Studies in Radar Cross- Sections -XI The Numerical Determination of the Radar Cross-Section of a Prolate Spheroid by K. M. Siegel, B. H. Gere, I. Marx and F.B. Sleator Project MIRO Contract No. AF 30 (602)-9 Willow Run Research Center Engineering Research Institute University of Michigan UMM- 126 December, 1953' Now Chairman, Department of Mathematics, Hamilton College, Clinton, New York

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 STUDIES IN RADAR CROSS-SECTIONS I Scattering by a Prolate Spheroid, by F. V. Schultz (UMM-42, March 1950). II The Zeros of the Associated Legendre Functions Pnm(/') of NonIntegral Degree, by K. M. Siegel, D. M. Brown, H. E. Hunter, H. A. Alperin, and C. W. Quillen (UMM-82, April 1951). III Scattering by a Cone, by K. M. Siegel and H. A. Alperin (UMM-87, January 1952). IV Comparison Between Theory and Experiment of the CrossSection of a Cone, by K. M. Siegel, H. A. Alperin, J. W. Crispin, H. E. Hunter, R. E. Kleinman, W. C. Orthwein and C. E. Schensted (UMM-92, February 1953). V A classified paper on Bistatic Radars by K. M. Siegel (UMM-98, August 1952). VI Cross-Sections of Corner Reflectors and Other Multiple Scatterers, by R. R. Bonkowski, C. R. Lubitz, and C. E. Schensted (UMM-106, October 1953). VII A classified summary report by K. M. Siegel, J. W. Crispin, and R. E. Kleinman (UMM-108, November 1952). VIII Theoretical Cross-Sections as a Function of Separation Angle Between Transmitter and Receiver at Small Wavelengths, by K. M. Siegel, H. A. Alperin, R. R. Bonkowski, J. W. Crispin, A. L. Maffett, C. E. Schensted, and I. V. Schensted (UMM-115, October 1953). IX Electromagnetic Scattering by an Oblate Spheroid, by L. M. Rauch (UMM-116, October 1953). X The Radar Cross-Section of a Sphere, by H. Weil (to be published).

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 XI The Numerical Determination of the Radar Cross-Section of a Prolate Spheroid, by K. M. Siegel, B. H. Gere, I. Marx, and F. B. Sleator (UMM-126, December 1953). XII A classified summary report by K. M. Siegel, M. E. Anderson, R. R. Bonkowski, and W. C. Orthwein (UMM-127, December 1953).

WILLOWY RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 TABLE OF CONTENTS SECTION TITLE PAGE List of Figures iii Nomenclature iv Preface x Introduction 1 I General Theory 4 II Prolate Spheroidal Functions and Recurrence 17 Relations III Exact Solution 28 IV Scalar Scattering 34 V Approximations 38 VI Mark III Computer 45 VII Machine Computations 46 VIII Results 52 Appendix A Recurrence Relations and Other Formulas for 57 Prolate Spheroidal Functions and Wave Functions Appendix B Formulas Used in Computations 61..._____.____________________ i ---------------------— __,

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 TABLE OF CONTENTS (Continued) SECTION TITLE PAGE Appendix C Computation of Bistatic Cross-Section 69 Appendix D Quantities Tabulated in Mark III Output 70 References 72 Distribution 75 -- ii,, _i

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 LIST OF FIGURES Number Title Page 1 Prolate Spheroidal Coordinates 17 2 Logical and Sequential Structure of Computations 48 3 Back-Scattering from a Prolate Spheroid, Nose-On 56 iii

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 NOMENCLATURE Symbol Definition First Used on Page Amn Separation constant for wave equation 19 An Expansion coefficient for incident wave 28 a Semi-major axis of spheroid, a = F4o 32 BNn Determinantal element 38 b Semi-minor axis of spheroid 48 CNn Determinantal element 39 c c = kF 19 cmn Expansion coefficient for wave functions 9 DNn Determinantal element 39 Duvw Range of variables u, v, w 6 mn dk Spheroidal coefficients 41 Ek Coefficient in recurrence relation for dmn 42 E Electric vector of electromagnetic field 5 e, eu, ev, etc., Unit vectors in directions of subscript variables 9 iv

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 NOMENCLATURE (Continued) F Semi-focal length of spheroid 17 F General wave function 9 — mn F Solution of vector wave equation 5 Fmn Coefficient in recurrence relations for dmn 42 f Solution of scalar wave equation 6 fmn Scalar wave function 7 mn Gk Coefficient in recurrence relation for dmn 42 k k gkmn Expansion coefficient in series for Amn 47 H Magnetic vector of electromagnetic field 5 hk Expansion coefficient in series for Amn 47 Nn Ik Boundary integrals of angular functions 50 i Imaginary unit, i = -T 5 Jn(C ) Bessel function of first kind 47 jn(C~) Spherical Bessel function of first kind 41 27T. k Wave number, k = = 4 Lu, Lv, L Linear differential operators resulting 6 from separation of wave equation ---- " -----------— v

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 NOMENCLATURE (Continued) XM( )YM )M Vector wave functions 12 n mn n -mn m Characteristic value of separation 7 constant p Nmn Normalization integral for angular 24 functions Pk+m(7]) Associated Legendre function of 43 first kind p Separation constant for scalar 6 wave equation Q-k+m( ) Associated Legendre function of 43 second kind q Separation constant for scalar 6 wave equation Rmn(c, ) Radial spheroidal function 19 r Radius vector 16 r Separation constant for scalar 6 wave equation S Poynting vector, S = ReE x H*) 15 S( i(c IT) Angular spheroidal function 19 t Time variable 4 U (u) Solution of equation Lu[U(u), p, q] = 0 6 UNn Determinantal element 39 ---------------------- v i I

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN -UMM-126 NOMENCLATURE (Continued) u Arbitrary independent variable 6 V(v) Solution of equation Lv[V(v), p, r] = 0 6 v Arbitrary independent variable 6 VNn Determinantal element 39 W(w) Solution of equation Lw[W(w), p, q, r] = 0 6 WNn Determinantal element 39 w Arbitrary independent variable 6 x, y, z Rectangular coordinates 7 an Expansion coefficient in series for scattered 30 wave an/, Normalized expansion coefficient, (a' = a /E'a 32 n n amn Coefficient in recurrence relation for R() 25 3n Expansion coefficient in series for scattered 30 wave 01 nNormalized expansion coefficient, fn =,/E'a 32 mn Coefficient in recurrence relation for R( ) 25'mn mn F(x) Gamma function 41 Y ~mn Coefficient in recurrence relation for R($ 25 7mn mn Yn Expansion coefficient in series for velocity 35 potential vii -I

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-126 NOMENCLATURE (Continued),a(,T) A(,7?) = 42- 1 2 18 6. Kronecker delta function 65 1,j e Dielectric constant 4 4 Arbitrary independent variable 20 T7 Angular spheroidal variable 17 0 Spherical variable 51 Kmn Ratio of radial to angular function 21 X Wavelength of radiation 46 I, ~Permeability 4 i, j Parity function, uij =2 -[1 + (-l)i+] 41 Radial spheroidal variable 17 Ir Coefficient in recurrence relation for Rn 25 Pmn Coefficient in recurrence relation for R ) 25 p Function of index k which tends to zero as k 43 approaches integral values ~() (T)()=1 - T2 18 O( Scattering cross-section 32 0r Coefficient in recurrence relation for R( 25 mn mn _____..........________ viii _____-.________________

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-126 NOMENCLATURE (Continued) T() T() = 2_ 1 18 0 Azimuthal variable 7 mn Spheroidal scalar wave function 59 X mn Spheroidal scalar wave function, 59 Xmn(xy,) = Xmn Spheroidal wave function 22 4.' Velocity potential 34 wc Circular frequency of radiation 4 -----------------------.. __ x...ix

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 PREFACE This paper is the eleventh in a series of reports growing out of studies of radar cross-sections at the Willow Run Research Center of the University of Michigan. The primary aims of this program are: (1) To show that radar cross-sections can be determined analytically. (2) To elaborate means for computing cross-sections of objects of military interest. (3) To demonstrate that these theoretical cross-sections are in agreement with experimentally determined values. Intermediate objectives are: (1) To compute the exact theoretical cross-sections of various simple bodies by solution of the appropriate boundary-value problems arising from the electromagnetic vector wave equation. (2) To examine the various approximations possible in this problem, and determine the limits of their validity and utility. (3) To find means of combining the simple body solutions in order to determine the cross-sections of composite bodies. (4) To tabulate various formulas and functions necessary to enable such computations to be done quickly for arbitrary objects. (5) To collect, summarize, and evaluate existing experimental data. Titles of the papers already published or presently in process of publication are listed on the back of the title page. K. M. Siegel....... —-- x — i

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 INTRODUCTION In May 1951 it was found necessary to determine theoretically the nose-on radar cross-section of a prolate spheroid of fixed dimensions as the wavelength of the radiation decreased from the Rayleigh region (wavelength greater than characteristic dimension) to the first maximum in the resonance region (wavelength approximately equal to characteristic dimension). It was known that a maximum was to be expected in this "resonance" region; the exact location and width of this maximum (in terms of wavelength) and its exact height (in terms of cross-section) were in question. The work of Mie (Ref. 28) on the sphere and of Schultz (Ref. 21) on the prolate spheroid supplied the basis for these computations. By 1 September 1952 the necessary numerical analysis had been done and the problem had been coded for the Mark III Electronic Calculator at the U. S. Naval Proving Ground, Dahlgren, Virginia; and the large scale digital computations were begun soon thereafter. By 15 November 1952, the computations described in this report were completed. This paper presents the solution to this problem. Although it was not possible to compute a sufficient number of points to determine the entire curve of cross-section vs. wavelength, the location, width, and height of the first resonance maximum are now accurately known, as is the entire curve for greater wavelengths; for shorter wavelengths only certain general features are known. This curve is presented as Figure 3, in Section VIII. The prolate spheroid problem can by no means be considered completed. This report contains an addition to our knowledge of exact solutions and can be considered a good start in our ability to predict the results of the scattering of sound or electromagnetic waves by a prolate spheroid. However, the numerical results obtained apply to only one spheroid (ratio of major to minor axis 10:1), to only one position of the transmitter (nose-on), and to only one direction of scattering (back-scattering). ---------------------,..... —-------

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 In addition to the solution of the scattering problem, this report includes recurrence relations and other identities that illuminate the mathematical structure of the spheroidal functions used for the solution. General results about the recurrence relations among solutions of linear second-order differential equations, obtained in the course of the present work and given in detail in References 19 and 20, may be of value in other problems of mathematical physics. The approach used to solve the problem of the spheroid is precisely that used by Mie forty-six years ago to solve the problem of the sphere. Anyone who has carried out the numerical computations necessary to determine the scattering from a sphere has probably felt that there must be a less cumbersome way of handling such problems. Until such a way is found, however, the present method must be used. When this method is applied to determine the scattering from a spheroid, there are added difficulties due to the fact that the only known representations of the spheroidal functions that are of practical applicability are representations in series of spherical or cylindrical functions. It is hoped that the theoretical results obtained for the spheroidal functions are a start towards simplifying the problem of scattering by a prolate spheroid at least so that it is no more difficult than the problem of scattering by a sphere. Many people cooperated in the work described in this report, and to give a precise assessment of each personal contribution is quite impossible. Aside from the authors, the major collaborators were: Dr. W. Bauer, Mr. R. Beach, Dr. D. M. Brown, Mr. H. Hunter, Dr. L. M. Rauch, and Miss I.Wyman. We wish to express our sincere appreciation to the Bureau of Ordnance, U. S. Navy, for its generous support of the numerical computation on the Mark III Electronic Calculator. We are further indebted to Mr. Ralph A. Niemann, Head of the Computation Division, Computation and Ballistics Department, U. S. Naval Proving Ground, Dahlgren, Virginia, and to Messrs. G. H. Gleissner, D. F. Eliezer, Karl Kozarsky, and A. M. Fleishman of the programming and coding staff at Dahlgren who spent many hours preparing the difficult machine program and checking the numerical results. 2

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN. UMM-126 Section I contains a discussion of the physical problem to be solved and the mathematical procedures employed. It may be omitted by those familiar with the method of Hansen. Section II, which is essentially independent of the remainder of the report, contains a discussion of a new theory of recurrence relations as applied to the prolate spheroidal functions. This contribution was made after work on the original problem had been completed, and the results of this section were not used in the computations. Section III reviews the work of Schultz in obtaining an exact expression for the radar cross-section of a prolate spheroid. Section IV carries out a similar analysis for the scattering of scalar (sound) waves by a prolate spheroid. Section V discusses the approximate expression which was actually evaluated by machine computation, and other approximations which made a machine program possible. Section VI describes the Mark III digital computer on which the calculations were carried out. Section VII outlines the program for machine computation. Section VIII lists the numerical results obtained, and compares the conclusions drawn from the data with various other results. Those interested only in the numerical results of this investigation may turn immediately to Section VIII. -------------------—.3 ------------

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN - UMM-126 GENERAL THEORY The radar cross-section of a perfectly conducting prolate spheroid is to be determined for an incident field due to a plane electromagnetic wave whose direction of propagation is parallel to the major axis of the spheroid. It is assumed that the medium exterior to the scattering body is free space, with fixed dielectric constant e and permeability AL and with zero conductivity. Furthermore both the incident and the scattered field are assumed monochromatic, with fixed circular frequency w; the solutions for polychromatic fields may subsequently be derived by Fourier series or Fourier integral methods. The mathematical problem of obtaining an expression for the scattered field from knowledge of the incident field, and deducing the radar cross-section, is first discussed in general terms. Either the electric vector or the magnetic vector alone suffices to determine the radar cross-section. In the monochromatic case, the electric vector may be written eiwt E and the magnetic vector eiWtH where E and H are complex vector functions of position. For this form of time dependence, Maxwell's equations combine to form the system of differential equations Vx VxF-k2F = 0, V F = 0 (I-1) for F = E and for F = H, where k2 = E t W. The method of solution to be described below is such as to ensure that the vectors obtained to satisfy (I-l) will automatically satisfy Maxwell's equations as well. On the surface of the scattering body, it is required that the total electric vector field be normal and the total magnetic vector field be tangential. At infinity, it is required that the scattered field behave asymptotically like a diverging spherical wave with center in the scattering body. In the present report the scattered electric field will be obtained. 4!

