THE semen OF THE {WM EQUATION FOR THE PROM}? SPHEROiDAL WWuW WSW fimhrifiobomodfih. D. WGAN STATE URIVERSITY Pafiicia James Wells 1955 -' 1W 0W WIMM/iriifl’ filflleIYIfi/Jlfl'fiiifiiiififill ; '0’ ’7 O O 9 t HE‘S 300627 7838 HERARY Michigan State University This is to certify that the thesis entitled The Solution of the Integral Equation for the Prolate Spheroidal Transmitting Antenna presented by Patricia James Wells has been accepted towards fulfillment of the requirements for Doc tor ' a degree in Mathematics Weutflu Major professor Date 3‘11! 28. 1955 0-169 )VlESI.l RETURNING MATERIALS: Piace in book drop to LJBRARJES remove this checkout from w your record. FINES will - be charged if book is returned after the date stamped beiow. THE SOLUTION OF THE INTEGRAL EQUATION FOR THE PROLATE SPHEROIDAL TRANSMITTING ANTENNA By Patricia James Wells ABSTRACT Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1955 Approved by Wat. P, Wiles... ABSTRACT This thesis thoroughly studies the problem of the prolate spheroidal transmitting antenna formulated as an integral equation. The Albert-Synge integral equation (Quarterly of Applied Mathematics, Vol. 6, 1948L,satisfied by the current on an antenna surface of revolution, is solved for the case when the surface is a prolate spheroid. As a part of this solution, expansions are found in terms of the prolate spher- oidal wave functions for the components of the field of an electric dipole. By means of these expansions recursion relations are found between the spheroidal wave functions of orders one and zero. Finally, the Hallén method of successive approximations, which has been used to solve the integral equation for the finite cylindrical antenna, is applied to the Albert-Synge integral equation for the spheroid. The approximate solutions so found are then compared with the known exact solution. THE SOLUTION OF THE INTEGRAL EQUATION FOR THE PROLATE SPHEROIDAL TRANSMITTING ANTENNA By Patricia James Wells A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1955 ACKNOWLEDGMENTS The author wishes to express her appreciation to Professor Charles Po Wells for his stimulating advice and frequent encouragement. She also wishes to thank Professor Alfred Leitner for his assist- ance and for the interest he has taken in this thesis. . Thanks are due to Professor R. D. Spence for the use of his unpublished tables of spheroidal functions. The writer is deeply indebted to the Office of Ordnance Research for the financial support which made this invest— igation possible. DEDICATION To Jacky and Mary Wells DEDICATION To Jacky and Mary Wells TABLE OF CONTENTS Introduction ....................... I. Background 1. The Prolate Spheroidal Wave Functions.................... 2. The Solution to the Boundary Value Problem.............o.. II. The Integral Equation of Albert and Synge l. The Solution to the Albert- Synge Integral Equation for the Prolate Spheroid......... 2. Recursion Relations.... III.Successive Approximations.. 1 6 10 16 25 34 INTRODUCTION The problem of the perfectly conducting, radiating antenna, when formulated mathematically, becomes a boundary value problem. Consider a mathematical surface S comprised of two parts: 1.) the actual surface of the antenna, and 2.) the gap, or source of radiated energy. Vector functions, E and g, are sought, which, on S, and in the infinite region exterior to S, are regular functions satisfying Maxwell's equations. Certain well-known boundary conditions are prescribed. The tangential component of the electric vector 5 5_ I fit? »’--- Fig. l §_must vanish everywhere on S except at the gap where this function is given; the behavior of the field (§,fl) is prescrib- ed at infinity. This condition at infinity, which will be stated more precisely later, is known as the radiation condition. For many problems it is assumed that the field (§,.fl) possesses rotational symmetry, which essentially reduces the vector problem to a scalar one, thereby simp- lifying the mathematics of the problem. ' Despite the fact that an antenna problem may be so simply stated, the number of antenna problems to which exact solutions have been found is few. The difficulty lies in the nature of the surface S. In general, the only antenna problems which may be solved exactly by standard techniques are those for which S coincides with a coordin- ate surface in a separable coordinate system. One is, therefore, immediately limited to such Surfaces as the sphere, the spheroid, the infinite cylinder, etc.. Although some two-part exterior boundary value problems have been solved exactly for LaPlace's equation (Karp, 9), the comp- arable electromagnetic problems have proven more difficult; There exists no exact solution, for example, for such a simple surface geometrically as the finite cylinder. Of the eleven coordinate systems in which Maxwell's equations separate, the spherical and spheroidal coordinate systems are the only ones having a family of finite coord- inate surfaces; for this reason the mathematical theory of the spherical and spheroidal antennas has been extensively studied. The prolate spheroid, particularly, has been the subject of wide investigation because for eccentricities nearly one it closely approximates the finite cylinder of small width. The wave functions themselves are basic to the difficult— ies encountered in prolate (or