THE UNIVERSITY OF MICHIGAN AFCRC TN-57-586 ASTIP Document No0 AD 133631 STUDIES IN RADAR CROSS SECTIONS XXIII A VARIATIONAL SOLUTION TO THE PROBLEM OF SCALAR SCATTERING BY A PROLATE SPHEROID!REDERICK Bo SLEATOR A-. disse'rtati6n in- th:'e De.par.tment of- Mathematics New York Ufi-vers.t^ ia Jrit'ted. to.the faculty of the Graduate.SchQool. Qof;Arts:.and Science in partial fulfi3lment.of.- th'e reqtuirements for the degree of Doctcori of E'Philosophy Degree conferred in February 1957 Scientific Report No0 1 UNIVERSITY OF MICHIGAN REPORT Noo 2591-1-T CONTRACT No AF 19(604)-1949 March 1957 The research reported in this document has been made possible through the support and sponsorship extended by the AFCRC under contract AF 19(604)-1949o It is published for technical information only and does not necessarily represent recommendations or conclusions of the sponsoring agencyo

THE UNIVERSITY OF MI CHIGAN _ 2591-1-T STUDIES IN RADAR CROSS SECTIONS I Scattering by a Prolate Spheroid, by Fo V. Schultz (UMM-42, March 1950), W-33(038)-ac14222, UNCLASSIFIEDo II The Zeros of the Associated Legendre Functions pmn' (/L) of NonIntegral Degree, by Ko Mo Siegel Do M. BrownM Ho Eo Hunter, Ho Ao Alperin, and Co W. Quillen (UMM-82, April 1951), W-33(038) -ac-14222, UNCLASSIFIEDo III Scattering by a Cone, by K. M0 Siegel and Ho Ao Alperin (UMM87 January 1952), AF-30 (602)-9, UNCLASSIFIEDO IV Comparison Between Theory and Experiment of the Cross Section of a Cones by Ko M0 Siegel, Ho A. Alpering Jo Wo Crispin, Jro, H. Eo Hunter, R. Eo Kleinman, W0 C. Orthwein, and C. EB Schensted (UMM-92, February 1953), AF-30(602)-9, UNCLASSIFIEDO V An Examination of Bistatic Early Warning Radars, by Ko Mo Siegel, (UMM-98, August 1952) W-33(038)-ac-14222, SECRETo VI Cross Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies, by Ro Ro Bonkowskiq Co Ro Lubitzo and Co E. Schensted (UM-106, October 1953), AF-30(602)-99 SECRET UNCLASSIFIED when Appendix is removed. VII Summary of Radar Cross Section Studies Under Project Wizard, by Ko Mo Siegel, Jo Wo Crispin, Jro, and Ro E. Kleimnan (UMM-1089 November 1952)9 W-33(038)-ac-14222, SECRETo VIII Theoretical Cross Section as a Function of Separation Angle Between Transmitter and Receiver at Small Wavelengths, by Ko M. Siegel, Ho Ao Alperin, Ro R. Bonkowski, J. Wo Crispin, Jro, Ao Lo Maffettq Co E. Schensted, and I. Vo Schensted (UMM-1159 October 1953), W-33(038)-ac-142229 UNCLASSIFIEDo IX Electromagnetic Scattering by an Oblate Spheroids by Lo M. Rauch (UMM-116, October 1953), AF-30(602)~9, UNCLASSIFIEDO X Scattering -of Electromagnetic Waves by Spheres, by Ho Weil, Mo L0 Barasch, and T. Ao Kaplan (225520-T, July 1956), AF-30(602)-1070, UNCLASSIFIEDo XI The Numerical Determination of the Radar Cross Section of a Prolate Spheroid, by K Mol Siegel, Bo Ho Gere, Io Marx, and Fo Bo Sleator (UUMM-126, December 1953), AF-30(602)-9, UNCLASSIFIEDo ----- ~ ~ii

THE UNIVERSITY OF MICHIGAN 2591-1-T XII Summary of Radar Cross Section Studies Under Project MIRO, by Ko M., Siegel9 Mo Eo Anderson, Ro Ro Bonkowskil9 and Wo Co Orthwein (UMM-127, December 1953), AF-30(602)-99 SECRETo XIII Description of a Dynamic Measurement Program by Ko MO Siegel and Jo M. Wolf (UMM-1289 May 1954)9 W-33(O38)-ac-142229 CONFIDENTIALO XIV Radar Cross Section of a Ballistic Missile? by Ko Mo Siegel, M. Lo Barasch9 Jo We Crispin9 Jro 9 Wo Co Orthwein9 Io Vo Schensted, and H. Well (UMM-1349 Septo 1954) W-33(038)-ac-142229 SECRETo XV Radar Cross Sections of B-s7 and B-52 Aircraft9 by Co Eo Schensted9 Jo Wo Crispins Jro9 and Ko Mo Siegel (2260-1-T9 August 1954)9 AF=33 (616)-25319 CONFIDENTIALo XVI Microwave Reflection Characteristics of Buildings9 by H0 Weil1 Ro Ro Bonkowskiq T. Ao Kaplan9 and Mo Leichter (2255-12-Tg May 1955)9 AF-30(602)-1070 SECP.So XVII Complete Scattering Matrices and Circular Polarization Cross Sections for the B-47 Aircraft at S-band9 by Ao Lo Maffett9 Mo Lo Barasch, Wo Eo Burdick9 Ro Fo Goodrich Wo Co Orthweing Co Eo Schenstedq and Ko Mo Siegel (2260-6-T9 June 1955)9 AF-33(616)-25319 CONFIDENTIAL XVIII Airborne Passive Measures and Countermeasures9 by Ko Mo Siegel9 Mo Lo Barasch9 Jo Wo Crispinq Jro9 Ro Fo Goodrich9 Ao Ho Haolpin Ao Lo Maffett Wo Co Orthweins Co Eo Schensted9 and Co Jo Titus (2260-29-F9 January 1956)9 AF-33(616)-25319 SECRETo XIX Radar Cross Section of a Ballistic Missile - II by Ko Mo Siegel9 Mo Lo Barasch9 Ho Brysk9 Jo Wo Crispins Jro9 To Bo Curtz9 and To Ao Kaplan (2428-3-T January 1956) AF-04(645)-339 SECRETo XX Radar Cross Section of Aircraft and Missiles9 by Ko M0 Siegel. Wo Eo Burdick9 Jo Wo CrispinS Jro0 and So Chapman (WR-31l-J 1 March 1956) SECRETo XXI Radar Cross Section of a Balli c ist Msile - III by Ko Mo Siegel9 Ho Brysk9 Jo Wo Crispin? Jro9 and Ro Eo Kleinman (2428-19T9 October 1956) AF-04(645)-339 SECRET. XXII Elementary Slot Radiators9 R0 Fo Goodrich Ao Lo Maffett No Reitlinger, Co Eo Schensted9 and Ko Mo Siegel9 (2472-13-T9 November 1956)9 AF 33(038)-28634; HAC-PO L-265165-F319 UNCLASSIFIEDo XXIII A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroids by F0 Bo Sleator (2591-1-T March 1957)9 AF 19 (604)-1949 UNCLASSIFIEDo iii

THE UNIVERSITY OF MICHIGAN 2591-1-T TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Formulation of the Variational Problem 4 3 Evaluation of the Integrals C/z-z/ 7 4 Evaluation of the Integrals Bl< 24 5 Determination of the Scattering Cross Section 26 6 Verification of Results 29 7 Convergence of the Solution 40 8 nNumerical Results 51 9 Conclusions 55 Appendixo Power Series Solution 58 Bibliography 66 I —------------------ --------------------

THE UNIVERSITY OF MICHIGAN 2591-1-T PREFACE This paper is the twenty-third in a series growing out of studies of radar cross-sections at the Engineering Research Institute of The University of Michigano The primary aims of this program are| 10 To show that radar cross-sections can be determined analytic callyo 20 Ao To determine means for computing the radiation patterns from antennas by approximate techniques which determine the pattern to the accuracy required in military problems but which do not require the unique determination of exact solutionso Bo To determine means for computing the radar cross-sections of various objects of military interesto (Since 2A and 2B are inter-related by the reciprocity theorem it is necessary to solve only one of these problems) 3o To demonstrate that these theoretical cross-sections and theoretically determined radiation patterns are in agreement with experimentally determined oneso Intermediate objectives ares lo Ao To compute the exact theoretical cross-sections of various simple bodies by solution of the approximate boundaryvalue problems arising from electromagnetic theoryo ------- ----— v, _______

THE UNIVERSITY OF MICHIGAN 2591-1-T Bo Compute the exact radiation patterns from infinitesimal slots on the surface of simple shapes by the solution of appropriate boundary-value problems arising from electromagnetic theory. (Since 1A and 1B are inter-related by the reciprocity theorem it is necessary to solve only one of these problems) 2o To examine the various approximations possible in this problem and to determine the limits of their validity and utilityo 30 To find means of combining the simple-body solutions in order to determine the cross-sections of composite bodieso 4o To tabulate various formulas and functions necessary to enable such computations to be done quickly for arbitrary objectso 5o To collects summarize, and evaluate existing experimental data. Titles of the papers already published or presently in process of publication are listed on the preceding pageo The major portion of the effort in this report was performed for the Air Force Cambridge Research Center under Air Force Contract AF 19(604)-1949 Ko Mo Siegel.____________________ii vi ______

