ELECTROMAGNETIC PLANE WAVE SCATTERING BY A PERFECTLY CONDUCTING DISK by George Russell Mattson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1970 Doctoral Committee: Professor Chen-To Tai, Chairman Professor Chiao-Min Chu Professor Albert E. Heins Professor Thomas B. A. Senior Doctor Piergiorgio L. E. Uslenghi Doctor Vaughan H. Weston RL-491 = RL-491

RL-491 ABSTRACT ELECTROMAGNETIC PLANE WAVE SCATTERING BY A PERFECTLY CONDUCTING DISK by George Russell Mattson Chairman: Professor Chen-To Tai In this research, the phenomenon of plane wave electromagnetic scattering by a perfectly conducting disk is studied for both low and high frequencies. Consideration is limited to only the far-zone backscattered fields for incident plane waves having either the electric or magnetic vector parallel to the plane of the disk. For frequencies near or below resonance, a method based upon Flammer's (1953) exact solutions to the plane wave scattering problems is developed for computing the far-zone backscattered fields for both incident 'plane waves. This method, which involves the computation of various oblate spheroidal wave functions and has been programmed for use on an IBM 360 computer, is used to compute direct and cross-polarized radar cross sections for integer values of the disk ka product (k = free space wave number, a = disk radius) that range from one to seven. These computed radar cross sections are compared with the same cross sections obtained experimentally. The high frequency investigation is based upon an approximate method proposed and applied by Ufimtsev (1958), which has as its foundation the approximation that for a perfectly conducting flat plate or disk of large radius of curvature the edge behaves locally like a half-plane. Known solutions to plane wave electromagnetic scattering problems for the half-plane are used to obtain explicit, though approximate, expressions for the surface current densities on the disk, which, in turn, are used to find approximate expressions for the farzone backscattered fields. These expressions, which are very similar to those

obtained by Ufimtsev for the disk scattering problems, represent a formal extension of his results to one greater inverse power of (ka) and to greater aspect angles for the case of backscattering. Several critical comparisons are made in order to test the exact and approximate solutions of the plane wave scattering problems. These comparisons are between the exact solutions and low frequency solutions to the disk scattering problems due to Eggimann (1961) for a ka-product of one half, between the exact and high-frequency solutions for an intermediate value of ka, and between the high frequency solutions and solutions due to the Geometrical Theory of Diffraction. The last comparison is not a quantitative one but is concerned with the forms of the two approximate solutions. By employing /the approximate results obtained in this research, an analysis patterned after that of Ross (1967) is used to find new expressions for disk and cone backscattering from expressions due to the Geometrical Theory of Diffraction.

ACKNOWLEDGME NTS The author wishes to express his appreciation to each member of the committee for his time and assistance. He is especially grateful to Professor C. T. Tai and Doctor P. L. E. Uslenghi for their guidance and suggestions in developing this research, and to Doctor R. W. Larson for his support and forbearance during the formative stage of this research. The aid of Mr. Alan Cole, who did the computer programming for this effort and Mrs. Mary Wright, who typed the manuscript, is deeply appreciated. Thanks are extended to all of the personnel of the University of Michigan Radiation Laboratory who contributed either directly or indirectly to the successful completion of this work. ii

! TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF ILLUSTRATIONS iv LIST OF APPENDICES v / I INTRODUCTION 1 1.1 General Discussion 1 1.2 The Scattering Problem 3 II EXACT DISK SCATTERING 7 2.1 The Curvilinear Coordinate System 7 2.2 Solution to the Scattering Problem 9 2.3 Calculations Necessary to Implement Equations (2.8) and (2.15) 15 2.4 Numerical Computations and Comparison with Experimental Data 18 in HALF PLANE APPROXIMATION TO DISK CURRENTS 43 3.1 The Basic Assumptions 43 3.2 The Half-Plane Currents 44 3.3 The Non-Uniform Currents on the Disk 51 3.4 The Far-Zone Backscattered Fields 55 3.5 The Far-Zone Backscattered Fields Due to the Unbounded Component 59 3.6 A Note on Extending Equations (3.83) and (3.84) 74 IV SOME FURTHER CONSIDERATIONS 76 4.1 A Comparison Involving the Exact Solution 76 4.2 Comparison Between the Exact and Approximate Solutions _ __77 4.3 An Application of the Geometrical Theoryl of Diffraction in Light of the Results of Chapter III 83 BIBLIOGRAPHY 88 iii

LIST OF ILLUSTRATIONS Fig. Page 1-1 (The Geometry of the General Disk Scattering Problem. 4 2-1 The Oblate Spheroidal Coordinate System. 7 2-2a RCS of a Disk for E-Polarization; c = 1.0. 22 2-2b RCS of a Disk for H-Polarization; c = 1.0. 23 2-2c Cross-Polarized RCS of a Disk; c = 1.0. 24 2-3a RCS of a Disk for E-!Polarization; c = 2.0. 25 2-3b RCS of a Disk for H-Polarization; c = 2.0. 26 2-3c Cross-Polarized RCS of a Disk; c = 2.0. 27 2-4a RCS of a Disk for E-Polarization; c = 3.0. 28 2-4b RCS of a Disk for H-Polarization; c = 3.0. 29 2-4c Cross-Polarized RCS of a Disk; c = 3.0. 30 2-5a RCS of a Disk for E-Polarization; c = 4.0. 31 2-5b RCS of a Disk for H-Polarization; c = 4.0. 32 2-5c Cross-Polarized RCS of a Disk; c = 4.0. 33 2-6a RCS of a Disk for E-Polarization; c = 5.0. 34 2-6b RCS of a Disk for H-polarization; c = 5.0. 35 2-6c Cross-Polarized RCS of a Disk; c = 5.0. 36 2-7a RCS of a Disk for E-Polarization; c = 6.0. 37 2-7b RCS of a Disk for H-Polarization; c = 6.0. 38 2-7c Cross-Polarized RCS of a Disk; c = 6.0. 39 2-8a RCS of a Disk for E-Polarization; c = 7.0. 40 2-8b RCS of a Disk for H-Polarization; c = 7.0. 41 2-8c Cross-Polarized RCS of a Disk; c = 7.0. 42 3-1 The Half-Plane Scattering Problem. 45 3-2 The Edge Geometry Used to Approximate the Disk Surface Current Density. 52 4-la Computed RCS of a Disk for E-Polarization; c = 6.0. 80 4-lb Computed RCS of a Disk for H-Polarization; c = 6.0. 81 4-lc Computed Cross-Polarized RCS of a Disk; c = 6.0. 82 iv

LIST OF APPENDICES Page Appendix A Appendix B Appendix C Appendix D Some Properties of the Oblate Spheroidal Coordinate System The Far-Zone Bistatic Scattered Fields The a (-ic) r Disk Scattering Program 89 92 95 97 v

Chapter I INTRODUCTION 1.1 General Discussion Exact solutions to problems of scattering of plane electromagnetic waves by perfectly conducting bodies of finite dimensions are few in number. ]For instance, if the class of body shapes known as oblate spheroids is considered, it is found that exact analytical solutions are known only for the two limiting shapes, the ideal disk and the sphere. The solution to the sphere scattering problem is the simpler of the two and is given in terms of the Mie series, which has proven amenable to many types of analysis. The nature of the disk scattering problem has resulted in several different formulations for the exact solution. Solutions have been obtained by Meixner and Andrejewski (1950) in terms of Hertz vectors, by Flammer (1 953) in terms of oblate spheroidal wave functions, by Nomura and Katsura (1955) in terms of hypergeometric polynomials, and Lure (1960) using sets of paired integral equations. Except for the special case of the direction of incidence normal to the plane of the disk, few calculations have been done using any of the above solutions. During the course of this work a means of carrying out some calculations of radar cross sections using'lone of the exact solutions to the disk scattering problem became desirable. The solution as formulated by Flammer was chosen for this work because it is given directly in terms of oblate spheroidal wave functions. Chapter I: is concerned with this computational effort and provides the theoretical development necessary to calculate the scattering cross sections of the disk as a function of aspect angle. Actual computations, however, are carried out only for, backscattering. The exact solution considered above is useful only for frequencies near or below resonance due to convergence properties of the functions involved, limitations of the existing tables of oblate spheroidal functions, and computer time 1

2 limitations. Consequently, it is desirable to obtain a solution or approximation that is valid and easily applied at high frequencies. Hence an asymptotic expansion of the solution is desirable. Jones "(1965)l (Heins and Jones (1967)) has developed and applied to the problem of electromagnetic scattering by a disk of a plane wave incident at normal incidence a systematic process which yields as many terms of the asymptotic development of the solution for high frequencies as one is willing to calculate. An intention to apply the process to the problem of oblique incidence has been indicated. Other asymptotic solutions have also been considered. These generally are based upon the principle that the edge of the disk behaves locally like a half-plane. Different degrees of approximation can be assumed. In the Geometrical Theory of Diffraction an edge diffraction I,_-_- _,_ mechanism, in which each point on the edge diffracts a cone of rays with the cone half angle equal to the smaller angle between the direction of incidence and the tangent to the edge, is assumed. This means that the scattered field at any point in space will generally be the superposition of the scattered fields from a finite number of points on the disk edge. Situations where all points on the disk edge contribute must be considered separately. An exposition of the Geometrical\Theory of Diffraction has been given by Keller (1962). Another approach has been suggested and investigated by Ufim tsev (1958). This method approximates the local disk edge currents by those that would be found on a half-plane tangential to the disk at the given edge point. The scattered fields are then found from the resulting current distribution. The method, of course, is approximate as it fails to account for perturbations in the assumed current distribution that arise because of the finite dimensions of the disk. Ufimtsev has found only the first term in each of the asymptotic expansions for the far-zone scattered fields. Also, his results are valid only I for small angles. An investigation of this method is undertaken in Chapter m. The solution is formally extended to obtain another term in each asymptotic

3 series and to improve the description of the dependence on aspect angle for the case of backscattering. Mention should be made of the fact that solutions for scattering problems involving the disk may be applied to scattering by a circular aperture by proper application of the rigorous form of Babinet's principle (Bouwkamp (1954)). Consequently,\an exact solution to plane wave scattering by a circular aperture may be found. Conversely, approximate solutions to problems of scattering by a circular aperture may be applied to problems of disk scattering. In any case, only disk scattering will be considered here. 1.2 The Scattering Problem Figure 1-1 defines the geometry of the problem to be considered. A plane wave F. is incident in the yz-plane with an angle 0 between the neg1.. - ative z-axis and the direction of incidence. F., which may be either the incident electric or magnetic field, can be resolved into components in the 0 -direction and the R -direction. Hence any incident plane wave F. can be expressed in terms of the following two incident fields: - A - ik(y sin 0 + z cos 0) -i -ik(ysin 0 + z cos0) 2 (1.2) - E0 = 0 H0 The constants 770 and k appearing in these equations are the characteristic impedance of free space and the wave number of the incident field respectively. The unit vector x has been used in place of - to indicate that the direction of incidence is confined to the yz-plane. An e timedependence has been suppressed here, and will continue to be suppressed throughout this work.

4 z Disk of radius a F. I / Disk of radius a FIG. 1-1: The Geometry of the General Disk Scattering Problem. P is the Point of Observation.

5 The incident field of equation (1.1), which has the electric field parallel to the plane of the disk is said to be E-polarized. Similarly, the incident field of equation (1. 2) is said to be H-polarized. In order to keep the computational effort tractable only the far-zone scattered field will be considered. Also, consideration will be further restricted to only the backscattered far-zone field in order to make comparisons with experimental data, even though several of the techniques used can be applied to the case of bistatic-scattering in the far-zone. The behavior of both direct and cross-polarized components of the far-zone backscattered field will be investigated. It can be shown that the cross-polarized component of the backscattered field will depend on the polarization angle ' as sin 2V when V varies and all other aspects of the incident field remain unchanged. In particular, when i is equal to zero or ninety degrees there will be no crosspolarized component of the backscattered field. Because of this, the crosspolarized component of the backscattered field can be expressed in terms of the backscattered fields due to the incident fields of equations (1. 1) and (1. 2). Since the cross-polarized component of the backscattered field attains its greatest value for a polarization angle of forty-five degrees, all measurements and theoretical calculations involving that component will be done only for this polarization angle. Calculation of the cross-polarized component of the far-zone backscattered field is straightforward for that polarization angle. Let the incident electric field be written as E E i - A +EGA (1. 3) The total backscattered field is just the vector sum of the scattered fields due to each component of the incident field. Neither component of equation

6 (1.3) will give rise to a cross-polarized term. In any case the backscattered field may be written as -s 1 5A E = (E0 + E (1.4) Then the magnitude of the cross-polarized component of the backscattered field may be found from IEc= E1. E (1.5) where Ai ~E + (1.6) If either sign is taken in equation (1. 6), equation (1.5) yields ES I= 1 S ESE | (1.7) This simple result will be used in Chapters II and III to find expressions for the cross-polarized radar cross section of a disk.

Chapter II EXACT DISK SCATTERING 2.1 The Curvilinear Coordinate System An ideal conducting disk of radius a may be modeled by the focal circle, = 0, of the oblate coordinate system of which a cut in the yz-plane is shown in Fig. 2-1. The system of confocal hyperbolae and ellipses shown is rotationally symmetric about the z-axis. Only two oblate coordinates, the angular coordinate r1 and the radial coordinate g are indicated, as the third oblate coordinate v is simply equal to cos 0, 0 being the angle of rotation about the zaxis measured in the xy-plane relative to the x-axis. For this reason it is convenient to adopt a hybrid coordinate system and use ( instead of v. Relationships between Cartesian coordinates and this hybrid coordinate system are given in Appendix 2-A. Also given are the metrical coefficients h, h, and h. - r-' FIG. 2-1: THE OBLATE SPHEROIDAL COORDINATE SYSTEM 7

8 The scalar Helmholz equation, 2 2 (V2 +k2)=, (2.1) is separable in oblate spheroidal coordinates, and its eigenfunctions are expressible as m (Cos mmO) em () S ()(-ic,)R (-iC, cos m (2.2) ml ml mlsin MO The functions S (-ic, r) and R (-ic, iM) are respectively the angular oblate spheroidal functions of the first kind and the radial functions of the ith kind, i = 1, 2, 3, 4 following the definition used by Flammer (1953) except that the subscript S used here is lhis subscript n. The quantity c is equal to the product of the wave number k = 27r/X and the radius of the disk (c = ka). (1) The angular functions S (-ic, rT) can be expressed in terms of associated Legendre functions by -ic, r) d P m (r). i(2. 3) ml n m +n mn O,1 The prime indicates the sum is to be taken over even or odd values of n according as (I -m) is even or odd. Quite extensive tables of the d have - n been published by Stratton, et al (1956). The angular functions are orthogonal over the interval (-1, 1) in rl. That is, Nm6 = S (-iC, r) S (1)-ic ) d r(2.4) m Ir r (-ic 1 -with (n+ 2m)! (d )2 ml E n!(2n+ 2m+ 1) (2.5) n = 0, 1

9 Further properties of the angular and radial functions will be developed as needed. In general, the notation used here will be that used by Flammer in Spheroidal Wave Functions (1957), which gives a comprehensive discussion of the properties of the oblate spheroidal functions. Numerical values for the functions are given for some values of c, r-, and; and tables for the coefficients in the series expansions of both angular and radial functions are listed. 2. 2 Solution to the Scattering Problem. - i If the field F in Fig. 1-1 is either the electric field E or magnetic field -i -i -s H, then both F and the scattered field F satisfy the wave equation, - 2 (2.6) VxVxYF-k2FO, (2.6) which is a special case of the vector Helmholz equation. The boundary con-t -i -s dition on the total field F = F + F on the disk surface is then either the - t Dirichlet (Ax F = 0) or Neumann (ftx (Vx F ) =0) condition for F equal to the incident electric or magnetic field respectively. The scattered field F must, in either case, satisfy Sommerfeld's radiation condition as the distance from the disk becomes infinite (e-. oo ). No vector function that is a solution of equation (2. 6) and that also satisfies either the Dirichlet,or Neumann condition on the disk is known, or is one likely to be found (Morse and Feshbach; Sec. 13.1, 1953). In spite of this, Flammer (1953) has shown that it is possible to solve exactly the problem of scattering of plane electromagnetic waves by a perfectly conducting disk in terms of even wave functions of the M and N type given by Mem/ r, ) = VW emn A(r (2. 7a) N em )(ri,, )= k 1 VM i)em) ( \(2.7b) u =x,y,z,

10 where pem )(r), ~, ) is given by the part of equation (2.2) that is even with respect to the angle (. The solutions to the scattering problems as given by Flammer are completely general and give the scattered fields everywhere in space for any angles of incidence and observation in Fig. 1-1. However, considerable simplification results if the point of observation is restricted to the far-zone and only terms of order (R) are retained. This will be done here, and calculations will be carried out only for the case of backscattering at arbitrary angles of incidence in order to compare with experimental data and various approximate methods for describing the far-zone fields of the disk for plane wave incidence. This restriction is not as severe as it might appear for all the essential features of the computational problem are preserved, the major differences being that fewer values of the radial functions need be found and that fewer terms are needed. All of the quantities dependent on c and 0 computed for this special scattering problem are also needed for the general one. Expressions for farzone bistatic scattering by the disk for the two incident fields F of Fig. 1-1 given by equations (1.1) and (1. 2) have been relegated to Appendix 2-B. Only backscattering will be considered in the following, and the far-zone scattered fields will be given in terms of spherical coordinates. The far-zone backscattered electric field due to the incident field of t equation (1.1) has a component only in the 0-direction, E, given by PE ikR oco 2D E J' (c) C 2 + 2 a m(c, ) E - Jm (S< 1(-ic, cos 0)) + m =m+2 m+lm, i = m m-1,...-,

11 where 60m = ( m )= Kroneker delta function, Om ^1, m = 0 c = ka,. 0= V77 = impedance of free space, and R is the distance from the disk at which the fields are measured. (1) The angular function S (-ic, cos 0) and normalization constant N have been discussed in section 2. 1. The J (c) and J' (c) are "joining" factors defined in terms of values of radial functions (or derivatives of) of first and third kinds on the surface of the disk. The a (c, 0) and E (c,0) are "weighting" factors that determine the contributions of terms in m the various vector wave functions to the scattered fields. A prime on a summation sign in this chapter indicates that the sum is to be taken over alternate values of the index. The joining factors JM (c) and J' (c) are defined by: ( (3 0 ( -m) even (C) (-iciO) (2.9a) 0 (U -m) odd o (.e-m) even J' (c)- (2.9b) nR (-ic,i0) 3) --- (-m) odd Rm (-ic, iO) It is also necessary to consider four other joining factors that relate the angular functions for cos 0 = 0 to the radial functions of the first and

12 third kinds. These are defined by the following expressions: (1) Sm (-ic,O) (1)1 S' (-ic,O) R m (-ic, iO) 0 Km I (3) r(3 — (-ic, iO) 0 K(3)') K -(c) m Q(1)' (3)' Rm l (-ic, io) ( -m) even ( -m) odd (I-m) odd (2. 10a) (2. 10b) ( -m) even ( -m) even (2. la) ( -m) odd (U-m) even (2.11b) (I-m) odd The primes on the angular and radial functions indicate differentiation with respect to the angular (ri) and radial (Q) variables respectively. The various kinds of joining factors are notlindependent, but are related by = J (c) (C), (2.12a) Kt () ' (c) = J ' (c (c). (2. 12b) M M.iM I

13 E E The weighting factors a (c,0) and 3 (c,0) have not yet been considered in detail. These are complex functions of and c given by in detail. These are complex functions of 0 and c given by IE (ce) = -a (c,), mI rn (2. 13a) a0 (c,0) (3) 21 b (c,0)K (c)On On 2 3 (2. 13b) bln(c, ) K()(c), ln in E() a (c 0) = 1 a (ac 0)K (c)- a 2n(c, )K2n(C) aOn 'On 2Lj32n 2n - mn (2. 13c) E n=m m (c ) c31 (3) 1 Veb (C (c)n - a (c,o) C)- ac- a,(co)K (c) mn mn 2 / m+l,n m+, 2, m-l,n( m-l,n n=m n=m+2 n=m m> 1, (2. 13d) where )n-l -1 -i (1) b (c,) = 2(2-6m )i N (sin 0) S (-ic, cos ), mn 0Om mn mn n-i - -i si(1) a (c, 0) = -2(2-6 -- )i N (coso) 0 ) (-ic,cos 0) mn Om mn mn - (2.14a) (2.14b) This completes the /expressions for far-zone backscattering for Epolarization. For H-polarization the situation is entirely similar. The incident magnetic field is now that of equation (1.2), for which the far-zone backscattered electric field will have only a 0- component.

