C gs Q (It ) (lJ) 1) rS +t l) - rt l) r(r + 1) - t + 1) 1 Qr1 d9 d~Ct(~,~l,~s)Q ~)= r(r+l)-t(t+i) 1 *4EES-Q~~~-1) 1( t(t ) dS1 I dP^?j> 1 2 )_r dtill 2 [) r 1 d9 1 -d' s t d } -d~ sldi Qt(g 2) p( I r2.r -Q,, dQ( d^) r(r+l)-t(t+l) d i t 1 Ir d(i)____ _ 2 _ dQ() 2Q _dQt _ __ Qt Q() sdP 2 s d J) r(r+l)-t(t+l) 1 - ) dS Q d r(r+ 1) - t(t+ 1) Ft 1 Q d)r - Q t() 2S - ti rs j- tQ s ts QS - 1)Lt (5,) Q QEQ) -Ps )Q() t s Lt s s J t s r(r+- t(t+l) P t I 9 Pt - 2 (-i)+ _____I d ______ +,F, (t+n - Q1) Q )( ) r(r+ 1- ) - t(t+ 1) Q(r )s1 + -- s ds i t s ~^s (cont'd) 74

THE UNIVERSITY OF MICHIGAN 7133-5-T It+PII r_ _ F _ (-1) t_ ) s> 2 - ) rQt (C 4) r(r+l)-t(t+ 1) (t-)! t ( 1)'Q r; t (C.4) Also, o0 C(?,,.g 1s)QI()d? = = for I >t (C.5) since Ct =0 for I >t. t...75 ---------------------— 75 ------------

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX D DERIVATION OF RELATION (3.21) In this appendix we give a proof of the statement of equation (3. 21), that is r, t ( t, r' Dr. t (is) = (+1) Dt,r (5sD. 1) The choice of sign is determined by the sign chosen in (2.18). The proof follows an inductive argument. First we show that (D. 1) is true for M = 0 and then that if it is true for any M it is true for M+1. Since r =t = 0 when M = 0, equation (D. 1) is certainly true for M = 0. Assume next that it is true for M. We can then integrate (3.19) and, following the same procedure as we did there, end up with the recurrence relations (3. 26a, b, c). We are interested mainly in (3.26a) since for r =t equation (D. 1) is obviously true. Repeating here (3. 26a) and subsequently employing it in (D. 1) which is assumed to hold for M, we obtain M+1 i 2 F2r(rt- ) t(t-) M, I D () D (t D r,t s r(r+ 1)-t(t+l) [2r-1 r-l, t(s) 2t- 1 r, t- s( + (t+l1)(t+ + 1) M, I _ (r+l)(r+_+1) M,I I ~ 2t+ 3 r, t+l( s 2r+3 r+l, t(S r(r+1)-t(t+l) L2r-1 tr-1 1 2 ~ >(+1- t ( - 1 (1) 2t-1 D (s1) D 2(+) t t- ) r+t-l M. I + (t+l)(t+1+1) +t+1 M, I (r+ l)(r+I+1) ( +1)r+t+ Mr+i l. (cont'd) 76

THE UNIVERSITY OF MICHIGAN 7133-5-T = (+1)r+t 2 (t-i) M, r(r-.) M, I t(t+l)-r(r+l) L2t-1 t-l,r s 2r-1 t,r- (s + (r+l)(r+ + 1)_(M,. (t+l)(t+l+1) M, I 2r+3 t, r+1 s)- 2t+3 t+l,r(' ()r+t D 1, ) (. (D2) t,r s So (D. 1) is true for M+ 1 if it is true for M. Since it is true for M = 0, it is true for all M (M=0,1,2,...). 77