WILLOWY RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 Mathematically, the boundary conditions on the electric field may be described as follows. Let u, v, w be the variables of a coordinate system in which u = uo, a constant, is the equation of the surface of the scattering body. Denote the incident electric vector by E', the scattered electric vector by E, and the total electric vector by E" = E' + E. On the surface of the scattering body it is required that the components E'(uo, v, w) and Ew(uo, v, w) vanish, or that Ev(uo, v, w) = -Ev(Uo, v, w), E(u,0 v, w) = -E (uo0 v, w). (1-2) At infinity it is required that on a sphere of radius R with center in the scattering body, and with exterior unit normal N, one has lim R (N.V)E + i kE] =0, (1-3) uniformly in all directions. Recent results of F. Rellich (Ref. 2), H. Weyl (Ref. 3 and 5), and C. Mueller (Ref. 4) show that the mathematical problem of solving the system of equations (I-1) subject to the boundary conditions (1-2) and (1-3) possesses a unique solution, provided that the scattering body is sufficiently smooth. In the present case the smoothness condition is satisfied, and to prove that the desired electromagnetic field has been obtained it is sufficient to show that the conditions (I-1), (1-2), and (1-3) have been met. If a vector F furnishes a solution of the system (I-1), each cartesian component f(x, y, z) of the vector must be a solution of the scalar wave equation V2f + k2f = 0. (1-4) Furthermore, if the vector F satisfies the "vector radiation condition" (1-3), each cartesian coordinate f must satisfy the scalar radiation 1. This form of the boundary condition at infinity is readily obtained, e.g.,by combining the results of pages 68 and 85 of S. Silver (Ref. 1) with well-known vector identities........ —---- ------------

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 condition: on a sphere of radius R, one has lim R[N-(Vf) + i kf] = 0, (1-5) uniformly in all directions. Suppose that the scalar wave equation separates in the u, v, w coordinate systemi, i.e., possesses solutions of the form f = U(u) V(v) W(w), where each of the functions U, V, W satisfies an ordinary linear differential equation of the second order. In this case powerful tools are furnished by the methods of functional analysis, as applied, for example, by P. M. Morse and H. Feshbach (Ref. 6) and by B. Friedman (Ref. 7). Corresponding to each solution f there are three separation constants, p, q, r (only two of which are independent), which appear in the separated differential equations. With Lu, Lv, Lw denoting the appropriate linear differential operators, one finds that U, V, and W respectively satisfy the equations Lu[U(u), p,q] = 0, Lv[V(v), p,r] = 0, and Lw[W(w), p,q,r] = 0. In the case to be considered (axial symmetry), Lw depends on p alone, and the notation is to follow this assumption. The further condition that, for each fixed set p, q, r, the scalar wave equation possess one solution which is finite and single-valued in the entire range Duvw of the variables u, v, and w determines sequences of separation constants m', qmn' and rmn, and corresponding sequences of solutions Umn(u), Vmn(v), and Wm(w) of the equations Lu[Umn' Pm' qmn] = O~ Lv[Vmn, Pm' rmn] = O, Lw[Wm, pm] = 0, respectively (m, n = 0, 1, 2,...). In the most frequently occurring case of an axially symmetric coordinate system, the variable w = 4 is the azimuthal angle about the axis of symmetry. In this case one has 6 ~

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 d2W Lw[Wm, Pm] L [Wm Pm]- m + Pm Wm() = 0~ d p2 and the condition that the solution be single-valued determines the separation constants Pm = m2 and the functions W0 = a0, a constant, and Wm = am cos mc + bm sin m 0, am and bm constants (m = 0, 1, 2,...). The functions fmn = U) (u) Vm) (v) Wm (w) finite in the entire range Duvw are called scalar wave functions of the first kind. A theorem found to be very useful is that every solution g(x, y, z) of the equation (I -4) which is finite and single-valued in the region Duvw can be expressed as a double series - m 0 n c f ( x, )( (I - 6 ) m=0n - 0 Cmn mn (x, y, z), and that the series is uniformly convergent in any closed, bounded subregion of DuvwIf the surface u = uo of the scattering body encloses the finite singular points of the differential equation Lu = 0, a more general series expansion appertains. Let U (2)(u) be a set of solutions of mn Lu[Umn, Pm' qmn] = 0 each of which is linearly independent of the function of the first kind with the same subscripts m, n. These are called functions of the second kind, and so are the wave functions (2) (2) (1) fmn Umn Vmn Wm In general the variable u will be infinite in range, and the functions of the second kind can be so chosen that of the two combinations (3) (1) (2) (4) (1) (2) fmn fmn mn fmn mn = fmn mn (1-7) one satisfies the radiation condition (1-5) while the other satisfies the "absorption condition" obtained from (1-5) if one replaces the imaginary 2. This theorem is easily deduced from the results about separable equations given in Reference 6 (1953), especially in Section 6.3, in conjunction with well-known theorems about Fourier series. - ~.. - 7,

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 unit i by its negative. In the present work, the functions of the fourth kind satisfy the radiation condition. In the region Du vw exterior to the scattering body, say described by u > u, the functions of the third and fourth kinds are finite and single-valued, and therefore any solution g' (x, y, z) having the same behavior in Du vw may be expanded in a double series of wave functions of the third and fourth kinds, which converges uniformly in any bounded, closed sub-region of the region. By choosing functions of the fourth kind one is certain that g' will itself satisfy the radiation condition, and conversely that any function satisfying the radiation condition will have such an expansion. In the usual case both VrJ1v) and Wm(w) form orthogonal sets over their respective ranges Dv and Dw, that is JDv fD fmn VWnI(v) Wmt(w) dvdw = 0 (I-8) v w unless both m = m' and n = n1 (or there is a similar relation with a weight factor under the integral sign). The standard Fourier series methods now enable one to determine the coefficients in the series for a function g or g' from the values of the function on u = u0 alone. The solution of the vector wave equation system (I-1) proceeds in two steps. First one must obtain an expression for the incident electric vector, and then given this vector one must satisfy the boundary conditions (1-2) on the surface of the scattering body by a vector function that is a solution of (I-1) and satisfies the radiation condition (1-3) at infinity. To this end a variety of methods have been devised for generating sets Fmn of vector functions that satisfy (I-1) from sets fmn of scalar functions that satisfy the scalar wave equation (1-4). Two properties of these sets are of importance: there must be enough functions to permit the expression of an arbitrary wave function in terms of them (completeness) and it must be possible to satisfy the boundary conditions (1-2). Previous treatments of such problems have generally assumed the first property without proof, and have concentrated on the objective of making the boundary conditions easy to satisfy. A brief discussion of these previously suggested procedures will be followed by a sketch of a completeness proof for the method of the present report, and this ---------------------- 8 ______________________

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-126 proof with minor modifications can be extended to any method in which differentiation operators applied to solutions of the scalar wave equation are used to obtain solutions of the vector wave equation. If Fmn (u,v,w) is the given set of wave functions, and if the incident and the scattered fields are respectively assumed to have expansions E = Z Cmn Fmn and E = Cmn Fmn where the coefficients cmn are known, then the boundary conditions (1-2) give Z Cmn Fmn (uv.w) * ev (u0,v,w) E E Cmn Fmn (uovw) ev (uo,v,w), E CCmn mn (u0o,v,w) e w (u0,v,w)v = - Cmnn Fmn (u'v'w)-w (uO'v'w)l where ev and ew are unit vectors in the directions of increasing v and increasing w, respectively. For the plane, the sphere, the cone, the paraboloid, and the wedge, wave functions can be found that permit evaluation of the coefficients cmn individually. For other scattering bodies it seems that an infinite system of equations for the infinite set cmn of unknowns is required. Some sets of wave functions that have been used may be briefly reviewed. Let fmn(x,y,z) denote throughout the set of solutions of the scalar wave equation (1-4) obtained by separation in the appropriate system of coordinates. In the more general case when div E 0, i.e., when the region surrounding the scattering body has a non-zero charge density, Maxwell's equations combine to form the wave equation V x Vx F - V(V-F)-k2F = O (I-10) for F = E or F = H, and the additional boundary condition div E = G(v,w), a given function, on the surface of the body, is needed to determine a 9

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 unique solution. For this more general problem the plane wave functions fmn a fmn a x b, fmn (a_ x b) x a may be used, cf. Reference 8, p. 395, where a is a vector in the direction of propagation of the wave and b is another constant vector. If a is replaced by the radial unit vector 2r, and if any one of these functions is multiplied by an amplitude factor that is a function of direction and integrated over the unit sphere of all possible directions, the result is a more general wave function. For a sphere, the method of P. Debye (Ref. 9) has been shown by A. Sommerfeld in Reference 10, Ch. XX, to be based on the use of the vector functions fmn fr alone. Since these have a zero component in the direction of increasing azimuthal angle (, in general both electric and magnetic vectors of this type are required for the construction of a field, see Reference 10. For axially symmetric bodies, Morse and Feshbach (Ref. 6) suggest the functions Uln(u) Vln((v) e, and the curl of each of the functions, where Uln and Vln are the functions obtained by separation of variables as discussed above and e0 is the unit vector in the direction of increasing azimuthal angle. However, the applicability of these vector wave functions is limited by the fact that the field expressed by them must have its component in the direction of e independent of'. For such a case they would greatly facilitate the satisfaction of boundary conditions on an axially symmetric body. A very general method can be obtained by extending the idea of W. W. Hansen (Ref. 11), described in detail in Reference 8. This method was originally devised for the solution of problems in spherical and circular-cylindrical coordinate systems. For the spherical case the sets Vfmn, Vxer fmn Vx( Vxer fmn), and for the cylindrical case the sets Vfmn, Vx fmn Vx ( Vxa fmn) 10

MWILLOTW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 are appropriate, where fmn denotes the solution of the scalar wave equation in the particular coordinate system and where a denotes a vector along the axis of the cylinder. The gradient vectors have zero curl but non-zero divergence, and it is expected that they would have to be employed for the solution of the more general equation (I-10). The other two sets have zero divergence. If the expansion of an arbitrary solution of the first equation in (I-1) in terms of functions from these two sets can be carried out, in other words if completeness of one or both sets can be established, the solution will automatically satisfy the second equation div F = 0, and the problem is solved. The more general method to be described includes, besides the spherical and circular-cylindrical cases, all scattering problems such that the scattering surface is a contour surface of a coordinate system in which the scalar wave equation is separable. Equations (1-9) will not, in general, reduce to a finite system, but completeness of the sets of wave functions employed will be an immediate consequence of the completeness of the set of scalar wave functions. As has been pointed out, a vector function which has been shown to solve the vector wave equation and to satisfy the boundary conditions on the body and at infinity is the unique solution (Refs. 2-5). The incident wave and the scattered wave are expanded by essentially the same method. The most general incident wave to be considered is a plane elliptically polarized wave. Both the electric and the magnetic vector may be obtained from the superposition of two linearly polarized waves lying along the axes of the ellipse, and it is intended to give an expansion for only one of these linearly polarized waves. This wave has electric and magnetic vectors of the forms E' = E' eik(r a) and H' = H eik(r. a) respectively, where a'is a unit vector in the direction of propagation and where E' and Ho are constant, complex vectors whose amplitude factors are real, mutually perpendicular vectors lying in the plane normal to a. First, in terms of the scalar wave functions fmn for the given coordinate system, write ik(r.a) = E Cftn f (1) m=0 n=0 -11..

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 The coefficients c'mn are determined from the relations rnn i rw (i) (1) C1n mn Vmn(v) Wm(w) dv dw v RW = Jf e ik(r a) V () W dv dw. v w This scalar expansion leads immediately to an expansion of the incident wave in terms of the vector wave functions of the first kind = V x e f(1) (1) Vx ey f e(1) x e f XMmn - xrnx Y YMm mn -z in n obtained from the second set of Hansen circular cylinder functions if one chooses the constant vector as a cartesian unit vector. The incident electric and magnetic vectors have the forms E' = E 0o E c mn f( H' = Ho E cmnf ). (I- l) E'rnn m) -E~ 1Z Cn mn_ The two vectors together form an electromagnetic field, and so Maxwell's equations E =-xH HI, VxE (I-12) kv — k g- - must hold for E=E', H=H'. Applying (1-12) to (I-ll), one obtains E -- iJJE r [mnrHox m) + Hy' M() + H Z (n)] (1-13) H' Ci "-inn (1)1 z Hl c E' y El) H' _ E: Cmnn[Eox Mmn Eoy -mn oz Mmnj m= 0 n=0 where E = El' e + E' e + El' e, H' =H' e + H e + H' e. 0 ox-x oy -y oz -z' -o ox-x oy-y oz -z -------------- 1~12..

