- THE UNIVERSITY OF MICHIGAN 4405-1-F 4405-1-F = RL-2113 AFCRL-62-108 STUDIES IN NON-LINEAR MODELING-IV: FAR FIELD SCATTERING BY SIMPLE SHAPES AT LOW FREQUENCIES by R. E. Kleinman, R. D. Low and F. B. Sleator February 1962 Project 4600 Task 46017 Final Report Contract AF-19( 604) -8030 Report No. 4405-1-F * Prepared For Electronics Research Directorate Air Force Cambridge Research Laboratories Office of Aerospace Research L. G. Hanscom Field Bedford, Massachusetts i

- THE UNIVERSITY OF MICHIGAN 4405-1-F Requests for additional copies by Agencies of the Department of Defense, their contractors, and other Government agencies should be directed to: ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA Department of Defense contractors must be established for ASTIA services or have their "need-to-know" certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to: U. S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 25, D. C. ii -- - -- -

THE UNIVERSITY OF MICHIGAN 4405-1-F TABLE OF CONTENTS I Introduction 1 II A Boundary Value Problem Relating the Sphere and the Prolate Spheroid 4 III Non-linear Modeling of Scalar Scattering by a Prolate Spheroid 8 IV An Alternate Modeling Procedure 12 V A Numerical Example 16 Appendix A: An Identity Involving Spheroidal Function 21 Appendix B: Some Comments on the "Far Field" 26 Appendix C: Modeling in Quantum Theory 31 Reference s 34 iii

-- THE UNIVERSITY OF MICHIGAN 4405-1-F I INTRODUCTION This is the final report on Contract AF-19(604)-8030 which was concerned with the theoretical investigation of electromagnetic scattering problems. The objectives of the contract were two-fold. One goal was to obtain the solution for scattering by a body formed by the union or intersection of shapes for each of which solutions were known. This solution was to avoid the problem of inverting infinite matrices. The second goal was the extension of the method of non-linear modeling to a particular and practical problem, namely, modeling the solution for scattering from a prolate spheroid by that for scattering from a sphere. This work is an outgrowth of previous investigations carried out by the Radiation Laboratory supported largely by the Air Force Cambridge Research Laboratories under Contract AF-19(602)-4993. The effort on the problem of scattering by complicated shapes is part of a long range program, the ultimate goal of which is the theoretical determination of the electromagnetic (and acoustic) scattering properties of an arbitrarily shaped target to any desired degree of accuracy. Though this goal may never be fully attained, considerable activity, both in the Radiation Laboratory and in the field in general, has produced a variety of approximate techniques as well as exact solutions, thus considerably enlarging the class of shapes whose scattering properties can be said to be known. Under the contract of which this is the final report, one concern has been refining techniques applicable for low frequencies (wavelength large with respect to scatterer) to extend their range of validity. This effort has met with considerable success. Darling (1960) developed a static approximation which employs probablistic techniques for solving Laplace's equation with Dirichlet boundary conditions on a I I 1

THE UNIVERSITY OF MICHIGAN 4405-1-F surface which is the intersection of two (or more) surfaces for which both the Dirichlet and Neumann solutions are known. Under the present contract these results have been extended to the scalar wave equation. In addition the Neumann boundary condition is treated. These results together with extensions of the low frequency expansions developed by Senior (1961) form the subject of an invited paper at the URSI Symposium on Electromagnetic Theory and Antennas, June 1962, and will appear in the proceedings of I iat meeting. In addition the work on Rayleigh scattering from complex targets described by Siegel (1958,1959) has been extended to include higher order terms. These low frequency approximations can be used to penetrate the resonance region, which for the present purpose might be defined as that middle range of frequency where both low and high frequency approximations are no longer valid. This region of theoretical inadequacy is narrowed by extending the range of validity of available approximations. In some cases, the resonance region can be eliminated in the sense that the ranges of validity of the low and high frequency techniques adjoin or overlap. The results of these investigations are reported in the literature; Siegel (1962), and Senior et.al.(1962), and it would appear redundant to further dwell on them here. Non-linear modeling as conceived by Ritt (1956) and developed by Belyea et al (1959), (1960), and Chen et al (1961) can be characterized as an attempt to transform by analytic methods a physical problem for which neither experimental nor direct theoretical techniques is feasible into another problem more amenable to either experiment or theory, or both. Under the present contract, two facets of the problem have received attention. One effort consists of a systematic investigation of the consequences of a basic assumption of non-linear modeling, i. e. there exists a functional relation (modeling function) of the form fi=fl(f2) when f1 and f2 are a 2