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX E DERIVATION OF EQUATIONS (3. 24a, b) In order to arrive at equations (3.24a, b), we start with (3. 22) and (3.23) which we repeat here M+1 M+1, t r V r\-9 Y1 2J-7 ir(r-i) M It M+i1(P) = -2 O r(r+l)-t(t+l) L2r- r-l,t s + t(t- ) DM, (f ) + (t+ l)(t+ +1) DM,Q ) + 2t- rt- 2t+3 r,t+l( s (r+l)(r+i +1) DM__ r -_ 2r+3 r+lt( L() Qt (1 ) Qr( Ptl)csl t s (E.1) M+1 M+1 t -M+1) E, rt ( s)Qr( i)Pt(r)cosl. (E.2) t=O r=0 l=0 When r t, a comparison of these two equations gives (3.24a). When r=t, we rewrite the above equations as follows. Equation (E.1): M+1 M+1, t v 1 r(r- ) M I M+P1 1 =2 L r- r(r+l)-t(t+l) t2r-1 Dr-l, t( t=O r=O 1=0 - t(t-lk) DM,( )+ (t+l)(t+k+l) DM,: I + 2t-1 r,t-'s' 2t+3 r,t+ls' (r+l)(r+1+1) DMI 7 Q( I — D (Q Q ( )P (r/)cos Io 2r+3 r+lt s 1 t 1 M+1 M+1, t -2 t rO Z Ir(r+l)-t(t+l) r- 1 r-t(s) -8 - (cont'd) 78

THE UNIVERSITY OF MICHIGAN7133-5-T - t(t-i) DM, ( ) + (t+l)(t+l+1) DM, I 2t-1 "r,t-1 s 2t+3 r, t+1 s) (r+l)(r+1+1) M, L r I V s (r+j+ D,t(sj Pt (21)cosipi. (E.3) Equation (E.2): M+1 t Iv (p=L E Et t (s)Q Q )P (tcosl 1 (E.4) A comparison of (E. 3) and (E. 4) gives M+1l M+1, I Mi P -2 r( r-) M, t( I t-1) Mt Vt 5"-' L rt= r(r+sl)-t(t+l)L2r- D Q 2t-1 "rt- 1 + (t+l)(t+ +1) DM I ) (r+l)(r++1) DM, Qr(s'2t+3 r,t+l~s 2r+3 r+l, t' ISS Q t s (E.5) Using (3.24a), the above expression can be written M+1, I Q EtM+l )=- E' (E) (E.6) tot s r=0 Qt() rt s t s which is equation (3.24b). 79

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX F THE SURFACE INTEGRAL FOR THE NEUMANN PROBLEM The integral to be evaluated in this appendix is that of equation (4.6) -ikc? +1 2r ui(p ) I(p1) = ( 1) e s d7 d0e+ikcr G(p, ps). (F. 1) 1/1 0 1 S By equation (3.3) ~s 1- -~~2 au (Ps) ~~', a- = -ik coso0 r+sineO - cos- u(p) r - -2 -ikc[cos n+sin -ri2 cos =-ikc cos 0 + sin s cosJ e s Substituting the above expression together with the appropriate part (Q < 1) of (2.22) in (F. 1) we have -ikcc cos ( n1l) n ) 4 n ( j n) n= m=0 d0 cos0 nr+ sin0 cos 1cos0 e n0. 8ns5 n X - c osm- e). (F.2)!.......________________________ 880 8

- THE UNIVERSITY OF MICHIGAN 7133-5-T Using (B.4) for the Wronskian and the expansion (B. 6) with."2 1/ 2 kp = -kc sinO - 1-, we can write for equation (F. 2) Os ik~ cOD 00 n m/ e ikc (p ) ikc ( E i) (2n+l) (n-m)' Pm( n 1 n s * cs +1 -jc d Pn rn1 ) - -ick(~scos 0 +l)r7 l ^ n -K \^ o0s.ojdr1P (n))Je kcsinO8 2 - cos -111 *\ dcos r+sinO i-r- cos 0 cos cosm0. (F.3) To perform the integration with respect to 0 we employ the identify cos cos W0= - cos(+ 1)0+cos( - 1) the result being ikc C Qnm ee ISp) - (-i () (n+m)! Pn(nl) m cosm + ikc (sP, c m(-i)m+ (2n+ 1) (n- m) q Cos -zn~l) Pn (r} 1 )'n s (p) ="n=O m=0 (n+ m) n s dr e 2m PQ(si)J i n n 20 1r sin e oo n m ikcs soi n(im+l( 2nl(n-m),' Q + 0- < / 2s' (2n+m1) -', P (r,) nm -- cois m 2 i(?-l Wfeo" Qn+m)- n(1 Js)'. dn sin 1 +1I____ _____________ 81 ________________ (cont'd) 81