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 The series (1-13) may be rewritten as series in the functions ar)n/aX, af(n/8ay, and afm)n/az, with coefficients that are precisely the coefficients of fmn in the expansions (I-11) of the cartesian components of E' and H'. In the analogous expansion in vector wave functions Mmn of the scattered field E, H (or of any electromagnetic field), again the compo(1) nents of the gradient of fmn would appear, and the coefficients would come from the expansions of the cartesian components of E and H in terms of the scalar wave functions fmn. (Of course functions of the first kind must be used for fields finite at the singularities of the separated functions, while functions of the fourth kind must be used for fields finite in more restricted regions but possibly having singularities outside those regions.) These expansions in terms of scalar wave functions are, like (I-11), absolutely and uniformly convergent in any bounded, closed subregion of space containing no singularities of the field. The derived series (1-13) or their analogs have the same convergence properties in all cases where the derivatives of fm), at a fixed distance from the origin, vanish asymptotically for suitably large m and n to at least as (1) high an order as do the functions fmn themselves. A brief computation will show that the spherical, the circular-cylindrical, and the spheroidal scalar wave functions will have this property. It is to be expected, in fact, that all wave functions Mmn derived by the Hansen method have the same sort of asymptotic behavior. It follows that the vectors xMmn YM lmn' ZMmn form a complete set of vector wave functions, if by completeness is meant the property that any solution of the system (I-l) has a convergent series expansion in these vector functions. For the spherical or the circular-cylindrical (1) scalar wave functions fmn, the components of the gradient can be explicitly expressed as finite linear combinations or as absolutely and uniformly convergent series in the wave functions fmn by use of the recurrence relations for the functions Umn (u) and Vmn(v), and for spheroidal functions analogous expressions are given on page 79 of Appendix A. Substitution of these expressions into the series (1-13) and application of well-known theorems on the convergence of double series is an alternative method for proving the completeness of the vector wave functions. ____________________________ 13

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 The radiation condition (1-3) at infinity is automatically satisfied since functions of the fourth kind are used for the expansions. To satisfy the boundary conditions (1-2) on the surface of the scattering body, one must solve the equations similar to (1-9) obtained by setting equal to zero the v and w components of the total field. In the most general case it will be necessary, in addition, to utilize the boundary condition for the total magnetic vector: H"(uov,w) = 0 or Hu(u, v, w) = - Hu(o v, w). The previously detailed theory shows that the magnetic vector H of the scattered field has an expansion 00 00 H= 2 X n _x + Cn e + e f(4) (I -15) m=0 n=0 lmn mn y mn z mn The first equation in (1-12) consequently gives i t x x.(4) _6(4) 4 E - E [cmn mn + cmnMn + cmn M]mn](I16) m=O n=0 When all the functions in (1-13), (I-15),and (1-16) are expressed in terms of the unit vectors e u, ev, and ew, and substituted into the boundary conditions (1-2) and (1-14), there result three sets of infinitely many equations in the infinitely many unknowns cmn, cnn, and czn. Another form of the solution may be obtained by first expanding the scattered electric field as in (1-15), then deriving the scattered magnetic field as in (1-16) by the second equation in (1-12). This procedure leads to three sets of equations for the three sets of coefficients in the expansion of the electric field. In all attempts to produce numerical results from solutions of this type, certain simplifying assumptions are made. If either the incident electric or the incident magnetic vector is linearly polarized so that the line of polarization lies in a cartesian coordinate plane, one set of wave functions and the corresponding set of coefficients cmn fall out entirely. If the incident wave lies along the axis of an axially symmetric scattering body, say the z-axis, the dependence on the azimuth angle 1 __________________________ 14 -----------------------

WILLOW' RUN RESEAkRCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 becomes very simple. Consequently the double sums in (1-13), (1-15), and (I-16) become sums over n alone, with only one value of m appearing in each sum and with one set of wave functions eliminated as above. There remain relations equivalent to two sets of infinitely many equations in infinitely many unknowns. An approximate solution of this system of equations is obtained in this report and applied to give an approximate expression for the electric vector of the field due to the scattering of a plane wave by a perfectly conducting spheroid. From the electric vector E of the scattered field, the radar crosssection o is derived as follows. Consider an electromagnetic field whose electric and magnetic vectors are given by eiwt E and eit H, respectively, where E and H are vector functions of position. With E* and H* denoting the complex conjugates of E and H, respectively, define the real numbers Eo = (E.E*). Ho = (H.H*). These "magnitudes" are in the ratio Ho =V/7ii Eo For a monochromatic, harmonic field of this sort, the mean (time average) intensity of energy flow per unit area, or mean power density, is described by the real vectors S = Re(E X H*) (1-17) that is, the direction of S is the direction of energy flow and the magnitude S of the vector is equal to the mean energy pr unit time crossing a unit area whose normal is oriented like S. A substitution in (1-17) and a simple computation show that I 1 2 S =- Eo0H =- 2 Eo. (I-18) 3. A proof of this fact appears in Reference 8, page 137. 15

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 As before, let E' and H' denote the complex, time-separated electric and magnetic vectors of the incident field, and let E and H denote the complex, time-separated electric and magnetic vectors of the scattered field, respectively. Besides Eo = (EE*) and Ho = (H-H_*),' ) introduced above, define Eo = (E' *E'*), Ho = (H1' H *) (the symbol * again denoting the complex conjugate vector). From (1-18), the mean power density for the incident field is equal to S' = 2 /7i (E)2, while for the scattered field it is equal to S = - 1 /7 (Eo)2. For the definition of the radar cross-section, one introduces a fictitious isotropically scattering body. The total mean power intercepted by this body is equal to the product of mean power density by the projected area of the body. If such a scatterer of cross-sectional area a is placed at the origin, the total mean power scattered is equal to a S', and the mean power density observed at a distance r from the origin is equal to o S'/41Tr2. The electromagnetic wave actually scattered from the given nonisotropic body behaves asymptotically like a diverging spherical wave, cf. (1-3). Consequently the mean power density actually observed at a point far from the scattering body depends (asymptotically as r —-oo) on the direction of the ray from the origin to the point of observation. In terms of a spherical coordinate system, let r = rer denote such a ray. The radar cross-section a of a scattering body is a function of the direction of er, defined as the cross-section a of an isotropic body of the type described which scatters the incident wave with a power density in the direction of er asymptotically equal to the power density of the wave actually scattered by the given body. In a formula, the radar cross-section is that value of a for which a S'/47Tr2 and S are asymptotically equal for arbitrarily large values of r, or a = lim 4T r2(Eo)2/(EO)2. (I-19) r —- oo.....____________________ 16.

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 II PROLATE SPHEROIDAL FUNCTIONS AND RECURRENCE RELATIONS The scalar wave equation separates in the prolate spheroidal coordinate system, in which the coordinate surfaces are confocal ellipsoids and hyperboloids of revolution, and halfplanes through the axis of the system, say the z-axis. Denote by 4 the azimuthal angle about this axis. The. foci of the system are placed on the axis, a distance F to either side of the origin. In each halfplane < = const., the curves q = const. are portions of hyperbolas, while the curves 4 = const. are portions of ellipses (Figure 1). It is customary to call T7 the "angular" Z=F 7=1 /:fo / n x+y2 Z=-F 7=-1 FIG. 1 PROLATE SPHEROIDAL COORDINATES and 4 the "radial" variable, by analogy to the spherical case, since....r/F, 7-.cos 0, as. - oo. ______________________ 17

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 The prolate spheroidal variables are defined by the formulas x = F(1 -712)1/2 (2_ 1)1/2 cos, y = F(1 - 7 2)1/ (2 _ 1)1/2 sin, (11-1) z = F 7 where -1 < 7] < 1, 1 < < oo, and 0 ~< q5 < 27r. The partial derivatives of the spheroidal variables with respect to x, y, and z are 1 1 11 87 1 T2T2 cos a 77 77 T2Tsin ai7 zI ax F' ay FA' az F' ax- -Fa' a = (II-2) ax' y F a- F-a 8 _sin af cos af ax FZTT, ay F1T F2:T T' Y FT'T " | where a(, 77) = ~2 - 72, Z (77) = 1 - 772, T() = 2 1. (11-3) (These three abbreviations will be used throughout Sections II and III.) The cartesian unit vectors are givenin terms of unit vectors of the spheroidal system by 1 1 A2 ex = - -'-1cos Q~er] + f -cos q e_ - sinq _~,, I 1 ely = sin T + i ~sin e + cos e, (II-4) 1 1 ez le +t I e1 ------------------- 2 18

WILLOWC RUN RESEEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 The scalar wave equation (1-4) has solutions of the form fmn = Umn(4) Vmn(77) (amn cos m 4 + bmn sin m ). The prolate spheroidal functions Umn and Vmn must satisfy the differential equations ( - ) d - _ - Amn - c2 ]Umn 0, (11-5) ~d~~d2 1 mn m d 1_72dV n m-2 (1 d ) mn - 12 + Amn + C2 2Vmn = 0, (11-6) respectively, where c = kF. The separation constants Amn are fixed by the requirement that the angular functions of the first kind (1) Vmn = Smn (c,?7) be finite for -1 < t7 < 1. The radial functions of the first kind Umn = Rmn (c,1) are solutions of the equation (11-5) for the same values of Amn; they are finite in the range 1 ~< < oo, and are normalized to behave asymptotically like the spherical Bessel functions: (1) 1 m + n + 1 Rmn (c,0)- cos-co (c - 1) (II-7) as c- -oo, or more precisely, if (c4)2 (2_ 1)>> m2 and (c~)2>Amn. The radial functions of the fourth kind are of interest for present purposes, as indicated in Section I; these are determined by the radiation condition (1-5) as the solutions of (11-5) with the asymptotic behavior Rmn (c, )~ exp -i (c - + n + 1 (11-8) (3) as c - oo. The function Rm)n has similar behavior with -i taking the place of i. The radial functions of the second kind, appearing as the imaginary part of those of the third kind [cf. (1-7)], thus behave asymptotically as follows: _(2),t)v 1 m+n+l. Rn (c,) — sin (c - ------- ). (11-9) 19

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 The asymptotic behavior of the derivatives of these functions may be obtained by differentiation:4 D(1)I1 M 1 m+n+l sin (c m + n + 1 (11-10) Rm)n(c,) -- sin (c - 2 X), (II-10) m n 2 (2)n 1 m+n+ 1 R()n (c,) cos (c - 2 ) (II-l) The angular functions of the second kind will not be needed in the present work. Both the radial and the angular functions of the second kind have singularities, at 4 = 1 and at 7 = ~ 1, respectively. Since the argument c will remain fixed except in the tabulated results of machine computations, it will be omitted in writing the spheroidal functions. The radial functions Wmn = Rmn and the angular functions Wmn = Smn are solutions of the same differential equation, md[(1 2) -dw +.->-A - AmnWmn =, (11-12) but refer to different portions of the real, -axis as range of the independent variable. This equation is similar to the Lame or the Mathieu equation, with regular singular points at4 = + 1 and with an irregular singular point at 4 = oo. The older literature about the theory of this equation is listed by H. Bateman (Ref. 13) and M. J. 0. Strutt (Ref. 14), the more recent literature by J. A. Stratton et al. (Ref. 15) and by C. Flammer (Ref. 16). In the present report, the notation of Reference 15 has been adopted, except that the subscript f has been replaced by n for typographical reasons. By multiplying equation (11-12) for Rmn by Rmn, multiplying equation (II-12) for Rmn by Rmn, and subtracting, one obtains a linear first-order differential equation for the Wronskian of these functions, and the value of the arbitrary constant is easily deduced from the asymptotic value of the Wronskian by use of (11-7), (II-10), and (II-11) 4. A proof of this fact appears in J. F. Ritt (Ref. 12). 20..

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 [cf. Reference 16, Formula (67)]: (R) R() R(l) R)mcT)n [2 T(II-13) Rmn Rmn - Rmn Rmn = [. (13) The Wronskian of the angular functions is similarly found to be (1) (2 )1 (I1)1 (2) Smn Smn - Smn Smn = [h ( )]-, (1114) and the constant h will now be determined. Since Rmn () and Smn(1 ) are both solutions of (11-12) regular at 4 = 1, their ratio must be a constant, and the functions of the second kind are also so selected that their ratio is constant: SO,(S - KM RM! S() (2 \- K 2R(2) / 5) mn() Kmn mn( Smn() - Kn mn-1) (in the notation of Reference 16, Formulas (68) and (69); but note that Flammer's index n coincides with n + m in the notation of Reference 15 and of the present report). If the resulting identity (2) (2) (1) (1) (2) (1) Kmn Rmn/Rmn = Kmn Smn/Smn is differentiated and compared to (II 13) and (11-14), one finds that Km/c[Rn] = - Km h[S(n], since 1() = - T (4). Finally, the first equation of (11-15) gives the desired evaluation: (K) (2) h = - c/K Kn. (11-16) In the literature about spheroidal functions and their applications one sometimes finds the statement that great simplifications could be achieved if recurrence relations for the functions Smn and Rmn were known. By a method employed for Mathieu functions by E. T. Whittaker (Ref. 17) the existence of such relations for the radial functions of the first kind was established by I. Marx (Ref. 18), and integral representations for the coefficients were obtained. Further methods, detailed in References 19 and 20, give alternative representations that permit deeper insight into the structure of these relations, and incidentally enable one to extend the recurrence relations to radial functions of the second kind and to both kinds of angular functions. The coefficients, in either representation, are at least as difficult to evaluate as the 21,

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN ------------ UMM-126 functions whose computation they are supposed to simplify. What is really needed is a representation of the solutions of such non-hypergeometric equations as (11-12), with the simplicity of the hypergeometric series —enabling one to identify the recurrence coefficients as simpler functions of r, n, and c — and the possibility of such a representation is still an open question. The methods used to obtain the existence of the recurrence relations and to evaluate their coefficients are sketched below. The spheroidal functions are factors of the spheroidal wave functions. The wave functions may conveniently be separated into a set even in f and a set odd in 0, and here only the former set /mn (x,y,z) = mn (r,, 1,) = S(n (i ) Rm (t) cos mO (II-17) will be needed. Since the scalar wave operator V2 + k2 commutes with all differentiations, the functions a4mn/a z and a4/mn/a x are likewise solutions of the scalar wave equation (1-4), and they are regular solutions which are even functions of <, as is easily verified by means of formulas (11-2) and the series representations given in Reference 16. Consequently the theorem related in the preceding section ensures that they may be expanded in double series (1-6) of the wave functions 44n X convergent in any closed, bounded subregion of space. Since both the function sets cos m 4, for m = 0, 1, 2,..., and Smn (Ti), for fixed m and for n = 0, 1, 2,..., are orthogonal as in (1-8) over the ranges of their respective arguments, the coefficients in the expansions may be isolated by double integration. The method of Whittaker consists in comparing the series representations of amn/z and am(')/ax with the expressions obtained from the differentiation formulas (II-2): (1)amn mn (1) a E E fpq pq (, 77 ) az p=O q=O cos m z Sn Rmn + TS4 Rn RmJ l 22FA L.......... 22..............