THE UNIVERSITY OF MICHIGAN 4405-1-F solutions of the same partial differential equation for different parameter values. That is, if Dk is the differential operator (in n dimensions) containing a parameter k, and Dko fi.(xl, x2,..xn) = 0 i = I or 2 does the assumption that f1 be a function of f2 constitute a sensible restriction and if so what can be said about the functional form? These questions have been thoroughly investigated for particular elliptic,parabolic and hyperbolic operators. The results are highly dependent on the choice of operator. In the case of the hyperbolic operator chosen, the time dependent wave equation, the assumption essentially did not constitute a restriction. Whereas,in the case of the Helmholtz equation (the elliptic operator chosern, not only is the restriction severe but explicit power series representations of the modeling functions can be found. This work is the subject of a separate technical report, Ruehr (1962). In view of the considerable progress in these general theoretical treatments, it was hoped that the techniques of non-linear modeling had developed sufficiently so as to be applicable to the solution of specific practical problems. To this end considerable effort was devoted to the modeling of the field scattered by a prolate spheroid with that scattered by a sphere. Although the prolate spheroid is a shape for which the theoretical treatment is known, numerical calculations are quite difficult and accurate experiments are quite expensive. The hope was to transform the problem to one involving a spherical scatterer which is much more amenable to theoretical treatment. Sections II, III and IV present the results of three different methods of attack. In Section II a precise formulation of the problem is given. With the help of a representation of spheroidal functions in terms of spherical functions presented in Appendix A, the problem is solved but unfortunately in an inappropriate form. 3

THE UNIVERSITY OF MICHIGAN 4405-1 -F In Section III, it is shown that non-linear modeling can be achieved after relaxing one of the requirements,and the modeling function is explicitly derived. By making a different assumption about the functional relationship between the sphere and spheroid fields in Section IV, a different modeling function is derived which is in fact linear. Since the coefficients in any expansion in spherical harmonics of the far zone term of any field are precisely the coefficients in a multipole expansion of that field, a result derived explicitly in Appendix B, this linear modeling function is shown to be an extrapolation from that case when two terms of a multipole expansion suffice to describe the spheroid field. A numerical example is presented in Section V which compares the spheroid field obtained from each of the two modeling functions with the exact result. The comparison does not provide a criterion for preferring either modeling function and suggests that their range of applicability is severely limited. Thus the promise of the more general results, another example of which (non-linear modeling of quantum mechanical systems,written by J.E.Belyea) appears in Appendix C, has not yet been completely fulfilled in the sphere-spheroid modeling problem. II A BOUNDARY VALUE PROBLEM RELATING THE SPHERE AND THE PROLATE SPHEROID In most previous work on non-linear modeling, problems were chosen to illustrate the technique rather than out of inherent interest. Here we attempt to go to the next stage of useful rather than illustrative application. The particular problem we shall consider is that of finding the field scattered by a rigid (Neumann boundary condition) prolate spheroid when a plane wave is incident along the axis of symmetry. In an attempt to avoid the use of series of 4