THE UNIVERSITY OF MICHIGAN 2591-1-T CHAPTER I INTRODUCTION The natural limitation on the variety of problems in scattering and transmission theory which can be handled conveniently by the technique of separation of variables has led to the development of methods of essentially different charactero Among these are variational methods, of which perhaps the most widely used is that developed by Schwinger and employed with considerable success by him and by numerous others in the solution of various problems in diffraction of sound and electromagnetic waves as well as quantum scatteringo Practically all the literature which has appeared so far in this field has concerned itself with problems in one and two dimensions9 primarily because of the difficulty in performing the required integrations0 Regarding the electromagnetic problem9 Mentzer remarks in his recent book on scattering of radio waveso "The formulation with three dimensional| scatterersq such as spheres9 leads to surface integrals which usually are completely unmanageable, the integration processes with simpler geometries are9 at best9 very difficulto" One exception to this is a solution to the problem of the loop obtained by Kouyoumjiano However9 similar statements are often found in the literature9 even for the scalar caseo The present paper represents another attempted break-through,into MMentzer' JO Rog "Scattering and Diffraction of Radio Waves 9" Pergamon Press, Ltdos po 45 (1955)o __________________ 1,____________

THE UNIVERSITY OF MICHIGAN 2591-1-T into the tunmanageable' third dimensions It is true that the spheroidal scatterer affords perhaps the simplest geometry next to the sphere of any three-dimensional body, and the setup possesses cylindrical symmetrt. nonetheless the problem is essentially three-dimensional and this result in addition to that of Kouyoumjian may help to dispel some of the pessimism noted aboveo The solution obtained may be of little value in itself, since the prolate spheroid has already been dealt with quite extensively by the separation technique. However, it does indicate that the integrations involved in some three-dimensional problems may be more or less manageable, and it may also shed a little more light on the value of the variational method in general, From a mathematical standpoint, some of the procedures used here, just as in many of the papers of Schwinger and others, are not rigorous No justification is presented for the numerous changes in order of integration, and some of the integrals -which appear are at least formally 2 divergento However, recent work of Bouwkamp has indicated that in some similar problems this formal divergence is only formals and can be eliminated without affecting the results at allo These considerations, together with the fact that the present solution agrees with known results exactly in the limiting cases of zero eccentricity and very large wavelength and extremely well for the case of a thin spheroid in the resonance regions seem to indicate that an attempt to introduce ZBouwkamp, Co Jo, "Diffraction Theory —A Critique of Some Recent Developmentst Reso Repo EM-50, No Yo UJ Insto of the Matho Sciences (April 1953)o ~_____________________ ii2 ______r____

THE UNIVERSITY OF MICHIGAN 2591-1-T mathematical rigor into this development would not be worthwhile at presento The principal physical quantity obtained in the present analysis is the nose-on back-scattering cross section of the rigid spheroido It is easy to see how the results and techniques could be extended to inelude certain additional information. Further analytical work on the forms already obtained might also be profitable under some circumstanceso Various possibilities are discussed in more detail in the final section of the papero The author wishes to express the deepest appreciation to certain colleagues9 in particular Messrso Ko Mo Siege19 Co Eo Schensted9 and Ao Ho Halpins for many illuminating and invaluable discussions of the problemo Credit is also due Mr Ho Eo Hunter of the Willow Run Laboratories for his meticulous work in computing the numerical results.contained hereo Finallys the utmost gratitude is accorded Profo Wilhelm Magnus of New York University for his infallibly prompt and considered advice and assistance, without which the work could not have been completed in the appointed timeo ----------------— 3 — __________ __ _

THE UNIVERSITY OF MICHIGAN 2591-1-T CHAPTER 2 FORMJLATION OF THE VARIATIONAL PROBLEM We assume a rigid prolate spheroid with center at the origin, major axis of length 2a in the Z-axis, minor axis of length 2b, and a plane sound wave approaching in the negative Z direction, The following integral equation for the velocity potential at any point exterior to the scatterer can be establishedo3 (S) e eik" / (St) -t G(SSt)da' (1) where S is the field point in space St is the point on the spheroid V(S) is the velocity potential at S ike G(S,St) is the Green's function of free spaces e ( p distance from Sto St) k 21Tr/7 ( - wavelength) - s the derivative in the direction of the exterior normal! nt and the integration covers the surface. Application of the boundary condition - 0 to equation (1) n St yields a ek |n'(S) 2 G(S,S') da' (2) a'n - 47^ 1 yn'n Z n 3Sollfrey, W. "The Variational Solution of Scattering Problems," Research Report EM-llg New York Univ. Inst. of Mathematical Sciences, po7 ff (1949)o..................................... II _.......

. THE UNIVERSITY OF MICHIGAN 2591-1=T Then it follows4 that ((S ) is the solution of the variational problem J [V] = 0 where Y _(S) a7._ G(SqS1) 0(S) d da da...3 a. S.... (3) S.JS Bn S ~f5 (s) neik da It is expedient now to introduce the prolate spheroidal coordinate system g ~ 9 which is related to the rectangular system by the formulas ____ x F/ 2 1)(1 FY2) csos cos y F)( 2 L 1)(1 ~2) sin Q Z Fo< 3 sin. z ( F Y1 where F is the semi-focal distanceo Then if 42 Y12 Y -we have da F2 o< d 4 d( (4) Dn FY ag The trial function 0(Sw) (for S' on the scattering surface) may now be expanded in terms of the Legendre polynomials P, ( e ) which form a complete, orthogonal set over the interval'-'L _ 1o Thus ~(st) A,(~) PA(A) (5) where the coefficients At (e t) are to be determinedo Then the variational quantity J [E ] takes the form 4Sollfrey, Wo "The Variational Solution of Scattering Problems"tt Research Report EM-11 New York Univo Insto of Mathematical Sciences$ po 13 (1949)o 5

THE UNIVERSITY OF MICHIGAN 2591-1-T L[,p~ - ~ C,,,(J H..2 (6) T: AAi, -~e T[ S 2. v)T G(SS (>) da aXda or if the integrals in the numerator and denominator are represented by CC/,i and B/(.arespectivelys J[ C].,A, A/ e Cv o (7) Alu B.4o2 It is easily shown (cf. Sollfrey, loco cito) that the stationary value of J is the negative reciprocal of the back-scattered amplitude9 from iwhich the back-scattering cross section is immediately obtainableo 6 -

TH E UNIVERS ITY OF MICHIGAN 2591-1-T CHAPTER 3 EVALUATION OF THE INTEGRALS CGv To accomplish the integrations appearing in the numerator of equation (6) it is advantageous to use the Fourier integral representation of the Greents function.fkP rff it0"-IK G(SS) - e Iikp d iK 10. 2 J ZIB1-00 i2k2 2 where K (Kx9KyKz ) dK dKxo dKydKz K KoK and ~ (x-xts yQyt z-zt) Now we have formally ___ (G(S,S,) = a G ^ (., G ) o 2nan 3n D n' Rn \Y F-Y' a' 4 Actually this quantity must be regarded as a limit6 as t.- f S in order to avoid difficulty with the singularity at S St and accordingly we preserve a distinction between S and 4 tf until after the crucial integration has been performedo Furthermore -Levine, Ho and Schwinger, Jo "ttDiffraction by an Aperture in an Infinite Plane Screen, Io" Physo Revo'_9 Po 961 (1948)o 6Morse, PO Mo and Feshbach, Ho, "Methods of Theoretical Physicss, Mc.Graw Hill, New York, po 1043 (1955)o 7

THE UNIVERSITY OF MICHIGAN 2591-1-T G - ZG 3 x t + G 3_yt 4- G 3 zt, ag r axt~ a t Dyt - f I zt 0 and -.> ->. ~K - e e P-'i(KX, KY, Kz) a(xt',y',s' ) so thats ignoring for the present the question of the legitimacy of differentiating under the integral signsp we can write G _ -i eKP (Kx t + K- Kz - 2, 2T j 2- K2' + K ) dK -00 and 00 02G iK._, ei_____ K ( e( ax1, t y_ _ - nt F2 —-— *22 (Kx F2. K -a + Kz - n-Z nt F2Y t.22 Jj ~ 2- yk 2 ( K -00 0* x(K.x x Ky y i Kz - ) dK 00 -"'2iK~ - -00 2w2- 1, _ JJk22 (KAx cos ~+Ky s J sn + Kz 7) *(Kx ~ t cos, +t Ky,t sinTt + Kz r,) dK. The divergence of this integral expression must be eliminated at a later stage. Then rearranging the integrations involved, we can write 00o +1 2<'"l'T C F4<4 dml i dK e iK () pP 21Y2 p' C i p(ylt). 2 K,, 2,j I I-Jk2 -o00 -1 0 (9) *(Kx L3 cost+Ky $fsin$+ Kz)) (Kx, Scos't+Ky 7$ isinoe 4+ Kz4')d0ddrtd d)'

THE UNIVERSITY OF MICHIGAN 2591-1-T or p4 4 C/,X, F 2 im I,2v 2T2 7 where I,- represents the above integral It becomes convenient here to introduce the transformation Kx ^ r cos w sin V 0 r_ oo K r sin Lw sin y O y'rr K I r cos Y/ 0 wU 2 77o (This new coordinate system rvY Jfw is similar to ordinary spherical coordinates but the family of spheres is replaced by a family of oblate spheroidso The lack of orthogonality is not importanto) The Jacobian of this transformation is D(KxR Kz) r2 sin (r/V ts W) 6 a 2 In terms of the new coordinates9 K e o Frc sin (o_ 3 coso4 - t 6 cos ) <,sin (cp < ~3j sing o(t s sinQ)~ co' 4 Fr 5 cos(w ) < cos( ) +?t KxR coso0 +tK- y sin tKz r | sin Y/sos(J4-)+ ~ cos- / and t ~ ( 2 co2 r Substitution of these quantities into the integral above gives I —------ 9 ------------