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 ~8_V/mn = ~ pqE g /pq(, ax p=O q=O _ cos cosm O r1/2T1/2 [ t Rm + tS Rmn l -, o,1mnmn mn mn m sin v sin m + (1) (1) + sFT/2 mn mn (11-18) FE 1/2 T1/2 n/~ cos(m-l) fil/2T1/2 _()' (1) + (1) (iy1+ m (1) (1) +F z'A l"Smnmn mn Rn n 1/2 T V2 Smn Rmn cos(m+l) JL1/2Ti/2 [-(i))'.(1) (1+ ( R Il m S(l) R,(1 ZF+ -— ^ - Lt- -- mn 7Smn mn1 1/2T1/2 mn mnj the primes denoting differentiation with respect to the appropriate variable. The orthogonality of the wave functions implies that only coefficients fmn mnwi cients fpq with p = m and coefficients gpq with p = m- or p = m+1 are different from zero, and that these coefficients are equal to integrals over the right hand members. These integrals, moreover, are zero whenever the integrand is an odd function of 77. Since Smn ( 7) is an even or an odd function of 77 according as n is even or odd, it follows further that only coefficients fmqn with n and q of opposite parity, and coefficients gpn with n and q of equal parity, are different from zero. A differential formula for raising the index n, a formula for lowering the index n, a formula for raising m, and one for lowering m are sufficient, since all other recurrence relations may be derived from (1)' (i) (1) them. Consequently relations among Rmn, Rmn and Rm),n1, among (l/ (l) (l) (lm1 (l) (l) Rm,n-l Rm,n-l and Rmn, among Rmn, Rmn, and Rm-1,n, and among (1)' (1) Rm-l,n Rm-_l,n and Rmn are taken as the standard forms. Multiplying both sides of (11-18) by Spq (r7) cos p f with a suitable choice of the indices p and q, and integrating both sides with respect to r7 and q, one obtains the formulas 23..............

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 mn W (1)7(1 (1) (1) F fm,n-1 Nm,n-l Rm,n-1 = T Rmn J f Sm Smn( d v- 1 (II-19) + Rmn mn Sn drS, m,n-1 (m) (1)n I 1 (1) (1) F fn Nmn Rmn = T Rmn-1 l Sm,n- Smn d7i (II-20) + Rmn-1 ( Smn,n- Smn dl, and 2 F gm-l,n Nm-,n m-ln = mn m -(1,n d( (II-21) (l) rll (l)/2 T 1/2 ( (x) - R(nn Sm-ln [ mn ~ (1 /2TJ2 smn d771 1l,n.. 1/2 n mn R T1/2 R( ( S d 2Fgmn NN mn (=TR S_ S dT( 2- F mn mmn - 1, m-lnmn (II-22) (l)' 1 f l)n Tl/2 r (l f m-l S (1) - -l. f m1) [n L -l,n V1/2 T1/2m-l,n where N n =, [sn () d 7. (II-23) - - - 24

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN -UMM-126 The constants in the left hand members of (II-19) —(II-22) may be evaluated by an examination of the asymptotic forms of these relations. If (11-7) and (II-10), with suitable choices of m and n, are substituted into the formulas, one finds from the dominant terms that mn m,n- 1 (1) F fm,n-l Nm,n1 F fmn Nm n = cj 71 S Smn,n. d7 (II-24) and mn m-l,n 1 (1) (1) 2Fgm-l,nNm-,n -2Fgmn Nmn = /2 S() S() - m= mn m-l,n dU? -1 (11-25) The recurrence relations are now seen to have the form T amn (4) Rmn + 3mn (4) Rmn Rmn-l. (I-26) T an () Rmln -1 + Ymn ()Rmn- = - Rmn (II-27) and ()1) + () (II- T rmn (4) Rmn + Pmn (4) Rmn Rm -l,n' (11-28) T mn+ (n) Rm, + mn ( II-29) Integral representations of the coefficients in (II-26) —(II-29) are given in formulas (II-19) —(II-22), with constants as in (II-23) —(II-25). For a further study of the coefficients it is adequate to concentrate on the recurrence relations (11-26) and (11-27) which change the index n. Differentiation under the integral sign in the coefficient amn, easily proved to be valid, leads to a verification that the equation Tdamn/d4 + Imn + ymn = (11-30) connects the three coefficients. Eliminating 3mn and Ymn from (11-26) and (11-27) by means of (11-30), one obtains the identity 25

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ------ ~~UMM-126 d/d [amn/Rmn Rm,n-1 = T{ [Rm1)l2 - [Rm)]-2} (11-31) mn m n,n- l]'; - With (11-31) and the Wronskian identity (11-13) it is possible to prove (2) (2) that the radial functions of the second kind Rmn and Rm,n-1 satisfy recurrence relations of the form (11-26) and (11-27) with the same coefficients amn, f3mn and Ymnn Consider the functions Fmn ( ) = [T amn Rmn + mn Rmn,n- l mn-1 (2)'n (2) (2) (1) Gmn (s) = [T mn Rm Rm,n-l + R,n ]/Rmn (2)' In view of (11-13), Rmn may be eliminated from Fm, and the function rewritten as = R (2) R (1) R (1) R(l) m R(1) Fmn =[R( mn/Rm)n Rmn-1] [Ta Cmn Rmn + imn -mn m,n-l] + CY /R() R (1 R(2).(1) R(2) /R(1) + a/ mn/CRmmn n-l + Rmn/Rmn m,n- /Rmn-l; (II-26) shows that the quantity in brackets is zero. Differentiation of the rewritten form of Fmn and substitution of (11-31), and of (11-13) for both the values n and n - 1 of the second subscript, gives the result Fmn - 0. Similarly Gmn is seen to be zero. It follows that there are constants a, b such that (2)' (2) (2) (1) a Rmn+mn n n Rm,n-l = a Rmn-1 (2)' (2) (2) (l) amn Rm)n-l + Ymn Rmnl + R) = b Rm Examination of the asymptotic form of these relations, in the light of (I-7) and (II-9) —(II-11), shows finally that a and b are both zero. Equations (11-26) and (11-27) remain valid if functions of the second kind are substituted for those of the first kind. A similar proof may be carried out for formulas (11-28) and (11-29). 26

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 In Reference 19 it is proved that recurrence relations having the property just established —validity for two linearly independent pairs of functions —have uniquely determined coefficients, which in the present case assume the form amn= c[Rm,.nl Rm - Rmn Rmn-1] (II-32).=cT[R R2 R R ] mn m,n - mn mn m,n-l -F R(1) ( R2R( ) R 02 (2)' (1) (2) (II-33) [R m,n.-l mn Rmn mn-l]/[Rmn mn - Rmn Rmn = - cT (1) ()' (1)' R(2) Ymn =,- 1 m - c l Rmn] (II-34) =- [Rm mn R n-l - mn-1 Rmn]/[Rmn- Rm,-l Rmn- Rn)n- ] The coefficients 7rmn, Pmn, and amn in (II-28) and (11-29) have representations identical with (11-32), (11-33), and (11-34), respectively, except that the subscript pair m, n - 1 is replaced by the pair m - 1, n wherever it appears. To obtain formulas similar to (II-26) —(II-29) for the angular functions S (T11) and S)n (71), one needs to make only the modifications evident from an examination of equations (11-13), (II-.14), and (II-16): besides replacing, in (II-32) —(II-34), T (~) by Z (7J) and each radial function R (S) by the angular function S (T7) having the same subscripts and superscripts, one must change the constant c to h = - c/K () K(n Equations (II-15) may also be used, together with previous results, to put into integral form the coefficients in the recurrence relations for the angular functions. In an obvious manner, recurrence relations among three functions with contiguous subscript indices m or n can be derived from the differential formulas just established. Certain new identities among the spheroidal functions are obtained from a comparison of the formulas (II-32) —(II-34) with the integral forms of the coefficients given in (II-19) —(II-Z5). A listing of formulas for the spheroidal functions obtained from the results of this section is given in Appendix A. 27

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 III EXACT SOLUTION The application of the theory related in Section I to the problem of scattering from a prolate spheroid was carried out by F. V. Schultz (Ref. 21). A plane wave moving in the negative z-direction along the z-axis is scattered by the prolate spheroid k = 0o. The electric vector E' and the magnetic vector H' constituting this wave are assumed (without loss of generality) to be parallel to the positive y-axis and the positive x-axis, respectively, so that E = El eikz Hy = H' e ikzex (III-1) where H' = /p E'. The expansion of a plane scalar wave of unit amplitude in prolate spheroidal functions was given by P. M. Morse (Ref. 22). Specialized to the present case, the expansion becomes co ikz = (i) (1) eik= E AnSo R () R () (III-2) n = 0 n=0 where An 2 in ()/N ()/NIII- 3) with Non as in (11-23). Substitution of (III-2) and (III-3) in the first of Maxwell's equations (1-12) gives for E' the expansion El= - H An Vx Son () Ron () (III-4) ky n = 0 As in Section I, the spheroidal scalar wave functions (11-17) are used to define spheroidal vector wave functions ____________________8_.28

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 XMmn = V Xex /mn ( 77, ) = V= 7mn Xex YMmn, ZMmn (III-5) In terms of these functions and of the amplitude factor E one obtains from (III-4) the expression E' E' An x 1on (III-6) k n=0 for the incident wave. Using formulas (II-2) —(II-4), one may write (III-6) in the extended form E' An dRn/d 4 sin 4 - adSon/dI Ron sin (III-7) - L (7 Son dRoon/d4 dSod Ron os n. According to (1-2), the electric vector E of the scattered wave must satisfy the boundary conditions ~00~1 E7 (~o ~ = kF An(^ iSon (77) dRon (io)/dk sin k (III-8) n=0 and iE* co (7 T'o (1) (1) Eb ( 7o' ) =k An( O Sn (T7) dRO (o)/d( (III-9) + ~o (i) (1) ) +. adSon (77)/d71 Ron()) cos, where Ao = 4o2 - 72, and To = 2 - 1. (III-10) These are identities in?7 and 4. 29

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 The electric vector E of the wave scattered from the spheroid has an expansion (1-16) in vector wave functions of the fourth kind. It is clear from the boundary conditions that the coefficients in this expansion are zero for all terms except those whose t and ( components depend on the variable < in the same way as the right members of (III-8) and (III-9), respectively. Accordingly one may write co r (4) (4)i E = [an xMon+ Pn ZM (III-11) n=0 LJ In extended form, the expansion (III-11) becomes co T l4 (1) (4) -an F (1 on dRon/d_-e (2dSon/d7 Ron sin E= -F A on ono n 0 - (1 on) on/d + C dS(1)/d R() cos ] [d 77, (1) (4). (I (I -1i2) - B _ e (Z2A)2- - e (TA)) s-n R sin +e( A (dS)/d7 Rln - S(dR)/d) cos The electric vector E in (III-12) together with the magnetic vector H obtained by the same method gives an exact solution to the problem of scattering from the prolate spheroid t = o, if the coefficients an, 3n are so determined that the following two equations are satisfied: E nToSn) ("r) dRo4n) (o)/ d - Pn - Sn)(T) Rn 1 ( (III-13) =i El An T Sn) (r7) dRn ( o)/d~, k Eo on Ao3 n= 0O,,__________ ~30

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 1 n[ 7T0o Son (r7) dRn (4o)/d4 + dZ dS(n (r7)/dr7 Ro (4o) +n(2To)2 L7dSin (77)/d77 Rin (0o) - o Sin (71) dRln (0o)/d4 i E'E A T S(1) (1) i c Co F () (111-14) = E S AnL7Ton S (77) dR on (~o)/d~ n= 0 + ( + t dSon(77)/d Ron (o) Once a solution of the system (III-13), (III-14) for the coefficients On and O3n has been obtained, equation (III-ll) with these values of the coefficients substituted gives an exact representation of the electric vector E of the scattered field. The electric vector E' of the incident field is given by equation (III-5). To compute the radar cross-section from formula (I-19), one must examine the asymptotic behavior of E, and therefore of the vector functions XMon and Z 4n) as the distance r from the origin becomes arbitrarily large. Since r is asymptotically equal to F4 = c4/k, the expression (II-8) gives the desired information. Combining this formula and the differentiation formulas with the definition (II-17), (III-5) of the spheroidal vector wave functions, one obtains the asymptotic expressions n(4) F_ (]) - and M F F — (77) (1 -_ 772)12 cos ] exp[-i(cO- 2 r)] l)] [_ e (1 - 772)1/2 cos ] exp [-i(- n+ - - )] _r 3....__________________________ 31

WrILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 which hold for fixed k as c4 —-ooor r —oo, provided that (c4)2 >> Amn and (c )2 ( 2_1)> m2. Using these formulas, one obtains from (III-11) the asymptotic expression for the scattered wave5 -ikr (1) E -_ Ein _7n Sn (77) sin 0 n= 0 (1) ( 1) (III-15) (si.-15) + [Cn 77Son (7) -i ~n (1 - 7 2)1 2 Sn) (?)] cos 0. It is no loss of generality to assume that the incident wave has zero phase, so that E' = (E E'*)2 = Eo, a positive real constant. The scattered wave will then in general have non-zero phase. In order to facilitate the division indicated in (1-19) and to display the radar cross-section as a function of a significant dimension of the scattering body, namely the semi-major axis a = F~o of the prolate spheroid ~ = o0, one may introduce a new set of expansion coefficients (n a= n/E a, n = fn/E a (III-16) into the asymptotic expression (III-15). The complex conjugates of at and a3 may be denoted by c/l and tB, respectively. Substitution of (111-15) and (III-16) into the formula (1-19) gives for the radar crosssection of a prolate spheroid the expression a (7 7,) 47raZ sin2{ in a S (1)] (~~ii~On on (T7 0 n (-i )a* Son (17) +coS 2 io I nS()0nLn = 0 c1['o n( n) 5. Since Amn increases indefinitely with n (cf. Ref. 16), the validity of this formula is not immediately clear because of the requirement that (C )2>>Amn. For analytical proof that the expression is actually valid, see Reference 29. 32

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN -UMM-126 2 1/2 (') c (1) - i Oin (1 - t77) Sin (77))] * [ E ( a i)1 7Son (t77) n = 0 + i an (1 h )/ sn(81)) \ (III-17) The formula (III-17) is an exact formula if the expansion coefficients are obtained from an exact solution of the system (III-13), (III-14). For the case of back-scattering, that is, the scattering in the direction opposite to the direction of propagation of the incident wave, the value 77 = 1 is substituted into (III-17) and the cross-section becomes independent of < as well. Since Sm) (1) = 0 for m>0, the back-scattering cross-section is given by the simpler formula ()l 2 a = a (1, ) = 47a2 in an Sn) (1) (III-18) n=0 It is this quantity which was computed in the work reported in later sections. The problem of obtaining numerical answers in the "bistatic" case, where the transmitter is located on the axis of symmetry and the receiver at an arbitrary point, has also been investigated. The results are discussed in Appendix C. 33

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 IV SCALAR SCATTERING For a wave derived from a velocity potential that satisfies the scalar wave equation (1-4) - such as an acoustic wave - there is an analogous scattering problem, which will be briefly described. Again a plane wave with direction of propagation parallel to the major axis of a prolate spheroid is incident on the spheroid. The body is assumed to be smooth and rigid, so that the entire incident energy is scattered. For each ray from the origin, the cross-section of an isotropic scatterer yielding, asymptotically, the same mean power density is defined as the scalar scattering cross-section ao of the spheroid. Because of the difficulty of computing the radar cross-section a, the scalar scattering cross-section af is sometimes used as an approximation to a. The order of magnitude of error in making such an approximation can be better understood if one analyzes Figure 3 in Section VIII of this report, where the scattering body is a prolate spheroid, or Figure VI-1 in Reference 23, where the scattering body is a cone. Similar comparisons are known for the conducting wedge, sphere, and paraboloid. All the numerical quantities needed to find the scalar scattering cross-section are computed in the process of finding the radar crosssection. Consequently the former has been obtained for the same parameter values as the latter, as reported in Sections V, VII, and VIII. A monochromatic scalar wave is derived from a velocity potential of the form it = /eiwt, where 4/ is a scalar function of position which satisfies equation (1-4) for f = 4. For simplicity, the function / alone will be called the velocity potential. Let 4/denote the velocity potential of the incident wave, let 4' denote the velocity potential of the scattered wave, and let /' = // + 4/ denote the total velocity potential. On the surface of the scattering body, the gradient of the total velocity potential must be a tangential vector. In terms of spheroidal coordinates and other previously introduced notation, this condition may be written ___________________________ 34 ----------------------

WILLOWd RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 V7( (77, 40) * e,(i7, 0) = 0 or a/,(r/, o,/)/a3 + al' (r], o,q)/a4 = 0. (IV-1) At infinity, the velocity potential of the scattered wave must behave like that of a diverging spherical wave, i.e., 4/ must satisfy the scalar radiation condition (1-5). The incident wave has a velocity potential described by the function 4/(7], ~,b) = 4 eikZ = b o An n (7) R() (7). (IV-2) n=0 In formula (IV-2), the expansion (III-2) for eikz has been employed, with coefficients An as described in (III-3). if zero phase is assumed for the incident wave, the constant 4/4 is the maximum amplitude, a positive real number. The velocity potential of the scattered wave must be independent of 0 if (IV-1) is to be satisfied: it may have mathematical singularities situated inside the scattering body; and it must satisfy the radiation condition (1-5) at infinity. Consequently, an expansion of the form 00 (4) oO Y(TL, C1 X non (4 T )= En Sn)(7T) R() () (IV-3) n = 0 n = 0 (4) is indicated, where 4Qon are the spheroidal scalar wave functions introduced in Section III. The boundary condition (IV-l) implies that oo (1) 0o (4) 440 An S$(1) ( dRi ( +) Yn Sn (l)dR4)() (IV-4) n=0 d0 n=0 dn - 0 and the orthogonality of the angular spheroidal functions of the first kind permits one to obtain the constants Yn directly from (IV-4). For any non-negative integer N, multiplication of both members of (IV-4) by 35

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 SoN (?7) and integration with respect to TJ from -1 to 1 gives 44o AN dRoN ()/d4 + YN dRON (4O)/dA = 0. (IV-5) One may solve (IV-5) for YN and substitute into (IV-3) to obtain an exact expression for the velocity potential of the scattered wave: (1) W (4) (IV-6) 4 = -0 E An Hn (0) Son (T7) Ron ((I-6) n=0 where the abbreviation Hn is defined by Hn (4) = dRn (4)/dc [dRon (4)/d,]. (IV-7) Analogously to the radar cross-section, the scalar scattering crosssection al of a scalar wave is defined by a = lim 47Tr2 1r/12/14'I2, (IV-8) r -- oo o and under the assumption of zero phase for the incident wave one may omit the absolute value bars in the denominator. To compute (IV-8) one needs an asymptotic expression for (IV-7) as r-oco. Besides the asymptotic formula (11-8) for the radial spheroidal functions of the fourth kind, one may conveniently use the fact that l/c4 = a/c, r, where a is the semi-major axis of the spheroid 4 = 40. Substituting into (IV-7) one obtains6... E An Hn (0o) S exp [-i(kr - 2 )] (IV-9) scg n =0 as c4 —-coor r —oo, with Hn as in (IV-7). Finally, equation (IV-9) is substituted into (IV-8) to give the formula 6. See footnote 5, p. 3Z. ______________________ 36

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 47Ta2 in+l * af C2 2 n+AnHn(tso)Sn ) S 1 (-i)n+1 AnHn( o) Son (71) n=o n= 0 (IV-10) (IV -1 0) for the scalar scattering cross-section a' of a plane scalar wave scattered by a prolate spheroid. Here An denotes the complex conjugate of An. It may be noted that the scalar cross-section is axially symmetric, i.e., depends on 7l alone. For the back-scattering case, the formula (IV-10) is evaluated for 1 = 1. 37

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 V APPROXIMATIONS The equations (III-13) and (III-14) represent an exact solution to the vector scattering problem, in the sense that they define, at least in theory, the exact values of all the expansion coefficients in the expression for the scattered field. However, as they stand they are obviously of no value in computing a finite number of the coefficients. Schultz (Ref. 21) suggests the possibility of obtaining simpler expressions for the quantities an and |3n by expanding the equations in powers of T7 and equating the coefficients of like powers, but also points out that the amount of algebra involved in this process is prohibitive. The method developed by Schultz, which has been adopted for the computations to be reported here, is to multiply both members of each equation by SN (TJ) and integrate with respect toT7 from -1 to 1, for each integer N in the range of the summation index n. The result is an infinite set of linear equations in the infinite set of unknowns an and 13n, with constant coefficients. In the notation of Reference 21, which will be followed with minor modifications in this section and in the computations, these equations are E (anCNn + 3nDNn) = E' a BNn (V1) n=O n=O ES ((anVNn + OnWNn) = Ea UNn (V-2) n=0 n=O0 (N = 0,,2,...), where BNn - iAn(2 1)/2 dRon (o) Son S () d 7, (V 3) __________38

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 CNn = (o- 1)'/2 dRn29), 1 S(n) SO (T7) dye (V-4) DNn = -RnN( ( 4) 1 (1 -71 2)/ sI () SN (71) dr - UNn = 2 _[(4_) on on(71) SoN(r) d7 (V-6) l DNn - RnN (o) Sin (7) SoN (r7) d l7, (V-5) (1 ( ) 1 i An 2 1 dR(,0) dS (/1) () VNn(~-1)d 11 S(I) s(1) +VNn A () - 1)' 1(1 So-n (TS) SoN (7) d77 (v-6) -.dSo (7)s( ) dr(, + 0 Rn(o) (l S 7)d7 J-}1. dr7 WN = - 1/2 dR (o) 1 (1 - R2)1/2 s(77)S() d(4 V + ( Nn 12 R(no) j1 ( 2)/2 dN So() d7. (V-2) is of course beyond the capacity even of a machine. It is physi(V-7) cally plausible that the exact solution is considerably more than is needed for practical purposes, and that an adequate approximation is obtained by truncation of the series. For any n, the quantities a7n and 7 3n are measures of the radiated energy contained in the n'th-mode oscillation, Wand this n i wto die out fairly rapidly with increasing n as long as n exceeds the characteristic dimension times the wave number. Therefore by taking only the first few terms of the series, and the same range -----.39 —-— (V-8) (, _)/ (s)) d 77. \C>0 in j7 o0 N The problem of finding the exact solution to the infinite system (V-I), (V-2) is of course beyond the capacity even of a machine. It is physically plausible that the exact solution is considerably more than is needed by truncation of the series. For any n, the quantities an and On are exceeds the characteristic dimension times the wave number. There

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 of values of N, one may derive a finite system of linear equations whose solution should approximate the exact values. The degree of approximation will of course depend on the order of the system. A fourth order system was actually used for the computations, the choice being dictated primarily by limitations on machine capacity, time, and money. It was hoped that the degree of approximation obtained in this way would be at least comparable to that afforded by the other approximations inherent in the general method. Also of course the problem was simplified in that conputations were made only for the case of backscattering, so that the coefficients O3n drQp out of the picture. An estimate of the exact degree of error resulting from the truncation of the infinite determinants, such as might be carried out by an extension of the method of Wintner (Ref. 24), was not made. It was originally planned to program the computation for both third-order and fourth-order determinants, and for the three fineness ratios (ratios of major to minor axis) 10:1, 5:1, and (1 + e):1. At that time it was felt that an adequate criterion of accuracy would be furnished by comparison of the results for third-order and fourth-order determinants for all three fineness ratios, particularly since it was believed that an exact analysis could be carried out for the (1 + E):l ratio. Unfortunately economic limitations made it impossible to obtain the desired values at the 5:1 ratio, and when theoretical results for the ratio (1 + E):1 were obtained and showed no unexpected behavior, further study of this case was likewise abandoned. Improved accuracy might have been obtained with less effort if, prior to programming, the minimization procedure used by E. Schmidt (Ref. 27) to determine the solution in Hilbert space of an infinite number of linear equations in an infinite number of unknowns had been adapted to the case of n equations in m unknowns for m> n. Among the approximations made in the computations, the principal one concerns the representations of the spheroidal functions Smn({7) and Rmn(0). Because of the lack of knowledge of the theory of these functions, and because of the limited range of existing tables, it was necessary to represent the functions Smn and Rmn by series of associated Legendre functions and of spherical Bessel functions, respectively. The particular spheroidal functions required here are represented as follows: ---------------- 0 ---------------

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 00 S() ( = E nk dk +m (T7) (V-9) ~mn k k+ k=O _R__) = (02-)in/210 /nkn-k n(V-) E nk dk (k+2m) l mk (c!)/k! k=Onkk=O R(mnn() = -n () iRm / n (c) (V-10) k=0 00 mn:lnk dk Qk+m ()(V-, R^ )=n Rc~m~~n) i ~ R~n~k d(0.(V-12) where Pk+m (7) and Qk+m(~) are associated Legendre functions, jk+m(c~) are spherical Bessel functions of the first kind, and the constants dmn are the spheroidal coefficients, essentially defined by equation (V-9) and discussed in detail in References 15 and 16. The symbol Ui j as used here and throughout the rest of this report is defined as 1 n i / i~i+Ji * (0 if i+j is odd li,j -2[1 + (1) i]; i.e., iij= if i+j is even' and K(c, m, n) is defined below: 2 m-1 r(n+2m+l) 2 ) (V-13) K(c,m,n) = 1, (n even), 1 mn m n r(-2 r(m - ) d-2m fn k dk (k+2m) /k! k=O,_______________________ 41 _____......___

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 _8cm-lr( n+zm+l -8Cm 2(n+Z (V-14) K(c,m,n) =3 c, m (n odd). r (- )r(m - ) dl-2m E n,k dk (k+2m) /k! k=O The expansion (V-ll), (V-13), (V-14) for R2)n(4) is preferred over an expansion in spherical Neumann functions analogous to (V-10) because of its superior convergence in the region 4 1. The spheroidal coefficients dkmn are tabulated for certain ranges of the indices in References 15 and 16. However, the ranges covered are inadequate for the purposes of the present investigation. For this reason and because of the possibilities of errors in existing tables, it seemed advisable from the outset to compute all the coefficients used. The computation is carried out in the manner described in Reference 16. First the series (V-9) is substituted into the differential equation for the angular spheroidal functions. By means of the differential equation and the recurrence formulas for the associated Legendre functions, all differentiations and all variable coefficients are eliminated. In the simplified series, each associated Legendre function appears with a constant coefficient whose value must be zero. The result is a set of recurrence formulas for the spheroidal coefficients: m mn mn mn m mn Ek+2 dk+ + Fk dk + Gk-2 dk-2 = 0, (V-15) where m (2m+k+2)(2m+k+l) k+2 - (2m +2k+3)(2m +2k+ 5)' (V mn (m + k)(m + k + ) + Amn 2(m + k)(m + k+ 1)-2m - 1 k c2 (2m+2k-l )(2m+2k+3) (V17) m k(k - 1) k-Z (2m + 2k - 3)(2m + k-) - ------------------- 42