oblate) spheroidal problems. A suitable representation for the functions is in itself a problem, because at one extreme they must approximate the spherical wave functions and at the other cylindrical funct- ions. The ordinary differential equations, obtained by separation of the wave equation, have series solutions whose coefficients satisfy a three term recursion formula, thereby making computation of the functions difficult. Although a variety of series representations for these functions have been proposed, those defined by Stratton, Morse, Chu, and Hutner (22) are used in this thesis because of their more general acceptance in the literature. Further difficulties arise because relatively little is known concerning the properties of the functions. The many relationships between the functions of different orders and their derivatives, which are known for the hypergeometric functions, have not been made available for the spheroidal functions. Because other methods have not proven fruitful, the problem of the finite cylindrical antenna has been formulated as an integral equation, (Hallén, 7; Brillouin, 3; Albert and Synge, 23) and the solution sought by various methods of 4 successive approximations. These methods, however, have been subject to considerable criticism and frequent revision. The series solutions found by these approximations are not readily computed, and nearly nothing is known concerning their con- vergence or divergence. This thesis studies the problem of the radiating prolate spheroidal antenna formulated as an integral eqiation. The original motivation lay in the hOpe that this study might elucidate the cylindrical antenna problem. In Part I the spheroidal functions as defined by Stratton et a1 (22) are outlined; the problem of the prolate spheroidal antenna is solved by use of a Green's function and Green's formula, which yields an integral equation in the usual way. After Infeld, (8) a step function feed is assumed over a gap of finite width. The solution obtained here becomes that of Stratton and Chu (21) if an infinitesimal gap is assumed in- stead of a gap of finite width. The integral equation of Albert and Synge (23) is taken up in Part II, where the main results of this thesis are to be found. This integral equation is an exact formulation of the antenna problem and, being stated for a surface of revol— ution, it may be readily specialized to either the cylinder or the prolate spheroid. The statement and theory of the Albert-Synge integral equation is summarized; this resume is followed by a solution of the equation for the case of the prolate spheroid. This solution is effected by the expansion of the kernel and the unknown function in series of spheroidal functions of order one. The unknown coefficients in the expansion for the unknown function are determined by means of the orthogonality of the angular spheroidal functions. The solution so obtained, again under the assumption of a gap of finite width, is identical to that found in Part I. Expansions in terms of the spheroidal wave functions of order one are found for the components of the field of an electric dipole. By means of these expansions, recursion relations are found between the spheroidal wave functions of orders zero and one. Finally, in Part III, a method of successive approximat- ions, comparable to the method of Hallén, is applied to the Albert-Synge integral equation for the spheroid. The approx- imate solutions found thereby are then compared with the known exact solution to the integral equation. I BACKGROUND rm v = conST‘otnT ‘ U =CO“$T0;nT ‘ ’1 VI fill )>2 V=J “ v2 -\ z! ‘ < u 0 f ‘ is Fig. 2 l.) The Prolate Spheroidal Wave Functions A cross-section of the prolate spheroidal coordinate system is illustrated above. The surfaces v = constant are confocal hyperboloids of revolution, and the surfaces u = constant are confocal ellipsoids. The third coordinate d is the azimuthal variable about the axis of the system, in this case the z-axis. The surface d = constant is a half plane terminating at the axis. The relations between Cartes-- ian coordinates and the spheroidal coordinates are as follows: x = f(1 - v2)§’(u2 - l)\:cos d 2.: 2 k . (1-1) y f(1 - v )‘(u - 1) Sln d z = fuv where f is the semi-focal length, and 1:5 u < - “15%? cos (kfu _ 9.511) Radial functions of the third and fourth kinds are defined: (1-10) 12“” = Rm + 1132‘ mn mn mn t R2) \3) - 1) . Rmn I 13run - 1 mn t The behavior of R3, and Hg; for large u is easily found from mn (1-7) and (l-9). . m+n (1-11) am Fin em‘leu " "7"“) The Wronskian of the two linearly independent solutions is YD; (1-12) wmmn, ‘2‘] = 1 R 2 * mn kf(u 3 l) Inasmuch as the parameter kf will remain fixed (except for the computation in Part III), it will be omitted as an argu- ment for both the angular and radial functions; For the sake ofcompleteness the series for the radial functions is included. 