THE UNIVERSITY OF MICHIGAN 4405-1-F products of prolate-spheroidal wave functions, we shall try to solve an problem exterior to a sphere. Specifically, let S be a prolate spheroid of semi-major axis the polar axis of a spherical coordinate system (p, 0, 0) (Figure 2.1). equivalent a, lying in Let5 be ikpcos 0 0. =e inc - z ~cccj. FIGURE 2.1 ikpcose the sphere p = a, and let the plane wave e be incident on S. Let 0(p, 0) be the solution of the following boundary value problem: 20+k20 = 0 an ' exterior to S, on S, (2-1) (2-2) (2-3) p( sp -ik 0O) - 0, uniformly in 0, p sp-.-oo ikpcosO where 0s = 0 -e. It is then required to find a function h(O) such that if 0 satisfies (2-1), (2-2), and (2-3) above, then ap+h 0, on,. (2-4) 5 - -- - -. __I_;____

THE UNIVERSITY OF MICHIGAN 4405-1-F It was hoped that h(O) could be determined from (2-1) - (2-4) above without explicit knowledge of 0. That such an h(O) exists and is unique follows from the fact that (2-1) - (2-3) define 0(p, 0) uniquely everywhere exterior to S. So that one has only to compute the ratio -( 3p )p, the result being the desired h(0) as can be seen 0 ap p=a from (2-4). By defining a function V(p, 0) to be (p, 0)= -rn0(p, 0),(2-5) then one has 0p(a, 0) b (a, 0)= - () = h(). (2-6) Thus if we can formulate and solve the boundary problem for /(p, 0), then (2-6) gives the desired h(0). From (2-5), we see that 0 = e- and then (2-1) implies that ~ be solution of 2 V7 ~ - ( )2 2 (2-7) where (V7 Q) means V7 * V '. The boundary conditions on ~ follow from those on 0, and we have = 0, on S (2-8) an and -- -ikpcos, uniformly in 0 (2-9) p*oo Unfortunately the only method that was found to solve this transformed problem depended on the use of spheroidal functions and represented no improvement over directly computing h(0) from (2-6) as a ratio of two infinite series. This expression may be of interest in itself since, with the help of the expansion derived in Appendix A, it is given in terms of spherical rather than spheroidal variables. 6

THE UNIVERSITY OF MICHIGAN 4405-1-F Hence we include it here. In terms of the spheroidal coordinates z = c r, where 2c = interfocal distance, the solution of the hard spheroid problem is, e. g. Flammer (1957),where the notation is that of Morse and Feshbach (1953), 0 F =12.e o...n.o.S ( )he (). (2-10) L 1n=o -A he'() on on on on o Now, as derived in Appendix A, co, S (r7)he (i)= i d (o,n)P (cos0)h (kp) (2-11) n oon -m m m m=o,1 where the summation includes only odd values of m for n odd and even values of m for n even. Recognizing that ikcgrY= eikpcosO e =e and substituting (2-11) in (2-10) we obtain oo.n., oo, 0( ikpcos, 0)=e -2 o )on i m-nd (o,n)P (cosO)h (kp) - A h'( ) m m m n=o n on o m=o, 1 = OD LOm d (o,n)j e' o) = [(2m+l)i j (kp)-2i h (kp) / - h ~ P (cos). mom m n he' () m m=o I n=o, 1 on on o 7

THE UNIVERSITY OF MICHIGAN 4405-1-F Hence, if one computes 80/ap and forms the ratio(-0p/0)p=a, there results OD t. I \ * h(0)=-k m o --- —----— nzo-2 -- on on o ---- d (o, n)je ( ) E (2m+l)im ( h(ka)-2i h (ka)P (cos) m m A. he () m m=o n=o,1 on on o h(0)=-k j [md(o,n)j e() (2m+l)imjm(ka)-2imh (ka) A h. P (cos) m. A h e' Qn m m=o n=o, 1 on on o (2-12) The fact that we were unable to find h(O) without making use of the spheroid solution suggests that the problem as formulated in this section represents too severe a test of modeling techniques in their present state of development. By modifying our demands, we are somewhat more successful in finding modeling functions relating to the sphere and the spheroid. These efforts are discussed in the following sections. III NON-LINEAR MODELING OF SCALAR SCATTERING BY A PROLATE SPHEROID In this section we make use of the known far field behavior of any scattered field to explicitly derive a modeling function relating the sphere and spheroid fields. Let us denote by 0s the field scattered by a prolate spheroid when a plane wave is incident upon it along the axis of symmetry. Then the Helmholtz equation, radiation condition and boundary condition on the spheroid serve to uniquely determine 0s. Similarly let us denote by Os the field scattered by a sphere where we again defer specifying the boundary condition. Q ~~II I