THE UNIVERSITY OF MICHIGAN 7133-5-T 2 1 )c(coD l)n m)- m( - )' 0 sine (n-m) m Q 1 ikce s o iZ p m+1 (2kcs + ) ( 1 m ) To perform the integration with respect to rn we employ the following recurrence relations (Magnus and Oberhettinger, 1949, p. 62) 2 +lP(n) =(n-m+l)Pm (n)+(n+m)!P (n), mn, n=O,l,... (F.5) leads to ikc c-o i; P (r)J sin m m + (n-m+l) m m(n) 2n+lr - cos m n m (-1 ), n-1 m ~ l'Q m(F.4) (2n+ 1 /,l Pn-, =(n -m )P (nl) +(n + m)Pm (n7), m <, n, n = Os, (F 5) n n cosn +1 Sde) 1 - r P =P PMd (r), m e, nc = 0i i, 2) (F. 6) n- n-.n+ m,<,n, n=0,1,... (F.7) Substitution of these expressions in (F.4) and a simple rearrangement of the terms leads to ikcg OD n Q m \'s~s, = ike /, V Vs.m (n-)m)'. m < n, -1 (51 nn- (n-m+l). m. n+l (n+ m)' n+1'1 Qm +r (rl)s= (n m 1+- "' —5 +1+l -ikc(2n os 0l) _ pm-1, le."' I (cont'd) 82

THE UNIVERSITY OF MICHIGAN 7133-5-T sin 0 oo n-1 m + ikc s o ) _(i)m+1 -m+l +1 1 (n-m-)'. jm 1)m1 (n-m+l) pm Qn+l() 2+- -=O m1 l (n+m+l)' n-+l( Qm, pm l~r)Jm 1 (kC sin0O om +) _______ n-i1 ss m lsint0 b i1 m+1 sikc( s ikc(n-m-l)' pmm Qn- ( l o- p ml( s Qn-(1s -1 41^ Qii r+1 -ikcgcosO ~1r (n-m+ )'m n+1 1 s o ~ n cosO d+1)e (n+ m- 1)' mn+ 1JC m s m k dso Qn+l(Qs) -1 M- 0 o n n Io 0 0 To perform the integration with respect to r- we employ (B. 9) in Appendix B, the result being ikc O iD I (p) =i cos70,1 (-i)n 2 + Jl(z)P ( 3) 1 m-80 z n+/2 n3 (n_-m)_ _ m _ n-_ 1 (n-m+l)_ _, _ _ n+l( ) 1 * 1 n+m-l)..!n-1 1 Qm, (n +m)! mn+1- 1 n 5 +- sin 2 Uijno ~1 +i n=l m=0 (cont'd) 83

THE UNIVERSITY OF MICHIGAN 7133-5-T Qm mIn (n-m+l). m Qn+(l) (n-m-1)'. l) m Qn I) n+m+')' Pn+) ), (n+m —1) n n)'- cosm Qn+1 (s'n- 1 s oo n+1 +- sinT8 s (- lz)pn 2 s__1 n=0 m=O Jn+/(z) 1m n-m+l): m Qn-(1) (n-m+1)' m Qn+(1)....7 - - (n.- cos n+m- 1)' Pn- -1 1 Q (n+m- -1) n+l m ) Q, (9 n-l V n+ s(F. 8) where z = kc(s ~cosO ) (F.9) S 0 6 cos t+1 f= _~+cose (F. 10) S 0 Equation (F. 8) is now put in the following form ikcI s ikc cos n+1 / Jm (n-m+1)! e p-e IS(pl )y %;os (_)i)n i 3)2 ~ e1 2 0Lo mn/_ m z nn+/2 n+ (n+m)! Qm(1 P(i) m cos m + 77m(-i) n-1 (z)Pml( (n+- m)! P nm() yz mn-2 n-i (n+m-1)I n'1(m n s x cos m0 (cont'd) --------------. 84