THE' UNIVERS ITY OF MICHIGAN 2591-1-T or Z2|o(, Vf ) 2r (PdryQ- id ie FrcosL(J- d'd t | ~r 2(2 cos2 2 2Tr o JieF,sz'- Cos(W-'0)=,4'/3'oCos( J-'1 r'22 Sf ^irFjo sin, os M=o)+ 2 cosos 0 sin V cos( ) cos do doI We deal first wiith the 0 and }? integrals9 which can be resolved almost immediately as follo-ws 2' eiFrS os( sin' cos (tJ-V)+ cosVJ do 2 2 Ji sinJf eiFra sin y/cos(J-)cos (W) do 0 2 1 ~+ - cosy,f eiFr sin1cos("r-4) do 0 = ei siny~o iWJl(Fr,3sin/)-+ 14 cos 3o Jo(Frsin/s)in and similarly 21T eiFrF t sin cos(.) F sin/cos(w -4W cos 1 do -= s2sn inOi )rJl (FrN^' in + cos o 7Joo-Fr'~o' sin) o~~ O(I o( $ 2f

THE UNIVERSITY OF MICHIGAN 2591-1-T Putting these expressions into In-t and rearranging slightly, we have T oo 2 I1-o v 4 2 f f 5 r2sinr'dcIjdrd sin' C082 W \^2 3 4 ~cos%/~k22 2$, + cos2 I +i sincosL r where +1 1 If^ ( pf(y) P (yQ) eiFrcosy(? rsin J Fsi -1 Consider first I Y)o The integral is () iFrcos J J(Fr(Fr ssinin)d)d _ _ _ _ __ UI _____________ u _____________.~~~==

THE UNIVERSITY OF MICHIGAN 2591-1-T 7 We have in general I iz7rl cos^ ( vi'sinyP ()d inmV^d p c mosr)J / () -1 (11) for 0- G 7,f so that the above integral becomes i Pr 2"pl(cos"t3)/J(Fr)- iPi(cos )J, 3/2(Fr)] In similar fashion for w7t we have PV (l) eiFrcos Jl( - Fr,3 O sn ) d( f Pv(>j eSr3o Jl(-FFresina-~) dyt 1~ol~~~~~~~ ~~(12) 2 it + l Fr L ^ r1(cos5 )J 1/2(Fr-1)-i P cos (cJou/3(Fr) 2 v+l Fr E -1 vwhere cosil e cosOC and G- sin l esinT Thus 2l z Tn' lp1i [p((cos)J2(Fr)+P (cosa3/2 (2,u + l)(2V +l)Fr JVT2 o p ( cos ) J/2 (Fre)+ P L(cosr)J (Fr) and in similar fashion 7Magnus, Wo and Oberhettingers F0o "tFunctions of Mathematical Physicsn Chelsea po 77 (1949)o The integral is given in terms of Gegenbauer functions It takes the form used here when these are converted to Legendre polynomialso 192.

THE UNI VE RSITY OF MICHIGAN 2591-1-T ~ (v+l)P 1(cos C)J /(Fre)-=~Pa (cos )Jv E (Fre) | | 2 (1l)Py ((cosfF )Jr7(Fr~)-vP (cosP )J1 (Fr)j o and 7 2,4v 1 --- - V+ 1 1 = +ll(cos It+3/2(Fr -l(cs -/2 (2/+t1)(2v + 1)Fr e 1 ) ~(,+)P (cos s )J /2 (Fre)t P+-l(co s )Jt3/2(Fre) v-t 7 co %/2 ] Putting these expressions into (10) and observing that the dependence of the integrand on W. has disappeared9 we have 16-1W\i' j r sin~drd^' si2 1w. ~I/ 2A=00 JCOY Jk2r)2 -+ n- P~(cosVo)J/32(Fr) -(2,ml)(2uFl)Fr oo S|-cosl&'S l +P1 (Fr)1 1( P(cos)Ju/(Fr)+P (cos )J (Fre) - P (cos_ +l)p l( cos t) j 3/2 (Frs)) p orl( rcos )) J l1/2()Fr) A 4 tl - l+)P )(cos( esr J )3(Fr)vP, ((cos) J (( 13) r *4 / 4 (13)v+3 (equation co2_nt on 13___^ ^_ ^__ ^_ ^ Inext page)

THE UNIVERSITY OF MICHIGAN 2591-1-T ~+ 2sin'lcos' (coW os/J (F) + P (oSo 3/(Fr) e (v 1)Pl( cos-)J( + 3/2(Fr e) -vP1( Cos J 12(Fr + (is (ore) ~ l|)Py l(cos)U63) / 2(Fr)'+ PV (colOS ) J ( 3/22 Fr ) Io Thiscan be rewritten as follows, considering that the r integration is to be performed firsts and observing that the distinction between # and E is essential only in the r integrals and thatd -* 1, cosY* — cosY) as' -, IiJi~ T k.,^^-t3v I,. ~ noLy [~12 1 1 1 limi~ -167717' $s p 1 p 1 P 1 0 ~v"+ P1 PP R+.%El+As-^^S 2i1P +11 -vP laP1B1+ (vt>P^^ l W PU+22 -r v t -i >1 ( ) sncs'r 1 -l c 1,,,2,+Q l)PlPR1f tl )Pitf+2 - l 22 -------- 14 A -1 Y-1 1-1 1 4lPV+l 12

THE UNIVERSIT Y OF MICHIGAN 2591-1-T where co Rll lr Jl' 1/2(Fr) J /2(Fr e) r2 dr 00 Rr 1 3 J (Fr)J (FrC) r3 dr 00 "r3 dr (15) c,m o R Mlint fJ 0 (Fr) J,/2(Fr -) r dr R2.27/.r0am c,/ (Fr) Jf/ 2(Fr&) r3 dr'22- j/ 4 A +3/2 J-~3/2 r2_c2 and c2 k2 ~242 2 _cos2 and the argument of all the Legendre functions appearing is cos o Now it may be observed that since Ri j considered as a function of 4, 9 is even about 1' Tr/2, and since P1(cos yP) is even or odd according as m-n is even or oddo the integrand in (14) is even or odd about I9 1/2 according as/t t V is even or oddo Thus Ilr. vanishes if/a + is odd. Furthermore since Jn(-z). (-1)n Jn(z), it develops that the integrands in the expressions for the Rij are even or odd about r = 0 according as At+Vis even or oddq and consequently only the even case need be consideredo Thus we can set J + 1/2(Fr) Jnl/ (Fr e) r dr 15., - -

THE UNIVERSITY OF MICHIGAN_ 2591-1-T The representation (8) used here for the Green s function is not completely defined until one specifies the manner in which the singularity at K = (k9 O 0) is to be avoided~ In the present case the proper procedure is to divert the integration path in the complex K -plane slightly below the point kg inasmuch as the scattered wave is to be outgoing~ This corresponds to a diversion below the point r = c (and above the point r = - c) in the complex r-planeo We proceed then to evaluate the integral.r3 dr = 1 3 dz ( +1/2 (Fr) J n 2(Frel r dr ( Jml/2(Z)Jn l//2(Z ) (1 d fm~/2 nt r2c2 F2 z2~b2 ( -Where z - Frq b Fc9 and the path C consists of the real axis from - oto + oo but with divsersions above the point z b and below the point z = bo This integral apparently encompasses the divergence which first appeared in the expression obtained previously for the nornal derivative of the Green.s functions since the integrand osillates indefinitely without decaying as z —Tooon the real axiso Howeverg we can obtain a formal values at leasta as follows First let (CZ n ( Z) |r Jndl/2([) ( 7) so that -e 0n( z) Jn+1/2 ( z) and Morse and Feshbacho loco cito po 818o.16.-...,

THE UNIVERS ITY OF MICHIGAN 2591-=-T z3 dz =~ z2 dz m+l /2(Z) Jn) z2- 2 1/2 (z ) Z z2b2 ) iL iz);(cz) 12' dz = Jm+l/2(Z)On(- ) z2b2 It is clear from the original integral equation (1) thatC must be less than or equal to.and consequently from the definition of 1 (12) it follows that E < lo Now we. assume for the time being that m < n + 3 and write (1}.2o J 25 (Z =2 L /2 (z)+ H (Z ~ It is well known that the Hankel function H(l)(z) vanishes exponentially for large positive imaginary zs and that H( (z) behaves similarly for large negative imaginary z9 and accordingly we should try to express the integral (18) in the following formo Jmel+/2(z)n{ Z z2 dz.ml/2Z) n z2 dz zZ b2 2 C1 z2 b2 (19) J (2) z2dz t "CrHm+l/2(z )n(E ( z -b) z 2 where C1 is C closed by an infinite semicircle in the upper half of the z-plane and C2 is C closed by one in the lower halfo It must first be shown that the integrals over the semicircles vanisho For this we can use Jordan's Lemma2 which states that the integral of the quantity elmz f(z) over an infinite semicircle in the upper half plane will vanish provided that m >0 and f(Rei0) — 0 uniformly in ~ as R- %D9 with - 17,-..

THE UNIVERSITY OF MICHIGAN 2591-1-T obvious modifications to handle the lower half planes Consider first the behavior of n ( z) for large zo Using the definition (17) in the range where r is larges we can employ the Hankel asymptotic form for J 12() to obtain ez ~ nZ 2 I n 1 - 1n-l en d< %n- e-2 d On(~ I J d i JJ dj e Setting EZ2 f00 r00 r eid ei d ei'~A id JI dr J, d e" dz'^: 1 J35 d where A1 = constants independent of z2 and putting g. z + % we have e oo e, i e I ej 1 dt e Using operatorial symbols we can write ff(T) d D1 f(t) and D1 g() ei(D+i)=l [g(t ] so that But (D +i)1 O ~i Di D.t2 D i 1+ iD+i2D2+ i 3D3+.. I'*D+" 18 -

THE U NI'VERSITY OF MICHIGAN 2591-1-T so that rIo l+~ r - ~ = e O0 O i n02n)S_3 = - -i-l n.0O 22nn;.(z +,.u)n 0 -lim- i e i (2n)~ - i( n)! n0 l22nnn n0 22nlnn(Ez)nl zi for Izl large and 0 < arg z TT o Similarly ez e lJ 2 e- d = i e-if (Di)l [7 1 -1. where A2 constant, or CZ-io ii2n) 6ZA j yf' /^r n=O 22nn.. 2 -iez.iYe "-l2 for i z largeo Then 1 ~Ai-n-1 ( ie i< in (ie i) -in ( )n eiZ + eiz 2tA 19 le1.....