09 I (1-13) an 2 T IN2‘ __gg (u - ll,lo .n-r mn (r + 2m); J (u) émn (u) " um ‘-zdmn (r + 2m); 1 dr r: nm+r(u) Z- r r! m+r Y T=Ot 10 Finally, the angular functions form a complete set and are orthogonal dfl.the interval (-l,1)} \ _ 0 (1-14) ‘(Smn(v) Smr(V) dv ”.N :33 ”Wk "5‘1 11111 ‘t 2. The Solution_tg the_Boundary Value Problem for the Prolate_§pheroida1 Antenna The boundary value problem for the prolate spheroidal antenna, as formulated by Stratton and Chu (21% is outlined in this section. The method of solution differs insofar as a Green's function is used instead of the standard boundary value techniques. Certain modifications are made in the Stratton-Chu theory regarding the applied field at the gap; namely, after Infeld (7), the width of the gap is assumed to be finite. ‘ Let the surface, S, described in the introduction, be, in prolate spheroidal coordinates, the spheroid u = u', which coincides with the antenna and covers the gap from which energy is radiated. Harmonic time dependence is assumed. That is, all components of the vectors E and E vary with time as e'th, where w = 2““ = kc = 2E3, withvthe frequency’,,\the wave length, and c the velocity of light. For mathematical 11 simplicity, it is also assumed that E and H are independent of d; this is equivalent, physically, to assuming that the voltage at the gap is applied symmetrically. Under this hyp- othesis, Maxwell’s equations, curl E = ikfl, and curllg = -ikE, consist of two independent sets. The one which is of interest here involves only Eu’ Ev’ and Hfiz . 2 2%. <9 2 ‘% ‘1M(“'V)Ev:gu(““1)H¢’ \ ikf(u2—v2)‘Eu = 525(1 - v2)‘H¢ (1-15) 5 (1 - v2)‘§;(u2— v2)“‘Eu - (u2— ll‘g—Emz- v2).‘/"EV N ile (112-- V2)H¢' ' If Ev and E11 are eliminated from (1-15) the following equation for H¢ is obtainedl 2 . 2 2 a a 2 o _ (11 “V ) __ a LUV} - [SEW an (1-16) + f2k2(u2 — v2)l H¢,= 0 Upon separation of variables and comparison with (1—4), Sln(v) and R1n(u) are found to be the eigenfunctions of equation (1—16). There are two boundary conditions, as mentioned in the intro- duction. First, the total tangential component of the electric field must vanish of the surface: (1-17) EV 4- EA‘ z 0 u‘u' 12 where Ev and EA are the tangential components of the induced and applied fields respectively. EA is given. The second boundary condition concerns the behavior of the field at in- finity: (1-18) lim R Uzi-mg - ik_E_1 = R-so- where'g is the exterior unit normal to S, and R the radius of a large sphere with center at the center of the spheroid. This limit must hold uniformly in all directions. A more general statement of this condition by Synge and Albert is given later. For the applied field EAa step function is assumed: on the gap EA is equal to a constant voltage divided by the length of arc across the gap, while elsewhere on the surface E is A zero. For a gap extending from -v to v1, we have 1 2V; _ V _V(1 — v 2) E_--—--—--- = \v|fv 'A 2Vlev 2v 1f(u2 — v ?)V’ 1 (1—19) 2 O \V|>-vl The function EA may be expanded in a series of the orthogonal angular functions. ‘l on (u2— v2)“ Em = - 11(- ZBnS ln(V) (1-20) M20 “(1 - v 2) B = 521 n(v) dv Stratton and Chu (21) assume instead an infinitesimal gap. Under this assumption the coefficients in (1—20) become 13 Bn = Sln(0). However, as is pointed out by Infeld (8), the solution obtained using (1-20) becomes logarithmically infinite as the gap width goes to zero. Infeld shows that the assump- tion of a finite gap width, long compared to the width of the antenna but short in comparison with its length, leads to a solution without this singularity. For this reason we assume a finite gap width. We now apply Green's formula, which, for P = P(u,v) and P = Po(uo,vo), reads: {[G(P, Po) 5950 H¢(P0 ) - H¢(Po) §~EOG(P, P) ] dS0 (1-22) H¢ (P) P outside 5 = 0 P inside S é) . . . wheres-fi-o is the normal derivative to S, and G(P, PO) is the Green's function satisfying the following conditions: 1. G satisfies L[G] = 0 except at P = P0° 2. G(P,?O) = G(PO,P). 3. lim L[G] dU‘= -l, wheref>is the radius Pfio of a small Sphereoz with center at Po' 6) IL — v 40 éfa(u2 -1) G(P, P O):= 0 when u - u . The particular Green's function which satisfies these 14 conditions was found by Leitner and Hatcher (13). k 29:81 1n(v) Sl no(v) Rln)(u)R(3)(uo ) u< uO (1-23) G(P P ) = - ’ ° 2" {33m > N1. Rii’w >R13’ u < u 11:0 ' I where the prime indicates differentiation, WI; (u) = (u2-1Y4R§;(u), and R({;(u) is the following radial function: ' In the Green's function (1-23), the angular functions of the first kind only are chosen to satisfy regularity conditions, and the radial functions of the third kind are chosen to satisfy the radiation condition (1-18). 0n the surface S: uo= u"5,u, we have . . ‘ asO = 21Tf2(u'2- 1)/1(u'2- (3),“de ' C I: 9 (W2 - ll 3 (1 ) ano ( ’ o) f(11.2 _ v2)": au' ( o) = - '1' . G(P,P ) f(u'2— l)'la(u'2- vi)’3 0 f(U'Z- l)‘IKu'2- VET/o. a I -— ' ' fiofliflu ,vo) - 1kEA(vO) - H¢(u ,vo) 15 Substitution of these expressions into (1-22) yields: u.- (1-26) H¢(u,v,u') \ \ r I = Zfiiku‘ITu'2- 1)A(u'2- v:)AG(u,v,u?,vO)EA(vO)de -n The substitution of (1-20) and (1-23) into (1-26), followed by integration making use of the orthogonality of the Sln(vo)’ gives 0" S (v) R(3)(u) (1-27) H¢(u,v,u') : ikV: B ln ln n=0 n 2 Y Nln .35. [(u' affaifi’w'n Infeld shows that the convergence of this series is assured if v1 is chosen so that f(u'Z-lfalog V1 is finite. Equation (1-27) becomes identical to the Stratton- Chu solution (21) if the coefficients Bn are taken equal to Sln(0)’ This summary of the work which has been done per- taining to the exact solution of the spheroidal antenna problem (with the modifications in method and gap assumpt- ions mentioned above) is included as general background. in order that it may be compared later with the solution to the Albert-Synge integral equation 16 II THE INTEGRAL EQUATION 