, THE UNIVERSITY OF MICHIGAN 7133-5-T +D n-2 *n Pn 1 m cosm n S O- - n m+ P' n(1) m' cosC m <<(S )' n z Jn+-3/2(ZJPn+lG (n+m)t n m o - Ss? (-i) n+ 1<(- P a (O ( m)! p:(< ) -cos m } n s +2 sine s 2 n_1 ) n+/. 3 ) (m-l (n-m1+2) n+2 z J n+3/2(z)n+l (n+m)l m - - - -- -n 5' &Q ()(F Mrn() fn 1 m' cosm-< n Sm l1 ml zn- ()P 1 () (n - m)- Pnm() COS n mOs), In, 2n-1 (n+m-2)! n In Substitution of the relations (Magnus and Oberhettinger, 1949, p. 16) 2n+ 1 d Jn-ii(z) - Jn+(z)+ ) z J z (F.n12) jn+3/2(z) 2n +1 d(F.13) 5n+~2z2 Jn+i/2(z) Jn+2(Z)F.13) in (F. 11) and a regrouping of the terms leads to....................85.........

THE UNIVERSITY OF MICHIGAN 7133-5-T ikcS 00 n sCs s kc o n 2n+l (n-m)! e I(p)=- cos0 z n+=O(m)'. Jn+1 /(z) + m)P (P)- (n - m +)Pn ( ) cosm +2a -i) (n + (1) n (' m m n s O n m n + Is1 - (-i)n(2n+l)pn P)J pZ) (n + m)' 0' (n (P1) -' cosm * n=O m=O mm(I) j ativewe have kc sn s _I' ( _i) _ n (2n+l) j (n- m)8 2sin o _1 2z Jn+/1 (n+m)' ln 2 r, 1P;m(ni)g m cos m~ Q m (9) n- 1 (/)+ mn+l (/3 P 1 n ()Qos), n s \7.(i (2n+l) (n+m)' 2n+ m m __ n S 00 n.Cosa m1 nl mZ(i)n (2n+ 1) T 1Z(n-m+2)". m-1,^ n-: -i 2 +z)L P () + 1 — 2A Jn+1n/2 (n+ m)m n+l (n - m)! 1 Q, I ~) + P PMn cos ( (n+m-2)' n-1 n'm n s This expression can be simplified using the properties of the Legendre functions mentioned above. After simplifying and collecting terms in Jn+/2(z) and its derivative, we have........... —--- 86

THE UNIVERSITY OF MICHIGAN 7133-5-T e s I(pl) = kc ( )n (2n+ 1) (n-m)! J' ke I ()= - 2 m (n+ m) n+ ps? Jl-32 Q| QP) -|. cosO + s sinOj P1 (3)Pn (r1) nm cosm3, o Ji2! ^ n 1 Q_ (s) J ss (_)n s kc n+in (nr) P ()n 2n+ 1 (n - m)! Ln=: m fi iEM(i.iZ) co (Fn)Pm1() B 2 utn ) rlzz 2z (n+m)by fl 12 L M s 2 m -cos0 (n-m+l)Pn+l (f)- sine - P/ (P3) n+1 +sin0 P_ P 3m+ (1)-(n+m)(n -m+)sine 0 0 (pF|P ~|1 8- 1 nPi m- n1 cos mOc. (F.15 n (1 " c Q s But by (F. 10) fcosO + s sine = 1. (F. 16 0 0 s Moreover, from the definition of P and the recurrence relations (Magnus and Oberhettinger, 1949, p. 62) Pn+l= - 1 [n-m)3n P(3) - (n + m)PFmi], (F.17 Pn yi-ri n-1 (8 -------------------- 87