THE UNIVE RSITY OF MICHIGAN 259-1-T whereA3 incorporates A1l andA-2, and consequently H 1 Z r z - /- - leiz ((l)nfei9Z+e'Z l + - m+1/2 n z2Tbh vTz L2z L Z re+1/2 2)n(6Z) z2_b 2 I 2 i-nieiZ L i L( (l)neiz-(l' e1z(1) I in1 0A3 e (lf i ) i^D1^ (2^ ) i —+ el/ 1. (20) z I/E TrzT vrz v3/ Each term in (20) satisfies the hypotheses of Jordants Lemma (since < 1) and therefore the integral over the semicircle is zero. The same technique can be applied to the lower semicircle, and the original integral (16) can thus be expressed in terms of the residues at the poles included in the two contourso (1,2) Since the functions Hm l/2 (z) Jn,1/2(rz) have poles of order m - n at z = 0, it is clear that as long as m< n + 39 the integrands (l12) z3 Hml/2 (z) Jn 1/2(ez) z2-3 will be regular at z 09 and the only singularities appearing will be simple poles at z b, one of which. is included in the upper contour and the other in the lowero Thus we have 21> (2) J Jmel/2(z)n(z) 2 2 Hm1-/2 (b)2 n(b)~b Hmb )(bob C and from (18) J (J +1/2 ( zz i2 -b = H1l/2(b)Jn*1/21/(b) Hm+l/2 n2(b)Jn l/20,,- --— n —- 220

THE UNIVERSITY OF MICHIGAN 2591-1-T (2) But H (-b) J ( b) -H (b) J (a b) for m+-n even, so mel/2 n+l/2 n- +l/2 n+l/2 that J Jm+ l/2(z)Jn+l/2(E z) z2 i (b) Jn l/2( eb) (21) z bb As for the question of what happens when m> n +-3 examination of equation (14) shows that m takes only the values/ — 1 and,-+l, and n only the values v- 1i, sl, so that if / r +29 this situation will not occuro Furthermore it is apparent from the form of the quantity Ca/r given in equation (9) that in the limit as ~ - this must be symmetrica in/k and zv, so that the situation for / )7r+2 need not concern uso The case/ z>could be dealt with in similar fashion if desired, but the development would be slightly more complicatedo When the expression (21) is substituted in the formulas (15) and the limit process carried out, there result the expressions R|1-r i c n22 (Fe) Jry1/2(Fc) <'ic2 (L) 12 -2 H ( l/2 (Fc) Jr+3/2(Fc) sic2 (i) 22 2 w,/ +3/2 v+3/2l and when these are substituted into (14), it develops that the bulk of the integrand can be separated into two factors, one of which depends _______ 21

THE UNIVERSITY OF MICHIGAN 2591-1-T only on,jy, k, / and the other only on rg k,, Thus lim y = V5i 3 2 fsim 2 d2+ (Sks,)*(gkg) (22) 5 9 Fo (2,+l)(2zr+1) (52_cos2yv)2 where k nHnl/2( kF )costPPn(cosP ) 2Pn(cosS) ] -(n1)H (4 kFo (COS - nH k 2+3/P( cos n( ) w = cos on -(n+l)Jn 3-/ C[c os2P (cos )- Pn(cosV) for n i/9 -ro These expressions can be put into slightly less cumbersome form by introducing the spherical Bessel functions of Sommerfeld and using standard recurrence relations for these and the Legendre polynomialso Letting kF C Jn +1/(P) i n( n e2 p 1n( we can write:n(~,k3) ~ (:2nnl)-1 cosPn1 ~2Pn) n(p)+(%2cos2Pngn 1l(p) (2n +1)y / r2 okP) with a corresponding form for n(,k,, 1'), and C/,v becomes 22

THE UNIVERSITY OF MICHIGAN 2591-1-T C, _ 81r F4~23k3 i1At3vlj r _ ) 2 k (,k,p) (,k.1A (/ 2c^-cos2i^ 0 (24) It is of course possible by expanding the products /7U l v and applying recurrence relations for the Legendre polynomials to write CIar as a linear combination of integrals of the form Pm(cos pn (cos)) ( r/()s+l/2 ( 2o2) y)2 and since both the Legendre and Bessel functions involved are expressible in closed forms the integrands here can be decomposed into expressions involving only elementary functionso Recurrence relations among the various terms can then be obtaineds and the only integrations remaining are a few initial values of fairly simple formo However the number of terms involved for any moderate values of/u and fr is so large that it was judged more economical for actual computation to use the form (24) and perform the integration by numerical methodso Alternatively, since the integrand in (24) can be expressed entirely in terms of Bessel and Legendre functions of orderL and if and derivatives of these with respect to cos /, some thought has been given to the possibility of performing partial integrations in order at least to reduce the complexity of the remaining integralo It is not immediately clear that anything can be gained by this approach, howevers and to date no thorough investigation has been madeo ----- 23. —-----— __

THE UNIVERSITY OF MICHIGAN 2591-1-T CHAPTER 4 EVALUATION OF THE INTEGRALS B/. Referring to equations (6) and (7), we have B f | P,>() D e kzda JS S =< % _'ikFS/ F 2gd\d 0 -1 S +1 - Tiick2F2k\ pz&) elKikF) Pd -12 or setting kF.' ka, cos ika cosO BA = 2T7rix<2F2k FP(cos )ea sin cos d 0 Using a recurrence relation to eliminate cos 09 we get 27TiF,2kF2k ( ika cosQ - l t (At l)j P ((cosQ) e sin 0 do -+t-t P -(cosQ) e ika cosQ sin 9 dQ. 0 The integrals here are of the form given in equation (11), with m O0 ) cos 0, z kas and//- Oo Thus -------------------- 24 ----------

THE UNIVERSITY OF MICHIGAN 2591-1-T Bu= 21i2F2k l)i/ J (ka)+ i 1 l2(ka) 3 /2/ l' 2 ka;2{tr) i k /2a Ju/2(ka) ( +1 — ) J (ka) In terms of the spherical Bessel functions used in the preceding sections this becomes B/C = 47rF2 21/Uk d V/ (ka)o (25) d(ka) 25

THE UNIVERSITY OF MICHIGAN 2591-1-T CHAPTER 5 DETERMINATION OF THE SCATTERING CROSS SECTION Once values have been obtained for the quantities C, v and B/ over a sufficient range of the indices, the problem of finding the back-scattering cross section is relatively trivialo The stationary value of J [J -is found by setting the derivative of J with respect to each A,, equal to zeroo This operation yields the system of 9 equations 00- C 2> A =BV Jo BJ =0 for allv, (26) ^ 0 B^ where Jo is the stationary value of J CU3 o Existence of a solution of this set of homogeneous linear equations in the unknowns A/i requires that the determinant of the coefficients vanish, ioeo, C/ttv Cv I Jo = 0 Jor I J (27) These are linear equations in J0o the solution of which can be written in the form J a _ (28) where C Bau BA A U = cofactor of a,,,- 9Cfo Sollfrey, loco cit, po 15o 26 26 -- - -- -^ ^

THE UNIVERSITY OF MICHIGAN 2591-1-T Or we can write a- av~ - C - ^-(29) o s p as a me where C is any constant, which is perhaps a more convenient formo Furthermore the fact that a,/a 0 for, v+v odd means that the coefficients of even and odd index are completely independent, so that if we let a,uv- ~ uv for the odd case and aj-r - saf for the even, we have 1 I,~-+Cl (+, cv +C2 1 J- - - =' 1o (30) Jo Cl oL> C2 (3r CK C v 2 v \ 1 2 / where C1 and C2 are arbitrary constantso Once a value is obtained for JOg the system (26) can be solved if desired for the coefficients A/o The back-scattering cross sections however, is given directly by JO which is inversely proportional to the back-scattered amplitude, as remarked in Chapter 2o Specifically, if (J is the total scalar backscattering cross section, then a 41T 1L | (31)'Jo The above solution may also be obtained without resorting to variational language. The method is due to Galerkin and has been shown by JoneslO to be exactly equivalent to the variational approacho If the expansion (5) is substituted into the integrand of equation (2), and both sides are then multiplied by PT (j) and integrated over the spheroid ~Jones, De So, "A Critique of the Variational Method in Scattering Problems," IRE Transo Volo AP=4, No0 3 (July 1956)o 27

,THE UNIVERSITY OF MICHIGAN 2591-1-T (in the manner of the above development) there results the system Z A AtC, = 47T B1, ur, 1, 2 --— o, (32) A =0 The solution | Am of this system is equivalent (within a normalization constant) to that of the system (26). The value of Jo is immediately obtainable from this system by application of the stationary condition JoB A Bl = " A x C r for all V derived from equation (7)~ Thus J -_47 0 Am Ban and Co =_ 1i Bc (33) This expression is probably more convenient for analytical and computation purposes than those derived above, and is used in the developments which follow. -------------------- ~~ ~~28 ----------