0F ALBERT AND SYNGE The work of Albert and Synge (23) leading to an integral equation which exactly expresses the antenna problem for a surface of revolution is summarized below in section 1. Also in section 1, the integral equation is specialized to the case where the surface is a prolate spheroid. This equation is then solved and the solution so found is identical to (1-27). In section 2 the main results of this thesis are found. The Albert-Synge integral equation is used to find the expansions for the components of the field of an electric dipole. By means of these expansions recursion relations are found be- tween the spheroidal functions of orders one and zero. 1. The Albert-Synge Integral Equation and its Solution for the Prolate Spheroid The integral equation of Albert and Synge (23) is a statement of the boundary value problem for the perfectly conduction, radiating antenna, described in the introduction. Its derivation does not make use of the scalar or vector potentials, which are used in the derivation of the Hallén equation (7) for the cylinder. Furthermore, the Albert-Synge equation is a statement of the complete vector problem. 17 Basic to the work of Albert and Synge is the following theorem: (2—1) (23) Let (E,H) and (E', H') be two electromagnetic fields, each satisfying Maxwell's equations in an infinite region R, bounded internally by one or more surfaces S; letneither field have any singularities in R or on S, and let each field satisfy certain conditions for outward radiation at infinity; then where n is the exterior unit vector normal to S. This theorem may be expected to hold in any region R for which Green's theorem is true, (Kellogg—10). Albert and Synge impose a more general condition on the field at infinity than (1-18); this condition, referred to in the statement of the theorem, is: A constant B exists such that at all sufficiently great distance5/O from some fixed point 0 we have |§| < B/p \Ifl < 13/1: l§*flxfll +9411; av = ikAx‘P 1kPPO _, e (2-2) Wherew- #15:;- This field satisfied both the above condition at infinity 'and Maxwell's equations everywhere except at the point P0 where it has a pole (of third order for the electric vector and of second order for the magnetic vector). If Po lies inside 8, then the dipole field is regular in R, and (2-1) holds; if, on the other hand, PO lies in R, then (2-3) fit-(E x E' - _E_' x EMS = 4“'iik_‘[3_°_.l_\.~ S where E on the right hand side is evaluated at P0' The proofs of (2-1) and (2-3) together may be viewed mathemat- ically as a proof that Green's formula for the vector wave equation, VxVx _E_ — k2 E = O, is valid in the infinite region R, when the fields in question satisfy the given condition at infinity. The field of the dipole plays the role of the Green's function for the vector wave equation. (Morse and Feshbach, 16) Equation (2~3) gives the field (E,H) (for appropriate choices of A) in terms of its tangential components on the boundary S of the region R. 19 Albert and Synge apply equation (2-1) to the special case of an antenna, whose surface, including again the ficticious surface over the gap, has rotational symmetry about the z-axis. Symmetric excitation of the antenna is assumed, as it will be throughout the rest of this thesis, so that E and g are independent of the azimuthal angle. Cylindrical coordinates (z,R,d) are introduced, and the dipole A is taken to be a unit vector in the positive 2 direction. It is located on the z-axis inside the surface S of the antenna. Fig. 3 For the sake of completeness, the final integral equation of Albert and Synge, satisfied by the current on the surface of an antenna of revolution is included: *2 2 (2-4) (k2 V(' 32$ 6%} ZO)I(z)dz = 2Tiikc 13(z)RZ[1+R'2 131; d 2 dr 11 gap 20 'where the antenna extends from z :{1 to z =£2; R = R(z) is the equation in cylindrical coordinates of S; E(z) is the tangential component of the electric vector at the gap, and E(z) = 0 outside the gap; I(z) is the current across a . ikr section 2 = constant. “P = e /r, where r2 = [R(z)]2 + (z - zo)2. Having now summarized-the theory of Albert and Synge, we apply it to the case of the prolate spheroid. Although equat- ion (2-4) may be transformed directly to spheroidal coordinates, it is easier as well as enlightening to consider equation (2—1)o A_remains a unit vector in the z direction and may be chosen to lie on the degenerate surface u = 12 F> 4 J A“ ~--— , >2 he 1: *4 Fig. 4 The components of;A are 1 l 2 '2’ 2 2 _ u(l - v ) _ v(u - 1) (2-5) Av -. ..1. Au - _ .1. (u2 _ v2) 2 (u2 _ V2)2 Again W'= elkr/r, where r2 = f2(u2 + v2 + v2 - l - 2uvvo). It is easily verified that the following identity holds. (2-6) 9.3.: _ V(U~2 - 1):)‘t‘ _ u(l - v2) 34’ 5V0 (u2 -v2) 6“ (u2 -55) av Using (2-5) and (2—6) the existing components of the field (_E_' I?) are found from (2-2). \ \ H; = .1f1£(12_ v2%:z(u2 _ 1)/2[u a”) _ VAT—1 (u -v 5—1: 3v __i_1_s_2 ‘l; 2\/1.:_l.‘d\P f(u-l) (1'V)rr (2-7) ‘I ‘P ,_ 1 (1 — v2)‘ 2 2 32 EV-Hi“§(u2 -v2)7"[fk u)? -3V3vol E. =1_ (uz - 1)!2 [12ng _ 92)” 11 f2 (u2 _ v2?“ auavo \ 1 1, Since 2 = inand d5 = 2T1f2(u'2- 1)"(u‘ -V )JV, with v ranging from -1 to l, the integral equation (2-1) becomes: 1 ‘I (2-8) (u'2 — v2) 1 (13", H?! .. EV dev = o -l The current I(fu'v) across any cross-section z = fu'Va may be found from H(fu'v) by v - '2 VJ. 2 v1 0 I(fu v) - 2T7f(u - l) (l — v) H(fu v) Since a symmetrically excited field is assumed H¢ satisfies L[H¢] = 0 defined in (l — 16). Therefore a series SXpansion for Hp’ of the following form may be formally a S sumed. 