14 It is -E, ikHC " ~ J (C) f~ ) \2 E 5=T1 H AO 0 ~ ( lm (-6 ) — (C,()O'1)2 N Sm (c, cosO)/ eH OH ikRL.O m m -_ ~ n + 2 a (c O) 1111 I OD/i (C) 2 a N k m+11 --m+1 00, Z P =m-1 m-1,~ JJ (2. 15.) All quantities are as defined before.,except for the weighting factors, which are given by AH 1(C.,) = 1 -ca H (C.,0) (2. 16a) OD1 0 H_ 0n (c. 0)K (c 2 a ln(cO0) K (C Zbln(C., 0) n7 (C) n =2 - (2. 16b) (2. 1 6c) '~1 - i 00 I OD l - - OD o 3)1 1 b (c 10) (C)+ a (c 20) (c)+- a (C'e) (c) ln n On 2 1 2n n = n= n = 2

15 b (C,0) (c) H n=m+l a = m Oo - 3 23) 1 3/ (c b ( o)Km()+ ~a (c,0)3) (c)+ a,(c 0m+) n mn mn 2 m-l, n m-,n 2 mI, mLn n=m+1 n=m- n=m+l m> 1 l - - (2. 16d) with b (c, ) and a (c,0) as given by equations (2.14a, b). mn mn 2. 3 Calculations Necessary to Implement Equations (2.8) and (2.15) In principle, equations (2. 8) and (2.15) allow computation of the far-zone backscattered fields for an arbitrary incident plane wave for any disk. Actual computations, however, are limited by the tables used and by convergence properties of the various series as c is varied. For this work the tables of Stratton, et al (1956) have been used. These contain the coefficients d for n 0< m, <K 8 and varying orders of n up to 23 for a range of c (g in the tables) from.1 to 8 in increments in c of. 1 or. 2. The coefficients are given to seven significant digits, which is less than modern computers can utilize, but which is ample for the computations done here. The tables do not contain values of the normalization constants or of the joining factors, so these must be calculated from known properties of the oblate spheroidal functions. Equations (2.3) and (2. 5) allow computation of the angular functions and normalization constants in a straightforward manner given a program to calculate the associated Legendre functions needed in equation (2. 3). Calculation of the joining factors presents the greatest challenge and requires some further development of the properties of the angular functions. The angular functions may be expressed in power series expansions of ( 1 - Y2 ) m oo S (-ic, r)=(1l-r2) 2 C k(1 -.n2), (Q - m) even (2.17a) k = O

16 M OD S(1 (-ic r)=)( r2)2~-Z CM (l2)k,1 ( -in) odd ml, ) =,k=O )2k ( - 2k m s are related to the d MIcoefficients by (2. 17b) ml 1 C = ~ 2k 2in m~) rOD (2m+2r) I 1 rig 2! (-r)k(m+r+- 2) dr I r k 2 i n) even, — (2. 18a)in) odd, (2. 18b) ml 1 OD (2m+2r+l)! C 2k 2i k (m+k)i! (2+)1 2 \r =k (-r)k(in+r+ 3) kd dlQ. where (n) - = 1, (n) = n(n+ 1).... (n+k -1) 0 k The CM — enter into the calculation of all the joining factors. For the 2k J (c and J -I (c) the equations are Jm (c) = 11 1-~-Q (-ic) JI (c)= (ic (i -in) even,,,(I -m) odd J (2. 19a) (2. 19b) where the function Q"'(-ic) has different expressions when (I -in) is even and odd. Briefly., inlM (1) 1-c2 [K M urrnv (2m-2r). (-1) a ic) i r = 0 r 1 [2mr-r)J (Ie- m) even,.. (2. 20a)

17 (-l)M Km(1)I (-ic)2 Ir /n (2m-2r+l)I - (-ic) (2m-2r! ( -m) odd, r C r 2m-r 12) 2 (2.20b) and a (-ic) is defined as r ca (-ic) C2k x } (2. 21) r ~ r...2k The calculation of ae (-ic) is straightforward, but requires tedious r differentiation. The series for a (-ic) in terms of the C will consist of a r _ K2k _ finite number of terms and is given for 0 < r < 8 in Appendix 2-C in terms of the normalized coefficients C defined by 2k 'n mnl/ mn - c (2.22) 2k 2k 0 By virtue of equations (2. 20a, b) and (2. 12a,b) it is only necessary to thK1 (1)' determine a means of finding values for the K (c) and K ) (c) in order to be able to calculate all of the joining factors. Reference to equations (2. 10a, b) indicates that only angular functions and radial functions of the first kind are involved. The radial function of the first kind and its derivative at 0 = 0 are easily found from Q -m R (-ic i0)=(-1) 2 cm ( —m) even, 0m - c (m) 2md (-ic)( -m) even, (2.23a) ml (I+m)! (2m+1) 0

18 I -m-1 m((-i, i 0)=(-1) 2 m+1 (i-m)' m' 2 m R1) c ( (2m3) d (-ic) ( -m) odd. (2. 23b) Values of the angular function of the first kind and its derivative at r = 0 may be obtained either from equation (2.3) or equations (2.17a,b). For the computations done here it was decided to use the latter equations, since the mi C 's must be found in anycase, and since the resulting expressions are quite 2k _____ simple. For the angular function and its derivative at rl = 0, equations (2.17a,b) yield (1) Ie ml S m(-ic,O) = 0 C2k (-m) even, (2. 24a) k=0 S (-ic, 0) = C ( -m) odd. (2.24b) l0 ZC 2k Test values of the angular function at r7 = 0 computed from equations (2. 24a) and (2. 3) were found to be in excellent agreement. / The series in equations (2.24a, b), however, sometimes converge rather slowly to small values, especially for large values of/ c. This results in an effective loss in the number of significant digits in the summations given by equations (2. 24a, b) compared with the input data. For the range of c considered_ this loss is considered to be no more than 3 digits. In any case, computation of the C 's was done as carefully as possible. This effort appears to have been 2k successful in keeping propagation of round-off and other errors to a minimum. 2.4 Numerical Computations and Comparison with Experimental Data Let the scattered fields of equations (2.8) and (2. 15) ibe written as ikR s c0H( ikR P EB = ~ ^~ ~' *y s-p. E8.

19 EikR Es s - p cP ) *.15) E H where P (c,0) and P (c, ) are complex quantities equal to the entire summations over m in equations (2. 8) and (2/.15) respectively. In terms of these quantities the radar cross-section (RCS-normalized to a square wavelength) can be found by means of the expression 2 jlim 47rR2 jf 2 lim 4R- 2.I (2.25) X2 R 0 —woo X2 - For the incident field of equation (1.1) the radar cross-section in dB is, 10 log l0 () 20 og0 P (c,0) (2.26) -~. Similarly, for the incident field given by equation (1.2) the result is 0log10 -) 20 log 1( P ( (c ) (2.27). o 10 (o2 o ). Finally, for the cross-polarized scattered field for a polarization angle = 45 the radar cross-section in dB is found from equation (1.7)to be (C H E o10) log 201 log P(C. 0) Po(C ) (2.28) A program has been written to compute the direct and cross-polarized radar cross-sections for any value of c for which input data are available for an angular range in 0 of 0< 0 <. The restriction on 0 is necessary be2 cause the program would attempt to divide by 0 at either 0 or -, even though the actual solutions to the scattering problems have well behaved limits at these points. The input data consist of the value of c, a list of the

20 d (-ic) 's, and the angular range to be considered. In addition to the various n radar cross-sections, the program also returns the real and imaginary parts of E HJ P (c,0) and P (c,0), the normalization constants N., J (c) and J' (c), K (c) and K' (c), and the C ' is. A complete listing and explanation of the ml ml 2k program is given in Appendix D. Also included are illustrative tables of the Nm K, K m.) J (c), JQ(c), and Cmk for c = 4. 0. Values of computed direct and cross-polarized radar cross sections given by equations (2. 26) through (2. 28) are compared with experimental results obtained at the University of Michigan Radiation Laboratory experimental facility at Willow Run for values of c of 1 (1) 7 and values of 0 of 2 0(20) 880 in Figs. (2-2a) through (2-8c). The experimental data are given by two sets oft solid lines which reproduce the data obtained as the incident field was swept through a 180 scan from edge-on incidence through normal incidence (0=0) 0 and on to edge-on incidence. The two 90 segments were thent plotted one upon the other as a function of 0. The computed values are indicated as points every two degrees. On the whole, agreement between the experimental and computed data is quite good for the values of c that are used. There are, however, several types of discrepancies between the experimental and computed data that require explanation. For instance, the cross-polarized radar cross sections tend to show greater differences than the corresponding direct radar cross sections. This behavior is not very surprising in light of the fact that the cross polarized return is "generally much weaker than the two direct returns. Indeed, a difference in peak magnitudes between the two types of cross sections of 20dB or more occurs several times. Consequently, the cross-polarized return is more likely to be affected by stray returns and equipment limitations. The stray returns would generally be small and would manifest themselves principally by displacing the locations and changing the amplitudes of the nulls in the experimental patterns. Evidence of this type

21 of behavior is found in all returns for c = 1. 0 where the experimental data are consistently higher than the computed values. Again, in Fig. 2-3a similar behavior is found for the E-polarized return for 0 near 900. The behavior of the computed return is more consistent with the observed behavior of the cross-polarized return than the experimental data are. Both cases; cited are for low values of c, and consequently low amplitude returns, which are, of course, just the situations where stray returns would cause the greatest errors. Because of the symmetry inherent in a disk, the cross-polarized return for a plane wave at normal (0=0) incidence should be zero. In practice, however, it proves very difficult to obtain this condition with the equipment available. The major limitation is the degree of isolation available in the receiving antenna for signals ninety degrees out of space phase with respect to the desired signal. The direct return at normal incidence is so strong that the receiving antenna will pick up a slight error signal. This error signal will increase with increasing c, an effect that is actually observed in the strength of the cross-polarized return at zero aspect angle. For large values of c and 0 there occur large discrepancies between the measured and computed returns for E-polarization and cross —polarization. These arelbelieved&to reflect errors in the computed returns due to poor convergence of the various series used in computing the radar cross sections. This effect must be observed eventually, because of the limitations inherent in the tables of Stratton, et al (1956) that were used to compute the oblate spheroidal functions. The actual differences do become worse with increasing c, which is consistent with this view. The given data then can be used to determine in a qualitative way the limits of convergence of the exact solution as programmed. The success of the programming effort as outlined in this section implies that there now exists a standard, the computed returns, against which both experimental results and results due to approximate theories can be judged. Such a procedure will, for instance, be used in Chapter IV.

22 uted Values 6 0* COMP Experimental Data — C14 1-11, bD 0 V-4 0 V-4 -Ilftam..OOP909 9900*000*0*00 a* 1! av 0 (degrees) Aspect Angle C = 1100-0 p isk for E-PolarizatiOll; FIG. 2-2a*. CS Of a D

23 Computed Values *e* Experimental Data -- C0I I-. b bD 0 - -1 CD T-q - I (. 0 0. -I1 0. 0 0 0 0 0 0 30 60 Aspect Angle 0 (degrees) FIG. 2-2b: RCS of a Disk for H-Polarization; c = 1.0. 90

24 Computed Values *O* Experimental Data - -10...*-ooo**0 *-. 0 0 c9,< //b // - o 0 re 0 o0, -25 - 0 -30 0 -35 0 30 60 Aspect Angle 0 (degrees) FIG: 2-2c: Cross-Polarized RCS of a Disk; c = 1.0.

25 Computed Values *00 Experimental Data -- I 0 0 00 SOS.... -C b bD 0 Io 2- 1 -15 -20 Aspect Angle 0 (degrees) FIG. 2-3a: RCS of a Disk for E-Polarization; c = 2.0.

26 Computed Values *00 Experimental Data -- c/ '-4 b,-4 -I -21 Aspect Angle 0 (degrees) FIG. 2-3b: RCS of a Disk for H-Polarization; c = 2.0.

27 Computed Values * * Experimental Data - -10 -15 -20 0 uu C\I b bD 0 C) r —4 0 0 0 0 -30 0 -351 Aspect Angle e (degrees) FIG. 2-3c: Cross-Polarized RCS of a Disk; c = 2.0.

28 Computed Values * * Experimental Data -- 0 -10 *. 0 -15 -20 - - I 0 30 60 Aspect Angle 0 (degrees) FIG. 2-4a: RCS of a Disk for E-Polarization; c = 3.0. O*0

29 Computed Values *ee Experimental Data - bO b 0 v-4 Aspect Angle 0 (degrees) FIG. 2-4b: RCS of a Disk for H-Polarization; c = 3.0.

30 Computed Values *eo Experimental Data — I — I,0 0 0 C\V. b bO O -.,-. 0 -30 30 60 Aspect Angle 0 (degrees) FIG. 2-4c: Cross-Polarized RCS of a Disk; c = 3.0.

31 Computed Values *e0 Experimental Data -- ol b 0 O. 30 60 Aspect Angle 0 (degrees) FIG. 2-5a: RCS of a Disk for E-Polarization; c = 4.0.

32 - 0 Computed Values *e. Experimental Data CCl b 0 0 P'I -10 -15 Aspect Angle 0 (degrees) FIG. 2-5b: RCS of a Disk for H-Polarization; c = 4.0.

33 Computed Values *e0 Experimental. Data 0 0 Ie -I Us b bo 0,-4 'I 0 -3( Aspect Angle O (degrees) FIG. 2-5c: Cross-Polarized RCS of a Disk; c = 4.0.

34 Computed Values * * Experimental Data - 14 c I bO bO 0 0 00 30 60 Aspect Angle 0 (degrees) FIG. 2-6a: RCS of a Disk for E-Polarization; c = 5.0. I*

35 Computed Values *00 Experimental Data - 15 10 ll~ b bo 0 0 0 0 0 FIG. 2-6b: RCS of a Disk for H-Polarization; c = 5.0.

36 Computed Values **e Experimental Data -I -151- 1 0 ) 3 60 b r/ -20 * -25 -30 I 0 30 60 Aspect Angle e (degrees) FIG. 2-6c: Cross-Polarized RCS of a Disk; c = 5.0.

37 20 c Computed Values *ee Experimental Data 10 b bO 5 - 0 00 I 30 60 Aspect Angle 0 (degrees) FIG. 2-7a: RCS of a Disk for E-Polarization; c = 6.0. I

38 Computed Values *00 Experimental Data - bD 0 0 T-4 -I Aspect Angle 0 (degrees) FIG. 2-7b: RCS of a Disk for H-Polarization; c = 6.0.

39 Computed Values 000 Experimental Data Q b 0 0 'I Il 0 0 Aspect Angle 0 (degrees) FIG. 2-7c: Cross-Polarized RCS of a Disk; c = 6. 0.

40 201 Computed Values 0.. Experimental Data -- 0 0 0 0 0 _ _ _ 0 b bf) 0 0 'I -51 FIG. -2-8a: -RCS of a Dis8-k-for -E —Polari'za-tion;. —c-= -7-. 0-.

41 Computed Values *e. 2 Experimental Data - 20 15 0 -0 5 -5L \ 0 30 60 Aspect Angle 0 (degrees) FIG. 2-8b: RCS of a Disk for H-Polarization; c = 7.0.

42 Computed Values *~ Experimental Data — ~ -5- / -100 ib -15 -200 -25 0 30 60 Aspect Angle 6 (degrees) FIG. 2-8c: Cross-Polarized RCS of a Disk; c = 7.0. *0090

Chapter m HALF PLANE APPROXIMATION TO DISK CURRENTS 3. 1 The Basic Assumptions One usually assumes for a first order approximation that the surface A -i currents on a disk are given by those of physical optics; namely 2n x H on the top surface and zero on the bottom surface. This approximation gives reasonable results only for large disks for near normal incidence (the angle 0 in Fig. 1-1 small) and can predict no cross-polarized return for any value of 0. The presence of edges on the diffracting body is neglected and edge conditions (Bouwkamp(1954)) for the surface current density are violated. These edge conditions require that the component of surface current density normal to the edge remain finite at the edge while the component tangential to the edge become finite as the reciprocal of the square root of the distance to the edge. A better approximation that can satisfy the edge conditions is based upon the observation that for a disk of radius much larger than a wavelength the edge appears locally to have the properties of a half-plane tangential to the edge at the given edge point. We may then add to the physical optics current density a perturbation term that gives at the edge of the disk the current density that would exist on the tangential half-plane. Since for distances greater than a wavelength from the edge the half-plane current density approaches that of physical optics, the perturbation current density along a ray proceeding from the given edge point through the center of the disk may be taken to be the difference of the associated half-plane current density and that of physical optics. This approach was followed by Ufimstev (1!958) and will be used here to extend his results formally to greater values of 0 and inverse powers of (ka). 43

44 3.2 The Half-Plane Currents Figure 3-1 gives the statement of the half-plane scattering problem. The incident field is a plane wave directed obliquely to the half-plane as shown. Because of our interest in finding the surface current density along a ray normal to the edge, the observation point P is restricted to lie in the x'y'-plane. The incident field F. will be restricted to lie in a plane parallel to the' half-plane and may either be the incident electric field or the incident magnetic field. - -iFor F. = H (H-polarization), the incident fields may be written 1 H ______ IAA -ikS iH i (3.1 a) HH =(00) (c) os00x t -coso a sin z'^ e (3.la) -i 2.2. E- + E ) sin a cos a sin 0o -' (cos +os a si2 0) - -ik + sin a cos 00 sin 00 z ' e. (3. lb) Similarly, for F = E (E-polarization) the incident fields are given by i E E (cos ' - cos a sin 0 ') e, (3.2a) E:(0,O Ca) Ox 0 -- -H- i 2 A 2 2 2 A HE p( a)\ [sin a cos a sin 0^' - (cos 0 + cos a sin 0)y + + sin acos n si] -, (3.2b + sin a cos 0 sin O l (3. 2b)

45 y'.P Half Plane AN XI FIG. 3-1: The Half-Plane Scattering Problem.