THE UNIVERSITY OF MICHIGAN 7133-5-T (n+m) 1-_3 n n+l we have that m'3 m cose (n+m)PM (/3)-cos0 (n-m+l)Pn+ ()-sin0O (3) +sin9?~~~?~~~s + sins0 P) ()P 7-(n+m)(n-m+l)sinO - p1() V? -l s-1 +sine + 0 2) t+( tcose ) S 2 o [(n-m+l)Pn+l(m)' sPn(P)- (n+m)pmn (p) (F.19) * 2- 1 n1 n n-1 S Substituting (F. 16) and (F. 19) in (F.15) we have e (- ( - )n(2n + (n- m)' nz)' PM () IS m/p1) n = 2 n Pni( 1) Cos mM 2 88 —'.ko, y (_i)n 2n+l (n-r.nl (- m)o a2 -amIn 2z (n+m)+ /2 5/1 S In [(n-m+l)Pi' (P)m( P -n+m)P In ncn<rn C1 s M n S (F.20) L —----------------- 8 ------------------

THE UNIVERSITY OF MICHIGAN 7133-5-T According to equation (B. 12), 2 r^,rz ^ O (-1)1(21+n+ 1 ) +n 2i^7r N -TZ 21 +n -n/1(z)'= - 22 I rIF(n+. + ) Substituting these expressions in (F. 20) and using the formula OD \ O [n/ 2] 2ny, n n a zn ( nz- =n- 2 k z n=O /n=0 n=0 k=0 we obtain e I(P ) =-E z):-e ([2n-2)+ l _((n- 2 1+ m)!o n2 n n- 1 Q( p ) I 1- n= i ml=0 2_ itr(n 2Bn28 m (n -2+m) n-2in-2i1 m o~o n-22~ n-2 kc 7 z (n- 21- m) -X" 2z E.__J,~ n - 2(-i)1n+ 3 n -2 (n -21+m)' n=OJ0 m=0 2 1!I'(n-l+ ) (+1)(s +cose) r m 2 Lnml - 2- m+ 1)Pn_ 2+() ~2_ 1 n-21 s5~~~~n 89

THE UNIVERSITY OF MICHIGAN 7133-5-T A simple inspection cf this expression reveals that it is zero for n= 0. We can therefore write -ikc ~00 (P1) = e (-ikc) M (p1) (F.21) M=1 where M I (p) \ ( cosO )M-1 \7' 21+1 i-m) (t-+)(s tcos ) r (2M+1)Pn (1)+ 2 (im+l)P () Pl {2 s - ((+ m)P. p ()3) m cosmO1, M+l even, (F.22a) - _j1 Q'' S IM(P1)= 0, M+i odd, (F.22b) where above we have substituted (F. 9) for z and we have rearranged the series. Equation (F. 22a) can be further simplified by taking into consideration (F. 10) for 3 and the relation (F. 5). In this way we can write M I IM(Pl) = S T~y (F.23) IM~p1) =_E. jA/l (~ )P~ (rl)Q Q )cosMO (F.23) l=0 m=0 s where ---—. ——.. --------—.. 90

THE UNIVERSITY OF MICHIGAN 7133-5-T +o + Q M-1 Mm s o0 ____ _1 A Mm( ) = - cM+1 (2 +1) m)! (M + 3 Mm 3 2 l~~~~~-" —+22 2 2 s L x tes^ Cos o 2 Ls 0 C os s 91cose +I s/ s cos o -(^m)(~c~os )Px- S e+cosj o, M+l even, (F.24) Am' (is)= ~, M+1 odd (F.25)......... 91 -,.

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX G THE INTEGRAL deKt (, 1' ~s)Q (~) s According to the results of Appendix C, d Q Kt Q: d' Qt MQ' dr QQ( )QrQ) rdKr(+,t,ts)Q () Q )t() dPt()Q ()+Pt() dQt()Q 1) = rdrPt()t~)()l) +l-Q Qtl )tQA s S I ), m + t (l) d Qt (t)Q () Qt s9 Q t tst -l,)QJ 2)) F I - r(r+ 1)- t(t+l) 1 L t in 2 ( I'Pt(7l 2 -(2 - 1) Pt ) ) t ( + 1) - L sr~ r~ tsiJ r(r+ 1) - t(t+ 1) 1 s Q ss P) ts Q QI()QI(Q)Q - ~ 2_ (1- 92 1) - r 1 PI t 71 1 I _ )P__ r(r+ 1) - t(t+ 1) (tst t jt (cont (d)