THE UNIVERS ITY OF MICHIGAN 2591-l-T CHAPTER 6 VERIFICATION OF RESULTS The lack of mathematical rigor in some of the preceding analysis makes it imperative that some sort of check be obtained on the validity of the resultso The most obvious means to this end is to examine the behavior of the solution in the extremes of wavelength and eccentricity, where the correct solutions are well knowno Considering first the eccentricity, we can see at once that as this becomes infinite the nose-on scattering cross section should vanish for any finite ka, and little information is to be gained on the forms in question. We examine rather the case of vanishing eccentricity, ioeo. where the spheroid becomes a sphere0 This transformation is accomplished by letting * co and F - 0 in such a way that the product c F - a, the radius of the sphere~ Geometrically this implies that the major axis of the spheroid remains fixed and the minor axis is increased until the two are equal0 From equation (23) and following it is apparent at once that /P-! ka and that the terms cos Pn+l become negligible in comparison to the terms ~ 2P o Thus, since cos2 5becomes negligible compared to 4 2, the integral in (22) reduces to P (cos) P(cos sind T H ( 1) P (cosI) P2r(cosy) sind o /1/2 (ka)- +l)H (1) (ka) 0 0 ok~ -1/2 (ka)(7r+l) 3/2(ka) / (equation con'd on next page) 29

THE U N I VER SI TY OF MI C H I GAN 2591-1-T ( <S 2 (2/+ 1)(2UtI) 2ka d; (ka) ____ ___ ___ 2k_ _ _ Iv (ka) (34) (2/c 1) T/ d(ka) d(ka) /0 forU 2 (k (2,* -1) d(k) (ka) d(Ti (kka) for, g Equation (24) then yields C = 0 for. (35 -16'T 2ik3a4 d ( d 2, - i) " d(ka)' (ka) ( for and equation (25) immediately becomes B =- 41'ra2i'/k d(k )t (ka). (36) d(ka) The velocity potential at a large distance R from the scatterer in the direction of the approaching plane wave can be written in the form V(R) = eikz + f(YI) eikR R and as stated earlier, f(%f) is equal to the negative reciprocal of J the stationary value of J. Referring to equation (32) we have for the sphere A, Cv =- 41TB or A/ 4=7B/, C3/tA 30 --------------— __

THE UNIVERSITY OF MICHIGAN 2591-1-T and 2 f( 1 E7 A, BB, 2 (37.) 4 iT /vZ C — Substitution of (35) and (36) here yields.f(0) = (-1)"2 ( 2 1) Hi (ka)/ (ka) (38) The classical solution for the sphere is given by Sommerfeldl in the form d~ka) d(ka) V' -S i (2~,l)P~(cos~) c(kr) d-T (ka)/ ( - - ~ (ka) where V is the velocity potential of the scattered wave at the point r, Qo If this is restricted to give the back-scattered field at a large distance RR it becomes'ikR V = ie Z (-l)/(2+) ) d (ca(ka) d (ka), and multiplication of this by R e to obtain f(iT) renders it identical to equation (38)o The variational solution is thus shown to be correct in the limit of vanishing eccentricity. We consider next the extremes of wavelength as compared to the dimensions of the scatterer0 The relation with the sphere solution exhibited above makes it quite apparent that the present solution should be most practical in the region of large wavelengths and we should lSommerfeld, Ao, ttPartial Differential Equations in Physicst Academic Press, po 164 (1949)o 31

, THE UNIVERSITY OF MICHIGAN 2591-1-T certainly be able to compare it with the result obtained by Rayleigh for this region. To this end we write the back-scattered amplitude as in equation (37) f(t) =- A, B/ and expand the quantities A/M and BA in powers of k. The terms of order less than or equal to k2 should then give the Rayleigh result. Setting B/ = b kj J (39) AM - ~. arkj k 3J we obtain immediately, to order k2, { o0 1 1 22 0o o 11 11 f(iT) - a 1 abo + b+ao b -k a O - bl a albo 10 0-0 0 0 1 o 0 o 2 2 2 2 0 0 o0 0o 11 11 -aobl1 -albo k ab2 +- alb1 a2bo -aob2 talb1 11 22 22 22 ( + a2b + a b + albl -Aa2b (40) The quantities b can be obtained easily from (25) by substituting the power series expansion for the Bessel functions. We have, to order k2, 32

THE UNIVERSITY OF MICHIGAN 2591-1-T Bo (27TF3/22 (kF )3/2 1 0 ^ 2 r(5/2) Bl_ 1/3(2w F)3/2i2 (kF 1/2 1 -B _3/2o3 2/ (kF, 3/2 i 2 12 P (5/2) These yield 0o 1o 2 b bl 0o b2 - 4/3 7tF3, o 2o( b1~413IF (41) bl - b2 ~a bl1= 4/3 7riF2,x,1 b2 b 0s b2 8/15 T F32 o To find the a. we refer to the linear system (32), substitute power series expansions for A 9, CIAlr, and B., and equate coefficients of like powers of ko Thus if we put bo > -,^lzl t j C^ s Z C, k J o o eTo find ta w r to stlr sut ~oo o o oo o ~Oo ot 0 00 2o C.~- a C a,) C~ aa 47 b~ C1 a 41rb~ I —-0 —---- 33 0 C11 1 11 1 (42) a1lt C a 417b1 22 2 2

THE UNIVERSITY OF MICHIGAN 2591-1-T Referring now to formulas (23) and (24) it is clear at once that C~~- C~= 0, while C22 4 0, and it develops easily that 0 1 F3 C-oo 8773 2 16/3 77 F3 2 2 -'(5/2)r'('"/2) We find also by the obvious procedure that o 3- y3'22-= 2 (2 2+1) 2_ -.... [ _2 % 2. {l = 161T1 Fo;2 2 Applying these relations and those in (41) to (42), we find that 00 0 20 2 o C2 a + C a 4 b2 C11 a1 0 0 0 11 1 11 1 1 Co al+ C1 ao 4W7bI C22 a2 0 0 0 which reduces immediately to 00 0 C ao - 47Tb2 2 a0 4 2 11 1 1 Co a1 = 4Tb b1 or ----------------- 34 ---------------— _

THE UNIVERS ITY OF MICHIGAN 2591-1-T a0 4=..T 4TF... 3 3-l167 2F 2.16.,2? ~ log; + L 3 Equation (40) now reduces to -rr. o 2 1 1 k- - 2 -2 t-V -\ ~ 43 2 2 ___ k t(43) 2 $ r3 2 - log In his original work on scattering by small obstacles, Rayleigh gives formulas for the case of a plane sound wave incident on a prolate spheroid which can be written as follows 2 f() 7T 2L (44 2 1-L where -L ~2 —- 1) 1_ log - e-1 I2 / 2e -e T 4/31ra3(l e2)o Upon substitution of the relations e; a. F', and A= 2Tr/k, equation (44) becomes identical to (43), and the variational solution 12Rayleigh (Strutt, J, WQ) "On the Incidence of Aerial and Electromagnetic Waves on Small Obstacles," Phil0 Mag., Vol. 44, p. 28 (1897)0 35

mo THE UNIVERSITY OF MI CHIGAN 2591-1-T is thus shown to be in agreement with the Rayleigh result, In the limit of small wavelength the analysis is more difficult, and the geometrical optics result has not been obtained from the variational solutiono It becomes apparent, however, that the situation is similar to that which prevails in the case of the sphere,:in that the number of terms used in the series in (33) must be of the order of ka@ This can be shown by the following analysiso We examine first the behavior of the quantity CP, as ka becomes large in comparison to, and v o In this range we can use the Hankel asymptotic forms for the Bessel and Hankel functions appearing in the expressions for /!. and AY (equation (23))o Thus 1/2 o=' 2, 2 2-( 2 r(Ck9 t3(X2,p) eiP i^ j/., cosyfX 1rP tl)(cosyjp; j (CD( )/2ei ix(2L 2+l)P,., ( oos ) (. s2 D cs ), VA\. (,k V) )l/2 Jv(coso~P 1 P) (v+l)(oc.coso l Pj) o 1 _-i? ef~ " e^ 2 )= e + 1 e i 2 2 cG. " 1 4~2F2d~ z"t 3~ * L, J p coso)P-(cos') o e 1 i ( cos ) sind ( 5) 36

THE UNIVERSITY OF MICHIGAN 2591-1-T An approximate value for the first term of the integral when ka is large can be obtained by means of the stationary phase formula. Letting cos4 Y 2 cos P (eos +) P, (cos )y O2 cos2) we find that if o0 is the stationary phase points defined by the relation 12' (Yo) = 0, then,~r j e21iPp (cos()Pu(cos') 2cos2' siny d a(1 1/ 2 ka1/ i7 ) li;i(sgn ()~ 4 i~2,o i('2kFo2 a! k / e 4 2 for., v even 2 (2' 2 2 (46) 0 fora/, J odd The second term in the integral becomes 37

THE UNIVERSITY OF MICHIGAN 2591-1-T 1iv PM ( 7) P v) /2 - 2,2 (47) -1 which is independent of k9 so that the dependence of Cv on k has the form C ~ - 4 1qF jit+3il{ 1+ (Fl)" c(F) k) e 2kF % C(F ) kj (48) for ka largeo Use of the same asymptotic forms for the Bessel functions in the expression for B, (see Chapter 4) gives the result 2B, 2 TFo(<2 ka() eika2 f cos ka for,/i even 4TF (4.9) i sin ka for /u oddo For large k, the second term on the right in equation (48) dominates, and combining (45)X (46), (47)9 (48)s and (49) we can write the linear system (32) approximately as Aenn i'cska veven /,e even (5o) * Z Act~ v ( sin ka aodd > A/.,4 "rtt-v roddo /A odd Application of Cramer's rule to these systems gives Ai K-cos ka, even A/-t /tt even ka 1 T!lv I ka T4 v c^ ck /Codd I —-------- 3

THE UNIVERSITY OF MICHIGAN 2591-1-T where Tv is the cofactor of Zv and I'ul is the determinant. Then A B,, T16Tio3F cos2 ka even BA z D/^ —---------- &L- ^yu~u /^ even Ika 1,"1 v Ai B, -l6lin3F sin2ka Z, Aodd. Ska iZv\ Assuming the quantities ~ ~,/ are bounded, each term of the series (33) with / < < ka is of order 1/ka. Therefore any reasonable approximation to the true cross section would require at least approximately ka of these terms, as in the case of the sphereo The dominant terms in this region of the spe-ctrum may be those with, ^ ka. -In order to obtain the approximate values of these termss the Debye asymptotic forms for the Bessel functions might be utilized; however these lead to much more awkward integrals than those in equation (45), and no further attempts have been made in this direction0 39