46 where in the above equations a)= os + cos a sin 0 S =x' cos asin 0 + y' sin a sin 0 + z' cos 0 -- iE= r/oH0. The exact solutions to the three-dimensional diffraction problems for a half-plane may be obtained in a straightforward manner from those for the two-dimensional diffraction problems (Born and Wolf (1964)).' The three dimensional solutions are given in terms of the following incident unit plane waves: Ai A -ikS (3. 3a) E = (-cos a cos 00 x' - sin a cos 00 y' + sin 00 z')e Ai. A.-ikS (3.3b) H =(-sin ax x'+cos a y)e 1 i A,(3.4a) E = (sin a x - cos a ye )ie lA A -ikS (3.4b) H = (-cos a cos 00 A - sin a cos 00 y' + sin 00 z)le The fields of equations (3. la, b) and (3. 2a, b) can be expressed in terms of those of equations (3. 3a, b) and (3.4a, b). For H-polarization we have: + H - 7 r(-) - [sin a cos 0O H + cos a H2J (3. 5a) i a c 0 E H r ) [sin acos 0 E +cosQ 2 (3.5b)

47 For E-polarization the expressions are: E -(e0,) [cos ai E - sin acos O Ej (3 E r(00 ) [cos a H -sin acos % H (3.6b) The solution to the two-dimensional diffraction problem corresponding to E is given by E =vz= [(2 cos-2) -U(2 cosUD ) ', (3.7) where 2j- cos -\ i i-r 2 U(2 cos -krcos) (1-i)e dr (3.8) - O 12 =1 + = and r is the distance of the point of observation from the edge. Of interest here is the tangential value of fH along the half-plane for arbitrary 0. -- -_._,1 values of 0. This value is found by forming - _. - U -ikzt cos 0 o V' V(k sin 0 )\eik 0 (3.9) 0 wr the nctin i obnerepla the function V(k sin ) is obtained by replacing the wave number k in V of equation (3. 7) by k sin 00 Utilizing equation (11.6.4) of Born and Wolf (1964) the desired value of H1 is given by -(1) -i v', V) _ (3. k -y / (3. ) ay ox 0, 2 =o z I 0

48 -(1) The components of H that are tangential to the half-plane surface are H1 = o 2 0 (3. 11a) 2 -ikxa sin0 ) (3nlbosa H(1) 0 -2 sinaF (kx' sin 00, O)e- 0 XI 0' 2 2 a - i(kx' sin + 7r) \ si0 sin e 0 4,(3. 11b) kx' sin 0 s2 - ar -ikx' sin 0 cos cr nkx' sin 0 2 The function F(kr sin 00, a) is a Fresnel integral defined as follows: ' sin 0 kr sin 00 2 \ cosa 2c —os F(r sn 2 |(T 2 (3. 12) F(kr sin 000 +a) The solution to the two-dimensional diffraction problem corresponding to the incident magnetic field H 2 is given by H = W [U(2 cos ) + U (2 cos 2)] Z (3. 13) -ikz cos 0 W' = W (k sin0 e k cs 0. (3.14) 0 /The desired solution to the three dimensional diffraction problem due to I _ - -.. -.Ai the incident magnetic field i2 may be written with the help of equation (11. 6. 5) of Born and Wolf (1 64) as

49 H(2) =-cos ay0 + sin ^, k 0 ax ay O =o. 2; z= 0 I:i' (3. 15) Only the x'- and z-components of H(2)along the x-axis are needed. Only the x1- and z'-components of H along the x1-axis are needed. They are: H,2 -ik x' cos a sin 0 - O' = 2 sin 0 e 0.-e (F(kx' sin, ) (3. 16a) Hz 0=2, = 2 sin 0 e 0 --- -ik x' cos a sin 0 ( - kx sin 0, 7r- ) 2~ (3. 16b) H x =1 - 2 cos a cos 00 e X0=0 -ikx' cos a sin 0 (F(kx' sin 00, )) - _/ i(kx' sin O + T)/ rkx'sin cS0 0 O 2e 0/ (3. 16c) (2) -ikx' H(2) = -cos 00 2 cos a e x = 27r 2 a. rT kx' sin0 COSIZ e 0 cos a sin O \0 (kx' sisn 00 r -2 ) i(kx' sin O0 + ) 04 / (3.1( 3d) The total surface current densities induced on the half-plane along the x'-axis by the incident fields of equations (3. 3) and (3. 4) may easily be found from equations (3.11) and (3.16) respectively. We wish, however, to find those nonuniform current densities that are defined as the difference between the total current densities and the current densities assumed by the method of physical optics for the two types of incident fields. The physical optics current density A - i is simply given by 2n x H on the illuminated side of the half plane and by 0 on the shadow side. If we limit a to the range 0 < a < 7r, n becomes y and the two physical optics current densities are given by

50 A A- -ikx' cos a sin 0 K) =2 ix H1 2 sin a e, (3.17) p.o.. -(2.. A -ikx' cos c sin 0 K 2) = 2y'xH 2(sinO0 x'+cosacos 00z')e/ (3.18) for 0 = 0,and by zero for 0 = 2ir. Subtracting these current densities from the total current densities obtained from equations (3. 11) and (3. 16) one easily finds the desired nonuniform current densities to be/ j(1)[= N1: 2 sin a/e -ikx' cos a sin 00 sin sin 0e KN [2 -=i =2 ae co a si 0 Ns i(~ kx'sin' sn0 0 0 4 Al -ikxwhere use was made of the easily proven relationship: F(kr sin 0, + c i(kx sin ecurrent densities, are the same on both sidecos 0of the half-plane. Since ane 0 fkxt sin 0 0 2 r The non-uniform surface current densities, unlike the total surface current densities, are the same on both sides of the half-plane. Since an approximation to the non-uniform surface current densities on the disk is going to be obtained from equations (3. 19) and (3. 20) by purely geometrical means, the same can be said of the approximate non-uniform surface current densities on the disk. Also, since an idealized half-plane has no thickness, it

51 may be replaced formally by an open current sheet having a total surface current density of K + 2KN without changing any field quantities. The p.o. N same may also be said for a disk with respect to the surface current densities obtained from the approximations used here. Such a replacement will be implicit in all that follows for both the half-plane and the disk. We are now in a position to write down the half-plane non-uniform surface current densities that result for the incident plane waves of equations (3.1) and (3. 2). According to equations (3. 5a, b) and (3. 6a, b1) these are KH = [sin a cos 0 K) + cos a K()], (3.21) a) osaK N sin a os K (2) (3. 22) 3. 3 The Non-Uniform Currents on the Disk Figure 3-2 shows how the half-plane geometry of Fig. 3-1 is to be used in approximating the non-uniform currents for the disk. The y'-axis is parallel to the z-axis of the disk. The x'-axis is in the plane of the disk in the -p - A direction, and the z'-axis is in the plane of the disk in the p-direction. Hence, we may immediately write =A Al A A, we may immediately write y = Z, x = - p, z =, and x' = a-p. The halfplane angles 00 and ae are related to the angles 0 and 0 of the unprimed coordinate system by cos 00 = sin 0 cos ~, (3. 23a) sin 0 = "(0,+-) = os e + sin 0 sin 2 (3.23b)

52 z Incident Plane Wave / p x FIG. 3-2: The Edge Geometry Used to Approximate the Disk Surface Current Density. y

53 os -sin 0 sin 0 (3. 24a) cos C, = (3.24a) r(0,0+ ) 2 sin a = os 0 (3.24b) r(0, 0+ +) Equations (3. 19) and (3. 20) also involve cos - and sin, which can be 2 2' expressed directly in terms of 0 and 0. It proves convenient, however, for later usage to express sin aand cos - in terms of two functions T (0, ) and Q(0, 0) which are defined by T (0, 0) = r(0, 0 + ) - sin e sin0, (3. 25a) 2 Q (0, 0) =r(, 0+ ) + sin sin. (3.25b) a. a Then cos and sin are 2 2 a T (0, 0) 1/2 (3 Cos - Lj(3. 26a) 2 2r(e, + 2 )_ sinc=rQ(00) l/ (3.26b) 2l 2e, 0+2 ) The phase reference for the disk diffraction problem will be taken to be the center of the disk while the phase center for the half-plane diffraction problem has been taken to be the origin of the primed coordinate system, a point that is on the edge of the disk. Hence, the phase of (3.21) and (3.22) must be shifted by [ka sin 0 sin 0] radians before the equations can be used to approximate the non-uniform currents on the disk. With this addition and the substitutions indicated by equations (3. 23a, b), (3. 24a, b) and (3. 26a, b) equations (3.21) and (3.22) give the following for the disk non-uniform currents:

54 0H --- -- [(o2s)osj K e )ika sin 8 sin, o Cos K( (0,) - sin0 K (, e (3.27) 2H - 2) -- O, F 1 -E=H O- - [s N N(e(,0)+ cos )(e,8 ] -ika sin sin (3.28) Finally, application of equations (3. 19) and (3. 20) allows these complex appearing relations to be rewritten after some manipulation as K 22H" ' 0 -1:e/ik p sine0 sin + +, cos0 ~ (0,0), i ka T (, ) -k pf,0+ n)+ (3.29) KE- 2H0- 2xcos [ F (?(ap)T(0), ) 0 - eik p sin 0 sin + E2 n sin 0 n,+ sin (,+ ) A S^ei 2 kaT(0,0)-.kpo[0,0+T)+] + rk(a-p) Q(0, 0 r(0e +2 2) - (3.30) Note that since p cannot be negative these choices of surface current densities require that the half-plane approximations be arbitrarily terminated at the center of the disk. These expressions complete the approximate descriptions of the non-uniform surface current densities on the disk.

55 Because of the restrictions placed on the half-plane problem these current densities are the non-uniform surface current densities on the disk current sheet that result when the incident electromagnetic field is a plane wave with either the electric vector or the magnetic vector constrained to be parallel to the plane of the disk. However, these are just the incident fields given by equations (1. 1) and (1.2) respectively. The total surface current density on the disk current sheet may then be written in terms of the above surface current densities and Ithe corresponding physical optics current densities as -E -E - KT =K.o. E (3.31) for the incident field of equation (1.1) and, -H -H - K K +K (3.32) T P.o. H for the incident field of equation (1. 2). These surface current densities will be used in the next section to find the far-zone backscattered fields for the two problems. 3.4 The Far-Zone Backscattered Fields The previous sections of this chapter were devoted entirely to finding an approximate description of the surface current densities induced on a disk by the incident fields of equations (1. 1) and (1.2). With such a description available the task then becomes that of finding the scattered fields. The scattered electric and magnetic fields in free space are related by Maxwell's equations, so only one need be considered here. A natural choice is the scattered magnetic field, which for the scattering problems of equations (1. 1) and (1.2) determined by the equation, Hs = (V X K 'dS' (3.33) S

56 where the surface of integration S is the open surface bounded by the disk rim that has a unit normal vector =. The surface current densities K and - H KT are given by equations (3. 31) and (3. 32) and are used for the incident fields of equations (1. 1) and (1.2) respectively. The function i is the free space Green's function given by - ikr =-, =r=R-R'|, -(3. 34) where R and R' are respectively position vectors from the center of the disk to the point of observation (OP of Fig. 1-1) and the point of integration. The prime over the V - operator in equation (3.33) denotes that the operation is to be taken with respect to the coordinates of integration. Equation (3. 33) gives the magnetic field anywhere in space. Only the far-zone backscattered field is of interest here. For this case r*, = R, and the integral of equation (3. 33) may be written in terms of the p'- and 0'components of the surface current density as 27r a -S -ik eikR -ikp sin e sin r K H K A 4irR J TTosLT 0 0 e+ sin0 [sin. K1H - cos 0KH]/} pd pdo, (3.35) where the primes have been dropped for clarity. The contributions of the physical optics surface current densities to equation (3.35) will be considered first. The two physical optics currenttdensities are defined as 2n x H ' and are: -H A 2He iky sin 0 (3. 36a) K.. y2He e

57 K =x 2H cose-iky (3. 36b) p.o. 0 Determination of the two scattered magnetic fields from equation (3. 35) is straightforward. The results, expressed in spherical coordinates, are ikR - S H iaH0 cos Ole J1(2ka sin 0) H tv- -_p.o. r 2R sin 0 (3.37a) and ikR S, E Via H0 cos O e J1(2k a sin 0) p.o. 2R sin 0(3.37b) Both fields have the same 0-dependence, a result that is not unexpected for the scattered physical optics fields from a regular body. Also, both fields have been found in closed form. This is a fortuitous state of affairs that will not be repeated in finding the scattered far-zone magnetic fields due to the non-uniform components of current. Instead an attempt will be made to expand the integrals for these fields asymptotically in inverse powers of the wave number k. This will be the primary approximation. A secondary approximation will consist, where appropriate, of expanding the resulting functions in ascending powers of sin 0. The objective is to seek a solution that is most accurate for aspect angles near normal incidence. When the non-uniform surface current densities given by equations (3. 29) and (3. 30) are substituted into equation (3. 35) the resulting expressions may be written as ikH ikR a a -i2k p sin 0 sin k - H- S cose IF k(ap) T(O, + HN 27R 2 I L'

58 + cos~i, [kaT (0,) -kpQ(, 0)+ ] r,+'..).Tk(a-p).. [c Cos 0 + sin - sin sin 0 pd pd, (3.38) -S 0 ik ikR 27 a r H N -2cosOe -i2kpsin sin[F k(a-p)T( 0).) ) 1 0 c0 - ___ __ EN 27rR 2 sin + sin Or(O, [kaT(0,0) -kp Q (0,) + 4] +,Ile X- 4 X + x k (a-p) Q( eA)0 r(0 + I[-cos (os s + sin ) + sin 0 sin pd pd. (3.39) A A A A The unit vectors x, y, z, and 0 are all constant with respect to the variables of integration. This mixed system was chosen to make similarities between the two equations evident and to keep the equations as compact as possible. Both equations/contain a term that is bounded for every value of p and an unbounded /. ----- - - -- -\ — 1 r-}: term that becomes infinite as (a-p) /. Furthermore, the bounded terms will give scattered fields of equal magnitude for the two polarizations, thereby \ never contributing to the cross polarized scattered far-zone fields. The unbounded terms have the same p-dependence for the two polarizations but different 0- and 0-dependencies, so that they will contribute to the cross-polarized scattered far-zone fields. Then, since the physical optics scattered_ fields given in equations (3. 37a, b) cannot give any cross-polarized component, it is only necessary to compute the contributions to the far-zone scattered fields in equations (3.38) and (3.39) from the unbounded terms if only the cross

59 polarized far-zone backscattered field is desired. Otherwise all the terms must\ be included. 3.5 The Far-Zone Backscattered Fields Due to the Unbounded Component The contributions to the far-zone fields from the unbounded terms in equations (3. 38) and (3. 39) will be considered first. Both have the same pdependence, which may be written 1l(ka, O, ) f-e -ikpQ(O, ) pdp (340) (ka e (3.40) This may be evaluated in terms of Fresnel integrals. Let t = a-p. Then, a a k eikaQ(, ktQ(0, ) adt f iktQ(0O) Idt} (3.41) I (ka, O, e e F t dt (3.41) -~-. - -eIntegrating the second integral by parts to bring it into the same form as the first integral and recognizing that the result is a Fresnel integral we obtain I ka 6 \-ikaQ( k0)a Q -+ - - - (3. 42) 1Q J0 (0- i2kQ(0,0 ri kQ e, ) O f. where the Fresnel integral is given by W i- T i' 2 '1W iir- 2 = 2 2 f(e) e - ei dT. (3.43) 0 o Inspection of equations (3. 23b) and (3. 25b) reveals that Q(O, 0) is non-zero for all values of 0 as long as 0 is not equal to ninety degrees. This means that the argument of the Fresnel integral in equation (3. 42) will be of the order of (ka).

60 Hence the asymptotic expansion of the Fresnel integral may be used in this angular range of 0. The first few terms of that asymptotic expansion are 1i e 2 1 3 15 1 2 d ~ (W) +e2 + (1+ )+ e 2 ir v iW 2W2 i. r - 6 (3.44) 5 The remaining integral in equation (3. 44) is of the order of (W), which is the order of the last term in the series. Substituting equation (3. 44) into (3. 42), that equation can be written +T7-1 i 1 -ikaQ(e, )+ I1 (ka, e, i=a )a3/2ikaQ0 ) + e a(0 ) i2kaQ(0, 1 2 ___12 +ikaQ(0, ) i2kaQ(0, ) 3 1 -3 4i7rka1Q(0,))3 ( + + ( ka), (O <.* (3.45) 4ika'Q(,))3 i2kaQ(O, ) ) Inspection of this equation reveals that the first term is canceled by another, so that Il(ka,, a) may be written as iI Il(ka,,0=) a /2 e 4 i2kaQ( '1 a ) i2kaQ(O., + ((ka)-2), (0< ). (3.46) -3/ Keeping terms to (ka) /2 the expressions for the components of the far-zone Keepin- -em -o -ka - -- backscattered magnetic fields for the two polarizations due to the unbounded surface current density are given by

61 -e ikR -s cos 0 cos 0 ( 1 + )-i2ka sin0 sin Hos N "0 (1+ 1 ),es x HN 2 JR Q( r( + i2kaQ(O, ) 2( -s -aH ikR 27r sin V+sin r(e, +) 1 )e -i2kasinOsin X HEN 1 2R 72 i2kaQ(O,) \e Ado (O < 7) (3.48) X [-cos 0 (cos p + sin 0y) + sin 0 sin p^] d, (<2). (3.48) where use has been made of the easily proven relationships i'T(0,P) Q(O, ) = cos 0, (3. 49) Q(0,e) - T(0,) = 2 sin 0 sin 0. (3.50) Both integrals are well behaved for the allowed ranges of 0 and j. In order to obtain an approximate solution to the integrals, the integrands will be expanded in ascending powers of sin 0. Two functions need to be considered, r(0, +) and Q(0, ). r(0, + ) is given by equation (3. 23b), which can 2.2. — 2 be rewritten in terms of only sin 0 by replacing cos 0 by 1- sin 0. When this is done ( r(0, + 2)) becomes 2 2 (sin cos +0 3= 4) 4= co ) 1+ -in20 cos + sin 0 cos + 2 8.. (0 <. (3.51) 92

62 -1 The function (Q(0, 0)) may also be expanded in powers of sin 0. The result is [Q (0,0)] = 1 - sin 0 sin + S2 (1 sin2) -sin 0 sin +.. ( < ) (3.52) The limit on 0 follows from the fact that 1 - sin 0 < Q(0, 0)< 1 + sin 0 for any value of 0. When the expansions given by equations (3. 51) and (3. 52) are used to find the scattered magnetic fields from equations (3. 47) and (3. 48), some terms will give zero contributions when integrated. These are of the form 27r -i2ka sin 0 sin 0 e (sin 0) cos 0 do - O, (3.53) which is easily shown to be 0 for any positive integervalue of i. Since equations (3. 51) and(3. 52) as given either are, or may be, written entirely interms of powers of sin 0 as well as sin 0, only the x-component of HHN1 and the y- and z-components -S A A 5 of H will be non-zero. In fact, since 0 = cos0 y - sin 0 z, HE1 will ENi EN1 have a component only in the 0-direction. By virtue of the above observations the far-zone scattered magnetic fields may be written in powers of sin 0 to 2 the order of sin 0 as - - aS H ea 2 c A r-i2ka sinO sin { 2 2 i H cos 0 X e cos (1+sin 0 - sinO sin) HN1 27rR f + i2-ka cs2 ( sin20 - 2sinO sin + sin20 sin2 d, (3.54) i1ika 2 2 *

63 - ikR 27rT aH0 e, -S 0 - A -i2ka sine sin.2.... 2 H 0 e ksin sinsin tsin+ sino (1-sin ) + EN1 2irR 0 2 +i2ka [in0sin l s si (1+ sin(1-2 sin2) d (3.55) All of the integrals over 0 are of the form 2 r ei2kaini (sin d. (3.56a) I~ - ik i i sn~ If x = 0, the value of the integral is a Bessel function: 27r 27T Jo(X) = *o -ix sin e d (3. 56b) Both sides of this equation may be differentiated with respect to x to give 2w 27r J (x) = -27T J(x) = -ir -e -ix sin 0 sin0 dd, ^ __ (3. 57a) as both integrands are continuous within the given limits. Continued differentiation followed by application of the recursion formulas for Bessel functions yields?T 0 W -ix sin 0 sin2 0 d -I (3. 57b) 3 1 -ix sin 7[2 J1() 2 J3( = i e ixsin 0 sin 0 d0, 0 (3. 57c)

64 27T [3[ J (x) ) + _ J(x)] = + X e -ix s.n sin d. 2 4(3.57d) The components of the far-zone scattered magnetic fields of equations (3. 47) and (3. 48) are then given in terms of Bessel functions of the argument 2ka sin 0 by _a ekR -S. 2 H aHN1 2R cos 0 (1 + sin 0) J0(2ka sin 0) + J(2ka sin 0 + LN i 2R J2(2ka sin 0) 1 15 + 1+ + k --- 2a (1 + sin 0) J(2ka sin 0) + ka 2ik 3 2 3 2 + (1 + - sin 0) J2(2ka sin 0) - - sin J4(2ka sin 0) + i2J (2ka sin 0)l + ka J(3.58) -S a ikR SEN i 2R -- (2ka sin 0) - J2(2ka sin e) - J2(2ka sin 0) 1 r -i ka + 2ika 1(1+ 8 sin 0) J(2ka sin 0) - 2 3 2 - (1 + sin 0) J2(2ka sin 0) + 8 sin 0 J4(2ka sin 0) + + i sin 0 [J(2ka sin 0) - J3(2ka sin 0)]. (3 59) While use of the recursion relations for the Bessel functions has resulted in the inclusion in equations (3. 58) and (3. 59) of terms that are of the order of -2 (ka), no claim can be made that the equations are accurate to that order, for