THE UNIVERSITY OF MICHIGAN 7133-5-T =_ _ (-1_ ) (t+l) Q g ()-1) (t+l)! r(r+1)-t(t+l) (t-")!'r 1)+ r(r+1)-t(t+1) (t-l)! Qr(9s)'l Q QI t ti ts = ( r(r+l)-t(t+l) (t-)! t' Q(1) Qr(l 1; <t (G.1) Also K0t(,"1,gs)Qr()dg = 0 if I >t (G.2) s3 --------— 93

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX H REDEFINITION OF Qn n The original definition of Q (,) as given by (2.26) is r/a (-1)m r(n+m+l)r'(1/2) (U2- 1)2 n+m+2 n+m+1 3 n - 2n+1 r(n+3|) n+m+l 2 1 2' 22; 2 n l ff(n+ 2) /2 /a u|> 1, larg(P-Dl 1)<. (H.1) Hobson (1953, pp 233-234) has shown that if z = ye+ {j-1, (H.2) then the function 1 2 3 U (/a_-i) (1 U( Z++l 2Fl(+m, n+m+l; n+3; 2) z z zl>l1, larg(u-1)1 <r (H.3) satisfies the associated Legendre equation. Using this expression we can define a new function Qm(Ia) which holds for I z > 1 or equivalently I|u| > 0, which is identical to Qnm(,) give&by (H. 1) in their common domain of definition, I\ I > 1. To do this it is sufficient to compare (H. 1) and (H. 3) for large values of I|| I. The resulting relation between the two functions is Qm() = (_l)m2m (n+m+l1)(1/2) ( H.4) n~or"~(n+) ~~9or49 ------------------------ 94.

THE UNIVERSITY OF MICHIGAN 7133-5-T 1 2 2m Qm() = (l)m2m r(n+m+l)r(l/2) (1. -1) F +m, + m+ln+3 1 Q 1- m 2m- -.(3 n+m+1 F1 2 n+ r(n+ 3) zz z z > 1, |arg(- 1)< 7r. (H.5) Letting'i = i,, >0, we have z = i(+ 2+1 ), and Q2+ 1)m/2 Qm(i)= (-2) r(n+m-l)r(l/2) (+ i1 r(n+-) m+1 X 2F1 + m, n+m+; n+ )2 2 1 2' > 0.(H. 6) Using the relation (Magnus and Oberhettinger, 1949, p. 8) c-a-b 2F (a, b;c;z) = (1 - z 2F1(c - a, c- b;c;z) we can write 3 1 (1 \-2m 2( 1 3 1 2F1+m, n+m+l;n+; —) = ( ) Fn-m+l, -m;n+; - 2 12 2 2 2 1 2 2 2 z z z (H.7) Letting z = i+ ) 2+) and substituting in (H. 6) we obtain m()= (-2)m r(n+m+l)r(l/2) ( 2+1)m/2 n(i) ".n+l - 3 n 1 r(n+ )+ 2 1 2m2 r7+2,3m+1 2F 1 (-m+l,-m;n+ 2;- ) F0 (H.8) 6 2 3 95 ------— 95

THE UNIVERSITY OF MICHIGAN 7133-5-T 1 3 which holds at 0 =0 also since (n-m+,l)+( —m)-(n+) = -2m 0 for 2 2 m = 0, 1, 2,... (Magnus and Oberhettinger, 1949, p. 7). Equation (H. 8) can be rewritten to read 1 mr. n _ (-1) r(n+m+l)r(l/2) (g +1) Qn = n+1 (m 3 2F1 (-m+l, -m;n+2;-)'2, e>~0. 2 i2 2 22/~Y (H.9).............. 96

THE UNIVERSITY OF MICHIGAN 7133-5-T APPENDIX I THE FAR FIELD FOR THE DISC In this appendix we give the first six terms in the far field expansion for the disc for both Dirichlet and Neumann boundary conditions with the incident wave at normal incidence. The far field in both cases is given by (5.25). This expansion involves the M coefficients A (0) given by (5. 23) and (5.24) which contain the Legendre functions of the second kind and their first derivatives with respect to i evaluated at e = 0. s s Their values are determined as follows. From equation (5.5), 1 r(n+l)r(1/2) 1 3 1 n i n+1'(n+|) ( iN 121 2 2 7F (+ 1 r(n+-) Now, Letting e = 0, we have () = n+ +l-1) (2) QQ(0) = n+ (1.4) ['(n+ F) (2i) Now, * See, for example, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series No. 55, p. 557 (June 1964)... 97