THE UNIVERSITY OF MICHIGAN 2591-1-T CHAPTER 7 CONVERGENCE OF THE SOLUTION The principal question remaining is that of the convergence of the series in equation (33)0 It will be shown that at least for small enough values of ka, this series must be absolutely convergento Some of the estimates used in the following are rather rough, and error terms are in general ignored, To make the proof absolutely rigorous a more careful analysis of these error terms would be necessary. However this would materially complicate the already tedious development, and we therefore limit ourselves to what might be called a strong plausibility argumento We consider first the behavior of the quantities B, L(equation (25)) as,/ increases with ka fixedo The asymptotic form of the Bessel function Jn(z) for z fixed and n large is easily found to be13 en +1/2 zn /n(Z) 32( 1 n +1 (51) 2 (n 32 1/2 which can be differentiated with respect to z to give n +-1/2 n-1l dW- nz r (52) dz 2n+3/2(nl/2 (52) 2 (n +_1/2)n Using this form in equation (25) immediately shows that for fixed ka, the quantities | By | ultimately die out as (eka) 0 It is clear then, 13 Cfo Watson, Go No, "Theory of Bessel Functions,n Cambridgeo po 225 (1952)o 40

THE UNIVERS ITY OF MICHIGAN 2591-1-T that as long as the A, do not diverge too rapidly, the series I A, B, will converge absolutely for any finite value of kao Due to the complicated form which still prevails for the coefficients Cu it is difficult to obtain a direct proof of the boundedness of the quantities I A,- A however, a constructive proof of the existence of a set of A/, which are bounded in absolute value and satisfy the system (32) can be given by the following line of argumento We assume for the moment that the first N values of I A/ are bounded, where N is a number which is large with respect to ka and unityo Discarding the first N equations temporarily and transposing the products A, CvC for / _ N to the right hand sides of the remaining equations, we can show that the resulting system of equations in the A for +u > N has, for some range of ka, a solution of which each member is bounded in absolute value and which can be determined in the limit by the usual method of truncationo Furthermore we can show that when this solution is substituted in the first N equations of the original systems the resulting system can in general be solved for the Ak with / _ N0 The result is the unique bounded solution to the original system (32), whose existence guarantees the convergence of the series in (33)o The first step is to show that there exists a number N such that if A I is bounded for all u z N) then the system oo N Z A, C,, 41TB - X A CIuv A v- N +-1, - (53) --------- 41 N

THE UNIVERSITY OF MICHIGAN 2591-1-T has a solution j A, expressible linearly in terms of the set { A/ such that |A/I M for some M oDand all A 7 N To accomplish this we make use of the following theorem, due to Pellet and Wintner: 14 Given the system Xv -v. = Cy, V 1 2, o0o 0, where oo if the quantities I C are bounded and the coefficients ayv are subject to the condition 00 Su V i a,lz, < 1, 1, 1 2, ooooD (54) then the Xv exist and are equal to the limiting form as m - ooof (in) the XV determined by solving the reduced system (m) m (m) X( )- a,,iv X CC, lr 1,s 2, ooo mo (55-) We first translate the notation of the theorem into that of the system (53), after dividing each equation of the latter by the corresponding quantity Cv for convenienceo This entails the relations Xv A N+v a,~, =CN ON+~,N+, /CN+ -, N+ NU a)v 0 (56) C v 4]BN v /CN4 v,N =- Ag CL+JN,/CNt UNt 14 Cf. Davis, Ho To, "Theory of Linear Operators," Principia Press, po 130, (1936)0 hL2

THE UNIVERS ITY OF MICH I GAN 2591-1-T Su E - CN+v,N+,N~t//C N+u,N+J Su.v SV CN,Y 9N+^/C NN-VN+U I The boundedness of 1 Cu follows immediately from the assumption on the /AI| for cN and from the forms given previously for B, and Cpt. It remains to show that S < 1 for - m 1, 2, ooo ooo Referring to equation (24) we can write the real and imaginary parts of Cu v for the range v < V / as )+34 -K C F =Re C, 1 ir2.F3o<2(_1) 2 k2 Pdf (^2q2)2 (57) */ p.P + (Y) +(P t 2pX,,()1)3, [tpY,(r)lb ( p)+82pv(r) vYu(19 0 ~~ and IC/ ImC3 3- 2 F2 2 2( k2 1( Pd.~YP~,(II~ I~(A'Sr -N~(~>~/C~i~d -~, P21-t-1 (58) Here i7 (P) is the spherical Neumann function/ N ul(lP) and the quantity kaoc ranges between kb and ka (b is the semi-minor axis of the spheroid) as Y1 goes from -1 to t-1 so that if N is large compared to ka and unity, then for all /, U, N we can use the asymptotic forms (51)9 (52), together with the corresponding ones for 43

THE UNIVERSITY OF MICHIGAN 2591-1-T 7n(Z) and Y/n(z), namely 2n+l n-1/2 n 2n n-1/2 n n( ) _ (-/) 2 (n+l/2), n( 2 (n-l/2) (ntl) nt l/2 ni- 1 en n+l/2 n-2 e z e z (59) R T to obtain estimates for C,.v and Cuv After some manipulation we arrive at the expressions -U — 3, - R -2172F(-l) 2 eka$) ( 1/2) (+ tl) f(Ac 1/2) (60) +1 0 I 2 2H2 d)i)(),, /r -TTF2c+ ek(-l) 2 eka U inequalities resullt: Here and hereater in this chapter the symbol t____ should n general (61) and the read ttless than oer apprdenominatorsely The following - 44 -- inequalities result:It

THE UNIVE R-S ITY OF MICHIGAN _ 2591-1-T RI vl 7T2F2ek(2+1_)2ak(aUt''l+)(............................- ( 6 3 ) o2(q + l/2) 1(V +_/2)+ Feka 4i /,ek a q 2 K 2(/t + /2)1 ( +//2v — +:'1 Bounds for the sunmmation of these quantities from /A =J -i — to oomay be obtained with the aid of the relations > eka | kI| 1 24 1- 4 eka ) eka( /2A. l 21/ k.-1/2 2-3 3l2V3/ - 3 (+23 z4(eka 1j- ~, I l 3,2 I eka -j The foresu nula corexpres ponding to (57) for the range is 45 c= U 1 R 3 616^o< 2 F 2 2. eka -^Tjand -i;- 4Tr2F(j 2+ 1)2( / c^ ~ -— 2 --— L2^j-^ ----, (65) ^^.*1"1 3^^'2 eka 1 2'1/+ 3 The formula corresponding to (57) for the range /At<'is I —---------------- 45 ----------------— I

THE UNIVERSITY OF MICHIGAN 2591-1-T CR 8 39 2F _ Z + L)3 (Ov k2 2 1 o~ PU I(-Q,(p) 2p) ()V (p) ] ) and the expression for Cft, is the same as beforeo Using the previous estimates and procedures, we obtains to order 1/v, 22+ 1)2eka Z | C.A - 2 - (66) Po0 f1 (2- /+.) and 2I i 22F( 2+ 1) (v+-)1)eka - eka (67 A =0vl - /2 ('2(2 ) + 1/2) Combining (64), (65), (66) and (67) we can thus write RD+C ^ C.F?2 ^ eka 2 2 0 (68),/vct~,2 3 An estimate is now required for Cry o This can be obtained from equation (60)o When / is set equal to V, the resulting integral can be evaluated exactly with the aid of the formula +1 -1 (z x)1 P (x) dx = 2 Pm(z) Qn(z) 15 R for m _ n, I|zl lo 5 The resulting expression for C-v is 15 Cf0Erdelyi, etoal0, "Tables of Integral Transforms" Volo 2, Bateman Manuscript Proiect~ McGraw-Hill p. 278 (1954)o 46

THE UNIVERSITY OF MI CHIGAN 2591-1-T R 4 Ty2-F( 1) )(u2) P 1()Q 2 p() Q - P () Q, (.^) 1H/2)/ 2 3 I2+3 ~- v j 3P7(P)Q~(~) (69) 3-U<~\(OQ^)|) ~ (69) For m large and. > 3/(2 2), the Legendre functions Pm(~) and Qm(f) may be approximated to order 1/m by the formulas (m + 1)cosh~1 pm()~ /_ (m -/2) e _1..... r9 (m+l) (e2Cosh _-)l/2 Qm( ) __sm+' l')e (m -l)cshl- r(m+3/2)(1 - e-2cosh11 )1/2 Substituting these into (69) we obtain, after some simplification, C _ 2T F 2 + o0() (70) Returning now to the theorem quoted above, the condition (54) requires, according to (56), that S, = ~' Li<l for all ur No (71) U =N l w CaV I Using the triangle inequality we can write 16Cfo Magnus and Oberhettinger, loco cito, po 73 -__________________ 47 _______________7_

THE UNIVERSITY OF MICHIGAN 2591-1-T A =N+1 ""'v JCvV C lcj v o~ iC o. (72) /=N tl (Here the quantity CI is ignored since its order of magnitude is clearly less than that of the errors in the other approximations.) Finally, combining (68), (70) and (72), we arrive at the relation 2 2 S 1l1 eka(I 2 1)2( +4 ) for all v N, (73) 3oR(2 4-2 30 - 1) so that for sufficiently small values of ka (and for ~ > 3/(2T2)) the system (53) must have a solution i A obtainable as a limit by means of the truncation technique. Construction of the remaining set [ At is accomplished by M ~N substituting the previous set into the first N equations of the system (32) and solving the resulting N x N system. To show that this is possible we must prove that the infinite series which appear as coefficients of the A/l converge. First consider the series oo C Re A/M, UvN. (74) y/ =N+1 Use of the approximations (51) and (52) in equation (57) yields the ___________________ 8 __________