22 oo (3) S (v) R (M) _ E 1 1 n=0 In Where the C(n are unknown coefficients independent of u and v. The Sln(v) are chosen since the solution must be regular at i l, and the radial functions of the third kind are chosen to satisfy the radiation condition. For EV = EA’ the expansion (1 - 20) may be used. It remains to find expansions for H; and El . An eikr expansion for /r is known, (Meixner—lS)’where‘r is the distance from the point P(vud) to the point Po(vouop6) ikr 00 5 (W8 (V ) e _ . mng_ mn 0 (2-10) r - 1k E em cos.m(¢ - #0) N7 m,n=O mn 1) 3 szm (u) lem)(uo) u 5 uO (D II II 2 for m = O 1 m # 0 Because Ré;)(l) = O, m # O, (2 -10) becomes for u0 = l . n(v) 80 (V ) ”1 V0 (1) (3) = 21k:5m1 Ron (l)Ron (u)3u > 1 R (2—11)e Using (2-11) and the expressions (2—7) for the components of the dipole field, series expansions for H3 and E; may be found in terms of the spheroidal functions of order zero and 23 their derivatives. Substitution of these series, as well as the series (2—9) and (l - 20) into the integral equation (2-8) leads to an infinite system of equations in infinite- ly many unknowns, because only a partial orthogonality relation exists between the spheroidal angular functions of orders one and zero. Since no recursion relations were known which would transform the functions of order zero into those of order one, or conversely, another approach was taken. The question of recursion relations for the spheroidal functions will be taken up later. The field (E', _I_{") of the dipole satisfies Maxwell's equations, and is independent of d. Consequently H3 will satisfy'L[H3] = 0. By the same reasoning that was used to form the series (2—9), we have: (p (2-12) H; = c n=0 111W) (3) R1n (u) n In where the coefficients Cn are independent of u and v. The distance r is symmetric in v and v0, so that-1r-g-ymu5t also possess this symmetry. Similarly, (l -'v§fh H; is symmetric in v and v0, so that ln( )81 n( ) (2 — 13) H' Ex v v0 R(3)(u) ¢= (l - vo fl;£ Nln where the coefficients Dn are now independent of u, v, and v0. Proceeding formally, we have from Maxwell's equations: 1 1 (2-14) - ikf (u2 — v2f(l — vgf'E; a) S (V)S (v ) -E 1n 1n 0 d 2 /J~ (3) _ Dn N111 EH11 - 1)R1 (11)] n=o Substitution of (2-13), (2-14), (2-9), and (1-20) into the integral equation (2—8), followed by integration yields: a) (v ) (2-15) :Dn4nifi‘f“ Rifi’m )—-— du. [(u'2 _ 1) R(3)(u' )1 a) S (v ) = ikV E B D —l§——2— R(3)(u ) 11:0 n n in Multiplication of equation (2— 15) by S ), followed by 1r(vo integration from -1 to l with respect to yo, gives us the unknown coefficients: Br m[u'2-1) YtR13)(u')] (2-16) 0(r = ikvd These coefficients are precisely those in the Stratton-Chu expansion. When they are substituted into equation (2-9) we have R(3)(u) CO V EE::I3 'lgl“‘ d (3) Mayo n In E'Hu'z ' libRin (u')] 25 The fact that this solution is identical to (lfifl) may be regarded as a verification of the exactness of the Albert- Synge integral equation. 2. Recursion Relations If the coefficients Dn in the eXpansion (2-13) for H3 could be found, then two series representations for the funct- ion Hg‘would be known. The first is the eXpansion (2-13) in terms of the spheroidal functions of order one, while the other is obtained from (2-11) and is in terms of the spheroidal functions of order zero. Equating these series would yield a relationship between these functions. Consider equation (2-3) where the dipole is taken outside the surface in question. For the case where S is a prolate spheroid,‘é =.l£’ and the field (§.g) is independent of the azimuthal variable, the only 'unknowns' appearing in the integral equation are the coefficients in the eXpansions for E; and Hg. By use of the orthogonality of the angular functions 26 equation (2-3) may be solved for these coefficients. It will be noticed, however, that the expansion for H3 will be somewhat different when PO is taken outside the surface u = u'. Fig. 5 In spheroidal coordinates, the z-axis is divided into three parts: 2 _<_ f, -f 5 z :5 f, and z. 3 1‘, which are resp— ectively the surfaces v = -I, u = I, and v = 1. In the eXpansion (2-13) only the case where P0 lay on the surface u = l was considered. Now let P0 = Po(il,u0). The expression 2 = f2(u2 + ug + v2 -1 — 2uuov) which is symm— for r becomes r etric is u and uo. It follows that (ui — 19&H3 is also symmetric in these variables. (The double prime notation is used to indicate that the point of singularity is P0(i1,u0) instead of P0(vg,l).) An expansion for H3 is therefore 1 a) Sln(v) Réi)(u)R§i)(uo) u 5 uO 2 V 2 N (1) (3) ¢ (uo- 1)‘ n=0 n 1n R1n (uO)R1n (u) uog u (2—17) where the coefficients Fn are independent of u, uo, and v; For P held fixed away from the z-axis, the field of the dipole is a regular function of P0' As a consequence, the expansion (2-13) for H3 eveluated at V0 = i1 must be identical to the expansion (2-17) for H3 evaluated at u0 = 1. Thus we have m 1n(V) Sin‘Y ) (39 ( ) Dn‘"R’"" " v2)tR1n “ _ (1 - 11-0 V3,” ( a) 3) - §;n(v) Rln (u) (3) “ Fn N1 2 'H74 R1n (u) n=o n (u - 1) Ju- \ Again multiplying by Slr(V) and integrating over the interval of orthogonality, yields ' (1) R1n (u) _ Sln(v) ”'1‘” F m " ”n I??? H:\ V:\ That these expressions are well-defined follows from (l-S) and (1—13) which define the angular and radial functions. The integral equation (2-3) may now be used to find the coefficients Fn’ and, by (2-18), the coefficients Dn’ For P0 = Po(1, uol, we write (2-3) in spheroidal coordinates. (2—19) ‘ 4fi1kEV(1,uo) = 2fif2 [tu'zv 2)&(u'2— 1)4{EVH3 E"H¢] dv IJsing Maxwell's equations and (l—27), the left hand side of (2-19) becomes: 28 (2-20) 4_._«____._ W R(i)(u%—[(1—-v2)7*s; M] 41111:}: (1’,u ) = - ‘1 :Bn v=1 v O f(uz- Dan [(u'f 1)‘/‘R(3)(u )']N1n %fiv If the apprOpriate series are substituted into the right hand side of (2-19), we have, after integration with respect to v2 (2-21)2Wf(u 2-_ 1) f: R(3)(u0 ) wm‘l’m )R(3)(u o] B F (110 - 1) n n Nlnn-(ii'E-J-.