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 The indices n, m, and k range over the following values: n = 0, 1, 2,..., m = 0, 1, 2,.., k = -24, -23,..., -1, 0, 1,..., 15, 16. The condition that the function represented by the series (V-9) be finite over the whole range -14 t ~ 1 implies that it is an entire function, since -1 and 1 are the only finite singular points of the differential equation satisfied by the spheroidal functions. The ratio test for infinite series of functions therefore shows that dk+2/dk must approach zero as k —oo. The convergent series (V-9) is truncated to give an approximate expression for the angular functions; to fix the number of terms used for the approximation, experience is the only guide available. Since Ef' in (V-15) vanishes for k = -2m, the coefficients dmn are all zero for k < -2m. For this range of values of k, however, the functions Qkm+m (4) are undefined. A detailed study of the spheroidal equation is carried out in Reference 15 for all values of the separation constant, not merely those that lead to the functions of the first kind. The coefficients dn as functions of the index k are defined for non-integral values of k, and it is found that the products dmn Qk+m (t) for nonintegral k have finite limit values as k approaches an integral value, even for k< -2m. The products are put into the form dmn n 1 m nm (V-19) d-(kl Q-Ikl+m d- kl Pkl-m-l (V19) for I|k>2m, where p is a quantity that tends to zero as k approaches integral values. The coefficients d/p are then obtained by means of the same recurrence formulas used for the coefficients dk when k>-2m. The expression (V-19) may be substituted into (V-ll) to give (2) -2m-1 mno R(2)n()=n,k dk Pmn-l (/P + 4n,k k On( (V-20) k=-o o k=-2m _____________________ 43

WILLOW RUN RESEARCH CENTER -U NIVERSITY OF MICHIGAN UMM-126 The constants dk /p also form a convergent series, and the first series in (V-20) is truncated at some finite negative k to give an approximation. The quantity Amn which appears in the formula (V-ll) for Fk is the separation constant of equation (11-12). This separation constant is a root of the transcendental equation formed by iteration of the recurmn ~mn rence formula (V-15) and application of the conditions E2m = 0 and dk+2/ k — O as k —oo, as outlined in Section VII. The actual values of Amn may be approximated by series in positive or negative powers of c = k F, according as this quantity is small or large; these series are given in Reference 16. In practice it is usually necessary to refine the approximation by an iteration scheme such as the one described in Section VII. The expressions for the angular spheroidal functions and for the derivatives of the radial spheroidal functions which are needed to obtain an approximate solution of the system (V-1), (V-2), serve at the same time to compute the scalar scattering cross-section 0( from the formula (IV-10). The results of this computation are compared to the results for the radar cross-section or, for corresponding values of the parameters, in Section VIII. Even after the infinite system (V-l), (V-2) for the radar crosssection coefficients is replaced by a finite approximation, the computations needed to solve the system are far beyond the practical capacity of manual computing machines. The principal difficulty is that a large number of previously untabulated functional values must be obtained and stored. At this point, therefore, the services of an electronic digital computer are indispensable. The next section is devoted to a deScription of the machine that was used. 44

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN -UMM-126 VI MARK III COMPUTER The machine employed for the present computations was the Mark III electronic digital computer. This computing machine was constructed for the Bureau of Ordnance, U.S. Navy, by the Harvard Computation Laboratory, and is now located at the Naval Proving Ground at Dahlgren, Virginia. The Mark III utilizes magnetic drum storage units, and at the time of its selection for this work had the largest capacity of any digital machine available. Previously, one value of a radar cross-section had been obtained at the cost of several man months of manual computation. To compute radar cross-sections with the necessary accuracy for the desired number of values of the parameters, it was necessary to use a machine with large storage capacity and large word length; this was the principal consideration leading to the choice of the Mark III. The difference in operating costs between this machine and faster machines with lower capacities was outweighed by the reduction in computing time made possible by the storage facilities of the Mark III. These facilities include channels capable of storing 4350 sixteen decimal digit numbers, of which 150 are permanently stored constants. In addition, the machine has a capacity of 4000 three-address instructions, and carries permanently stored routines for elementary operations and computational procedures. In this problem floating arithmetic sub-routines were employed using fifteen decimal digit coefficients, four digit exponents and signs. The machine operates with a coded decimal system, in which each decimal digit is represented by four binary digits. Input is supplied on magnetic tapes, and results are printed in decimal numbers by five electric typewriters. Because of the use of moving drums and tapes and of mechanical relays, the Mark III does not operate as rapidly as machines of some other types. Even with the storage capacity available, the various components of the problem could not be coded for simultaneous computation. It was found necessary to split the problem into five individually coded runs, described in some detail in the following section. 45 _

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN ---------—. UMM-126 VII MACHINE COMPUTATIONS The wavelength X of the radiation is considered to be in the "resonance region" if it is nearly equal to the "characteristic dimension" of the spheroid. In Section VIII it will be indicated why the quantity 2 W a should be considered the characteristic dimension, a being the semimajor axis of the spheroid. The range of the parameter 2ra/\ from which the computed values were selected was determined by a numerical analysis likewise described in Section VIII. The objective was to investigate how the radar cross-section changes from the region 2Zra/X< 1 but not too far from 1, where the values of the cross-section are close to those predicted by the Rayleigh approximation, to the resonance region where 27ra/ nearly equals 1 (and actually somewhat exceeds 1). In the latter region, it was first required to make a "guess" as to the location of e s the first maximum and the width of the arc containing it. In the range indicated by the guess, it was necessary to obtain enough values to determine the location and height of the first maximum and the width of the arc on which it occurs, and outside this range it was necessary to obtain enough values to permit rectification of a slightly incorrect initial guess. If the guess had been wrong in the direction of smaller wavelength by as much as a factor of five, a completely new computation would have been needed. Fortunately the original guess was adequate, and the choice of parameter values listed below permitted determination of the desired quantities. Once it was clear that the peak region had been correctly guessed, the values for which computations were to be made could be selected with a certain amount of freedom. This fact was exploited to side-step values for which it was felt that certain difficulties inherent in the method of computation might arise (e.g. the divergence of two quantities whose quotient is finite). The values of 2 Ta/A for which the radar cross-section was computed are listed on the following page: -______ - ~ ~~~~~46

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126.094,.105,.157,.236,.262,.314,.377,.524,.754,.785,.942, 1.26, 1.57, 1.89, 2.10, 2.51, 3.14, 3.77, 4.71, 5.89, 6.28, 10.5. In the computations a system of four equations in four unknowns replaced the infinite system (V-1), (V-2). For purposes of comparison, the system of three equations in three unknowns was computed for the 27Ta value = 1.89. Figure 2 gives a schematic diagram of the subdivision of the problem, indicating the interdependence of the various intermediate steps. The distribution of these steps among the five runs of the calculator is shown by listing in a separate column the quantities computed in each run. The process runs from left to right, with the first column showing known information. The order in which items appear in their column is not significant. A table of Bessel and associated Legendre functions and their derivatives was first computed, since existing tables were inadequate in the ranges required for this work. The values of the Bessel functions were obtained from an expansion of Jn+1/2(c ) in powers of (ct)-2, and further values and values of the derivatives were obtained from the usual recurrence formulas. The associated Legendre functions of the first kind and their derivatives were expanded in powers of t-2 and those of the second kind obtained from the functions of the first kind. The actual formulas used are listed in Appendix B. The approximate values of the separation constants Amnwere first obtained from the infinite series given in Reference 16, which after truncation may be written 4 | A mn 2k Amn = gk c (c<5), (VII-1) k=0 An 4h ck (c>5), (VII-2) ~ k4?..........

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-126 l ASSUMED RUN 1 RUN 2 RUN 3 RUN 4 RUN 5 Amn (Exact;) Amn (approx.) Ek+2 Fmn GM k-2 or or01'~~,mn dk o a,b,A k dmn' "=~~ ~ Nn 3,4 Nn 1,2,5,6 Pm-mnl ) r [ Nn'CNn Qm ( M DNnlUNn m+n Rmn () n(") --— ^Rmn (R) —-( FIG. 2 LOGICAL AND SEQUENTIAL STRUCTURE OF COMPUTATIONS

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 mn rnn with coefficients gk and hk given explicitly in Appendix B. In the course of manual computations it was discovered that for most values of c used, and particularly for those near c = 5, the number of coefficients given in (VII-1) and (VII-2) was not enough to determine the separation constants to the required degree of approximation, while the formulas for additional coefficients were prohibitively complicated. A small error in the value of Amn led to errors in the recurrence formulas for the spheroidal coefficients dk of the order of magnitude of the quantities computed. For this reason an iterative procedure was devised that served to refine the values of Amn sufficiently in all cases computed. The procedure is outlined as applied for even values of k; for odd k there are minor modifications: 1) The quantities Ek+2, F, and Gk2 are computed as in (V-16), (V-17), and (V-18), with the value Amn = Amn determined by the series (VIII-1) or (VII-2). 2) The value of Kk = dk+z/dk is found for k= 14 from the mn m mn formula K14 = - G14/F16, obtained from (V-15) with k = 16 if mn imn d1 /d14 is replaced by its limit zero. 3) The values of Kk for k = 12, 10,..., 0 are found successively from the recurrence formula (V-15), used in the form mn mn m mn m 1 / Kk-2 = - (Fk + Ek+2 Kk ) / G-2 ~n _ mmn m m nn 4) The quantity F0 + E K = - G-2 /K2 is computed. As follows from (V-18), G = 0 is the exact value that should be -2 obtained, and so G_ / K n | is a measure of the error in the first approximation Amn to the separation constant. If the error is not considered excessive, the values Kk obtained in step 3) mn may be used to compute the spheroidal coefficients dk. 49

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 5) If the error exceeds the bounds of required accuracy, a second (2) approximation Amn = A(m to the separation constant is derived from the formula (V-15) for k = 0, which may be written Kon = - [c-2 (m2 + m + Amn) + 1/(2m + 3)] / Em; the value Kmn used is the one computed in 4). An average of the two approximations Amn and Am) is then used to begin the process again at step 1). If the procedure is stable, the error in 4) will decrease to within the desired range of accuracy after a sufficient number of iterations. If the error does not decrease, the method is inapplicable; in the present computations, this difficulty did not arise. The procedure described above is similar to a method developed by C. J. Bouwkamp (Ref. 26). The method of Bouwkamp is theoretically superior in improving the approximations to Amn, converging more rapidly and for a wider range of parameter values. However, it is also much more difficult to program for a digital machine like the Mark III, and this fact led to its rejection in favor of the procedure outlined above. In the second and fourth runs of the machine, the quantities Ikn were computed. These are integrals involving the associated Legendre functions, and arise from the integration of equations (III-13) and (III-14) to Nn derive the system (V-l) and (V-2). The integrals Ik have been reduced in by Schultz to series in the spheroidal coefficients dk, as shown in the mn appendix. If enough values of the dk are known, computation of the integrals offers no difficulties. The computation of the remaining quantities listed in the table presented no special problems. The specific formulas used are given in Appendix B. The scalar cross-sections were computed manually from the tables listing the results of the first three machine runs......_________________________ 50 -

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 An examination of the formulas in Appendix B, particularly of those referring to runs 4 and 5 of the calculator, gives some indication of the increased labor that would have to be expended to compute the next approximation by means of a system of five equations in five unknowns. Besides the additional values of the Bessel and associated Legendre n functions, of the quantities Am,, dk, and Ik, and of the elements BNn' CNn; DNn; UNn, VNn, and WNn of the determinants a, it would be necessary to compute an additional determinant with complex elements, and all determinants computed would be of the fifth instead of the fourth order. Conside -ation of the estimated costs, using the computing facilities and techniques at present available, ruled out any effort to improve the results by such an extension. In Appendix C it is demonstrated that the quantities already computed for the back-scattering cross-section can also be used to compute the bistatic cross -section a ( 0, 0), where the transmitter is on the major axis of the spheroid and the receiver is on the ray with spherical coordinates 0, 4. 51

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 VIII RESULTS The problem at hand was to determine the nose-on radar crosssection at the Rayleigh side of the resonance region, and to determine the magnitude of the cross-section at the first maximum, for a prolate spheroid with fineness ratio of 10:1. Since it was known a priori that the computations would be limited by economic considerations to only a few points, it became necessary to predict the location of the resonance region and to see that most of the points chosen for computation were in the Rayleigh region near the first maximum. It was decided, however, that as a safety measure a few points should be computed at fairly wide intervals in the region of larger characteristic-dimension-to-wavelength ratio, to be used if the predicted location of the resonance region was inaccurate. The margin of safety was cut sharply after the second run of the machine computations, when it became necessary to reduce the number of points being computed. The reasoning behind the prediction of location of the resonance region is outlined here for the benefit of anyone who may have to make similar predictions in the future. It was expected that, for wavelengths small compared to 27a, the vector cross-section would oscillate about the approximate solution given by geometrical optics, Tb4 a G.O. = a2 It was expected also that, for wavelengths large compared to 2ia, the Rayleigh approximation would yield an asymptote to the graph of the vector cross-section. The Rayleigh solution for the vector case is 64 w3 T2 k4 TRay. = N2 (4T - N)......_________________________ 52