THE UNIVERSITY OF MICHIGAN 2591-1-T expression R -< rI2F1e k2a2(a 2+t l) eJka I'I 2,4 2 (2At 1) 1-i - | l P| ( 2P(1 (e)J d q with a similar formula obtaining for C | O It is obvious that the integrals here are bounded in absolute value for any vu N, and consequently it follows at once that the quantities C^ die out with increasing FJ at such a rate that for any set of bounded |AAI the series (74) converges absolutelyo Since the coefficients of the AM for / t N are formed by rearrangement of these series, it follows that they must exists and the system possesses a solution provided their determinant does not vanisho The latter is a function of k, a, and ~ 3 and while it is conceivable that it might have zeroes in one or another of these parameters, it cannot vanish identicallyo This completes the argument. As remarked at the beginning of this chapter, the error terms in the estimates used here are not taken into account and the proof may not be considered rigorous until this is done0 The criterion for convergence obtainable from equation (73) is probably of little value, due to the rough character of some of the inequalities usedo However, in view of the rates of which the critical quantities die out with increasing index it seems clear that the approach outlined here could be made to 49

, THE UNIVERSITY OF MICHIGAN 2591-1-T yield a rigorous proof of the convergence and a more significant criterion for the range of ka over which it holdso 50

T'H E UNIVE RS ITY OF MICHIGAN 2591-1-T CHAPTER 8 NUMERICAL RESULTS On the basis of the forms developed in the preceding chapters a value was computed for the nose-on back-scattering cross section.of a particular spheroid at a single wavelengtho In order to obtain a comparison with the exact solution, parameter values were chosen for which the latter had previously been determinedo17 The axis ratio of the spheroid was taken as 10:1, and the wavelength ratio ka was given the value 1o40, which is very near the location of the first maximum in the curve of cross section vs. kao The integrals in equation (24) were evaluated by means of Simpsonts rules using intervals of o5~ in the range O W<4 100 and 2o50 in the range 10 _L /90Oo (The smaller intervals near the origin were necessary because with the value of 4 very near unityD, the denominator of the integrand is small in this region and the value of the integrand in general rises quite sharplyo This effect would be less pronounced for a fatter spheroid), The linear system (32) was then solved, under the usual truncation assumptions, as an N*N system,'and the order N was given several values in order to obtain some indication of the convergence rateo Values of the scattering cross section ao were computed from equation (33) and divided by the geometric 17 Siegel, Ko Mo, et alo, "Theoretical and Numerical Determination of the Radar Cross Section of a Prolate Spheroid" IRE Transo Volo APo 4, Noo 3, July 1956, po2660' ——!- 5 -1"

THE UNIVERSITY OF MICHIGAN 2591-1-T optics result r b4/a2 for convenience in comparing with the known solution, The value of the latter at the point in question, to five significant figures, is ~-_ = 1o1022 r b4/a2 The following table contains the values computed from the variational result, listed as a function of the order N of the linear systemo N= 1 2 3 4 5 O- - 07337 1.833 1o025 lo110 1o05'm b /a2 The fifth order result is seen to agree with the exact answer within about.3% at this point. Some of the intermediate quantities used in obtaining these figures may also be of interest and are tabulated here for the sake of completeness. We list first the coefficients C/-t and B,/, removing certain common factors for convenience: 52

THE UNIVERSITY OF MICHIGAN 2591-1-T T " — "C ^ll l6 rr a k -— /- -Yj o0 4 1 l559O 106 l37306 l 2 ~lo93905 / 213 2 2 o5 | 2e4664 Im 3 650311 4 1 1 680845043 14 lo01933764. 0.. 1_ — 3 4o66476 12 -57312 i 12 9o3634 3515 16o38906 O-2 ~ 222699. l 4 0 158590 -536 3730 0- 2 -- 93905 10-3 | 2 | o29 297046674 1071 O 101 -4 1 3 1 1~31725 1 60819 10-2 4lo61 41940 10" 4 35211642 10 -2 -437o564 105 1 1 1.450434 loo4 1,01764 ~ 10/o1166+2 ~ 10-2 ---- 1 o --------------------- --------- 3 i 6,9536 ~10' -5,k~n T~ a53 3i3 3053351 " 10.- 3069719' 10~1 T hese quantities yield the fol alues o the A/<as the sol ution to the fifth order linear system6.92 536 "'101 -544415'10-3

THE UNIVERSITY OF MICHIGAN 2591-1-T Several possibilities might be considered in an attempt to improve the accuracy of the variational results The discrepancy between this and the exact answer must be attributable primarily to three factors: 1) round-off error in the numerical quantities, 2) approximations inherent in numerical integration, and 3) truncation of the serieso The first of these. appears to be most significant in the present caseo The accuracy employed throughout the computations was six decimal places, and a fifth-order answer was also computed using intervals of twice the above specified lengths in the integration processo This yielded a value of l1l025 for the quantity in question, which is somewhat more accurate than the value obtained with the shorter intervals, indicating that the point of diminishing returns had already been passed in the direction of refining the integration intervalso' Regarding the third factor, it seems that the successive orders of approximation form an oscillating sequence, and since the fifth-order answer is between the fourth and the correct value, it is to be expected that a sixth-order result, employing the same decimal accuracy, might be worse than the fifth0 The problem of maintaining greater accuracy in the numerical quantities might be troublesome for hand computation, due to limitations in the accuracy of available tables of the special functions and the necessity for complicated interpolation methods; however for a large scale computing machine it should not prove difficult o I —-- -54

THE UNIVERS ITY OF MICHIGAN 2591-1-T CHAPTER 9 CONCLUSIONS As remarked previously, some of the procedures employed in the foregoing are of extremely doubtful mathemati.cal character and are justified here only by the results they yieldo It would not be hard to reformulate the problem in mathematically rigorous fashion, after the manner of Bouwkamp and others, but it seems likely that the resulting integral forms might be even more difficult to handle than those incurred in the present approach, and since the latter apparently gives the correct result there is little reason to change it at this stageo As for the risks involved in proceeding in the above manners they are perhaps better left unexamined0 There is little doubt that the solution obtained here is correct, but the question of why it is correct might bear considerable discussiono One is led to the conclusion that, although it is not obvious at first glance, the various operations of differentiation, integration, and passing to limits have actually been performed in their proper sequenceo Some of the formal expressions used are thus incorrect, or at least misleading, but given proper (or mathematically improper) interpretation and handling, they can be made to yield a valid resulto ------ -^~~ -— 55

THE UNIVERSITY OF MICHIGAN 2591-1-T From the standpoint of accuracy and economy the present form of the variational solution still leaves something to be desiredo For cases where the available tables of spheroidal coefficients do not applys it is probably considerably superior to the wave-function solution, at least for hand computation, and the best available estimates indicate that it may be competitive even where these tables; are useful for the lattero However it must be admitted that the numerical evaluation of the remaining integrals is tedious when done by hand, and the cross section depends very sensitively on the values of these integralso Further analysis of the forms derived here might produce some means of facilitating the computation process or even eliminating the numerical integrations entirelyo For example9 the power series expansions of the Bessel and Hankel functions appearing in equation (23), together with the explicit representations of the Legendre polynomials9 leave only elementary integrals to be evaluated9 and the resulting forms might be handled fairly simply by a computing machineo Alternatively something might be gained through integration by parts9 a certain amount of which is possible after suitable manipulation of the integrandso For values of ka up to about o0O (for the 10Q1 spheroid) an excellent approximation to the cross section is given by the three-term power series, described in the appendix which followso 56

THE UNIVERSITY OF MICHIGAN 2591-1-T The expressions given there yield the cross section of an arbitrary spheroid in this region of the spectrum much more easily than do the variational formso It seems possible that one more term in this series might give a good approximation to the first maximum in the curve of cross section vs, ka, but the algebra involved in obtaining this would be rather formidable, and since the series is expected to diverge somewhere in the near vicinity of this maximum there is considerable uncertainty about the value of the result Another factor in the practicality of the variational solution is of course the rate of convergence. Nothing specific has been determined about this yet except that in the case computed it seems to be comparable to that of the wave-function solutions At higher values of ka the convergence would almost certainly be slower, though it is not obvious how fast the rate changes. This probably depends on the eccentricity of the spheroid in some manner which is difficult to predicto Although the back-scattering cross section is the only physical quantity computed in the foregoing it appears that more information could be obtained without too much troubleo The values of the A/ listed in the preceding chapter yield immediately the potential layer on the scattering surface, through equation (5)o Furthermore once these are known, the integration of equation (1) to give the scattered field at any point in space should be feasibleo 57

THE UNIVERSITY OF MICHIGAN 2591-1-T APPENDIX POWER SERIES SOLUTION The development in Chapter 5 indicates that the variational forms may be used in deriving a power series representation for the scattered fieldo The procedure and results in the case of the circular aperture problem have been discussed by Magnuso For the case of an electromagnetic wave striking an ellipsoid, the first two non-vanishing coefficients have been derived by Stevenson19 without reference to any variational expressionso Whether the latter offer any material advantage in deriving these and subsequent coefficients in the scalar problem is not immediately clearo At any rate they have been utilized to obtain the second and third coefficients for the prolate spheroidq and the results are given hereo The derivation proceeds along the lines described in Chapter 5o It is straightforwald but tedious, and the details, which are contained in an unpublished memorandum (2591-509-MO 11 June 1957), will not be included hereo 18Magnusq W "tInfinite Matrices Associated with Diffraction by an Apertureot" Research Report EM-32, New York Univo, Matho Research Group 1957o 9Stevensony Ao Fo "Electromagnetic Scattering by an Ellipsoid in the Third Approximationst Jouro Applied Physicso Volo 24S No0 99 Sept0 19530 -----------------— 58 _________