THE UNIVERSITY OF MICHIGAN 4405-1-F Assuming that the wavelength of the incident field is the same in both cases, then, exterior to both the sphere and the spheroid, both 0s and /s satisfy the same differential equation - (r2 + 2 (sin0 )+ku= (3-1) Lr2 ar r2 r sin 9 O where there is no angular variation in planes perpendicular to the axis of symmetry. We now make the assumption that after specifying boundary conditions on the sphere and spheroid (not necessarily the same condition on both) which uniquely determine the fields, there exists a functional relation between the two scattered fields, i.e. 0 = 0 ( ). (3-2) ~s s ( s Denoting a0S/a C by 0' it is clear that a0 a as a s and 0 (3-3) ar s ar ao a o Making use of (3-3) in the differential equation (3-1) satisfied by 0 we have s 1 a s 1 a, s 2 r(r2 0 a )+ n -2 (sin00 a )+k2 0. (3-4) r ar s ar r2 sin0 a ( s a 0 s Upon further differentiation and rearrangement of terms, (3-4) becomes a( a 4 ab ao 0s Lr 2 J r2 r r + r2sinO a (sin ao +ks (3-5) 9

THE UNIVERSITY OF MICHIGAN 4405-1-F However, since ib is also a solution of (3-1) this becomes s 2 S 2 0s (3ar r ( ) * (Finding the coefficient of 0 as a function of V is the point of departure s s for the exact analysis of Ruehr (1962). However, since the behavior of V for S large values of r is known, we can approximate this coefficient as follows: ikr " '.s- - f(0) (3-7) r Hence to this order in r a8 = ik V ar s (3-8) and 1 s =, r 80 thus a 52 52 2 (+,( s)2 -k2 (3-9) ar r 80 s and (3-6) becomes 2 t 0 +( 0s - 0 = ~0 * (3-10) The general solution of (3-10) is written immediately as s =A + Bs (3-11) T m -iwt The time dependence e is suppressed. 10

THE UNIVERSITY OF MICHIGAN 4405-1-F where A and B are constants to be determined. Unless B = 0, this is indeed a non-linear modeling function relating the spheroid field 0 and the sphere field '. The question remains, exactly which sphere and spheroid fields are related, or to put it more explicitly, what boundary conditions on the sphere and spheroid validate the assumption that the scattered fields are functionally related? Certainly it is valid when we take the example of Section II, i. e. apply a Neumann condition on the spheroid and the appropriate impedance type condition on the sphere so that the fields are identical. Then, of course, the modeling function (3-11) reduces to its simplest form since A = 1 and B = 0, bearing in mind the results of the previous section. Now we wish to determine whether there are cases other than this one for which the modeling function (3-11) is applicable. Although a simple argument makes this appear unlikely the question remains open. This follows by substituting (3-7) and a corresponding far field expression for 0, ikr 0 = e g(O), (3-12) s r in (3-11) obtaining -2ikr g(0)=A f(0)+ B r2 e, (3-13) f(A) From (3-13) it is clear that B must be zero or a function of r,hence not constant. Itwouldbe erroneous however, to conclude that B must be zero and hence that (3-11) only leads to the case mentioned above. By making the far field approximation we have essentially said that ikr e = 0, >1 (3-14) thus if 2ikr B = B1 r2 11