65 the approximations that were made earlier to the asymptotic series involved only the first two orders. The bounded integrals of equations (3. 38) and (3. 39) must now be considered. Again, the integrals over p are identical for both polarizations, so only one need be considered. It is 2 0 Comparison of equations (3, 12) and (3.43) reveals that this may beirewritten as (ka, 2ksinsin ) f/2(a-p) pdp, (3.61) which is the form that will be considered here. This integral can also be solved exactly in terms of Fresnel integrals. The solution, however, is complicated and expansion of it in terms of inverse powers of (ka) and ascending powers of sin 0 is at best very difficult. Direct expansion of the integral in inverse powers of (ka) appears to be a better procedure. As before, the integral must be transformed. To this end let a variable W be defined by W 2k (a-p) T (00) Then in terms of W, 12 (ka, 0, 1) may be written as -i2ka sinO sin 2kaT ) iZrWsin0sin0 ieka e T(O,) Ik- 2kaT,,) 2,. e (.x )f a - ]( dW, (3. 62)

66 This will be written as the sum of two integrals, the first and simplest of which is -ika sin sin2kaT(O,0) i7rWsin0 sin0 --. -- -- I (kao 0 )= 2kT(,)(O,,a 2kT(O.. J _ _ _ _ _ _ -0 [(#) f (W-2]dW - -i2ka sine sin oo i7rWsinO sin0 (a fe T ) 2kT(0,0 ) J 2[( a)e- e T(O, 0) 1 -i - ---- - o i7W sine sin0 e T(0, ) () f ( VWT ) - 2kaT(O, ) 7r f (W) - 2] dW - dW. (3.63) When the integral from zero to infinity in equation (3. 63) is integrated by parts the result is 00 e 0 i7rW sin0 sin T(, 0) [j f(1W) -1 dW T(, 0) X 2J i~rsin 0 sino x{e irW sin0 sin0 T(0, ) fIo /oo ii7TWQ(0, 0) _ o Jo 3 ( f(i^) - 2 21w e.V/ * (3.64) By virtue of equations (3. 43),(3. 44) and (3.50) this is expressible as

67 oo iTW sinO sin0 -e T(,0) 0 ) f ( W ir (Q(O., ) - (, )) X 1 1 /Te3_= T(= )+ -E1 +* (3. 65) } 2 Q(O,6) iQ (O.,) Q(0,:)J (3.65) Evaluation of the second integral in equation (3. 63) can be done asymptotically by applying equation (3. 44). This yields 00 irW sinO sin0 | e T( [(2 f. )f( -) - 21dW 2 2J 2kaT(e, 0) 7T iwrQ(O, 0)W~ ir - i e2T(0 ( 1 3 2kaT(O, ) -... W... n~ 00 + (-i 15 1 2kaT(0, 0) 7T iTrW sine sin0 - T(O, 0) e x 00 ir 2 2 d' dW 6 7 (3.66) -3/2 where the second integral, which will be neglected,is of the order of (ka) / 2 Asymptotic evaluation of the first integral is straightforward. The lowest order terms of the resulting series may be written as Oa 2kaT(O, ~) 7T irW sinO sin0 e T(O, ) [( ) f ( ) — ] dW =

68 e -i1/4 T7 A)-i, r 1OI0 I e ( ) 1 1 + (1 + ) i2kaT - Q(e,0) V irka 1 (+ (0, ) kaTO, )] 3_ +,(ka) 2J. (3.67) Then I (ka, 0, ) is given by -i2kasin0 sin cos -1 ae 2. cos 0. (+. I (ka., 0,0)- _ aka 2~ —kae-2 - Q.(0, ) + jkae4 e ikaQ(0O,)] + O[(ka) 5/2] (3.68) + g/rkaT(0T, J The second part of equation (3.62) must now be considered. This is an integral given by 1 2 2kaT(0, ) irW sin sin0 -i2ka sino sin. T(, ) Ib(ka, 0, 0) e) X 2kT(, ~) X ) f(V) -1 WdW. (3.69) This integral does not converge when the upper limit becomes infinite. Hence, it must be evaluated by somewhat different means than I (ka, 0, ). Integrating the exponential term by parts transforms this integral to a sum of integrals that can be expanded asymptotically. The resulting expression is

69 — i2ka sin 0 sin 0 Ib (ka, 0, ) = -.e.- X i2k2 T(0, ) (Q(0, ) -T(0, 0)) r i7W sin 0 sin 0 2kaT(0,0) X T(0, 0) f (aW)T 7r 0 2ka T(0,0) irWQ(0,0) I 1 i e l 2T(0,0) dW 2kaT(0, 0) irW sin0 sin0 e T f (v-W) - d W, (3. 70) where use has been made of equation (3.50). Of the two remaining integrals the first is further reducible to a Fresnel integral while the second is proportional to I (ka,0,0). So Ib(ka, 00) reduces to -i2ka sin0 sin ) Ib(ka, sine sin 0 )f Tr i2k T(O, )(Q(0, 0)-T(0,0)) [i 2kaT 7r 2kaT(0,0) (1-i) 2T(, 0) 2kaT(0,0) eikaQ(0,j) t "7 4 i7[Q(0,0) 7 I+ (0, l ) T(0, ) f 2kaQ / 2 i7QT(0,p) TQ(0,0) 2 ) a(ka,0, 0) ika (Q(0, ) - T(0, 0)) The asymptotic expansions of all of the functions contained in this equation have been considered before. Substitution of the appropriate expansions allows Ib(ka, 0, 0) to be written as

70 -i7T I ~a \-i2ka sinO sin0 aeT- _______L D ikaQ(0,0) + 2 Q (0O. )} [(k) 0/2 (3.72). + 2 Q(O~o 2(1+ +0 (ka) [Q(o, )1 The first term in this equation is identical to the second term in equation (3. 68). Therefore, when the two equations are combined to give I2(ka, 0, 0), those terms will cancel. From equation (3. 62) I2(ka, 0, j) is found to be -i2ka sin0 sin0 I2(a0= Ika, 0,) = I(ka, 0) Ib(ka, 0 ) = - 2k X + cos0 (0.0)+cos+ k 0 o ]2+O [(ka)2] (3.73) (01,e iF +K }- (3c73O ka [Q(,0) +c This is the desired expression for the integral over p of the bounded term in equations (3. 38) and (3.39). In order to carry out the indicated integrations over 0 in those equations, it is necessary to expand I2(ka,O, 0) in powers of sin 0. Both terms in equation (3. 73) contain inverse powers of the expression G(0, ) = Q(0, ) + cos 0. If 0 is required to lie in the interval containing zero and ninety degrees, it is easy to verify that the maximum and minimum values of G(0, ) occur for 0 = and - respectively and are?r cos O + sinO -1 G(0, ) =1 + cos 0 + sin = 2 (1 + cos + sin ) (3. 74a) 2 2 G(O, ) = 1 + cos 0 - sin0 = 2 (1 + si0 -. (3.74b) G 2 The particular form of these expressions was chosen to correspond to a general form which will be used in obtaining expansions for the two inverse

71 powers ofG(e, 0) in question. The function G(0, j) may be written explicitly as G(0,0) = sin0sin + /-sin 2cos 2+ 1 - sin (3.75) Both radicals may be replaced by their power series expansions for 0 <. 2' Hence, G(0, ) = 2 + sin 2sin0 - sinin (1+(1-sin2 )2) +... G(-.1 2 + sin0 sin0 1 2 12s L) 2(1 + sin si sine 2 )- sin i (1+(l-sin)2I....), < 2 4 16 sin (3.76) This has the form 2(1+a(0, )). By virtue of equations (3. 74a, b) and the condition on 0, |c(0, 0) is always less than unity. Therefore, convergent expansions for [G(0, )]-1 and [G(0, )]-2 can be found. They are [ 2 3 21 [, 1- 1 - sin 0 sin0+ sin 0 sin 0 sin0 (1 + 3 cos 0) 2 2 2 s 8 '(3.77a) +......, O<i, (3.77a) 2 and r ^\1~2 3 2 sin2 3sin3 0sin0 2 G(, = - sin0 sinp + sin 0 (1 ( + sin21 - +(l+o 2 + +... (3. 77b) cos 0 An expansion is also needed for 1 + 2 Q (, )the determination of which is straightforward, since (Q(0, 0)) has been found before and is given by equation (3.52). If cos 0 is replaced by its power series expansion in powers of sin 0 and the two series multiplied, the desired expansion is found to be

72 2 2 3 cos 0 3 sine sin n s in sin sin 2Q(0, )=2 2 4 - 4 +.. (3.78) Straightforward multiplication yields the desired expansion for I2(ka, 0, ), which is just the integrand of the remaining integral over 0 of the bounded terms in equations (3.38) and (3.39). Taking terms only to sin 0 the desired approximation to that integral becomes I (k) =a -a -i2ka sin0 sin0e i sino sin0 sin0] + a 2k 2 2 2 + 8 [3 4 sin0 sin + 3 sin (1+ sin2)] d 0<. (3.79) Unlike the integrals over 0 in equations (3.47) and (3.48) every term in the integrand will give a non-zero contribution to the integral, as none of them has the form of the integral in equation (3. 53). Indeed, all terms are of the form of equation (3. 56a), and hence, the integral can be evaluated as combinations of Bessel functions. For this integral, only equations (3. 57a, b) and (3. 56b) apply. The result is I3(ka,0)= - 2k (1+ 2) J0(2ka sin) - s (2ka sin) + 1 [ 33 2 +ka ( 3 sin ) J0(2ka sine) - 9 2 32 sin 0 J2(2ka sinO) + i sinO J (2ka sin 0) - (3. 80) (3 80) 32 2 2r

73 The components of the far-zone scattered magnetic field due to the bounded components in equations (3. 38) and (3. 39) are expressible in terms of I3(ka,0) as HHN I - cos 0 I3(ka,) (3.81) ikR -S A ikH0e H EN -' 8- R cos0 I3 (ka, 0). (3. 82) EN2 ~r R ' When these scattered fields are added to the physical optics scattered fields of equations (3. 37a) and (3. 37b) respectively and the scattered fields due to the unbounded currents of equations (3.58) and (3.59) respectively, the desired approximations to the two total backscattered far-zone magnetic fields are obtained. The resulting expressions can be simplified somewhat by making use of the series expansion of cos 0 in powers of sin 0 where appropriate and by applying the recursion relations for the Bessel functions in order to combine terms. The resulting approximate expressions, again 2 -1 valid to the orders of sin 0 and (ka), may be written as H rae i iJJ (2ka sin 0) 2 SA a".e"' sin02 0HH 2R cos 0 sin0 + (1+ s2 )J2(2ka sin0)+ i sin s q 2 + i J(2ka sin0) + 2i- + sin 0) J0(2ka sin 0) + 25 2 2 + (1 + sin 0) J(2ka sin0) - 2 sin (2ka sin 0) 16 (2ka sin0) - (2ka sin 0 < (3. 83) -i sin0 J (2ka sin0) -J (2ka sin 0 0< (3.83) 3 2 (3.83) ~~~~.~. —' ~- ~r.8~ < I

74 aHe i cos 0 J1 (2ka sin0) HE sn + J2 (2ka sin ) + E 2R sin 0 2 2 (2ka sin 0) sin 0 +ka +2 J1 (2ka sin j) + 1 1 1 2 + 2ika ( +16 sin e) J (2ka sin 0) + 7 2 3 2 + (1 + - sin 0) J2 (2ka sin) - sin (2ka sin 0) + 16 2 8 4 2i 7r + a J2 (2ka sin 0), 0 <. (3.84) ka (3.84) 3.6 A Note on Extending Equations (3. 83) and (3. 84) It may, at some time, become desirable to extend equations (3.83) and -1 (3. 84) to higher powers of sin 0 and (ka) by using the methods considered in this chapter. The extension to higher powers of sin 0 is straightforward and requires nothing fundamentally different than was done here. The extension to higher powers of (ka), however, will introduce an integral over 0 that has not been considered heretofore. This integral arises when the asymptotic expansions of the various integrals over p are taken to greater powers of (ka) than was done in deriving equations (3. 83) and (3. 84). It has the general form i7_ka Z (, os I (ka, 0) = e ika cos 0 sn (3. 85) where m may be restricted to be either zero or one with no loss in generality and may assume any non-negative integer value (consider for inege on stance the consequence of retaining more terms in expanding equation (3.42)).

75 When m = 1 this expression, like equation (3.53), can be shown to be identically zero. When m = 0, however, the integral is in general non-zero and is a function of both 0 and (ka). Because of the nature of the solutions to the scattering problems that were obtained in this chapter it would be desirable to expand I (ka, 0) in a c series of Bessel functions of the argument (2ka sin 0) and inverse powers of (ka). It is not obvious how to proceed, nor has this been done in this work, since, fortunately, the integral did not arise in the derivations of equations (3. 83) and (3. 84). Any attempt to extend the results of this chapter to higher orders of (ka) will necessitate either an exact or approximate evaluation of I (ka, 0). c

Chapter IV SOME FURTHER CONSIDERATIONS 4.1 A Comparison Involving the Exact Solution In Chapter II it was found that for c = ka = 1.0 all computed radar cross sections were consistently lower than the corresponding ones obtained experimentally. The regularity of this phenomenon indicates that some effect other than stray radar returns may be predominant. In particular, the computed values may be in error. Consequently, an independent verification of the accuracy of the programmed formulation of the exact solution for small values of c would be most helpful. This can be done by applying a solution to the problem of electromagnetic scattering by a disk of a unit incident plane wave for small values of c that was developed by Eggimann (1961). His solutions to the farzone scattered fields are given as power series in c. For the case of backscattering they may be written as ES Hs ( 3c){2 + sin2 0 + [16 - 15 sin 0 - 5 sin 0 E 0 0 kR 3 r 15 E E (4.1) ikR _ __3 2+\ F 2 1 E0 Hb +; c ^T 16 - 9 sin 80 (4.2) H 0H \kR ( 3c 15 L ( The dependence on c in these two expressions is quite simple, for even 5 though the solutions have been carried out to the order of c L only two terms in each of the resulting power series have non-zero coefficients. For small 3 c both backscattered fields behave like c, which implies that the measurement problem will become quite acute for small values of c, in accordance with actual experience. In order to compare equations (4.1) and (4.2) with the exact solutions considered in Chapter Hi it is necessary to choose a suitable value of c for which to compute the various radar cross sections. What is a reasonable 76

77 choice depends on the anticipated effect of those terms in the power series expansions for the far-zone backscattered fields that have not been taken into account in equations (4. 1) and (4. 2). Clearly, the choice of c = 1. 0 would not be expected to yield good results. Choice of c = 0. 5 might be acceptable, how6 ever, as the first missing term in the series, which is of the order of c, would be of the order of 1/8 as large as the first term. On this basis it was decided to calculate the three radar cross sections as a function of aspect angle using the exact solution and equations (4.1) and (4. 2) for c = 0. 5. The agreement between the two solutions was extremely good, as is shown by the comparisons given in Table 4-1. The slight differences in cross section encountered in that table can be attributed to neglect of terms of the order 6 of c or higher in equations (4. 1) and (4. 2). On the strength of Table 4-1 it is reasonable to conclude that the discrepancies found in Chapter II between the experimental and calculated cross sections for c = 1. 0 reflect effects other than errors in the computational effort. A possible effect not mentioned before would be differences in the measured scattering cross sections due to the finite thickness of the actual disk. 4.2 Comparison Between the Exact and Approximate Solutions In this section some radar cross sections for the disk as predicted by equations (3. 83) and (3. 84) will be compared with the same cross sections as predicted&by the exact solution. Since the exact solution gives best results for low frequencies (low values of c = ka), while equations (3. 83) and (3. 84) are by nature high frequency solutions, a compromise in the choice of c must be effected so that both solutions can be expected to give reasonable results. Choice of c = 6 appears to be good, for the exact solution agrees very well with the experimental data for aspect angles as large as seventy degrees. Also, the terms of highest order in (ka) in equations (3. 83) and (3. 84) will be quite small compared to the leading terms (except perhaps near minima of the scattered fields) so that these equations may also be expected

78 TABLE 4-1 Some Radar Cross Sections for c = 0.5 as Computed Using Eggimann's Solution and the Exact Solution due to Flammer (in dB per square wavelength). Radar Cross Sections From The Exact Solution (dB/X ) E-Pol. H-Pol. X-Pol. I Radar Cross Sections From Eqs. (4.1) and (4.2) (dB/k2) E-Pol. H-Pol. X-Pol. e I s 20 -29.32 -29.33 -90.78 -29.39 -29.40 -90.79 10~ 20~ 300 40~ 500 60~ 700 80~ 88~ -29.24 -29.01 -28.67 -28.29 -27.91 -27.58 -27.33 -27.18 -27.13 -29.61 -30.47 -31.98 -34.21 -37.37 -41.84 -48.52 -60.35 -88.24 -62.91 -51.19 -44.68 -40.43 -37.50 -35.48 -34.15 -33.39 -33.16 -29.30 -29.06 -28.72 -28.33 -27.94 -27.61 -27.36 -27.20 -27.15 -29.67 -30.54 -32.03 -34.26 -37.41 -41.87 -48.55 -60.38 -88.27 -62.94 -51.22 -44.71 -40.45 -37.52 -35.50 -34.17 -33.42 -33.18

79 to perform well, particularly for small aspect angles. Computed radar cross sections using both methods are shown in Figs. 4-la through 4-lc for E-polarization, H-polarization, and cross-polarization respectively. Qualitative agreement is good throughout, and quantitative agreement is generally good for aspect angles that are less than thirty degrees. Oddly, agreement is best for the cross-polarized case with the two methods predicting practically the same radar cross section for aspect angles as large as forty-two degrees. The case of E-polarization shows the best agreement of the two direct returns with equation (3. 84) and the exact solution predicting very nearly the same radar cross section for aspect angles as large as thirty-four degrees. Only the minimum at nineteen degrees shows any sizeable discrepancy between the two methods for this angular range. Finally, equation (3. 83) and the exact solution agree well only to aspect angles of twenty degrees for H-polarization. While it is expected that equations (3. 83) and (3. 84) will perform poorly for large aspect angles because of the approximations made in their derivations, it is odd that equation (3. 83) would fail for such low aspect angles. Since Fig. 2-7b indicates that the exact solution is valid in this case, equation (3. 83) must be in error, or must fail to account for some effect. Some deliberation reveals that both equations (3. 83) and (3. 84) were derived without taking into account the possibility of multiple diffraction by the disk. Furthermore, the effects of multiple diffraction will be greater for H-polarization than for E-polarization. This view is consistent with the actual behavior of equations (3.83) and (3.84). Introduction of the effects of multiple diffraction, which will not be considered in this work, is one means by which one can seek to improve the performance of the approximate solution. Another would be to keep still higher powers of sin 0 in equations (3. 83) and (3. 84). It would be extremely difficult to predict how successful these undertakings would be, but there is certainly reason to expect at least partial success, as equations (3. 83) and (3. 84) already predict the proper qualitative behavior for the radar cross-sections for large aspect angles.