THE UNIVERSITY OF MICHIGAN 7133-5-T Or, we can write Q (0). (2n)'. n = 0,1,... (I.5) 2 (2i) nn+l' Turning now to the derivative of Q, denote by Q(0) the derivative of Q (it) with n n n respect to ig evaluated at = 0. From (I. 1) Q1(0) 1 r(n+1)r(/2) { +1) F (+1 1; 3 n A n)L21... 2 2 2n+13 2F (n+2, 2;n+ 51 (I.7) 2 2 2 Employing (1.3), (1.4) and the relation,F (a,b;a-b;-1) =2a U (b-1) r(a-b+2) [ i i 3 1 r( L a)r(r+ a - b) (+a)r(n+2a-b) (158 2 2 n 21,;n+;-(I. 8) which can be found in the same reference and page as (I.3), equation (I. 7) becomes Q'(0) = i(n+l)Q (0) - n -I-+ n+ 1 2. (1.9) na n n 2 [F( +1r(D a) r( )r(n +a- 2)J Finally, with the help of (1. 5) and (1. 6), we obtain for (.9), - _ - 98

THE UNIVERSITY OF MICHIGAN 7133-5-T Q' (0)= r(2n) 2 n=0, 1,... (I.10) (-4)n [r(n )] 7r(2n+ 1) 2n+l = i2n+1' n = 0,1,... (I. 11) (2i) n!nt We now employ equation (5.26) and write: For the Dirichlet Case sf 2 Ul (p1)= ( p o(r1) sf 4 u2 (P1)- 9r (r3 o(r1) u(p1) = -- P2(r;)-(-Jr) 7T 4P) = 525ir P4(1)+(3+ ) +21( - 3 +~) o 10 sf 2 A 8 + sf 4 (16 2 1 525r2 1 97r4 5677r2 1 P (77 + + P ( +. ) u4 (P 52 4 3105? 2 93 7r (r1) (7r 9r4 20257r 1 Substituting these results in (5.25) we obtain 99

THE UNIVERSITY OF MICHIGAN 7133-5-T ikc1 uf = e P (r )+ ikc P (rll)+k 20 - 2() 1 L. 7r 0 2( +Q_ 4- p ( + ik3c3 L P2(1)+ ( 4+ 2 Po +k4c4 If 049l4+(e83+l;)22 )+( + 93 - 754r)Po(*'l 9- 97r 7r 97r - 5 F 32 32 4\ 92 + ik c 5 52_" 2P(r1+,2 5 1 __ +k 525c7 2 4 1 ( r4 5672) P2 1 525i9 567r (64 80 + 508 p P() + O(k6c6) (I. 12) 6- 4 0 1 \6 9f 2025 ~ J For the Neumann Case Sf uf (P) = 0 sf u1 (p) = 0 Sf 2 U2 (P1) = 37r P3(Tl) sf u3 (P1)= 0 sf 2 8 sf 4 u5 (P1)= - 2P( 277 2 and.............. 100

THE UNIVERSITY OF MICHIGAN 7133-5-T ikc - sf, = e Jk2 2 2 k 4 42 8 u lsf(p: C 32 Pl(r 1)-k CL75 3(r1)+ 77 11 - ik5c 4 P1(r1l)+O(k6c6 (. 13) 27 7r The results given by (I. 12) and (I, 13) are in complete agreement with those obtained by Senior (1960). -.. —................-..-..- 101