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 where 27r 4 k = T = -rab,,., Za2 (2 a a2 ^2^ and N = a _ a- a a- a - ~aTa- b a For -= 10, this yields b URay. = 7.117( a ) (.G.O. It was also believed that an upper bound on the abscissa of the first maximum could be obtained from the approximate solution given by physical optics, Tpb4 i -2 cos k sin ka\ sin ka2) (Tp 0 = a2 1 - 2 cos ka -ka-) + \ka -) and that a lower bound on the abscissas of most of the succeeding maxima could be deduced from the thin-wire theory discussed in Reference 25, which gives the following formulas for the locations of the maxima: 27ra cot -for the odd maxima 2 log t+log 1.84 4{ X 1 - 2ia -tan - for the even maxima. For the spheroid considered, with a/b = 10, the left hand side of this expression reduces to r{4.12 -3 log ra } All the relationships listed seem clearly to point to the choice of 2la as the characteristic dimension of the spheroid. _______________________________ 53 ---------------------

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 Use of this theory in predicting the locations of the maxima is vindicated by the results of the computations. The predictions seem to be very accurate indeed. In the interest of completeness the scalar solutions have been plotted for comparison with the vector solutions. The Rayleigh solution for the scalar case is / 47T3 T2 (2 - L\2 or X4 1 L' where 4 T = -Ta b2, and T b2 a ga +Va2 - 1 L - - 1F a2-b2 2Va2b - lo a - a- b2 For a/b = 10, this yields G.O. = 1.815 X' Some particular values of the scalar cross-sections of prolate spheroids were previously published by Spence and Granger (Ref. 26). Those values which correspond to the present problem are compared with the Mark III results. Figure 3 is a graph of all the results mentioned above. The ordinate in each case is the nose-on cross-section a divided by the geometricalirb4 optics result 2-, while the abscissa is 2 T times the ratio of semimajor axis a to wavelength X. The exact vector and scalar curves were extended to the points where the question of reliability made further plotting undesirable (partly because of the limited number of points obtained and partly because of the convergence question). The abscissas of the maxima as predicted by the thin-wire theory of Reference 25 are shown at the top of the graph. These serve to emphasize further that the number of points obtained was inadequate to define the curves in the 54

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 range beyond the first two maxima. However, this range was not of particular interest in the problem under consideration. Several results based on third-order systems have been included for purposes of comparison and to give some indication of the convergence situation. Numerical results for the nose-on radar cross-sections of prolate spheroids with fineness ratios other than 10:1 can be obtained in exactly the same way as those presented here. The algebraic results have been generalized in Appendix C to cover the case of transmitter on the axis of symmetry and receiver at an arbitrary point. The additional computations needed for this case are also noted in Appendix C. The algebraic expressions for the oblate spheroid problem are similar to those for the prolate spheroid (see Reference 28). The tabulated results obtained in the present investigation may be of use in solving the above-mentioned problems. In Appendix D the printed results available from the Mark III computations are listed. Possibilities are being investigated for the publication of these tables elsewhere; in any case, numerical values of any quantity tabulated will be furnished upon request. 55

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 100 60 - - Abscissas of Maxima 40 - -- (Thin Wire Theory) 20-_ —10 — — _.6- I r.4 ~ i / 1 / 1 t1 | /, 0 I | 7 r b ~ /2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ /.1-' // I n III II I 0/ I /I 0004n~A/ Sar I| Odp.04-, ___4_.002.04- -- y -f I —- - - --- _ __ Vector (Electromagnetic)-Mark ll Results - - - --------- Scalar (4th. Order).0 1 - I I I I I —-- --- -------- ---- Vctor } Rei La.1..006 — 1 2~~~ ~ Vector.Mark III Results */ul f / B Scalar (3rd. Order).0006 -^ --- -/- ---- -- -- --- -- - - -56.000 A Results of Spence.0002 —-.0001-.1.2.3.4.5.7 1.0 2 3 4 5 7 10 27 a FIG. 3 BACK-SCATTERING FROM A PROLATE SPHEROID, NOSE-ON --------------------- I

WILLOW' RUN RESE;ARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 APPENDIX A RECURRENCE RELATIONS AND OTHER FORMULAS FOR PROLATE SPHEROIDAL FUNCTIONS AND WAVE FUNCTIONS Differential recurrence relations for radial prolate spheroidal functions (j = 1 or 2): (2 -1)amn(k) dRn /d + 8mn() Rmn = R(,n-1 mn mnQ) Rmn = m,,n - (2 - 1) mn() dRm)n-1/d + Y mn() Rmn-1 = - Rn; (42 - 1 )mn(T) dR) /d4 + Pmn(Q) R(n = R(mj,n) (42 1)mn(4) dRmln/d + (mnR() Rm ),n = -Rmn 1 1( amn = Sn(7) S n(T) dT/c TSmn(T) Sm.n-l(7) dT (i) (2) (1) (2) cRm,n-1 Rmn- RmnR,n-1 ]; mn 2 dS2 m()/dSm),n (T) dT/ c TS()(T S(m () dr = c(1 -1) (Rmn)-1mn [ R)mnn- 1 R 1] c(- R (2n R_ (m.1 )lR R (2lm R (2) m),n - -1 R(1) R(2) (1)n R(2) [R(1) R (2)n R(1)' R(2) - mn-l mn - Rmn Rmn- mnmn mn mn 1 - T2 dS( ) fl (1)''m^n 11- t2 ^.2Sn(T) dSmn^.l(T)/dTdT/cjf TSf 1n(T) Sm,.n 1() dT 57 -

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 _C( _2 (1) R ) (2R' (R)' (2) -c( - 1) [RmnRm,n-1 - Rm nn-l R1n] (1) (2) (1) (2) / (i) (2) (1) (2) [RmnRmn-1 Rmn-l Rmn]/ [Rm,nRmn- -Rn- Rmn-1Rm,n-.'mn,(7)Smin(')drT 7mn 1 c 1 (1 -Tr2)1/2 S (T) S n() dT c -(1) [R (2) (1) R (2) [Rm- 1,n Rmn - Rm Rm,n-1 ] 1 1 Pmn {(42 1)1/2, (1 2)1/2 Smn( Sm 1,n () dT (4 )2 (1 -12)1/2 S(1) () S (T1) dT'] JcTldSmn(s)/dTSmrns(T) 1 - C ( 1) [RmlnRmn - Rmn m-l,n] [Rml,n Rmn Rmn Rm- 1 [Rmn Rmn Rmn R')mn I" (Trmn 1 f ( _,712)1/2 (1) (1)s 1,n (7 m 7i2_ J(1 -2r2)'/2 Sm (T) Smn(T) d -^.22 mn() -l,n(-R)/drd]. [ (1 )/2i S$(r) s- 1,n ()M dr]/_______________________________ 58

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-126 c(2l) (i) R(2)' ()' (2) - C( 2 m-1) m[l,n - Rm-l,n Rmn] () (2)' (1' (2) / (l) (2)1 (l)' (2) -[RmnR m-l) 2)n- l nRmn]' Rm-l,n Rm -ln- Rm- -,n R 1,n] Differential recurrence relations for angular prolate spheroidal functions (j = 1 or 2): (j) (j) (S) (1 772) amn(1 ) dSmn/dl/ + bmn(?) Smn = Sm,n (j) (j) (n)mn-1 (1 - 7 2) amn(7) dSm),n /d + cmn(77) Sm,n- = - Smn; (1 - 2) Pm (7) dSmn/ + rmn() Smn m-l,n' (j) (j) ( 0 (1 -r 2) Pmn(77) dSm)l,n/d?7 + Smn(7) Sm-l,n = - Sn; amn() = -amn(4), bmn(4) = - mnn(T), Cmn(S) = - Ymn(Q); Pmn() = - ( n( = - Pn( n) = mn(), Smn(t) = -Ormn()Recurrence relations for three contiguous functions: am,n+l Rm,n-l +(Ym,n+l mmn mn m,n+l ) Rmn + mn Rm,n+l = 0; rnm,n+l Rm-l,n + ((m+l,n 7mn-Pmn -rm+l,n) Rmn + 7rmn Rm+1,n = 0; and similarly for angular functions. Series for derivatives of spheroidal scalar wave functions: XXmn(x,y, z) = Xm)n(, ) = Smn() Rm n(e) eim; the symbol ln,j is defined on p. 41; aX) /az = fn+l j fj X; aXn/8ax + i ax()/a j=o i(-) (l) () mn (l-) =Lnj gj m+1 j; amn/a - i aX /ay n h X m-l,j; j=o j=o --------— 59 —--------

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 ~mn WI 1 (1) (1) fj = k sin (j-n)f 7 S) (T) S( (T) d / [Sj (rT)]2 d7; 1 1 j = -k cos (j-n)7r (1T S) (r) S (T)d [Sr 1j(T)] dT; j= -ko(n)2J mn m+ij(d hn = k cos1 (j-n) 7T (1 -r)2S ) S i( [Sm j( dT. Series for components of spheroidal vector wave functions: mn = X( X e = X, X e Mmn = V X e; Mmn = mn |(i) mn =mn (y) mn + - e; XMm [xn + im -nn) (e- i e "z' i ax ax (a()ni X m (e i; substitute series rom \ ax ay / preceding set of formulas........_____________ 60.......

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN - UMM-126 APPENDIX B FORMULAS USED IN COMPUTATIONS The essential formulas used in the computations and not given elsewhere in this report are listed here: 1) Legendre and Bessel Functions: these were computed from the following polynomials and recurrence relations. The symbol [x] denotes the largest integer not exceeding x: _ 2. (-1)k(n + 2k)! - 2k nc k=O (2k)!(n - 2k)I +2cos(ct -(n7-1) ) (2-1) k(n + 2k + 1)! 2c ~o~(c~-~nf) ~(2c'~ 1)~+ cos(cS -f1 n) t (2k + 1)! (n - 2k- 1)! (2c/)-k- (n = 2, 3,..., 17). k!(-n-k-l)I(-n-2k-l)( (n = -1, -2,..., -16). d(n nl (-kn-l)] (-)k(Zn2kZ) Zk P[(-n-2 ( =)l k!(-n - - 1))(-n-2k -2k- )) P1 -2n"+2 /2 _,1 /2 -] (-l-k-2n - 2k 2kIk- k-n-2 -k=O k!(-n-k-2)!(-n'-2k- 3) (n = -3, -4,s..., -16). dpn P 2[-(l2)]2 (-1)k(-2-n-2k- 2)1k dPn1)5 B. 2 k=0 ki(-n -k- 1)!(-n- 2k- 1)i n] (_-l )k(-2n-2k)I -2k - k — I (-n - k)!(-n - 2k)I 61

WILLOW/ RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 d pl = 2n+2 (2_ -1/2 d P-n —2 (4) 2 1 + [ -2(n+2)] (1)k(-2 -k-2) -2k 2 -(_))k(_2n -2k- 2)! 2k k=O +[- I2(n+) k3-n- )!(-n + [-(n"'l 2k(-)k ( -2n - 2k -4)! -2k + (n+l) E k!(-n 2)(-n-k-2)l(- -2k-3)l k=O (n = -3, -4,..., -16). 1 +1 [2(-1 2n- 4k - 1 Qn )' P() =1 g - - (1 +2k) (n-k) n-l -2k ()' k=O (n = 1, 2,..., 15), Qo() =4 loge Q1l ( = I +1 1 41 (+ 1 12/2 Qn+l (4) 2= Pn+ lge - (e - 1 ) n+l (4) [n] 2n - 4k + 1 - 2 (1 + 2k)(n - k + 1) n-2k k=O (n = 0, 1,..., 15), 12 - 1)1/2, Q1 16 Qo ~) = -(2-1l)/2 Q_'(4) - =-4(- )-1/2 d Q n +Ln+l 1 {_ d-TQn() -2- n1 [Pn+l() - Pn()] l~oge -4 [2n (2n_- 4k + 1) )Pn ] -.....[in] (2kn-4k -1)]nk() (2n-4k+l) Pn-2k-) -- (1+2k)(n-k+l) k=-2k0 (1 + 2k) (n- k) (n =1,2,..., 1-5), ______________________ 62 ----------------------

WILLOW RUN RESECARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 -d=- Qo (2_1 )-3/2 d/ d- Q-+l (~) = ( -l)-1 |(n + 1) [Pn+2 (l) 1~ge: _ 1 - (~z-)1/2 Pn+Z (~) d (1) + 2) -1 + ) P 1 -(n+2 nn)l 2 )L loge 2 ( )t2 n+21/ [l(n+l )] [ kn)2 (2n - 4k + 3) p1 ( k- (1 + 2k)(n- k + 2) n-2k+ -(n+ 2) { [1_-Pn+l (n ) loge- 2 1)17~ Pn+l( ) [2nl (2n - 4k+l) + j " (1 + 2k)(n - k + 1) n-2k (n = 0, 1,..., 15), d Q1()=(42)-3/2 Qll2 ()= 2)-3/3. 2) Separation Constants: The separation constant Amn was obtained from a) or b) below, depending on the value of c: 4 a) c > 5.0: A'mn hmn-k k= -1 where mn mn + Zn hmn = -(2n + 1), ho = (2n2 + 2n + 3 - 4m2) 2, hi = (2n + l)(n2 + n + 3 - 8m2) 2-4 mn 4 h2 = [5(n4 + 2n3 + 8n2 + 7n + 3) - 48m2 (2n2 + 2n + 1)] 2-6, h3n = [66n5 + 165n4 + 962n3 + 1278n2 + 1321n + 453 - m2(2368n3 + 3552n2 + 4448n + 1632) + m4(256n + 128)]. 210, 63