THE UNIVERSITY OF MICHIGAN 4405-1 -F we not only satisfy the requirement that B be zero, to the order in h/r with which we are concerned, but also obtain the non-linear modeling function g(0)=A f(O)+ 1 (3-15) relating the far field pattern factors. From (3-15) it is clear that by measuring or computing two pairs of points in the polar diagrams of the sphere and spheroid far fields, the constants A and B can be determined and thus the entire spheroid polar diagram can be produced from the corresponding sphere data. Just when (3-15) is valid is of course a crucial question. In Section V a calculation is carried out where Dirichletboundary conditions are assumed on both sphere and spheroid and the spheroid pattern factor predicted by (3-15) for a particular wavelength, sphere, and spheroid is compared with the exact result. The agreement is quite good but this mightbe attributed to a fortuitous choice of parameters rather than the efficacy of (3-15). Since the lengthy computations involving spheroidal functions are precisely what we are striving to avoid, we still have the problem of determining under which, if any, boundary conditions other than those of the previous section the modeling function (3-15) provides significant results. IV AN ALTERNATE MODELING PROCEDURE The importance of knowing the far field behavior of scattered fields is clearly illustrated in Appendix B. There we find that a knowledge of the far field essentially determines the field everywhere. In the previous section we used the known functional form of the far field to derive a modeling function relating the sphere and spheroid fields. In this section we again make use of this knowledge to derive a modeling function. This time we do not assume the functional relation between 12

- THE UNIVERSITY OF MICHIGAN. 4405-1-F fields scattered by sphere and spheroid until after deleting the factor e /r. What results then is an assumption of functional dependence between the pattern factors. A modeling function relating pattern factors is derived which in fact is linear. Again denoting the sphere and spheroid fields by ib and 0 where both s s satisfy equation (3-1), we can write these fields as convergent series expansions (see Sommerfeld (1949) ) in inverse powers of r as follows eikr co f(0) r n() er n O= e n (4-1) n=o n=o r where fo(0) and g (0) are the pattern factors of Section III appearing here with subscripts. For convenience we denote the sums as follows fn() C g ()2 n(0 g n -no, n = 0 (4-2) r r n=o r n=o r and derive the differential equation satisfied by ~ and 0. Substituting ikr ikr 0 e 0 S r into the scalar Helmholtz equation (3-1), we find, since ikr ikr 1 (r e )= e ( + 2ik -k2 ), r2 ar r r r r ' that a2 2ik + a (sin0 = (4-3) a r2 3r r2sin00 0 a' Whereas in the previous section we assumed a functional relation between i and 0, now we assume that i and 0 are related. That is, assume ~S I 13 13 I i

THE UNIVERSITY OF MICHIGAN 4405-1-F 0 = 0(Q), and denote -P by 0' av hence ar ar a0 a0-, and 0 = " ( + ar2 (4-4) Substituting (4-4) in (4-3) we obtain 0" (a2 1 ( ) + 0' ar2 arrS 1 a a~ + 2ik- -+ rn (sin o ar r2sine ao ao = 0. (4-5) However, since i satisfies the same differential equation as does 0, namely (4-3), the coefficient of 0' vanishes and we have simply (a / 2 +1 a 2 ~,, (~.O ) (4-6) and 0" or its coefficient must be zero. From (4-2), however, we find that 0D )+ ( 2 n= o n=o 1 rn o m(n-m)fm(0)fnm(0)+ f (0) fn m(0) m=o (4-7) In par and if this factor is to vanish the coefficient of 1/rn must vanish for all n. ticular, for n=o, we see that f(0) = 0 or fo(0) is constant. But, as shown in Appendix B (equation (B-4) with no 0-dependence), nl 2ik()1 fn (0) =2ik(n+l) asin - aO in 0 sin0 ao80 n (0) +n(n+l)f (0) 30 n (4-8) 14 --