80 20 20 Exact Solution - -- - Equation(3.84) — 15 10 I I 0I I -5- 1 j I! \ -10 0 30 60 Aspect Angle 0 (degrees) FIG. 4-la: Computed RCS of a Disk for E-Polarization, c = 6.0. 90

81 20 15 10 5 0 -5 5 Exact Solution - - - - Equation(3.83) ---\ \ \ - -- -, \ \ \ \ \ \ ~V... \ \. - b 0 / / - I I 30 60 Aspect Angle 0 (degrees) FIG. 4-lb: Computed RCS of a Disk for H-Polarization, c = 6.0:. 90

82 0a -5 -10 -15 Exact Solution - - - - Equations (3.83) and (3. 84) I \ / \ I I \ \ \ I I I I I I I I\ b bIf 0 0.1 - -20 -25 -30. 0 SO I I I I I / / / I I I I I I I 30 60 Aspect Angle 0 (degrees) Computed Cross-Polarized RCS of a Disk; c = 6.0. 90 FIG. 4-lc:

83 4.3 An Application of the Geometrical Theory of Diffraction in Light of the Results of Chapter m The important question of how the approximate solutions developed in this work agree with or differ from other approximate solutions to the problem of backscattering from a disk will be considered in this section. The approximate solutions that will form the basis for discussion will be those obtained by application of the Geometrical Theory of Diffraction, which was mentioned in Chapter I, and which can be expected to give reasonable results for disks of large c for aspect angles away from normal incidence. Since equations (3. 83) and (3. 84) purport to be most accurate for normal incidence, a non-rigorous procedure will be developed to obtain a continuation to normal incidence of the range of validity of the solutions obtained from the Geometrical Theory of Diffraction. Different arguments have been advanced in order to continue the results of the Geometrical Theory of Diffraction into the caustic region that occurs in the disk or cone backscattering problems for 0 equal to zero. The argument which is probably of most use here is that given by R. A. Ross in "Investigation of Scattering Center Theory" (1967), as his treatment attempts to account for depolarizing effects. Ross considers backscattering by a perfectly conducting flat-backed cone of arbitrary cone half angle, which problem, in principle, includes the disk problem, for the disk can be considered to be the limiting case of a flat backed cone as the cone half angle approaches ninety degrees. As it happens, Ross' results are at variance with equations (3. 83) and (3.84). In fact his results, given by his equation (B-10), become infinite for the limiting case of the disk. This not very satisfactory state of affairs may be eliminated by modifying Ross? analysis, which is not valid for the case of the disk. His analysis begins with his equations (B-3) and (B-4), which are reproduced below.

84 sin m /n a ikR r 3 - 20 e'12nR 2 ksin Ae n- n os — cos el 2n R r k sin 0 _ n n - 7 i (kR + - 2ka sin ) (4.3) n J t — 1 sin 7r/n / a -ikR J + 2 0 e2 2n R k sin 0 en ncos i (kR- -+ 2ka sin 0 ) + cos — 1 e 4 +2ka(4.4) n J --- These are the fields singly diffracted from two edge scattering centers on the base of the cone. The upper signs are to be used for E-polarization and the lower ones for H-polarization. Also, ikR A e represents the incident plane wave by a different convention than used previously, n = +, where y is the cone half angle, 2 2 k is the wave number. The total far-zone backscattered fields for 0 < y are just the algebraic sum of el and e2 and may be written as. ei(2ka sin e -- -- - sin r/n a i2kR Le 4 Aeiek el e2 2nR r k sin 1 37r + 20 (cos - - cos ) n n --— i (2ka sin 0 - — /4),,,. i (2ka sin - - 7/4) i(2ka sin 0 - r/4) +e.i (2ka sin 0-7r/4 (o T T 37 - 2 ( 7r (cos -- cos - ) (cos - 1 ) n n n (45) (4.5)

85 Ross seeks continuations of these expressions that are valid for values 7r 3f ~ 20-1, — 7 3~ + 20)-1 of 0 near zero by expanding (cos - - cos - ) for near zero and n ~ 2, n n which makes his results invalid for the disk. One could also take the point of view that since these expressions are valid for large 0, continuations could be fu b e (o - cos_3 4+20 -1 be found by expanding (cos - - cos - )1 for sin 0 large and n close to two. Actually, either expansion encounters difficulties since both terms in 3 7r~+20 -1 (cos — - cos ) go to zero for certain values of n and 0. Consen n quently, an alternate approach which requires no troublesome expansions will be used here. The total backscattered fields may be rewritten as T -- i2kR a A sin -e..-.. nel e2' n R 3T 3r 20 37r 20 C (cos- -cos cos- )C- i sin- sin C 3 -2/L7( 7r 3r7 20 2/37r. 2/20 (cos - 1) Cos I- 2 cos - cos cos + cos -- smI -j n n n n \ / J (4.6) where / 2 e i(2ka sin 0 - ) -i(2ka sin0- ) 1 3 2ka sin 0L 2 2 i(2ka sin0- ) e-i(2kasin0- 0 - - (4.7) 2 27rka sin0 2 (47) The term C can be recognized as the large argument expansion of the - i... x -. —_ Bessel function (-1) J (2kasin ), where =0, 1, 2... SimilarlyC2 is the large argument expansion of (-1) J 2(2ka sin 0). The actual choice of the Bessel functions must be made such that the resulting expressions for the backscattered fields be well behaved as functions of 0 for each value of S. Consider

86 first the coefficient multiplying C1. For any permissible value of n it can be shown to be finite. Hence the lowest order Bessel function corresponding to C1, J (2ka sin 0), is the appropriate choice for that term. Note that for n = 2 the coefficient of C is identically zero and that the term disappears from consideration, a feature that will also appear in the final expressions for the back-scattered fields. Next consider the coefficient multiplying C2. The lowest order Bessel function corresponding to C2, J (2ka sin 0), will suffice to keep its contribution finite. Indeed, the forms of equations (3. 83) and (3.84) dictate this choice. Finally, the Bessel function corresponding to C3 is taken to be-J (2ka sin 0) so that the polarization dependent term 3- 2 will behave in agreement with equations (3. 83) and (3. 84). The final form for the total backscattered fields is sm7 -ei2kR i a A sin-ei 4el!+!e2 n R 7r 37r 20 37r 20 - (cos - - cos cos ) J (2ka sin 0) - i sin - sin - J (2ka sin 0) fF n.c.n n n _ n n 1 \ 27T\ 7T 37T 20 2/37T 2[20\ cos - 2 cos cos cos - + cos - sin ) jn n n n n n + (cos -- 1)1 J2 (2ka sin 0), (4. 8) where again the upper sign is to be used for E-polarization and the lower sign is to be used for H-polarization. This equation is valid only for 0 less than the cone half angle 7y. For n = 2 the coefficient of J0 (2ka sin 0) becomes identically equal to zero and the expression reduces to a form which is in agreement with the leading terms of equations (3. 83) and (3. 84) for 0 near zero, the desired result. The method of approximation used in Chapter III

87 is seen to be in agreement with the Geometrical Theory of Diffraction. In fact, consideration of the results of Chapter III has resulted in a new form for expressions for backscattering by a finite cone for aspect angles near nose-on which differs radically from that obtained by Ross. The simplicity of this result suggests the possibility of applying the method of Chapter III to cone scattering in the same manner as was done for the disk. Such an undertaking certainly bears consideration.

BIBLIOGRAPHY Born, M. and E. Wolf (1964), Principles of Optics, MacMillan, New York. Bouwkamp, C.J. (1954), "Diffraction Theory,' Reports on Progress in Physics, 17 (35-100). Eggimann, W.H. (1961), "Higher-Order Evaluation of Electromagnetic Diffraction by Circular Disks", IRE Trans. on Microwave Theory and Techniques, MTT-9, (408-418). Flammer, C. (1953), "The Vector Wave Function Solution of the Diffraction of Electromagnetic Waves by Circular Disks and Apertures,I: Oblate Spheroidal Vector Wave Functions, II: The Diffraction Problem", J. Appl. Phys. 24, (1218-1231). Flammer, C. (1957), Spheroidal Wave Functions, Stanford University Press, Stanford. Heins, A.E. and D.S.Jones (1967), "Note on Diffraction by a Disc", Proc. Camb. Phil. Soc. 63, (851-853). Jones, D.S. (1965), "Diffraction of a High-Frequency Plane Electromagnetic Wave by a Perfectly Conducting Circular Disc", Proc. Camb. Phil. Soc. 61, (247-270). Keller, J.B. (1962), "Geometrical Theory of Diffraction", J. Optic, Soc. Amer. 52, (116-130). Lure, K.A. (1960), "Diffraction of a Plane Electromagnetic Wave on an Ideally Conducting Disk", Soviet Phys. JETP. 4, (1313-1325). Meixner, J. and W. Andrejewski (1950), Ann Phys, Lpz., 7 (157-168). Morse, P.M. and H. Feshbach (1953), Methods of Theoretical Physics, Technology Press, M.I.T., Cambridge. Nomura, Y. and S. Katsura (1955), "Diffraction of Electromagnetic Waves by Circular Plates and Circular Holes", J. Phys. Soc. Japan,10 (285-304). Ross, R.A. (1967), "Investigation of Scattering Center Theory" Cornell Aeronautical Laboratory Report AFAL-TR-67-343, AD 826872 (U). Stratton, J.A., P.M.Morse, L.J.Chu, J.D.C.Little, and F.J.Corbato (1956) Spheroidal Wave Functions, MIT Press and John Wiley and Sons, Inc. Ufimtsev, P. Ia. (1958), "Approximate Calculation of the Diffraction of Electromagnetic Waves by Certain Metal Objects-fl: The Diffraction by a Disk and a Finite Cylinder", Soviet Physics-Technical Physics, 3, No. 11, (2386-2396). 88

APPENDIX A SOME PROPERTIES OF THE OBLATE SPHEROIDAL COORDINATE SYSTEM The variables x, y, z of the!Cartesian coordinate system are expressible in terms of, rl, and of the oblate spheroidal coordinate system of Fig. 2-1 by 2 2 x =a,(1 - n 1+ 2 )C ( +A. 1. a) I -- 2 2 y = a (1 - r12) (1 + 2) sin 0, (A. l.b) z =ar7. (A. l.c) The gradient in the oblate spheroidal coordinate system can be written (h)- +(h)+(h)-l (A. 2) where hg, h, and h are the metrical coefficients given by r2 2 h =a I (A.3.a) 1+2 2 2 h a (A. 3. b) a 2 2(A. h =a (I + 2) (1 7 -2 (A. 3.c) 89

90 Expressions relating the unit vectors in the CartesianS and oblate spheroidal coordinate systems can easily be obtained with the help of equation (A. 2) by taking the gradient of both sides of equations (A. la, b, c). The results are A 1 +- A 1-r A A X - 2-2 r cos r+ 2+ 2 cos e- sin i, (A.4.a) y =-r 2 2 sinor+e 1-2 sin +cos, (A.4.b) + K2 + r2 2 2 (A.4.c) As ~ becomes large, the surfaces of constant [ 1approach spheres and the hyperbolae of constant 7r asymptotically approach lines of constant 0, where 0 is the spherical angle defined with respect to the z-axis. We may write, a v ~R, (A. 5. a) r AV COS 0. (A. 5. b) where v is to indicate "in the limit as ~ becomes infinite". Because of this asymptotic behavior of the coordinate system for large values of the radial argument, the radial functions behave asymptotically like the corresponding spherical Bessel or Hankel functions. For instance, for the normalization used by Flammer (1957) or Stratton, et al (1956), the radial function of the third kind behaves identically with the spherical Hankel function of the first kind for very large A. That is,

91 (3) i(kR _n+- 1 ) 1,R mn( ' e 2 (A. 6) The components of the curl operator in the oblate spheroidal coordinate system are given in terms of the components of a vector A by (hA_ ) D(h A,) (VxA) h= hh[ - - (A.7. a) (Vx A) = hhI [a - J (A.7.b) 1 [(hL A) a(h A (Vx A) h L a (A. 7. c)

APPENDIX B THE FAR-ZONE BISTATIC SCATTERED FIELDS For the incident field F. of Fig. 1-1 given by equation (1. 1) the far-zone 1 scattered fields may be written: S = -r? H -j ovCsm( 0 0 E ikRcom + M = 0 2(26 Om) x ____I (1) AN~ - m (-ic.P — (1) Cos y) 5in (-ic,9 Cos 0) + 2caE (c 0) Cos - + Cos 0 / Jit N+ s(1) (-ic Cos -Y) s(1) (-iccoso)+.0 M+l, I + (1-6Om) OD i = m Nl 1) s(1) (-ic, Cos (B. 1) S E = 0 O ikRZ in W" 2a (cl,0) Cos O = 0 S(1 (-ic, sin mn ( + -r) x 2 (1) Cos y) S3, (-ic, Cos0) - I =in+2 iN +i, e - (1 60m)ZI N-1)ls OmNi1 - (-ic, Cos 'y) (1) f(-i~c, Cos 04 (B. 2) 92

93 For the incident field F. given by equation (1.2) the far-zone scattered fields are -E ik7r o-D m (c,e) sin y oD EeH oOH ikR 2 Om E0=iHcos m (s + in) s 2(2-60 x m =0 = m+1 Tlm J, x N S( (-ic cos y) S( (-ic, cos 0) + +N + m Ni mr a (c, 0) cosT y J + m C +l SM (1) (1)(-i c, cos os)+ c[ 1Nm+lm-icos m m+ -m-l I (1) + (1-6 ) S (-ic, cos y) S (-i cos0) _ - -- S1) X ikRl 00 H (B. 3) e Z 2a (c) p bm sin th + 7lr EH -- 0 8H- o ikR cos sm 2 x m =0 n C L m Jm+l,i (1) (1) m+iN m - * lm6 (-ic Cos JS 1 Om Nm~.- ml (-is (B.4) All quantities in equations (B. 1) through (B.4) have been discussed and defined in Chapter II. Equations (B.4) and (B. 2) give field components that are orthogonal to the respective incident fields. These cross polarized components become identically zero when OP lies in the yz-plane (~ = + ), and, 2

94 for the special case of backscattering, equations (B. 1) and (B. 2) reduce to equations (2. 2. 3) and (2. 2. 10) respectively with (B. 2) and (B. 4) both giving zero contributions. The program for calculating the backscattered fields can easily be extended to give these bistatic fields. It is only necessary to compute products of angular functions for two different values of the argument instead of for single values of the argument and to introduce the 0 dependence in the sum over m in equations (B. 1) and (B. 3). Computation of equations (B. 2) and (B. 4) will then be a trivial extension, as the summations over I in them are also found in the respective direct returns. The apparent singularities due to cos 0 and sin 0 in equations (B. 1 - B. 4) can be eliminated by considering appropriate limits. In some cases ratios of the weighting factors and the singular function will be well behaved, while in other cases the angular functions will cancel the singularity.

APPENDIX C THE a (-ic) r Series representations of the coefficients a (-ic) as defined in equation r (2. 3. 5) are given below in terms of CO, and the normalized coefficients C2b = Cm/C for arange inthe index r of 0 <r<8. Series for the 2b lb i0,...... a (-ic) for 0 < r < 4 have also been derived by Flammer (1957) and agree r -. - with those given here with a slight change in notation. The argument (-ic) has been omitted here for brevity. ml (cm)-2 a0 nl ml -2,ml a=. -2(C ) C ait )-2 (c2)2 -2 ] a3f -2(3!) (C)-2 L2(C2i) -3C4 C2+C a. C )-2 ( )4-12 C (C C +3(C 3) >0 2 4 2 6 - )(C) 2 (C ) -10C (C ) +6(C ) + 4 4 a nil ml,mI,mt Mil,In,int +6CV (C ) -3C C -3C C +C aml mil -2 r ml 6 nil iml 4 nl ml2 mil3 " 6!(C r (C _30C (C ) o30(C' C1.)24^ 0 2 4 2 4 2 4,ml ml 3 3ml.ini,iM ml 2 +20oc (C ) -24C C C2+3 (C') -12C8(C 2 +6C C4 m +6C -2Cl 95

96 mal ml -2r tnt7 Malt,in5,m12 im1 3 U-(7) c -21C~ (02 +30(C 4 )(C2),Iml 3 rint mln tint 4 mA,it inC -0(4 2 +15 (0 30 4(2 )644,mlEtmlt2,mit2 mal tmlInt tn3 + 660 6 (04 )+6(06 )C (2 )~08(C )+ ml mt intil vl,mI mtnt l +280402 38 06 +60 l(2 30i 04 312 Sq2 +14j ml t U)2[( tInt B fmi nt if6 mlnA2C 0tint)4460 3,, m,2 8 u'8!C 2 -64 (2 )+5(4 2 (4*)"2 m4 nil mint5 -1200 int tn Int3 +5(04l ) +42CI (02 -126 04 (C2 + tInltmft2 fmi Iml~2,m12 xnil 2,i +6006 (04 )0 +30(06 C (02)2(06 ) 04 mln,mil 4 'ml tm tint 2-2 tit Cn -300 )~ +600 0 C 8 (2 8 4 (2 )~28(0)~mat mln minl a n2 titmm3 Ammn 248 06 02 +(8 ) +00o(2 24i04 min Imn tIrd 2 fmit min tn Ima i20fl C06 ~12C (2 ) 1C C4 +C4C2 1

APPENDIX D eC ****D********************************* * **** r * **8**D I SK OOO 1 I I I J I —. C *::DISK0002 C * D I S K S C A T T E R I N G P R O G R A M *DI SKOO03 C*.*D I SK0004 C * DISKOO C * PURPOSE: D I SK0006 C * TO COMPUTE THE FAR ZONE BACKSCATTERING FROM A DISK. *DISKOO07 C * *DISKOOOR C * METHOD: *ODISK0009 C * THE EXACT SOLUTION OF FLAMMER. *DISKOO10 C *-"DISKO011 l C LANGUAGE: *DISK0012 C * FORTRAN IV G-LEVEL. THE PROGRAM HAS BEEN SUCCESSFULLY RUN ON *DISKO013 C * AN IBM 360/67 UNDER THE UNIVERSITY OF MICHIGAN TERMINAL *DISK0014 C * SYSTEM (MTS ). *DISK0015 C * *DISK0016 C * SUBROUTINES REQUIRED: *DISK0017 C * THE. ONLY SUBROUTINES REQUIRED IN ADDITION TO THOSE LISTED HERE*DISK0018 C * ARE STANDARD FORTRAN IV LIBRARY ROUTINES (E.G., SIN, COS). *DISKO019 C *DISK0020 C *...**-***A*****************D*************** ***** ******' * *4* * *44.****4*D I SK0021,'q~.,~ ~~ ~II

The description of the main program is followed by descriptions of the I required subroutines in alphabetical order. In all descriptions, standard FORTRAN IV variable naming conventions are used, unless otherwise indicated. The program as listed was tested on an IBM 360/67 under the University of Michigan Terminal System (MTS), but as no system or machine specific subroutines are required, it should be directly transferable to any computer with a standard FORTRAN IV G-level compiler. MAIN PROGRAM The main program reads in from I/O unit 4 the namelist IN, containing Co the variables: Xi the double precision value of the initial angle in degrees (default is 2. 0), X2 the double precision value of the final angle in degrees (default is 88. ), and DX the double precision value of the step size (default is 2. 0). The program then calls on the routine RCS to compute and print the values of the radar cross section for the indicated range of incidence angles, for a value of ka read in (together with the Stratton-Chu coefficients d (-ic)) from I/O unit 5.