[(u'2 -11)"R§i) (u' )1 where‘W[R [R(1) R§3)] is the Wronskian of the radial functions of the first and third kinds and may be found from (1-12), Setting (2-20) equal to (2-21), and equating coefficients of the R(3)(uO ), yields: 2 -1 “ kf(u'2-l) ‘ The unknown coefficients Fn and Dn are therefore found to be (v) ] Rmms 2 ln 2 1n (2-22) F‘ = 2k 1‘ A D = 2k /f , . n / [(1 - V25} n (u2- l v=1 u= Dropping now the double prime notation, we have, finally, for the expansion for the d component of the magnetic vector of the dipole field: 29 (2-23) . 2 ‘I 2'& , =1k (u -l)"(1-v ) W 9* Hy: T (,2 42¢ [“S‘E'VS‘G] 2 ‘0 1n(v) 512%) 1 R(1)(u)R(3)(uO ) u 5 no ‘ 2“ /f 2:8‘11121, '17" (1)1(3) n=0 (1- v0 ) (ug - 1) ln Rl (uO )R1 (u) uo_ u where P0 is understood to lie on the z-axis. That is, either u0 = I, or vO =71, or both; I The function H3 when multiplied by either (u2 — 1)Aor 2 ‘A. by (1 - v0, depending upon the location of P 0, satisfies all the requirements for a Green' 5 function for the infinite domain except that, at the point P = P0 it possesses a singularity of higher order than that specified for the Green's function; The series (2—23) of eigenfunctionSmay nonetheless be expected to converge uniformly in any closed and bounded domain which does not contain the singular point P0; We now equate the series (2-23) for'Hg with u0 = l to the series for H; obtained from (2-11): a) R(1) (u S (v)S (v ) (2-24) 2k2/f E 1“ 1” g V R(3)(u) n=o N1n (1 ' v0) (uz- v2) an '1 ‘ a) (1) = 2k2/f(u'2" l)t(1 — v2)!l Z Son(V) Ron (1) n=o X [u Son(v)Rgi)h(u) - v sgn(v)R§:)(u)] 30 Multiplication of (2-24) by Sor(vo)’ followed by integration with respect to v0, removes the summation sign from the right hand side. Further multiplication by Slm(v) with integration with respect to v removes the summation from the left hand side and yields the following recursion relation. \ (1) (2-25) + (”(u) R113“ f 27‘ (112-1% (1 - v §)S lm(V)Sor(V)dV u=l“' ‘I; = (u2 - 1) 12mm [uR(3)' (u) (_.l__-__\L;.250 (v)Slm(v)dv u - v 2'la (3 v(l - v 2) a -Rorzu) (u2 - v2) Sbr(V)Slm(V)dv where the primes denote differentiation and (m - r) is even. The integrals in the coefficients vanish for (m - r) odd. Equation (2-25) gives the functions R§:)(u) in terms of Régzu) and its derivative. Because Riizu) ==R£31(u) + iR£31(u), it follows that (2-25) holds for the radial functions of all four kinds. Equation (2-25) may be somewhat modified. Substitution of the asymptotic forms (1-11) for the radial functions into the terms of (2-25) which are dominant as u‘? ax yields: (2- 26) . .kf.ftl - M)IS (v)SO r.(v)dv uzl 31 Ill-1“ 2 18 odd or The plus or minus sign is chosen according as even. Equation (2—25) may now be written: I I (2—27) ikfRi:)(u) _[ (1 - v2)h§1m(v)sor(v) dv “I \ | 2 /2. = (u2— 1)Vl[uRgi)'(u)‘j-(1 ' v2) Slm(V)Sor(v)dv -'(u2-'v ) \ V2 - Rgfi’(u) [ ::§_‘v§)l 50 (v) Slm(v)dv] Another relationship between the spheroidal functions, although this time only involving functions of order zero, may be found by using the identity (2—6). If the expansion (2-11) is substituted into this identity we obtain, after multiplication by Som(v) Sor(vo) and integration with respect to both v and v0: | (2-28) — Rgl’(1) R(3)(u) j—s (v)s'm (v) dv + R(3)(u)rJ:tu(1 ' v 2L So r(V) Som(v) dv] (u2 - v 2) where (m - r) is odd. By use of the asymptotic forms (1-11) we obtain an identity similar to (2-26): I I [Sor(v)55m(v) dv == ikf (v Sor(V) Som(v)dv (1) (2-29) Rom.(1) 1 Ror <1) -' 32 In (2—29) the upper or lower sign is chosen as m :Zr - 1 is even or odd. The recursion relation (2—28) may now be re- written; I :kf Rgg)(u) v30r(v)50m(v)dv - (3): v(u2- 1) -Ror.(u)Jfl;—§———§ Sor(v)80m(v)dv (2-30) (u _ V ) (3) u<1 _ v2) , + ROr (u)[(u2- 7v?) S‘or(V)Som(V)dV Recursion relation of the same form as (2-27) and (2—36), but with certain coefficients unknown, were shown by I. Marx; (14) to exist for all orders of the functions Réi)(u). Later, after the above equations had been derived independently by this author, it was discovered that complete recursion relations had been published in a report by the University of Michigan Willow Run Research Center (20) for the radial functions of all kinds and orders. It is shown in this report how these relations may be modified to hold for the angular functions or both the first and second kinds. When the Willow Run recursion relations are specialized to the present case, they become identical to (2-27) for m = r, and to (2-30) for m = r+1. The derivation of the recursion relations which is employed by Marx (15) and (20) is based on a method used by E. T2 Whittaker for Mathieu functions. It is altogether different from the