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 hn = [252n6 + 756n5 + 5885n4 + 10510n3 + 18478n2 + 13349n + 4425 - m2(14720n4 + 29440n3 + 64000n2 + 49280n + 17280) + m4(6144 n2 + 6144n + 3072)] * 2 -12 4 b) c <5: Amn C gk k k=0 where ~mn,./ mn /2 [ (2m - 1)(2m+ 1) o = -(n + m)(n + m + 1)2 1, + 2m - 1)(2 mn 1/2 [ (n+ l)(n+2)(n+ 2m + 1 )(n+2m + 2) 2 / [(2n+ 2m + 1)(2n + 2m + 3)3 (2n + 2m + 5) (n+2m)(n- 1) n (n+2m- 1) (2n + 2m - 3)(2n+ 2m - 1)3 (2n+ 2m + 1) gmn (4 2 1) F ____ (n- 1) n (n+2m- 1)(n+2m) m = 4m - 1 (2n+2m-5)(2n+2m-3)(2n+2m- 1)5 (2n+2m+1)(2n+2m+3) (n + 1)(n + 2)(n + 2m + 1)(n + 2m + 2) (2n+2m-1 )(2n+2m+1 )(2n+2m+3)5 (2n+2m+5)(2n+2m+7)' mn /42 12 r(n + 1)(n + 2)(n + 2m + 1)(n + 2m + 2) 94 ^(4m -1 )(2n+2m-1 )2(2n+2m+l )(2n+2m+3)7(2n+2m+5)(2n+2m+7)2 (n - 1) n (n + 2m - 1) (n + 2m) (2n+2m-5)2(2n+2m-3)(2n+2m- 1)7 (2n+2m+l)(2n+2m+3)2 + 2 4 r(n+l)(n+2)(n+3)(n+4)(n+2m+1 )(n+2m+2)(n+2m+3)(n+2m+4) + (2n+2m+l)(2n+2m+3)4(2n+2m+5)3 (2n+2m+7)2(2n+2m+9) (n-3)(n-2)(n-1) n (n+2m-3)(n+2m-2)(n+2m-1 )(n+2m) (2n+2m-7)(2n+2m+5)2(2n+2m-3)3(2n+2m-1)4(2n+2m+l)j + 2-3 (n-1 )2n2(n+2m- 1)2(n+2m)2 (n+l )2(n+2)2(n+2m+1 )2(n+2m+2)2 2 (2n+2m-3)2(2n+2m-1 )7(2n+2m+1)2 - (2n+2m+l )2(2n+2m+3)7(2n+2m+5) (n -1) n (n+l )(n+2)(n+2m-1 )(n+2m)(n+2m+ l)(n+2m+2) 1/2 (2n+2m-3)(2n+2m-1)4(2n+2m+1)2(2n+2m+3)4(2n+2m+5), __________,64

WILLOW RUN RESECARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 3) Integrals: the integrals involving the angular spheroidal functions, which arise in the process of determining the expansion coefficients for the scattered wave, have been reduced in Reference 21 to expressions equivalent to the following formulas. Here, 6j,k is the Kronecker symbol, which is equal to 0 if j A k and equal to 1 if j = k. The symbol Uij is defined on p. 41. Nn S)(17) S(1)( d 7= Nn Nk(d ) 2/(2k + ). on onN,n N /-1 k=0 I2 -= (1 - 2)-1/2 S()(7) Son (n,) dt I -in on -+ 1 In oN oN in = - 2 N,n+l nk 2k + 3 dk dk+l + N,k n,j dk d k=0 k=0 j=k+l T1NNn 1 (i) ( n1) 00 2k + 2 en,oN.on oN INn (1 (772), SoN (1) (n) dt1'/(7')' dT'_ = S N nl (2)k+ o)(k+3) k Nnk+l 1 (2k+- 2k+ d kk+l dk+ INn — O'N a) w 6N(1) on 2 IN,n+l 0 k 1 k+ 1dk on o N kdOn dON Nn+l (Zk+l)(2k+2) k+k k k k+1 k=0 I5S (1 - 72) 1/2S ) SoN(7) dr; N,n (2k+l)(2k+3) L k-1 dk+l - (k+2) dk dk....____________________.65 6....

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 Nn N d (1) d l 17 -1 _ 27(-2) 1/2 SW (8 >7 SW1 ) d() nk oN oN k(k-l) in (k+d)2N d in _ Zklkd [(k I )2 dk2 + (Zk+3) dk + k dk ].(k+lT 1kr-)2 z^r [ - nk nj dk d d + N n ol d) Rn )( 2k+ldk 2k-1 k-2 2k+3 k NnZNn d oN d in k=VNn j= (nk-1) n dN) I o () 1 4) Elements of Determinants: these are given in terms of the above integrals as follows, with An as defined in (III-3)' i 2 _ d 1) / n BNn -~ An (l_)( Roo) il = o(2_1)1/2 d (4)(o) INn + R/(4) oiNn DNn 6_in 2_ Nn 3 - A n 0 3 on 4 _____ d (4) Nn_(4 Nn VNn = (Ro2_1) d40I R4n (o)I4 0 d+ 3. W 421)d (4) () Nn _ t2 _1)/2R(4) Nn WNn.- ( d 2-1) c0I 5 Rn (~o) I 6

WILLOWV RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 5) Cross-Section Coefficients: The coefficients in the expansion for the back-scattered wave (r = 1) are quotients of the following determinants,whose elements are described in 4): Boo 0 Dol D03 B00 1 0 B22 C D D GY | B22 22 21 23 - G U1o + U12 V12 Wll W U30 + U32 32 W3133 Bl 0 Dlo D12 1 B33 C33 D30 D32 Uol + Uo03 V03 Woo W02 U21 + U2 2 3 W20 22 C00 Boo Dol D03 1 0 B22 D21 D23 2 G V10 U10 + U12 Wll Wl3 V30 U30 + U32 W31 W33 C 11 B11 D D1 BCl lDo D D a 1 B33 0 32 0/3 - V01 U01 + Uo3 Wo W02 V21 U21 + U W W,__________ 6______7....._________67

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-126 where Coo 0 Do1 D3 C 0 Dlo D12 C22 D2 D23 C33 D30 D32 G = V and H = V V W W v v W W V30 V32 W31 W33 V21 V23 W20 W22 6) Radar Back-Scattering Cross-Section: This is computed from the formula a=47ra2 Z in dn E 2n n lk n=0 k=0 --------------------- 68.....

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 APPENDIX C COMPUTATION OF BISTATIC CROSS-SECTION Formula (III-18) indicates that of the two sets of expansion coefficients an and Bn, only the an are used in computing the back-scattering cross-section. On the other hand, to obtain the bistatic cross-section Ua(0,) —where the transmitter is on the major axis of the spheroid and the receiver is on the ray with angles 8, ~ —one requires both the an and the 13n, as follows from formula (III-17). The only additional quantities required which are not used in obtaining the back-scattering crosssection are the angular functions Smn (cos 0). These are given by the expression (n) n mn m Smn (cos 0) = 1 n,k dk Pn+k(cos 0) where the Pn+k(cos 0) are associated Legendre functions. The spheroidal coefficients dk have been computed as described, and the associated Legendre functions are tabulated. It might be observed that in solving the systems of equations (III-13) and (III-14) for the an on a large scale machine, very little additional expense is involved in obtaining the 3n also. In the case where 0 = 7r/2, i.e.,the receiver is in the plane of the minor axes, the computation of O'(0,0) becomes particularly simple, since cos 0 = 0 and (1) (-l)n(2n) I S0,2n (0) 2()2 22n(n 1)2 ( l ) (-1)n(n+l)! 0,2n+l (~0) 22n.2 22n(n -)2 69

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 APPENDIX D QUANTITIES TABULATED IN MARK III OUTPUT A list of the parameter values computed is given on page 47 of the 27Ta text. For each value of listed, the following quantities are recorded: 1. Associated Legendre functions P-n -m (1) and Qm+n ( and their derivatives with respect to 4, for m = 0, 1, n = -1, -2,... -16. 2. Bessel functions Jn+i(c4), for n = 1, 2,... 17. 3. Separation constants Amn, for m = 0, 1; n = 0, 1, 2, 3. 4. Spheroidal coefficients dk (or dk /P in range of k < 0, where mn ) dk =0), for m = 0, 1, n = 0, 1, 2, 3, k = all necessary values in range -16 to +16 (see Appendix B, Part 3). 5. Radial spheroidal functions Rmn(m) and Rmn(2 ) and derivatives of these with respect to 4, for 4 = 1.005 m = 0, 1, n = 0, 1, 2, 3. Nn 6. Boundary integrals Ik for k = 1, 2, 3, 4, 5, 6, all combinations of N and n in range N = 0, 1, 2, 3, n = 0, 1, 2, 3. - __________________ 70

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 7. Determinantal elements BNn, CNn, DNn, UNn, VNn, WNn, for all combinations of N and n in range N = 0, 1, 2, 3, n = 0, 1, 2, 3. 8. Radar cross-section a. With the exception of a, all quantities are given to 15 significant figures. The values of a are rounded off to 5 significant figures....._____________________ 71

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 REFERENCES 1. S. Silver, Microwave Antenna Theory and Design, McGraw-Hill Book Co., Inc., 1949. 2. F. Rellich, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 53, p. 57, 1943. 3. H. Weyl, "Kapazitit von Strahlungsfeldern", Mathematische Zeitschrift, Vol. 55, p. 187, 1952. 4. C. Mueller, "Zur Methode der Strahlungskapazitat von H. Weyl", Mathematische Zeitschrift, Vol. 56, p. 80, 1952. 5. H. Weyl, "Die natUrlichen Randwertaufgaben im Aussenraum fair Strahlungsfelder beliebiger Dimension und beliebigen Ranges", Mathematische Zeitschrift, Vol. 56, p. 105, 1952. 6. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, (lecture notes), Technology Press, Mass. Inst. Tech. (1946); Methods of Theoretical Physics, McGraw-Hill Book Co., Inc., (1953). 7. B. Friedman, Techniques in Solving Partial Differential Equations, Institute for Mathematics and Mechanics, New York University, 1950. 8. J. A. Stratton, Electromagnetic Theory, McGraw-Hill Book Co., Inc., 1941. 9. P. Debye, "Der Lichtdruck auf Kugeln von beliebigem Material", Annalen der Physik, Vol. 30, p. 57, 1909. 10. P. Frank and R. v. Mises, Differential und Integralgleichungen der Mechanik und Physik, Vol. 2, Mary S. Rosenberg, 1943. 11. W. W. Hansen, "A New Type of Expansion in Radiation Problems", Physical Review, Vol. 47, p. 139, 1935. 72

WILLOW' RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-126 12. J. F. Ritt, "On the Differentiability of Asymptotic Series", Bulletin of the American Mathematical Society, Vol. 24, p. 225, 1918. 13. H. Bateman, Partial Differential Equations of Mathematical Physics, Dover Publications, 1944. 14. M. J. 0. Strutt, Lame'sche-Mathieusche-und verwandte Funktionen in Physik und Technik, Springer, 1932. 15. J. A. Stratton, P. M. Morse, L. J. Chu, and R. A. Hutner, Elliptic Cylinder and Spheroidal Wave Functions, Wiley, 1941. 16. C. Flammer, "Prolate Spheroidal Wave Functions", Technical Report No. 16, Stanford Research Institute, February 1951. 17. E. T. Whittaker, "On the Recurrence Formulae of the Mathieu Functions", Journal of the London Mathematical Society, Vol. 4, p. 88, 1929. 18. I. Marx, "Recurrence Relations for Prolate Spheroidal Wave Functions", Journal of Mathematics and Physics, Vol. 32, p. 269, 1954. 19. I. Marx, "On the Structure of Recurrence Relations", Michigan Mathematical Journal, Vol. 2, p. 57, 1953. 20. I. Marx, "On the Structure of Recurrence Relations II", (to be published). 21. F. V. Schultz,"Studies in Radar Cross-Sections I-Scattering by a Prolate Spheroid", Willow Run Research Center, University of Michigan, UMM-42, March 1950. 22. P. M. Morse, "Addition Formulae for Spheroidal Functions", Proceedings of the National Academy of Sciences, U.S.A., Vol. 21, p. 56, 1935. 73

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 23. K. M. Siegel, H. A. Alperin, J. W. Crispin, H. E. Hunter, R. E. Kleinman, W. C. Orthwein, and C. E. Schensted,/'tudies in Radar Cross-Sections IV-Comparison Between Theory and Experiment of the Cross-Section of a Cone", Willow Run Research Center, University of Michigan, UMM-92, February 1953. 24. A. Wintner, "On Hill's Periodic Lunar Orbit", American Journal of Mathematics, Vol. 60, p. 937, 1938. 25. J. H. Van Vleck, F. Bloch, and M. Hamermesh, "Theory of Radar Reflection from Wires or Thin Metallic Strips", Journal of Applied Physics, Vol. 18, p. 274, 1947. 26. C. J. Bouwkamp, "On Spheroidal Wave Functions of Order Zero", Journal of Mathematics andPhysics, Vol. 26, p. 79, 1947. 27. E. Schmidt, "Uber die Aufl6sung linearer Gleichungen mit unendlich vielen Unbekannten", Rendiconti del Circolo Matematico di Palermo, Vol. 25, p. 53, 1908. cf. also treatment in F. Riesz, "Les Systemes d'Equations Lineaires a une Infinite d'Inconnues", Gauthier-Villars, Paris, 1913. 28. G. Mie, "Beitrage zur Optik triber Medien, speziell kolloidaler Metall6sungen", Annalen der Physik, Vol. 25, p. 377, 1908. 29. H. Weil, "Studies in Radar Cross-Section X - Scattering by a Sphere", University of Michigan Engineering Research Institute, (to be published). 74

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-126 DIS T RIBU T ION Distributed in accordance with the terms of the contract........_________________________ 75

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