The program then writes on\I/0 unit 6 some of the auxiliary quantities wyhich have been computed: the (unnormalized) C (-ic), the N _______ 2Jk me K( (c) and K 1) (c), and the J (c) and J' (c). This process is repeated for each succeeding set of data, until an end of file is encountered. If it is desired to dispense with the printing out of these auxiliary quantities, the main program as listed may be replaced by the following shorter program: DOUBLE PRECISION X1, X2, DX / G NAMELIST /IN/ X1,X2, DX 1 READ(4, IN, END=500) CALL RCS(X1, X2, DX) GO TO 1 500 STOP END

c MAIN\ PR-OGRAM DISK0022!03 C DISK0023 IMrLICIT COMPLEX(Z) -____ ________0 ---____ 4DISKOO24 REAL KKA DISK00251 M- (C N M _I'LJAR_ AYff___ICi i2iL__ ____ _ ___ ______ _ SKI0f2ft, COMMON /ARRAYN/ NMAX(9,9) DISKOO27 COMMON /ARRAYJ/ ZJ(9.9) DTSKOO2R COMMON /ARRAYK/ K(999) DISK00291 COMMON JARRAYO/ ORTHL4B(9-999) n7 rK no nn --- DIMENSION/RMl(9) DISK0031t - DATA Ml / 0f'1t2,v3, 425q6 7, 8/ -DSI SK 2 REAL'V8 X1,X2,DX DI SK0033 NAMELIST /IN/ X19X2-,DX DI SK0034 1 X =- 2.DO DISK0035 X2 = 88.DO DISK0036 DX = Xl DI SK0037 READ(4, IN, ND=501 DISKOO38. CALL RCS(X 1X2,DX) DISK0039C) WRITE(6I,901) DISK0040 DO 20 M = 1, 9 DISK0041 DO 20 L = M, 9 DISKOO4 Li = L - 1 DISK0043 LIM =(NMAX(MgL) + 1) 2 DISKOO44 DO 15 KK = 2, LIM DI SK0045 15 C(MLKK) = Mi *C(MgLgKK) DISK0046 20 WRITE(6,900) M1U4),Ll,(C(MLKK), KK = 1L LIM) DISK0047 __WRITE(6t902) __ __._D1 SK004& WRITE(6,903) ((Mi(M), (ORTH(ML), L = M, 9)), M = 1, 9) DISK0049 ___WYRITEA 6.904) n T ____ ________ __ __oDSKK&5DQ WRI TE (6,903) ((Mi(M), K(M,L), L = M, 9) ),v M = 1, 9) DISK0051 WRTTF(6,9905) DT SK0O DO 40 M = 1, 9 DISK0053 __ DO 40 L = Me 9 - ______DLSKO_5_4_

C LISTING OF MAIN PROGRAM, CONTINUED. DI SKOOQO C DI SKOOQO ____L L- = __ —____ _.. __ fI LSK 0055 40 WRITE(6,906) M1(M),L1,ZJ(ML) DISK0056 500 GOULL _______ ____T O- - DI SKQOa7 500 STOP DISK00587 900 FORMAT( 'OM =',13,5X?'L ='13,l 5XqlP6E15.5/23X,6E15,5) DISKOO59 901 FORMAT('1THE (IJNNORMALIZED) CONSTANTS C(M9L92K)...') DISK0060] 90 F OR MAII IT-H E —NORM L L141 ION-CON ST ANTSJ. NML. ikL - D _ I K1K 61 903 FORMAT( /14, 1P9E14.5//I4, 14X,8E14.5//I4,28X,77E14.5//I4,42x,6E14.5//DISKoO62 &14_9__C14,56X,.5E 5 70 E 14.5//I4- 70X,5//I 4,84k(, 314.5//I4998X,92E 14, 5DISK00632 &14,112X9E14,5) DISK0064 904 FORMAT(O1THE JOINING FACTORS KML(1) AND KML(l)''..') DISK0065~ 905 FORMAI('1THE JOINING FACTORS JML(1) AND JML(1)'',,') DISK0066 906 FORMAT('OM =',I252 'L =',j2,5XlP2El5,5) DI SK0067 END DISK0068

ROUTINE: ABMEH USE: CALL ABMEH (M, KA, X, ZAE, ZAH, ZBE, ZBH) where M is m+ 1, KA is the real value of c = KAJ E ZAE is the returned complex value of a (c,X), ZAH is the returned complex value of a (c,X), m E ZBE is the returned complex value of 13 (cX), m ZBH is the returned complex value of 3H (c,X). COMMENTS: These quantities are described by equations (2. 13a, b, c, d) and (2.16a, b, c, d).

SUBROUTINE ABMEH( M, KA~X v ZAE, ZAH, ZBE, ZBH) DI SK0069 C DISK0070..I McPL GL._;_LLDI CPLEXA1L __) __. ~.__ ___._~_.__ _____.-___._.. ______. __. Dl~.DSW0017I_ REAL*4 KA DISK0072 ~REAL*8LX ____-___DSKf3 DATA ZERO/(0.,0.)/, ZHALF/(O.5,0,0)/, ZONE/(1.00,O)/, DISK0074 ZTWO/(2v.O,.O )/ DI SK0075 ZlE = ZERO DISK0076 I= ZERO DISK007____ 7____ ___ __________ I1OLLL DO 20 N =M, 9 DISK0078 -C ALL -B-MJNM, NXJ9 ZABI _______ _____ - DLSI(QOI9H CALL KML3(MNvKAZK) DISKOO0O IF(MOD(N-Ms2) )?lv24s2l DI SKOOi 21 ZTERMH = ZAB * ZK DISKOO082 Z = 71HZlLTER M H DI__________________-IB3 GO TO 20 DISKOOBA 24 ZTERMF = lAB * 7K DTSKOO85 ZIE = ZIE + ZTERME DI SK0086 20 CONTINUE DI SK0087 Z2E = ZERO DISK0088 ___ Z2H = ZERO DILSKf9 IF(M.EQ. 1) GO TO 40 DISK0090 M DI S0092 DO 30 N =M, 9 DISK0092 CALL AMN(M1,NXZAB) DI SK0093 CALL KML3(M1,N,KAZK) DISKO94 I F ( MOD(N-M1,2)) 31,34,31 0______ ___ ___ ____ _0_ 12J$KQ995_ 31 ZIERME = ZAB It ZK DI SK0096 Z2E = Z2E + ZTERME _ _ DISKOO97 GO TO 30 DISK0098 34 ZTERMH = ZAB * ZK DISK0099 Z2H = Z2H + ZTERMH DISKO100 30 _ ~DNIL _____-_______ CONTINE____ -D 1SK.I 0101 H"

C LISTING OF ROUTINE "ABMEH", CONTINUED. DISK0102 C DI SK0103 4Q -ZE __ 3 -= ZFR - _ ___ ---- -DI SK Z3H = ZERO DI SK0105 - ___ IFLM G_9) G TO 6 _______ __ _.. — __ DIS.1S 106 MP1 = M + 1 DISK0107 DO 50 N = MP1, 9 DISKO108 CALL AMN(MP1,N,X,ZAB) DI SK0109 ___CALL LP1 N., K __________________ D. T SK IF(MOD(N-MP1,2)) 51,54,51 DISKOl11 51 ZTERME = ZAB ZK OD SK0112. Z3E = Z3E + ZTERME DISK0113 GO TO 50 DISK0114 54 ZTERMH = ZAB * ZK -. DISK0115 Z3H = Z3H + ZTERMH DISK0116 50 CONTINUE DISK0117 IF(M.NE. 2) GO TO 60 __ __-__ DISK0118 Z2E = ZTWO * Z2E DISK0119 Z2H = ZTWO * Z2H DISK0120 60 ZAE = Z1E / (Z1E - ZHALF * (Z2E + Z3E)) DDISK0121 _ Z A lH =Z1H Z 1 H / 7LLH 3HZH)ALE_ _1___L Z2L _ _ Z 3 ) __D ISKg122-_ ZBE = ZONE - ZAE DISK0123 _._ ZBH = ZQNE - ZAH__ D__ _ ________ SKTI2 RETURN DI SK0125 END DI SK0126

ROUTINE: ABMN USE: CALL AMN(M, N, X, ZA), or CALL BMN(M, N, X, ZA) where M is m+ 1 N is n;+ 1, X is a double precision angle in degrees, ZA is the returned complex value of a (KA, X) or _mn b (KA X) as AMN or BMN is called. mn COMMENTS: o These quantities are described by equations (2. 14a, b).

SUBROUTINE ABMN DISK0127 C DI SK0128... COMQL_/ARRAYfl_ QRTH(9,)s__ _ _,_9 ) SKO129 -COMMON /ARRAYS/ S(9,9) DISK0130 -..COMPLEX _ _ ZA: 7 —.___ --- -__I-SKOi3 -REAL*8 X DISK0132 DATA RAD /.1746i32F-1/ DI SK01!33 ENTRY AMN(M,N, X,ZA) DI SK0134 _ — XC = RAD SNGL(X) - L____ SK135 XC = -1. / COS(XC) DISK0136 GO TO 1 - __-LSKL 37 ENTRY BMN(M,N,X,ZA) DISK0138 XC = RAD * SNGL(X) DISKO139 XC = 1. / SIN(XC) DISK0140 1 IF(M-l) 5,10.5 -DISK014.1 10 DEL = 2.0 DISK0142 GO TO 15 DISK0143 | 5 DEL = 4.0 DISK0144 15 IPOWR = MOD(N+2,4)+ 1 DISK0145 GO TO (20,21,22,23), IPOWR DISK0146 20 Z = CMPLX( 1.0.0) _____________ __ D47 GO TO 30 DISK0148 21 Z = CMPLX(O 0,.0,I) - _______ ---_ _ — -- __ — DISKQI49 GO TO 30 DISK0150 22 Z = CMPLX(-1.0,0.0) DISK0151 GO TO 30 DISK0152 23 Z. CMPL.X (Q.O,,- _ OJ......._. DISK0153 30 ZA = Z * CMPLX(XC * S(M,N) * DEL /ORTH(M,N), 0.0) DISK0154 RETURN __________._- DI SKO1 55 END DI SK0156 I.C)

ROUTINE: ALGNDR USE: CALL ALGNDR(N, M, X, P) where N is n+ 1, M ism+ 1, X is a double precision value in degrees, P is the returned single precision value of the associated Legendre function, Pn (r), where r = cos X. n i COMMENTS: Intermediate calculations are in double precision; the returned value P is single precision.

SUBROUTINE\I ALGNDR(Nv Mt vP) DISK0157 1 C DI SK 5 8 I.MAPL ICLZ.REAL*8!(A-Ht1 ULZi..~ ___~_~_~____ ~_~ __. ~_ _~. DLSKQ159x~_ REAL44 p DI SK0160 COMMON /DFCTRL/ FACT(57) -- _QISK_1Q6 ETA= DCOS(X *174532925199432960-i) DISK0162 ABSETA =DABS(ETA) DISK0163 IF(ABSETA.LE. 1.D-12) GO TO 201 DISK0164 LE(AftSLTA GE~ 999999999999 L GO Jt 25Q- DIS__K _16__ ____ __ DISK MI = M + 1 DI 5K0166 NMO2 = N + ML2-_ — __ _.2__ __ __ ____ ___1 ASK6L SUM = 0.DO DI SK0168 DO 90 K = NM02v N DISK0169 K2 = 2 * K DISK0170 TERM=FACT(K2-1)/FACT(K)*ETA k*(K2-N-M)/(FACT(N-K+1)* FACT(K2-N-M+ 1))DISKOiJ7 IF(MOD(K,2).EO. 0) TERM -TERM DISK0172 90 SUM = SUM + TERM DI SK0173 A=(i.DO-ETA ETA)**(DFLOAT(M-1)/2,DO0)/2.DO**(N-1) DISK0174 IF(MOD(N,2).EO. 0) A = -A DI SK0175 P = SNGL(A*SUM) DI SK0176 500 RETURN DISK0177__ ____ ____ __ __ __DISK 17 201 IF(MOD(N-M,2)) 202,203,202 DISK0178. 202 P = 0. __ DISK0179 GO TO 500 DISK0180 /~<203 K = (N+M) / 2 DI SK018. A = (FACT(N+M-1)/FACT(K))/(FACT(N-K+1)*,2.*(N-1)) DISK0182l IF(MOD(N+KI2)_.NE 0) A = -A DISK0183 P= A DISK01841 GO TO 500 DISKO185r 250 IF(M.EO. 1) GO TO 260 DI SK0186 P = 0. DI SKO187 C70 TO 500 260 K. = GO TO 500 E-D-__ - DI SKO1 88 UISKO01P9 DISKOi9O DI SKO191 -

ROUTINE: ALPHA USE: CALL ALPHA(M, L) where M is m+ 1, L is I + 1. COMMENTS: This routine returns the values 0 ml r (-ic) / r' for r = 0, 1,..., m in the common array COMMON /ARRAYA/ ALF(9). These quantities are described in equation (2. 21)..... - -I..

SUBROUT IN E "'ALP HA(M, L) D I SK019 2 C )DI SK0193 COMMON /ARRAYA ALL9.I ___ __... )D1K0194 COMMON /ARRAYC/ C(9,9912) DISK0195 COMMON /ARRAYN/ NMAX(9 9IS(_K 9 ) _-__ __-___ -D1S COMMON /FCTRL/ FACT(57) DISK0197 CO = C(MtL1) DISK019 C2 = C(ML92) DISK0199' C4 = C(IM91 3) ____ __ ____S _2__ _.___ C6 = C(ML,4) DISK0201 C8 = C(M-LL5) -D__ __ _ ___ I-SK2__ ___ _- ___- __0_ -- LQ.20 C10 = C(ML,6) DISK0203 C12 = C(MtL,7) DISK0204 C14 =C ( ML L8) 9 DISK0205 C16_ - M LL9 __ _ =C_ __________ ___ PSKI02-0 CSOR = 1. I. (CO * CO) DISK0207 ALF(1) = CSR ___ - ISK2 IF(M.EO. 1) GO TO 500 DI SK0209 ALF( 2) = 2. *CSOR *C? 01 sKo2i0 IF(M.EO. 2) GO TO 500 DISK0211 I C2P2 = C? * CDISK ALF(3) = CSQR * (3.'C2P2- 2.*C4) DISKO2131 - IF(M ~ E )GOLOi0__ — __ ___ __ ____ _ _E)_GLS O2Tik C2P3 = C2P2 * C2 DISKO2L5 ALF(4) =-2. * CSOR R (2.*C2P3 3.*C4*C2 + CC6) DISK0216 IF(M EO, 4). GO TO 500 DISK0217 C2P4 = C2P3 * CZ, 2 _ _ _ _ _ _ ___ DJSJ(02]J& C4P2 = C4 * C4 DISK0219; CSOR (5. C2P4 12.*-C-C-4C2P2 + 6.0C4P2 6.*C6*C2 iSKQ22Q - 2.*C8) DISK0221 -IF(M -EQ. 5) GO TO 500 01SK0222 C2P5 = C2P4 * C? DISKO223 AC SQ0R *3o*- 2P-5 ---- (3*C 23P5 -- -1Q.*C4.C2P342 b*C-4.P2P2*CZ.. D ISK 0224. + 6*C6*C2P2 - 3. C6*C4 - 3*C8B*C? + C10) DISK0225 I.A

C LISTING OF ROUTINE "AL-PHA" CONTINUED. IK02 C DI SK0227 IF(.E0)BIL~f500DI-SK022-8 C2P6 =C2P5 *C2 DI SK0229 C4P3G4.-AP.2-.C4 --- ILS-KQ2-?3Q0 C6P2 = C6 * C6 DI SK0231; ALF(7) = CSOR (7.*C2P6- 30t *C4*AC2P4 + 30.*,'C4P2*C2P2 4.*C4P3 DISKO22 & + 20,'-C6*C2P3 - 24.*"C6*"C4*"C2 + 3.*'C6P2- 12.*"C84C2P2+ 6.*-C8*"C4 DI SK0233' &,C0*2-,*J __- _-_____ 1SKD2341 IF(M.EO. 7) GO-TO 500 DI SK02351 _ C2P7 C2P6 _C2 DISK0___ -PL.Q~23 ALF(8) 2 -. 4CSOR *(4.",C2P7 -21.*,C4*CP5 + 30.*,C4P2*C2P3 DISK0237 - O.*lC4P3*-C2 + 15.*"C6'C2P4 -30.*C6*C44*C2P2 + 6.*AC6*'C4P2 DI1SK02381 &+6.4C6P24'C2 -1O.*C8-C2P3 + 12.*,C8*C4*C2 - 3.*8*C6 DISK0239 ~ 6.*ClO*C2P2 -3.*VC1O*4"C4_ 3.'Cl2""C2 + C14) DS04 IF(M.EO. 8) GO TO 500 DISK0241 C2P8 = C2P7 * C2 DISK0242 C4P4 = C4P3 * C4 DI SK0243 C8P2 = C8 * C8 DISK024 ALF(9)= CSOR (9.*C2P8- 56.4C4*C2P6 + 105.*C4P2*C2P4 DISK0245: ~ 6.C4P3*-C2P2 _+ 5.*"C4P4_+ 42.*C6*C2P5 - 120 QC6*C4*2 2SG4 ~ + 60.4-C6*"C4P.24C2 + 30.*',C6P2*1C2P2 - 120.*C6P2-*C4 - 300.*C8-*C2P4 DISK0247 -- + 60.*C-84'C-4C2P2- 12,*C8-4C4P2 - 24.*-C8*C6*-C2 +3,*C8P2 __D S KD2A8 & + 20.4,ClO'AC2P3 - 24.::-Clo*C4*"-C2 + 6.*-CIO*'C6- 12.,-Cl2"C2P2 DISK0249 ~+ 6.*-,C12*-'C4 + 6. -C.14*C2 2 2.*1-6 DI SK0250 I.I.4 I.A!UU KtJ URN END DI SK0251 -DI1 SK.O2-5Z

f ROUTINE: CMLK USE: CALL CMLK(M, L, K, ANS) where M is m+ 1, L is i + 1, K isk+ 1, ANS is the returned double precision value of C2k I I. COMMENTS: The computations are described by equations (2. 18a, b).

SUBROUTINE CMLK( M,L,K,ANS) DI SK0253 C DI SK0254 p...PLIC-L-T.PEAL*8LA-hQ-. -..._ D —.1............. SK0255 REAL*4 D DISK0256i..... COMMON./ARRAYN/-..-NMAXL9_ 91 -.....-.................. - - - — DISKQ25_ COMMON /ARRAYD/ D(9,9,12) DI SK0258 COMMON /DFCTRL/ FACT(57) D SK025 M2 =2*M DI SK0260! Ml '1 M - I-_ ---S -- -- -- - K Q2_61 LIM = (NMAX(M,L) + 1) / 2 DISK0262! IFLG = MOD(L-ML2).___ --. -DISK 0263 IF(IFLG) 11,12,11 DISK02641 11 HALVES = -0.5DO DISK02651 GO TO 13 DISK0266' - 12. HALVES = -1.5DO - ____ ____-_______ _ —DI SK0267_ I 13 KK =K -.2 DISK0268! SUM = O.DO D ISK0269 DO 10 N = K, LIM DISK0270 N2 = 2 * N DISK02711 IF(KK) 2,3,4 DISK02721 2 T1 = 1,D00 _____-_ — DI SK0273, T2 = 1.DO DISK0274| - GO TO 6 S —0___ — ___ ____1LSQ2Ll5 3 Ti = DFLOAT(M+N) + HALVES DISK0276| T2 = l.DO - DFLOAT(N) DISK0277 GO TO 6 DISK0278 4. 1LE_=_FLU.ALM+N_+ _HALLES - - --._.____Dl....DSK0279 F1 = T1 DISK0280 J2.. - _.DQ _DEL QAI t N ). --- - - ----- - D I SK 02 81 F2 = 12 DISK0282I DO 5. = 1, KK DnT Kn?2R, F1 = Fl + l.DO DISK0284'.- -E2_ f- E2_+- 1.DQ0 _ DISK0285 T1 = Tl * Fl DISK0286.-A C3

C L I STING OF ROUT INE "ICMLK",9 CONT INUIED. [)I SK0287 - 5 12 =T2EF2___6 IF(IFLG) 8,7,8 7 13 =.FACT(M2+N2-3)/FACT(N2-1)____ GO TO 9 8 T3 = FACT(M2+N2-2)/ FACT(N2) 9 SUM = SUM + T1IT2*-T3"DBLE(D(MPLN)) 10 CONTINUE__ ANS = SUM/ (2.DO"-*"Ml FACT(K) FACT(M1+K) RETURN _ _ _ __ _ _ _ END r)ISK0288 D _-RI _SKQG2_A9 DI SK0290 ___-DI SK01 DISK0292 DIS5K0293 DIS5K0294 — AD5KQ_2_9_5_ DIS5K0296. ----— DI SJ( Z9 DI SKO298! I.