derivation employed in this thesisfi 33 The comment is made in.the Willow Run report (20) that the coefficients in the recursion relations "are at least as difficult to compute as the functions whose computation they are supposed to simplify}" So much is certainly true. On the other hand, the problem treated here, that of solving the Albert-Synge integral equation for the prolate spheroid, is witness to the fact that problems do exist to which the re- cursion relations may be directly applied. The application of the recursion relations to the infinite system of equations in infinitely many unknowns, mentioned on page 23, leads directly to a linear equation from which the unknown coeff- icients (in) and hence the solutions may be obtained. Other relationships may be derived in similar fashion if the method is modified by changing the position and orien— tation of the dipole. 34 III SUCCESSIVE APPROXIMATIONS An integral equation satisfied by the current on the surface of a finite cylindrical radiating antenna was derived by Hallén (7) in 1938, and a solution based on a method of successive approximations was advanced by him at this time. This method has been the subject of much revision and some criticism. Among the many authors on this subject other than Hallén are Bouwkamp (2), Middleton (12), King (11) and (12), Harrison (11), Gray (6), Brillouin (3), Schelkunoff (l8), and Dike (5). Hallén's integral equation is one of the first kind: i (3~l) J(M(z,zo) I(z) dz = f(zo) 0 where I(z) is the unknown function and the kernel and f(zo) are known. The general scheme of Hallén's method of approx- imations, as outlined by King and Middleton (12), depends upon two functions defined by I(z) (3-2) g(z,zo) ='f(;;) and k W(zo) ={K(z,z0) g(z,zo) dz 0 Using this notation, equation (3-1) may be written: 35 4.. I(zo).rK(z,zO) g(z,zo) dz M (3-3) f(zo) o L + jflK(z,zO) [I(z) - I(zo)g(z,zo)] dz I(zO)W(zO) +.fKKz,zo)[I(z) - I(z0)g(z,zofld27 This formulation of equation (3-1) is an attempt to write the equation as an integral equation of the second kind so that an iterative process of approximations may be applied. It is now argued that for a good choice of g(z,zo), the difference integral will be small, and if equation (3-3) is solved for I(zo) _ f(zo) 1 f... (3-4) I(ZO) ”W - wag) K(Z,ZO)[I(Z) - I(zo)g(z,zo)]dz 0 then the quantity f(zo)/W(zo) may be taken as a good first approximation, 11(zo)f‘ The second approximation, I2(zo) is found by substitution of 11(z0) into the difference integral, and similarly for each successive approximation. In this manner, I(zo) is‘eXpanded'in a series of powers of l/W(zo). It is a matter of some controversy whether or not the choice of the expansion parameter W(zo), which of course depends upon the choice of g(z,ZOL is important. However,it seems reasonable, as is contended by King and Middleton, that at least the first few approximations will be better if g(z,zo) sin k (h -IZ)):- sin k (hr-\zd)’ is carefully chosen. On this basis, they use instead of elkrused by Hallén, because sin k(h -|z|) is known 36 to be a close approximation to the current. Other authors have tried various other functions in arriving at an expansion parameter} A number of difficulties with Hallén's method of solution arise. First, even the first few approximations are difficult to compute owing to the complexity of the functions involved. Furthermore, the computed solutions do not agree as closely with experimental values as might be expected. (Dike: 5) These discrepancies may be due to the fact that so few terms of the series can be practically computed, or to the unknown behavior of the series solution with respect to convergence} In the hope that some light be shed upon the dependability of the method, a comparable series of approximations is sought for the integral equation for the prolate spheroid. In this case the exact solution is known, and the approximated series solution may be compared with if? To this end equation (2-1) is put in the form of (3—1)? (3_5) J{K(vayo) H(v)dv = ikV M(VO) S (v) (3) where M(v) =»Z Bn-—lfl—— R ln (u) R53)(u) _. N "T“? 1‘ n—o ln (u - l) (D S (v)S (v ) mm =;: e n=0 Nln d 2 “7(3) KEN“ -1)R1n (u)l as can be verified using the results of Part II. If it is 37 considered that the dominant term of the series (l—27) is a reasonably good approximation for H, (the subscript ¢ is drop— pe) now inasmuch as numerical subscripts will be needed) then S (v) (3-6) g(v,vo) : ~19...— 1n(vo) S with the choice of n increasing with kf, n = O, 2, 4, ... For values of kf which are not too large, the function 510(v) may be used. Because 810(v) is an even function ranging from O to one as v goes from 1 to 0, it is seen to behave similarly to sin k(h -\z\) = sin kf (l -|V|). For this choice of g(v,vo) we have: (1) R (u) I 10 d 2 (3 (3-7) W(u) Elm] mun - 1/‘R10(u)1 u=l which is independent of V0 and constant for any given Spheroid. If equation (3-5) is written in the form of (3—4) we have I IJIIMV’VO) [H(v) - H(vo) g(v,vo)1dv ‘1 (3-8) Tit-em“) = 3%- H(vo) - The first few approximations are given: a) ..1.. - - if ikV H1(Vo) " M(vO)/W _ W n=o - its! H2(v0) - 2H1(v0) - W J2(v0) H3(v0) =H1(vo) + H2(v0) - ;:~ J2(v ) + W3. J3(v0) 38 where a) R(1)(u)n S (v ) 1 J (v ) =E B —-:-L-—--—— 43—3-— R§3)(u){d [(u2 - 1) R(3)(u)§n o r ’ r du n r=o (“2 " 1f; Nlr -- u=l In general Hn (vO ) contains a term-la