THE UNIVERSITY OF MICHIGAN 7133-5-T REFERENCES Ar, E. and R. E. Kleinman (1966) "The Exterior Neumann Problem for the ThreeDimensional Helmholtz Equation", Arch. Rational Mech. Anal. 23, 218-236. Darling, D. A. and T. B. A. Senior (1965) "Low-Frequency Expansions for Scattering by Separable and Nonseparable Bodies", J. Acoust. Soc. Amer. 37, 228-234. Hobson, E.W. (1953) The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Co., New York. Kellogg, O.D. (1929) Foundations of Potential Theory, Springer, Berlin. Kleinman, R. E. (1965) "The Dirichlet Problem for the Helmholtz Equation", Arch. Rational Mech. Anal. 18, 205-229. Magnus, W. and F. Oberhettinger (1949) Formulas and Theorems for the Special Functions of Mathematical Physics, Chelsea Publishing Co., New York. Morse, P.M. and H. Feshbach (1953) Methods of Theoretical Physics, McGraw-Hill Book Co., Inc., New York. Senior, T.B.A. (1960) "Scalar Diffraction by a Prolate Spheroid at Low Frequencies", Can. J. Phys. 38, 1632-1641. Senior, T.B.A. (1961) "The Convergence of Low Frequency Expansions in Scalar Scattering by Spheroids", The University of Michigan Radiation Laboratory Report No. 3648-4-T. Senior, T.B.A. (1966) "The Scattering from Acoustically Hard and Soft Prolate Spheroids for Axial Incidence", Can. J. Phys. 44, 655-667. Sleator, F.B. (1964) "Diffraction and Scattering by Regular Bodies III: The Prolate Spheroid", The University of Michigan Radiation Laboratory Report No. 3648-6-T. Spence, R. D. and S. Granger (1951) "The Scattering of Sound from a Prolate Spheroid", J. Acoust. Soc.Amer. 23, 701-706. Whittaker, E.T. and G.N. Watson (1952) A Course of Modern Analysis, Cambridge University Press, Cambridge. Wilcox, C. H. (1956) "A Generalization of Theorems of Rellich and Atkinson", Proc. Amer. Math. Soc. 7, 271-276. 102

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA- R&D (Security clatsification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author) 3 a. REPORT SECURITY C LASSIFICATION The University of Michigan UNCLASSIFIED Dept. of Electrical Engineering, Radiation Laboratory 2b. GROUP Ann Arbor, Michigan 48108 3. REPORT TITLE LOW FREQUENCY SCATTERING BY SPHEROIDS AND DISCS 4. DESCRIPTIVE NOTES (Type of report and inclusive dtees) Interim Scientific Report 5. AUTHOR(S) (Laet namte, first name. initial) Asvestas, John S. Kleinman, Ralph E. 6. REPORT QATF'. 7a. TOTAL NO. OF PAGES 7b. No. OF REFS June 1967 102 14 8a. CONTRACT OR GRANT NO. *^. ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-4328 7133-5-T b. PROJECT NO. Task, Work Unit Nos. 5635 - 02 56350201 ____ Scientific Report No. 4_ c. b. OTHER R PORT NQS) (Any other nabere that may be alned DoD Element 61445014 this repot AFCRL- 67- 0360 d. Dod Subelement 681305AFCRL-6 10. AVAIL ABILITY/.IMITA'I!ON NoT!CES. Distribution of this document is unlimited. It may be released to the Clearinghouse, Department of Commerce, for sale to the general public. 11. SUPPIEMIENTARY NOTES. 12.1PONSeRING MIlITARY ACTIVtrv Air Force Cambridge Research Laboratories L. G. Hanscom Field __i..____| _Bedford, Massachusetts 01730 13. ABSTRACT The problem of scattering of a scalar plane wave by a spheroid of revolution is solved for either Dirichlet or Neumann boundary conditions, arbitrary major to minor axis ratio, and arbitrary incident direction. The solution is obtained using an iterative method applied to solutions of the corresponding potential problem and is expressed as a series of products of Legendre and trigonometric functions, and ascending powers of wave number. A recursion relation for the coefficients in this series is derived. These results are employed to calculate scattering cross sections for 2:1, 5:1 and 10:1 prolate spheroids. DD 1JAN64 1473 UNCLASSIFIED Security Classification

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UNIVERSITY OF MICHIGAN 3 9015 02493 844411111 II 3 9015 02493 8444