ROUTINES: FILLJO, FILLK, and FILLS USE: CALL FILLJO(KA) CALL FILLK(KA) CALL FILLS(X) where KA is the real value of c = KA, X is the double precision value of an angle in degrees. COMMENTS: These three utility subroutines fill the common arrays I, COMMON /ARRAYJ/ ZJ(9, 9), COMMON /ARRAYO/ ORTH(9,9), COMMON /ARRAYK/ XK(9, 9), and COMMON /ARRAYS/ S(9, 9) with the quantities I J (c) or c) N (c), K (c) or K (c) ime J'me(c), me rKm (c) mle '_m(c), and S (-ic, r7), where ri = cos X, by calling the routines JML, NML, KML, and SML, respectively. Since, for a given combination of m and 1, either Jmi or J1 is zero, computer storage is conserved by storing both quantities in the same array; a similar statement holds for K) and K1)' ml ml'

SUBROUTINE FILLJO(KA) DISK0299 COMMON /ARRAYJ/ ZJ(9,9) DISK0300.._.C OM._0N /-JA R RAYOkMHL 9ORT 9J _ __ D ISKOD30O 1 -COMPLEX ZJ DISK0302 ~.._REAL~~__ DI SK03D_3_O3 DO 100 M = 1, 9 DI SK0304 DO 100 L = Mg 9 DISK0305 CALL JML(M,LKAZJ(ML)) DISK0306 10 L B___P L LLrrUL~,JROT1~UIIW_________ _.___-.~ ______ ~_._ _.__ _ DIL L5KfiIOJ& RE TURN DISK0308 END_ _ __ _ ___ _ DISKQ3Q9 -~___ SUBROUILNfFILLKIJ~AI___~* ___~___ __ — __. _-__ __-~ -_ -___DlS41oSflJ1 COMMON /ARRAYK/ XK(9,9) DISK0311 REAL*4 KA K____ ____ __ ____ D SK03 I2 DO 100 M = 1, 9 DISK0313 DO-100 L = M. 9 D1SK0314 100 CALL KML(MLKAXK(ML)) DISKO315 ___ REf-TURN ____ -_ D I SKO 31 6 END DISK0317 SUBROUTINE FILLS(X) DISK0318 COMMON /ARRAYS/__S(9 _ ___ _S( 9__ DI SK0319_ REAL*8 x DISK0320 __DO 100 M = 1, 9 DI SK0321_ DO 100 L = Mg 9 DISK0322 100 CALL SML(M, L X S(ML) TERM) DISK0323 RE TU RN DISK0324 END DI SK0325

ROUTINE: GETCN USE: I CALL GETCN \ COMMENTS: This routine obtains the coefficients C2k by calling on the routine 2k ml ml ml CMLK. The coefficients are then normalized to give C 0 C 2 /C m- mf m9 mS C / CO,...... C / C. These normalized values are then conver 4 0 2k / 0 to single precision. Ml S.. - - st The (normalized) value of;Ck is returned in the (m+l, 1+1, k+1) 2k element of the common array COMMON /ARRAYC/ C(9, 9,12). ted -. h-^z

SUB ROUTINE GETCN DI SK0326 c DI SK0327 DIMENSION DC(12) DI SKO329 CDMMON /ARRAYC/C99I2 LK3I COMMON /ARRAYN/ NMAX(9,9) DI SK0331 DO 100 M = 19 9 -DISK0332 DO 100 L = M, 9 DISK0333.~~._ JJM___= (M=INMAXLY(.M~tLJ ~+. 1J).L2___.__~~~_.._.. _.. ____ ___ _._.._.[LSK~L~33A. J LIMi = LIM + 1 DISK0335 O 20 K =, LIM ___ DISK0336 20 CALL CMLK(M,LK,DC(K)) DISK0337 DO 30 K = 2, LIM DISK0338 30 C(MLK) = SNGL(DC(K)/DC(1)) DISK0339 C(MtL,1 = SNGL (DC(1) - __1 _ _ __ DISKO340 IF(LIM1.GT. 12) GO TO 100 DI SK0341 DO 40 K LIMi, 12 _ ______ DISK0342 40 C( Mt LK) = 0 DI SK0343 100 CONTINUE DISKO344 RETURN DI SK0345 END ___ _ -D-____ _ _ ___ __ ____ ILSK03A416 - I.A

ROUTINE: GETD USE: CALL GETD(KA) where KA is the returned real (single precision) value of c = KA. COMMENTS: This routine reads the coefficients d (-ic) from I/O unit 5, and returns them in the common array COMMON /ARRAYD/ D(9, 9,12). The value of dm (-ic) is stored in the array element D(M, L, [(n + 2)/ 2] ), where M = m + 1 L = + 1, and[x] denotes the greatest integer less than or equal to x. The input format is described on the following page, "INPUT FORMAT FOR THE SUBROUTINE GETD".

INPUT FORMAT FOR THE SUBROUTINE GETD For each value of c = KA there will be one group of cards, as follows. The first card of each group should contain the floating point value of c (corresponding to g in the Stratton-Chu tables) in card columns 1-10. Columns 11-80 may be used for comments. The last card of each group should contain a 9 in card column 1. Columns 15-80 may be used for comments. The body of each group is composed of one set of cards for each combination of m and I. The first card of each such set should contain the value of m in column 1, and that of x in column 2, where 0< m < < 8. In columns 3 and 4 should be punched the maximum value of n, with leading zero, if any, as read directly from the Stratton-Chu tables. In columns 5-14, the floating point value of c = KA may be punched; if punched, it is checked against the value of c given on the first card as an aid in detecting out of order cards. Columns 15-80 may be used for comments. The remaining cards in each set should contain the values of the coefficients, which will be read by the format "( 6E12. 7 )". Columns 73-80 may be used for comments or identification. The sets may be arranged in any order within the group. _ --- —. --

SUBROUTINE GETD(C) DISK0347 i C DISK0348.... CO~MMiN IARRYD_ D~ (,9,12 L 9,l1__4___ _-_ __-.___DI S 039 COMMON /ARRAYN/ NMAX(9,9) DISK0350. o- nn.LQQ -.................... L__I —__,-_.' Kn?3.. DO 100 J = 1, 9 DISK0352 100 NMAX( T..) = n.- DTsKOiA _ READ(5,900,END=500) C DISK0354 900 FORMAT( F10.0) ___D__- - -! DSK0 3545 1 READ(5,901) M1, L1, N1, Cl DISK0356 901 FOR MAT(211, I F2 O)_Q_. 2.TLDSK0 357 IF(M1.EO. 9) GO TO 500 DI SK0358 IF(C1) 3.4,3 DI SK0359 3 IF(ABS(C1 - C).GE..001) GO TO 501 DISK0360 4 IF(MOD(LI-M1,2) -,OD(N12)) 501,7,501 DISK0361 7 M = Ml + I DISK0362 L = LI + 1 DISK0363 NLIM = (N1 + 2) / 2 DISK0364 NMAX(ML) = N1 + 1 DISK0365 READ(5,902) ( D(M,L,I), I = 1, NLIM) DISK0366 902 FORMAT(6E12.7) DISK0367 GO TO 1 DISK0368" _ 501 _ WRITE(6,903) C,M1,L1,N1,Cl DI SK0369 903 FORMAT(' —***ERROR**** C = ', F5.2, ' LAST CARD READ WAS:' / DISK0370 & IX, 211 I2,F10.2) DISK0371.-A 1-A STOP 500 RETURN END DI SK0372 DI SK0.3 7 3 D-I SK 0374

ROUTINE: USE: CALL where M L KA ZJ JML JML I,(M, L, KA, ZJ) is m+ 1, is m + 1, is the real value of c = KA, is the returned complex value of J (c) or J' (c) as (L-M) is even or odd is even or odd. \ J-A I\D) COMMENTS: These quantities are described in equations (2. 9a, b).

SUBROUTIINE JMIL(MLKAZANS) DI SK0375 C -COYLP-L.EX-1A-N-S______ REAL KA DAIA PI02LU77961 ____ CALL QSTAR(MLKAO) ZANS = ClMPLXI..X 1*0 DI SKO0376 - D-LSKfLW7J7 DI SK0378 -—.-I SK03JS9 DI SK0380 DI SKO3R DI SK0382 -_D1 SK QD3-8 DI SK0384 ZAINJS = CMPL X(1I.,O. e) / ZANS RE1UR N__ END CAD

ROUTINES: KML and KML3 USE: CALL CALL where KML(M, L, KA, ANS) and KML3(M, L, KA, ZK3) M ism+ 1, L is + 1, KA is the real value of c = KA, ANS is the returned value of K(1)(c) or K( )(c) as (L-M) is even or odd, \ (3) (3) ZK3 is the returned value of K' (c) or K ' (c) as (L-M) is even or odd. COMMENTS: These quantities are described in equations (2. 10a, b) and (2. la, b).

SUBROUTINE KML(MLKAANS) DI SK0385' CO M MONi /ARRAYC/ C(9,99,912) DI SK0386 cM nli ON /\1ARR AYNI/NMAXL9 9L) ALS03SJ REAL*4 KA DISK0388 LI M = NMAX XMLt1L2_ _ IS 38 SUM 0. DISK0390 DO 2 K 2AIM f)T DI/l SK392 20 SUM= SUM + C(MLK) DISK0392 RM LL~~LAML.SKAtR) D I SKD L3-93 ANS =C(ML,1) *(SUM ~1.0) 1R DI SK0394 RETURN __ ~ l Q95 END - ~ DISK0396 SUBROUTINE KML3(MLKAZK3) DI SK0397 COMMON ARRAYK/ KI9~9 ____ __ _ D ISK 0 39 COMMON /ARRAYJ/ ZJ(9,t9) DIS5K0399 REAL K2. KA 0 100 U, COMPLEX ZK3,ZJ Z K3 = QIPA. X:le.LL0? AJIL _________ RE TURN E N D_ _ _ _ __ __ _ _ _ _ _ _ DISK0401L.-_- — D-S 1SK4O02 - DIS5K0403I 11~T4h0f4AOA

ROUTINE: NML USE: CALL NML(M, L, ANS) where M is m+ 1, \ L is x + 1, ANS is the returned value of the normalization or orthogonality constant, N (c). COMMENTS: / These constants are given by equation (2. 5).

SUBROUTINE NML(M,LANS) C....C OM 01_..A RR A Y D L. D ( 9, 9,12_}................................... COMMON /ARRAYN/ NMAX(9,9)..OM.....C.MON __/_FCTRL/ AC_5.2_1.......__... __. Ni = 1+ MOD(L-M,2) DISK0405 D I SK 0406 -.. -... -.......-... DI -S K-4O -7 — DI SK0408...__ -JiSK 0_409 -DISK0410 LIM = NMAX(ML) DISK041 I =0 DISKO412 _ SUM = 0__ __ DTSK041I3 M23 = 2*M - 3 DISK0414 M22_= 2*M - _ 2_ _ _ -..........-.-. —..-..- -. -.- - _.....DLSKO l AI41_ DO 10 N N1, LIM, 2 DISK0416 N2 =2N DISK0417 I = I + 1 DISKO418 D1 = D(ML, I) DSO_419j 10 SUM = SUM + (FACT(N+M22)/FACT(N)) * Di*D1 / FLOAT(N2 + M23) DISK0420 ANS = 2. * SUM DIS__K042_1 _ RETURN DISK0422 END DISK0423 -.

ROUTINE: PEH USE: CALL PEH(KA, X, ZPE, ZPH) where KA X ZPE is the real value of c = KA, is the double precision value of an angle in degrees, is the returned complex value of PE (c, X), ZPH s te reurnd cmple vaue o P c X ZPH is the returned complex value of P (c, X). H ' COMMENTS: The returned quantities are given by the sum over m in equations (2. B) and (2. 15). All otherwise undefined quantities (e.g., J1 1 (-ic))are set to zero. -l - t. 00

SfSUKROLJI NE PEH(KA, X, ZPE, ZPH) DI SK0424 C DISK0425 IJ-P.LICIT C OMPLEX._____ _______ ____..__. __-DISKO426 RFAL*4 KA DISK0427..... REAL*B X -__ ---___ -. — ____ _ ___ — DI SKQOA28 COMMON /ARRAYJ/ ZJ(9,9) DISK0429 COMMON /ARRAYO/ XN( 9,9) I SK0430 COMMON /ARRAYS/ S(9,9) DISK0431......_DATA__ZERO_/Ji.,O _ZJWD _- 2.,-.) /!~_I_ Z -4 0ISK0432 ZPE = ZERO DISK0433 H.._____ERO ---— DSK043A4 DO 100 M = 1, 9 DISK0440 ilV = M - 1 DT CKO(441 IF(M1) 5,5,10 DISK0442 - 5 ZD E L ZTWO __ — _______DISKA3_ ZDEL3 = ZERO DISK0444 GO TO 15 D_________ SK0445 10 ZDEL = ZFOUR DI SK0446 ZDEL3 = ZTWO InTSKn447 15 CALL ABMEH(M,KA,X,ZAE,ZAH,ZBE,ZBH) DISK0448 Z Z1E = ZERO 44____ DTK_4_4 Z1H = ZERO DISK0450 - DO 20 L = M- 9_ 9__ —_-_ TISKO_45_L S2 = S(M,L) DISK0452 S2 = S2 * S2 DI TKn453 IF(MOD(L-M,2)) 21,24,21 DISK0454 i 21 I. =Z1.-ZH.+_ ZJ(M,L) * CMPLX (S2/XN(M,L),-O. —.- — '_.- -.. —. DISK0455-5 GO TO 20 DISK0456 24 -LE_- Z1E_-+ ZJ (M_,L L * CMPLX (_S2/XN {M,L),..0 — __-.. DI SK04-57-. 20 CONTINUE DISK045R 72F = 7FRn DI' SK04Kn9 Z2H = ZERO -I F(M GE.9)GO TO 40 DI SK0460 DI-SKO461 -

c LISTIT ING O)F RO(IT TNE " PE H", CONT INUF D, D'ISK0462 C DI SKO463 MP1= M + 1 DISK0464 DO 30 L = MPl,. 9 DISK0465 S2Z$ThPUL) __ _____ _____ __ S(_DlJ')DIK 0466 S2 S2 * 52 DISK0467 TF(MOD(L-MPL,2) ) 31,34931 DISK04 31 Z2E = Z2E + ZJi(PlL) ~ CMPLX(S2/XN(MPlL), 0.0) DISK0469 & D L _ i L3 0O _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ Y S K A T 34 Z2H = Z2H + ZJ(MP1,L) CMPLX(2/XN(MP1L),t 0,O) DISKO471 30 ___CONMJhVE ______ __ CO_ _ - _ D~SKQAffi 40 Z3E = ZERO DISK0473 73H = Z SK0474 IF(M.LE.-1) GO TO 65 DISK0475 DO 50 L = Ml, 9 D SK0476 S2 = S(MLL)L DI SK0477 _ 2 S= S S2 i 2~DISK0478 IF(MOD(L-M192)) 51,54,51 DISK0479 51 Z3E = Z3E + ZJ(M1,L) * CMPLX(S2/XN( M1,L, 0.0) DISK0480 GO TO 50 DI SK0481 54 Z3H = Z3H + ZJ(M14 * CMPLX(S2/XN(MlL), 0.0) DISKO48 50 CONTINUE DI SK0483 65__ ZTERME ZDEL4,LZB E ZAE~+ZAE O ZW 2 EtEiEL3DZS3E) 4 ZIERMH = ZDEL*ZBH'Z1H + ZAH (ZTWOC)*Z2H+ZDEL3*Z3H) DISK0485 SIGN 1.0 DISK0486 IF(MOD(M,2).EO. 0) SIGN =-1.0 DISK0487 ZPE =ZPE + SIGN —,- * ZTERME______ __ ______ ____ _ ZPH = ZPH + SIGN ZTERMH DI 5K0489 100 CONTINUE 5_ __ ____-__ _. -- D SK04-9 END DIS K0493

ROUTINE: QSTAR USE: CALL QSTAR(M, L, KA, Q) where M ism+ 1, L is + 1, KA is the real ( single precision) value of c = KA, Q is the returned value of Q*:(-ic). COMMENTS: The Q*.(-ic) are described in equations (2. 20a, b). mc\

SUBROUtTINE OSTAR(M,L,KA,Q) D I SK0494 C DI SK0495 COMMON /FCTRL/.FACT(57) DISK0496 COMMON /ARRAYA/ ALF(9) DISK0497. COI'MOGN /ARRAYK/ K (9,9).. D I SK0498 REAL KA,K DISK0499 INTEGER R DISK0500 CALL ALPHA(M,L) DISK0501 _ SUM = 0. - __________ _- — DISK0502 IF(MOD(L-M,2)) 21,10,21 DISK0503 10 DO 15 R = 1, M_____ ________ __DISK0504 15 SUM = SUM + ALF(R) * FACT(2*M-2*R+1) / ((2**(M-R) * FACT(M-R+1)) DISK0505 & **2). DISK0506 GO TO 100 DISK0507 21 DO 25 R = 1, M __' - -- -__ ____ DISK0508 25 SUM = SUM + ALF(R) * FACT(2*M-2*R+2) / ((2**(M-R) * FACT(M-R+1)) DISK0509 **2) ________________________________SK51 100 XK2 = K(M,L) DISK0511 XK2 = XK2 * XK2 DISK0512 0 = XK2 * SUM / KA DISK0513 IF(MOD(L-M,2).NE. O) = -0 ____ _ _ DISK0514I RETURN DISK0515 __END __ __ ________DISK 05 16 I.3 t%3

ROUTINE: RCS USE: CALL RCS(X1, X2, DX) where X1 is the double precision value of the initial angle in degrees, X2 is the double precision value of the final angle in degrees, DX is the double precision value of the step size in degrees. COMMENTS: RCS is the main routine of the disk scattering program. It first performs all necessary initialization, and then computes the radar cross sections (Ch 10 log, 10 log10 and 10 logand writes' 10, and writes2 these values on I/O unit 6. The angle X1 must be greater than 0. The program has been tested only for values of X2 less than 90.

SUBROUJTINE RCS(Xl9X29DX) D I K 05 1 C DISK0518 IMPLIJjCIT M PL E X (I i __ _ _ ______ _____ ___ _9_ ____ __ ____ r ISJag51 9 j REAL*4 KA DISK0520 R A L 8 X1., 2,Q X D X X__ - __________ _____21_ DATA SOTPIR /0.5641896/, HSPR/0.2820948/ DI SK0522 DATA ISET /0/ DISK 0 523 IF(ISEI) 150,100,150 DISK0524 100 CALL SET D-I SK0525 ISET = 1 DISK0526 150 CALL GETD(KA) _ DI SK0527 CALL GETCN DI SK0528 CALL FILLK(KA) DI SK0529 CALL FILLJO( KA) DISK0530' IRITE(6,9901) X1,9X2,DXKA DISKO531 X = X1 DISK0532 LINE = 0 DISK0533 WRITE(6,900) DISK0534 1 CALL FILLS(X) DI SK0535 CALL PEH(KA,X,ZPE,ZPH) DISK0536 j~_ ~___ ~_P E = AC AI~_____ ______ _______ __ _____DISIft5 PH = CABS(ZPH) DISK0538. _. ______Z2I._l;R = lIE_~ - Z PH___ ____ i1ISK0_ __539 C PCR BCA(ZPCR) DISko54o E = 20, * ALOG10 SOTPIR * PE) DISK0541 H = 20. * ALOG1O(SOTPIR.* PH) DISK0542 CR 20. *ALOG1O(HSPR_* PCR) _DISK 0543 IF(MOD(LINE,5)) 15,10,15 DISK0544 10 WRITE(6,90D) - DISK054+5 15 WRI TE( 6C,903) X, EH,.CR ZPE, ZPH, ZPCR DI SK0546 I903 FORMAT(F6.2,3F9,2, 3(3X,2G13.6)) DISK0547 LINE = LINE + 1 DISK0548 X = X + DX DI SK0549 IF( X-X2.L E 1.D-2) GO TO 1 -DISK0550

C LISTING OF ROUTTINE "RCS", CONTINUtED. DISK0551 I.. -I K.-.-..r U I 1 DISK0552 - RETURN_____ ____ ____ SKQ553 900 FORMAT('1ANGLE E-POL H-POL CR-POL RE(PE) IM(PE)DISK0554 — RE ~ PH) _. IM(PH).. REPEPH-HJ. M...LIE-PHL'L) DTS<K0555 ~ 901 FORIAT('I',////// '-RADAR CROSS SECTION OF A DISK' // 6X,'INPUT PADISK0556 &RAMETERS:' / 10X. 'INITIAL ANGLE IS', F7.2, ' DEGREES' / lOX. DISK0557.'FINAL ANGLE IS', F7.2, ' DEGREES' / 10X, 'IN INCREMENTS OF', DISK0558 - F__.IF7.2, ' DEGREES '__.X, O. 'ALUE OF _C _._S ___2'1_.__ DTSKOL55 59 902 FORMAT(' ') DISK0560 FND rnT CVr% I I- --- -L - ---~ ---L --- — u I.f\ u o 0 I C-1 cn

ROUTINE: RML USE: CALL RML(M, L, KA, R) where M ism+ 1, L is x + 1, KA is the real (single precision) value of c = KA, R is the returned value of R() (-ic,0) or R ) (-ic,O), depending on ml ml whether (L-M) is even or odd.j COMMENTS: These quantities are described in equations (2. 23a, b). - - - -------- aCn

SUJBROtITI NE RML(M,L,C,ANS) DISK0562 C DI SK0563 C-_ —CMMOII /FCTRL/ FACT(57) DISK0564 COMMON /ARRAYD/.D(9,9,12) DISK0565 - X = FACT(L-M+1) * FACT(M) / FACT(L+I1M-_1) * 2**(M- _l____. DISK0566 IF(MOD(L-M,2)) 21,10,21 DISK0567 10 ANS = C*.(M-1) * X / FLOAT(2.*M-1) DISK0568 IF(MOD((L-M)/2,2).NE. O) ANS = -ANS DISK0569 GO --- - TO 500 DI SK057_Q0 21 ANS = C**M * X / FLOAT(2*M+1) DISK0571 - I F ( MOD ( ( L-M-UL2.2_L_ NE. 0_)-_ANS -ANLS L DSK0572 500 RETURN DISK0573 END DI SK0 574 I C-<1

ROUTINE: SET USE: CALL SET COMMENTS: This routine calculates factorials for use in other routines. Single precision factorials are returned in the common array COMMON /FCTRL7' FACT(57) and double precision factorials are returned in the common array COMMON /DFCTRL/ DFACT(57). O: In both cases, the value of n' is contained in the n+1 st array element. The value of n is restricted to 0 <n < 56.