Jn(vo ). ‘W The approximations H1, H2, and H were computed for kf = 2, 3 various values of u, and for appropriate gap widths. The value given for v in each case was chosen so that the series involved would converge faster. These results are given below. The numerical computation is not very satisfactory, because, although the series for H1 and H converge, and converge relatively rapidly, the convergence of the series for J1 and J2 is open to question. The sums were taken to four terms: n = O, 2, 4, 6. The odd terms vanish because for a symmetrically located gap the Br for odd r are zero. H = ikV( .911 - 1.682i) u = 1.005 H1 = ikV(l.73O - 2.470i) v1= .25 H2 = ikV(1.453 - 5.295i) V = '25 H3 = ikV(—10.616 — 12.769i) H = ikV( .318 - 1.105i) “ = 1°020 H1 = ikV( .624 - 1.486i) v = .3 1 _ . . v z .25 H2 — ikV( .348 - 2.2201) H = ikV(-1.272 - 2.7011) 39 u - 1.044 H = ikV( 0127 r .8431) v1= '35 H1 = ikV( .287 - 1.0171) v = .25 H2 = ikV( 2052 - 1.288i) H3 = ikV( -.284 - 1.1561) u = 1.001 H = ikV( .053 + .064i) V1: '2 H = ikY( .068 + .0671) v = .2 1 H2 = 1kv( .017 + .7651) H3 = 1kv(2.242 + .6681) No positive conclusions, of course, can be drawn on the basis of these figures. They are, nonetheless, suggestive. The first approximation is in most cases also the best approx- imation. It is not too surprizing that the H be good approx- 1 imations since the first terms of the series for H and H1 are identical. In both cases the first terms of the series is also the dominant term. This is an immediate consequence of the choice of g(v,v0). If one assumes that the series of successive approximations converges, it is to be expected that the second and third approximations be better than they are. How- ever, the terms of the approximations should be computed for other 40 values of kf, before more can be said about the possible divergence of the approximation series. The basic difficulty with respect to this method seems to lie in the fact that the integral equation is one of the first kind. The initial choice of E(V,VO) appears in every term of the series, and is never affected by the iteration. Difficulties are in general to be expected with integral equations of the first kind. (Courant and Hilbert : 4) If the problem could be formulated as an integral equation of the second kind, the extensive theory dealing with such equations would be available; at least the usual methods of successive approximations could be applied. 10. 11. 12. 41 REFERENCES Bergman. S. and M. Schiffer. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. Academic Press, New York, 1953. Bouwkamp, C. J. Hallén's Theor for a Straight Perfectly Conducting Wire, Used as a Iransmitting or Receiving Aerial. Physica 9: 609-631, 1942. Brillouin, L. The Antenna Problem. Quart. Appl. Math. 1: 201-214, 1943. Courant, R. and D. Hilbert. Methods of Mathematical Physic§ Vol. 1. Interscience Publishers Inc.,fiNew York, 1953. Dike, S. H. Difficulties with the Present Solutions of the Hallén Integral Equation. Quart. Appl. Math. 10: 225—241, 1952. Gray, M. C. A Modification of Hallén's Solution of the Antenna Problem. Jour._App1. Phys. 15: 61-65, 1944. Hallén, E. Theoretical Investigations into the Trans- mitting and Receiving Properties of Antennas. Nova Acta (Uppsala) 11: 1-44, 1938. Infeld, I. The Influence of the Width of the Gap upon the theory of Antennas. _Quart. Appl. Matho 5: 113-132, 1947. Karp, S. N. Separation of Variables and Weiner-quf Technique . Research Report No. E”M”-25, New York University, New York, 1950. Kellogg, O.D. Foundations of Potential Theorr. Frederick Ungar Publishing Cempahy, New York, 1929. King, R. and C: W: Harrison Jr. The Distribution of Current along a Symmetrical Center-Driven Antenna. Proceedings_gf the IIRIE. 31: 548-567, 1943. King, R. and D. Middleton. The Cylindrical Antenna: Current and Impedance. Quart. Appl. Math. 3: 302-335, 1946. “" Leitner A. and E} C. Hatcher Jr. Radiation from a Point Dipole Located at the Tip of a Prolate Spheroid. Jour. A221. Phy. 25: 1250- 1253, 1954. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. Marx, I. Recurrence Relations for Prolate Spheroidal Wave Functions. Jour. Math. and Phys. 32: 269—275, 1954. Meixner, J. und F. W. Schéfke. Mathieusche Funktionen und Spharoid-Funktionen. Springer4verlag, Berlin, 1954. Morse, P. M. and H. Feshbach. Methods of Theoretical Physics. Academic Press Inc. New York, 1953; Myers, H. A. The_Unsymmetrically Fed Prolate Spheroidal Antenna. Unpublished Ph. D. Thesis, Michigan State College. 1954. Schelkunoff, S. A. Advanced Antenna Theory. John Wiley and Sons, New York, 1952. Schultz, F. V. Scattering byda Prolate Spheroid. UMM-42 Willow Run Research Center, Univ. of Mich., 1950. Siegel, K. M., B.H. Gere, I. Marx, and F: B? Sleator, The Numerical_Determination of the Radar Cross- Section of aiProlate S heroid. Willow Run Research “Center, TERM-W6, ni LIT" of lch.,' 1953. Stratton, J. A. and L. J. Chu..Forced Oscillations of a . Prolate Spheroid. Jour. Appl. Phys. 12: 241- 248, 1942. Stratton, J. A.Q 2. M. Morse: L. J. Chu, and R. A. Hutner. Elliptic.Cylinder and Spheroidal Wave Functions. John Wiley and Sons, New York, 1941. Synge, J. L. and G. E. Albert. The General Problem of Antenna Radiation and the Fundamental Integral Equation, with Application to an Antenna of Revolution. Quart. Appl. Math. 6: 117-156, 19482 . ._ l1: « "7'11111111141143