THE UNIVERSITY OF MICHIGAN 2591-1-T We are concerned with an expression for the scattered field f(tr) of the form f( ) Z nRnkn (75) It was shown earlier that the coefficients of index 0 and 1 vanish identically0 Stevenson has shown:that in the electromagnetic case at least the coefficient R3 also vanisheso It develops that this is also true in the scalar problems and that here R5 vanishes as wells though R7 apparently does noto We will limit ourselves to determination of expressions for R4 and R6 When the power series expansions of the quantities A 9 B-z and C/,z> are substituted into the linear system (32) and the expression (37) for the scattered fields it is easily shown that the nth coefficient in the series (75) can be written as an inner product Rn - b2 A o Bn (76) where A and B are ectors ose components are proportional where An and Bn are vectors whose components are proportional to certain coefficients in the series (39) for Ar and B> respectivelyv and that furthermore An 2 Cn Bn (77) 3/2 a where Cn1 is the inverse of a matrix Cn whose elements are proportional to certain coefficients in the series for C/~o 59

THE UNIVERS ITY OF MICHIGAN 2591-1-T Combining (76) and (77) we can write Rn as a quadratic form Rrn ba) 0 Bn Bn. (78) \a/ It can also be shown that the dimension of An and Bn should in general be 1(n+2)2 for n even3 or (.(n+l) (n-3) for n odd, though in the present problem a number of the components vanish9 so that the dimension is actually less than the specified value in each case consideredo Moreover, because of the vanishing of C/rz/ for7z + odd, it develops at once that the matrix Cn is the direct sum of two submatrices, one deriving from the even values of 0 andz)and the other from the odd2 so that the transformation is considerably simplifiedo For the computation of R, and R6 the essential forms are as follows o The quantities B,;and Co, are expanded in the power series B r 2b2 S b kr? s r' CJ = -4 r2b2a o3 fc, ki7,Ai+s (79) Ths not on dffers sgty f:ro h n hapte 5 tat.(This notation differs slightly from that in Chapter 5 in that for the Eake of economy we have removed common factors from the coefficients b~ and C<) The vector BL may then be written aB b (t boi g a2tc,.' b and the corresponding matrix C is ------------------- ~~ ~~60 ----------

THE UN IVERSITY OF MICHIGAN 2591-1-T Coo coo Co2 cU, U11 4 2 ~o 2 o C.4 cj0 0 0 -' c- 0 c~2 022 2 22 / \' Inversion of this matrix and substitution in (78) gives finally 2-0-2 02 2 b'. 00` R b2 (b22 bb CO22b2(b2)2 2: 0 + -a c co 2 2c i 1 2^ " -~ (80) e11 (c1( 2', )2 Similarly we can write P (b, b2 b0. b2, b, b1 bi b3 b1) > 29 29 4k 4 6l 19 3s 3 5 and. -, 0 0 0 0 cO 0C 0 0 c l 2 1~ 0~o o 0 0 c^2.1o2 ~ 0 c'l 0 ell C^ Q 0 l~ c0'0 0 2 o C6: 0 0 C2 o 0 o 2 co 22 o2 C11 13 c11 2 2 o 2 o 4 * Co Co2 ~00 Co 2 C,0 e;. ~'~2 c., which yield R6= b2! I _ t|)2 [2 cL2 oog 02 222)2 a2 | (co)3(c22)2'*u.,c,.(-oo~,:,)2...c o2 o,: oo22 ( 6 2 C (C 22. -...2 0)3 2 C2 o2)22 2 )3 C.)( 2C02 22.2 bb oo00 22)2 oo 2 022()2) bb2c02 ) +22..... 6,1 — _ —

THE UN I VE RS ITY OF MICH I GAN 2591-1-T t 2 bobg(coo)2(c2)2 (b2)2 (coo)3c22(Coo-2)2 -2 2 b b0 ( e c2 b2co2 b c)22+)22 bb (c ~0)3c22+(b2)2(c2oC22)0 2 b 2CO0 02b4)O)C2g1. - _n_9_ fL(bl)2 cl133 c1 cl 32 (c L)2c331 (bl)2(jl)2c33 11)3c33 o (b3)2(cl)3-2 b c lc33+2 bb3(c )2c32 bb l)2c33 (8) The values of the b'/and c'e/are obtainable from the forms derived in Chapters 2 and 3o The general expressions are tabulated belowo Values of bz'.. r.. _ __ \1 1 2' 3 k4 5 6 0' | ^ 2a a3 a5 F _| I' 3 - 15 420 2 1 Ia2 ia4, r ---- - | 3 5 384 5 2 a a 3 1 35 ______________ _ ~ 62

THE UNIVERSITY OF MICHIGAN 2591-1-T \Y( z2I sj Values of c/ - i 0 0 2 4/3 o 2 0 J2 K~2 2(32-2)+3 72.ogI iO 2 2 14a 2lo2 L( (+32) 2 log*i'j. 5a2 2 _=.. ^2 2 2 a K(6464- 6O(23 )+.3 (2(9^-4) log~tA] ______ 105 ____~"1j __a2_ I 3a lJ.i-.\. 1o' 1 2 _2 K2)(3r2) (3 l)2 logl ~1 3 20 r2(454 log2)3l 2 2 ____ 2 _.__2___54 4.60 2t,l _ ~ +l 29 2 ). ^ ~~~3 3 0 3 t(5g2-)1 f-~+62 log^-1 +- 4)- log il 0 2%"'t~-"L~_._..... r._.. 2 13 2 og %~z,~_1_~2~.)_;. )+(), ~ o. 1a2 2 -1 These formulas along with equation (43) may be used to obtain the first three coefficients'in the power series solution for an arbitrary spheroid0 For the 10~1 spheroid the values of 4 andO( are 10050 and ol005 respectively~ and substitution of these 63

THE UNIVERSITY OF MICHIGAN 2591-1-T in the above gives the following values of the R n n R2 60736 a3 o 10~3 R, -2838 a5 ~ 10~3 R6 2o922 a7 o lo4 For comparison with the exact solution it is convenient to obtain the corresponding series representation of the quantity CT.. - Sn (ka)n -r-b4/a2 we obtain as the first three non-vanishing coefficients (b;) b a5 S6 s 48(afr L' R2 -1,5296 79, aS aa The accompanying graph shows how the exact curve is approximated by the power series representationso It appears that the three-term expression gives an excellent approximation to the correct curve out to about ka l= oOo Using the three coefficients derived here it may be possible to develop an expression which gives a still better approximation in some range of ka> 1 but this has not yet been thoroughly investigatedo 64

THE UNIVERSITY OF MICHIGAN 2591-=-T 10.o0__ _____ (b) / lOiS//. (a) Exact Solution lo-0- - e/(b) Power Series, 1 Term (Rayleigh) /() "t 3Teims \ (c) o001 -.01l ~2 o3 o5 o7 lo0 2o0 ka BACK-SCATTERING CROSS SECTION n M _________________________ 65 -----— / —--- (R4aylegh 65 ~ ~ ()Pwr eis em

THE UNIVERSITY OF MICHIGAN 2591-1-T BIBLIOGRAPHY Bouwkamp, Co Jo0 "tDiffraction Theory-A Critique of Some Recent Developmemis"t, Research Report EM-509 New York Univo Institute of Mathematical Sciences, Division of Electromagnetic Research, (1953)0 Davis, Ho Tog "The Theory of Linear Operators", Principia Press, Bloomington, Indo (1931)o Erdelyi, Ao, Magnus, W.O, Oberhettinger, Fo, Tricomi; Fo Go0 "Tables of Integral Transforms 9t Volo 2, Bateman Manuscript Project, McGraw-Hill, New Yorks (1954)o Jones, Do So. "A Critique of the Variational Method in Scattering Problems", URSI-Michigan Symposium on Electromagnetic Wave Theory, IRE Transo Vol. AP-49 No0 3, July 1956, po 297o Levine~ H0 & Schwinger J o, "Diffraction by an Aperture in an Infinite Plane Screen. ItP Physo Revo Volo 74, (1.948), po 96lo Magnus, Wo "Infinite Matrices Associated with Diffraction by an Apertures" Research.Report No0 EM-32, New York Univ0 Matho Research Groups (195l) Magnus, Wo & Oberhettinger, Fo S "Formulas and Theorems for the Special Functions of Mathematical Physics", Chelseaq New York, (1954)o Mentzer, Jo Ro "tScattering and Diffraction of Radio Waves", Pergamon Press, London & New York (1955) 0 Morse, PO Mo & Feshbach, Ho, "Methods of Theoretical Physics", McGraw-Hill, New York (1955)o Rayleigh, (Strutt, Jo Wo). "On the Incidence of Aerial and Electromagnetic Waves on Small Obstacles", Philo Mago Volo44, (1897), po 28o0 Siegel, Ko Mo, Schultz, Fo Vo9 Gere, Bo Ho, Sleator, Fo Bo, "The Theoretical and Numerical Determination of the Radar Cross Section of a Prolate Spheroid", URSI-Michigan Symposium on Electromagnetic Wave Theory9 IRE Transo Volo AP-4, Noo 3, July 19569 po266o ---- t.... -66 r

ITTHE UNIVERSITY OF MICHIGAN 2591l1-T Sollfrey, Wo, ttThe Variational Solution of Scattering Problems", Research Report EM-11, New York UniVo Institute for the Mathematical Sciences, (1949) Sommerfeld,'A., "Partial Differential Equations in Physics", Academic Press, New York, (1949) Stevenson, Ao F., "tElectromagnetic Scattering by an Ellipsoid in the Third Approximation," Jour, Appo Physo Volo 24, No, 9, Sept. 1953. Watson, Go N., "Theory of Bessel Functions", Cambridge, (1952)o 67

UNIVERSITY OF MICHIGAN I3 90I1111 03521 035 3 9'015 03525 0375