SiBROUTINE SET DISK0575 DC DI SK0576 COMMON /DFCTRL/ DFACT __ DI SK0577 COMMON /FCTRL/ FACT DISK0578 REAL*8 DFACT(57) X___ __ _____ _ - __ _ -_ I)I SK0579 REAL*4 FACT(57), Y DISK05RO D[FACT(1) = 1.ODO DISK0581 FACT( 1) = 1. DISK0582 X = O.ODO DISK0583 Y = 0.0 DISK0584 DO 10 I = 2, 57 DISK05-85' X = X + lODO DISK0586 Y = Y + 1.0 DISK0587 DFACT(I) = DFACT( I-) * X DISK0588 10 FACT( I) = FACT( -1) * Y _ DISK0589 RE TURN DI SK0590 END __ __ ____ _DISK 0591 CD CD0

ROUTINE: SML USE: CALL SML(M, L, X, S, TERM) where M is m+ 1, L is + 1, X is a double precision value in degrees, S is the returned value of S m(-ic, a7), where r] = cos X, TERM is the returned value of the last term in the truncated infinite / series for S (-ic, ri) \ COMMENTS: These quantities are described in equation (2. 3).

SOBROLJ T I NE SML ( M, L, X, S,T ERM) DISK0592 C DI SK0593 COMMON /ARRAYN/ NMAX ______ DISK059' COMMON /ARRAYD/ D DISK0595 DIMENSION D(9,9,12)vNMAXA9,91j _____ _DL&K05-96f IV = M 1 DISK0597 Ni 1 + MOD(L-M,2) DISKO598 SUM = 0. DISK0599 I ___ DISK06 ~ LIM = NMAX(ML) DISK0601 DO 10 N = Ni, LIM, 2 DISK0602 =1+ 1 D1DSK0603 CALL ALGNDR(N+M1,MqXP) DISK0604 A = D(MLI) * p DI SK0605 10 SUM = SUM + A DISK0606 S = SUM D1SK0607 TERM = A S____ D15K0608 RETURN DISK0609 END DISK0610 I'

TABLE D. 1: Computed Values of the N,; c= 4.0. m 1 m =m+l 1 m+2 =m+3 I=-m+4 I =m+5 I=m+6 1Im+7 =m+8 m 0 3.01944 -01 2.86300 -01 5.35684 -01 2.78851 -01 2.28613 -01 1.83212 -01 1.54446 -01 1.33615 -01 1.17794 -01 0 1 1.35184 -01 7.93609 -01 2.31898+00 3.17825+00 4.32010+00 5.43325+00 6.53625+00 7.62515+00 1 2 1.31711+00 1.13096+01 4.16236+01 9.55735+01 1.82917+02 3.09306+02 4.80918+02 2 3 4.02492+01 3.99638+02 1.82122+03 5.55766+03 1.35837+04 2.86467+04 3 4 2.29307+03 2.58174+04 1.40912+05 5.25775+05 1.55329+06 4 _ 5 2.05663+05 2.61883+06 1.66664+07 7.30644+07 5 M 6 2.66495+07 3.81786+08 2.77795+09 6 7 4.71717+09 7.54895+10 7 8 1.09507 +12 8

TABLE D. 2: Computed Values of the J.; c = 4.0. When (U - m) is Odd Jm is Identically Equal to Zero. m Re Im 1Re Im 2 Re Im 3Re Im 4Re Im 5Re Im 6 Re Im 7Re Im Re Im Im!Q=m \ 9.99211 -01 2.80788 -02 9.36914 -01 2.43120 -01 5. 56934 -01 4.96748 -01 1.47108 -01 3. 54213 -01 1.57472 -02 1.24496 -01 6.47630 -04 2.54403 -02 1.08692 -05 3.29682 -03 8.71379 -08 2.95191 -04 3. 82533 -10 1.95585 -05 ' =m+ 2\ 3.42953 -01 4.74696 -01 2.02982 -02 1.41018 -01 2.96440 -04 1.72149 -02 2.16126 -06 1.47012 -03 9.88357 -09 9.94162 -05 2.96546 -11 5.44561 -06 5.97020 -14 2.44340 -07 II =m+4 8.77859 -05 9. 36900 -03 2.03551 -07 4.51166 -04 3.48214 -10 1.86605 -05 4.50728 -13 6.71363 -07 4.40904 -16 2.09977 -08 'L =m+6 1.23782 -10 1.11258 -05 6.67737 -14 2.58406 -07 3.08583 -17 5. 55503 -09 I g=m+8 '\ 1.31852 -17 3.63114 -09 in Re Im Re Im Re Im Re Im Re4 Im Re Im Re Im Re Im Re Im 7 8

TABLE D. 3: Computed Values of the J'; Equal to Zero. c= 4.0. When ( -m) is Even J' is Identically ml m 0Re Im 1Re Im 2Re Im 3Re Im 4Re Im 5Re Im 6Re Im 7Re Im Re Im /i=m+l 9.99117 -01 -2.97216 -02 8.60676 -01 -3.46285 -01 8.90760 -02 -2.84854 -01 1. 71770 -03 -4. 14096 -02 2.17955 -05 -4.66851 -03 1.69702 -07 -4.11950 -04 7.97290 -10 -2. 82363 -05 2.33184 -12 -1. 52704 -06 l =m+3 2. 34236 -02 -1. 51244 -01 1. 23395 -04 -1. 11076 -02 4.82366 -07 -6. 94526 -04 1. 33709 -09 -3. 65663 -05 2. 62702 -12 -1. 62081 -06 3. 68423 -15 -6. 06979 -08 -=m+5 1. 54044 -07 -3.92484 -04 1.59806 -10 -1. 26414 -05 1. 35059 -13 -3. 67504 -07 9.14819 -17 -9. 56462 -09:I=m+7 5.29698 -14 -2. 30152 -07 1. 62796 -17 -4.03480 -09 ~ m+9 m Re Im Re Im Re Im Re Im Re Im Re Im Re Im Re Im Re Im 0 1 2 3 4 5 6 7 8 PP.

(1) (1) TABLE D. 4: Computed Values of the K; c = 4.0. When (. - m) is Odd K is Identically Equal to Zero. m I_ _ am 0 1 2 3 4 5 6 7 8 2.67504 -01 4.49342 -01 2.24319 +00 2.17631 +01 3.58824 +02 9.09732 +03 3.29894 +05 1.62300 +07 1.04112 +09 -1.87742 +00 -1.09092 +01 -1. 59099 +02 -4.01265 +03 -1.47948 +05 -7.40904 +06 -4.82454 +08 I =m+4 1.64856 +01 3.63493 +02 1.26205 +04 6.14827 +05 3.95432 +07 m -4. 78416 +02 -2. 19328 +04 -1. 36436 +06 2. 64819 +04 0 1 2 3 4 5 6 7 8 en c-I CO

(1) (1) TABLE D. 5: Computed Values of the K '; c = 4.0. When (- - m) is Even K ' is Identically Equal to Zero. m 0 1 2 3 4 5 6 7 8 2.75232 -01 1.61564 +00 1.76146 +01 3.11178 +02 8.22116 +03 3.05946 +05 1.53127 +07 9.94077 +08 -4.05493 +00 -5. 65928 +01 -1. 36485 +03 -4. 92092 +04 -2.44114 +06 -1. 58419 +08 I=m+5 \ 8.05489 +01 2.65734 +03 1.26106 +05 7.99710 +06 i. =m+7 -3. 32631 +03 -2.01991 +05 I/ -=m+9 m 0 1 2 3 4 5 6 7 8 IC)

TABLE D. 6: Computed Values of the C2 for 0<,m<8; c = 4.0. 2k 0 2 4 6 8 10 12 14 16 18 m=0,: =0 1.00000 +00 -1.71230 +00 1. 09088 +00 -3. 62524 -01 7. 36102 -02 -1.00450 -02 9.81254 -04 -7.07229 -05 3. 33636 -06 m=O,. = 1.00000 +00 -1.24357'+00 6.09385 -01 -1.62934 -01 2. 75326 -02 -3.20476 -03 2.68510 -04 -1.45616 -05 m=0, l =2l 1.00000 +00 -4.05535 +00 3. 59071 +00 -1.42549 +00 3.23501 -01 -4. 76286 -02 4.91521 -03 -3. 68945 -04 1. 78998 -05 m=O, x=3 *.9 1.00000 +00 -4.58477 +00 3. 38955 +00 -1.12806 +00 2.18829 -01 -2.80492 -02 2.56038 -03 -1.72903 -04 7. 77808 -06 m=O, 11=4: 1.00000 +00 -7.09304 +00 1.09179 +01 -5.69152 +00 1.51755 +00 -2.48568 -01 2. 77103 -02 -2. 24284 -03 1. 35936 -04 -5.49739 -06 m=0, =5 1.00000 +00 -9.05448 +00 1.58369 +01 -7. 63939 +00 1.83664 +00 -2.71226 -01 2.73837 -02 -1.99184 -03 9.43467 -05 2k 0 2 4 6 8 -., -zI 10 12 14 16 18

TABLE D. 6 continued 2k m=O, =6 0 2 4 6 8 10 12 14 16 18 20 1.00000 +00 -1.25381 +01 3. 55994 +01 -3. 53896 +01 1. 34079 +01 -2. 73303 +00 i 3.53856 -01 -3. 19344 -02 2.09928 -03 -8. 98435 -05 m=0, =7 1.00000 +00 -1.55281 +01 5.15756 +01 -5. 57729 +01 1.99636 +01 -3. 75558 +00 4.47602 -01 -3. 72988 -02 2.28105 -03 -9. 26557 -05 1.00000 +00 -2.00216 +01 9. 37084 +01 -1. 65304 +02 1.21800 +02 -3. 62982 +01 5.99303 +00 -6.41861 -01 4. 87320 -02 -2. 73616 -03 1.01829 -04 1.00000 +00 -1.38635 +00 7. 29676 +00 -2.05070 -01 3.59428 -02 -4. 30613 -03 3. 74505 -04 -2.43340 -05 1.05028 -06 m=l, 1=2 3.\00000 +00 -3.04880 +00 1. 25433 +00 -2.87974 -01 4.25530 -02 -4.40015 -03 3.36444 -04 -1.95707 -05 7. 79559 -07 m=1l, =3 6.00000 +00 -1.45544 +01 9.70406 +00 -3.11402 +00 5.94224 -01 -7. 55256 -02 6.85953 -03 -4.60427 -04 2.03285 -05 2k 0 2 4 6 8 10 12 14 16 18 20

TABLE D. 6 continued 2k 0 2 4 6 8 10 12 14 16 18 m=l, 1=4 1.00000 +01 -2.82859 +01 1. 68363 +01 -4. 73857 +00 7.98281 -01 -9.04938 -02 7.40692 -03 -4. 53708 -04 1.87494 -05 m=l, -=5 1.50000 +01 -6.83366 +01 8.53019 +01 -3.77888 +01 8.76867 +00 -1.27136 +00 1.27120 -01 -9.32721 -03 5.17384 -04 -1.93699 -05 m=lot f=6 2.10000 +01 -1. 16343 +02 1.60985 +02 -6. 66883 +01 1.41252 +01 -1. 86682 +00 1. 70659 -01 -1.13503 -02 4.96890 -04 m=l, =7 2.80000 +01 -2.17850 +02 4.92877 +02 -4.24231 +02 1.42286 +02 -2.60232 +01 3.05508 +00 -2.52183 -01 1.52846 -02 -6.08990 -04 mxl 1=8 3.60000 +01 -3.33854 +02 8.61278 +02 -7.96539 +02 2.53801 +02 -4.31565 +01 4. 69673 +00 -3. 60232 -01 2.04182 -02 -7.74821 -04 m=2 1=2 3.00000 +00 -3.25690 +00 1.40103 +00 -3.32090 -01 5.02592 -02 -5.29413 -03 4.10690 -04 -2.41062 -05 9.54314 -07 2k 0 2 4 6 8 10 12 14 16 18 c.

TABLE D. 6 continued 2k 0 2 4 6 8 10 12 14 16 m-2, =3 1.50000 +01 -1. 25576 +01 4. 37868 +00 -8.70801 -01 1.13381 -01 -1.04721 -02 7. 23226 -04 -3.83750 -05 1.41088 -06 m,2,. =4 4.50000 +01 -8.87741 +01 4.93351 +01 -1.35373 +01 2.25384 +00 -2.53927 -01 2.07066 -02 -1.26225 -03 5.13217 -05 m=2, x 5 1.05000 +02 -2.37376 +02 1.20217 +02 -2.94968 +01 4.40556 +00 -4.48538 -01 3.33182 -02 -1.86934 -03 7.15361 -05 m=2, I =6 2.10000 +02 -7.84548 +02 8.60895 +02 -3.35926 +02 6.94083 +01 -9.05733 +00 8.22766 -01 -5.52937 -02 2.83096 -03 m=2 x =7 m2, - g7 3. 78000 +02 -1. 65802 +03 1.98358 +03 -7. 27629 +02 1. 38348 +02 -1. 65881 +01 1.38907 +00 -8.64613 -02 4.12951 -03 m —2 =8 6.30000 +02 -3.91131 +03 7.70994 +03 -5.99037 +03 1.81546 +03 -3.02084 +02 3.24980 +01 -2.47456 +00 1.39257 -01 2k 0 2 4 6 8 10 12 l14\ 16 cn 0 18 -9.88037 -05 -1. 36890 -04 -5. 19657 -03 18

TABLE D. 6 continued 2k 0 2 4 6 8 10 12 14 16 m=3,; l 3 1.50000 +01 -1. 30307 +01 4. 66416 +00 -9.45793 -01 1.25005 -01 -1. 16838 -02 8.14537 -04 -4.34587 -05 1.58623 -06 m=3, =4 1.05000 +02 -7.38281 +01 2.21587+01 -3.86504 +00 4.47923 -01 -3.72622 -02 2.34055 -03 -1.13952 -04 3.88676 -06 m=3, =5 4. 20000 +02 -7.43359 +02 3. 59758 +02 -8. 68360 +01 1.28748 +01 -1. 30608 +00 9.68166 -02 -5.41369 -03 2.04320 -04 m=3, J=6 1.26000 +03 -2.49614 +03 1.10913 +03 -2.41940 +02 3.25024 +01 -3.00571 +00 2.04486 -01 -1.05881 -02 3.77607 -04 m=3, =7 3.15000 +03 -1.04929 +04 1.05193 +04 -3.69014 +03 6. 88871 +02 -8.18291 +01 6.81,560 +00 -4.22739 -01 2.01050 -02 -6. 57678 -04 mx3, Qg8 6.93000 +03 -2.64307 +04 2.85573 +04 -9.45236 +03 1.63408 +03 -1. 79496 +02 1.38630 +01 -8.00562 -01 3.56744 -02 2k 0 2 4 6 18 10 12 14 16 I1. 18 -1.11166 -03 18

TABLE D.6 continued 2k 0 2 4 6 8 10 12 14 16 18 m=4, -=4 1.05000 +02 -7. 54405 +01 2. 30075 +01 -4.06271 +00 4. 75435 -01 -3.98638 -02 2. 51984 -03 -1.23097 -04 4.16688 -06 m=4, =5 9.45000 +02 -5.69636 +02 1.49518 +02 -2.31626 +01 2.41355 +00 -1.82213 -01 1.03527 -02 -4.01232 -04 m=4, — =6 4. 72500 +03 -7. 78369 +03 3. 35014 +03 -7. 22717 +02 9.66326 +01 -8.91706 +00 6.05958 -01 -3.12994 -02 1. 10284 -03 m=4, =7 1.73250 +04 -3.13929 +04 1.24721 +04 -2.45176 +03 2.99425 +02 -2.53696 +01 1.59218 +00 -7.65497 -02 2.55736 -03 m=4, s-8 5.19750 +04 -1. 60504 +05 1.50282 +05 -4. 80370 +04 8.18784 +03 -8.93013 +02 6.86917 +01 -3.95639 +00 1. 75704 -01 -5.41116 -03 m=5, =5 9.45000 +02 -5. 77352 +02 1.53116 +02 -2.39146 +01 2. 50855 +00 -1.90571 -01 1.10122 -02 -4.95899 -04 1.56525 -05 2k 0 2 4 6 8 10 12 14 16 18

TABLE D. 6 continued 2k m=5, =6 0 2 4 6 8 10 12 14 1.03950 +04 -5.46951 +03 1. 27328 +03 -1.77169 +02 1.67528 +01 -1.15762 +00 6.06797 -02 -2. 19290 -03 m=5, =7 6.23700 +04 -9.74069 +04 3.77923 +04 -7.36992 +03 8.97122 +02 -7.59066 +01 4.76071 +00 -2.28451 -01 m-5, -=8 2. 70270 +05 -4. 58623 +05 1. 65061 +05 -2.95423 +04 3.30751 +03 -2. 58567 +02 1. 50595 +01 -6. 75736 -01 m=6, 1=6 1.03950 +04 -5.51829 +03 1.29368 +03 -1.81034 +02 1.71994 +01 -1.19401 +00 6. 35247 -02 -2.65276 -03 m=6, =7 1.35135 +05 -6. 30032 +04 1.31642 +04 -1.66128 +03 1.43703 +02 -9.15021 +00 4.45032 -01 -1.50679 -02 m=6, =8 9.45945 +05 -1.41645 +06 5.00458 +05 -8.90266 +04 9.94320 +03 -7.76576 +02 4.52105 +01 -2.02547 +00 2k 0 2 4 6 8 10 12 14 en GO wA 16 7.55073 -03 2. 12398 -02 7.84514 -05 6.30543 -02 16

TABLE D.6 continued 2k 0 2 4 6 8 10 12 14 m=7, g=7 1. 35135 +05 -6. 33916 +04 1.33114 +04 -1.68678 +03 1.46411 +02 -9. 34866 +00 4. 55080 -01 -1.52873 -02 m=7, -=8 2.02702 +06 -8.47796 +05 1.60596 +05 1-1.85338 +04 1.47683 +03 -8.71681 +01 3.95357 +00 -1.25937 -01 m1=8, I x8 2.02702 +06 -8. 51555 +05 1. 61899 +05 -1.87415 +04 1.49728 +03 -8.85625 +01 4.01891 +00 -1. 27080 -01 2k 0 2 4 6 8 10 12 14 U.