THE UNIVERSITY OF MICHIGAN 4405-1-F hence if f (0) is constant, f (0)= 0 for n > 0 o n We have thus established that if ad 21 2 ar +2 ()7 2 = 0, then / is constant. On the other hand (4-6) can be satisfied if 0" = 0 which implies the linear relation 0= A +B, (4-9) where A and B are constants. In the far zone this clearly becomes (compare equation (3-15) ), g () = Af (0) + B. (4-10) o o spheroid pattern factors is a severthe assumed fustriction and can be metion byrequiring either that the sphere fieldbe constant or that (4-10) be valid. Both choices can be shown to be very unlikely (not to say impossible) unless the scatterers are very small by examining expansions of the far fields in series of Legendre polynomials. Thus by writing = co =o f ()= a P (cos0) and g (0) = b P (cos0) (4-11) o n n o - n n n=o n=o we see that if f (0) is constant then a =0, n > 0. Furthermore, if f and g can be o n o o approximated by the first two terms (rather than just one) of the series we have, since P =1 and P(x) = x, f = a P (cosO)+ a Pi(cos 0) 0 0o0 (4-12) =a + a1 cos 0 0 15

THE UNIVERSITY OF MICHIGAN 4405-1 -F and g9 b P (cos0)+ bi Pi(cos0) = b +bi cos. (4-13) We can immediately derive a linear relation between f and g by eliminating cos 0 in (4-12) and (4-13), obtaining b, b a -bl a g=-f+ 0 (4-14) o a, 0 a, Except for a different notation of the constants, this is precisely the same as (4-10). These results strongly suggest that the assumed functional relation between sphere and spheroid fields is valid only when the pattern factors can be approximated by the first (one or two) terms of a Legendre function expansion. This effectively limits the discussion to very low frequencies, the Rayleigh region, where other techniques are available (Siegel, 1958,1959). Although it may be possible that for some particular boundary conditions on sphere and spheroid, the linear relation (4-10) can be used outside the very long wavelength region, no method for determining these conditions has been found. In the next section a calculation is carried out for a particular case using (4-10) and the results are compared with the exact pattern as well as that predicted by the non-linear modeling function (3-15). V A NUMERICAL EXAMPLE In this section we carry out an explicit calculation of the pattern factor of a prolate spheroid with length-to-width ratio a = 10 for ka = 1 where a is the semib major axis. For this example, Dirichlet boundary conditions are assumed on both sphere and spheroid and the primary excitation is taken to be a plane wave incident along the C xis of symmetry. Both real and imaginary parts of the far field are I I 16

THE UNIVERSITY OF MICHIGAN - 4405-1-F calculated from the exact series as well as the modeling formulas developed in Sections III and IV. The constants in equations (3-15) and (4-10) are evaluated by using the exact results for forward and backscattering. Thus the pattern factor g(0) for the spheroid is calculated from the formulae: g(0) = A f(0) + f() (5-1) where A = g(0)f(0)-g(7r)f(7r) and B = f(0)f(r) f(0)g( 7r)g(0) = f2(0) - f( f2(O) -f2(r) g(0) = a f(0) + b (5-2) where g(0)-g(7r) f(0) g(7r) -f( 7r) g(0) f(0) -f(7r) ' f(0)-f(7r) and S (c, -1) R((c ) g(O)= 2i o con 0 S (c, cos 0) (5-3) zC N (c) (3)on n=o on R (c, ) ) on o where the notation and tables of Flammer (1957) are employed. Equation (5-3) was also used to calculate g(0) andg(7r) of (5-1) and (5-2). The sphere pattern f(0) was calculated from the formula CD j (ka) f(0) = i e (1) (2n+)- - P (cos 0), (5-4) (1) n P ( n=o h (ka) n using the tables of Gumprecht and Sliepcevich (1951). The results of these calculations are shown in Figures 5. 1 and 5. 2. It is seen that the linear relation (5-2) yields better results for the imaginary part while the non-linear relation (5-1) is 17

-.12 -. 14 ______ exact result (eqn (5-3) ) 1- - - - -non-linear modeling (eqn(5-1) ) - - linear modeling(eqn(5-2) ) -.18 --.20 \ -.22 X.2 8 \ -.243 \ \ -.32 - 2 6 a 34-.38 -.36 -. 3 4- I ( 36 0 20 40 60 80 100 120 140 160 180 0 (in degrees) FIGURE 5. 1: Real Part of the Pattern Factor for a Soft 10:1 Prolate Spheroid (ka=l) H z ci -.4 I C?_4 H 0) old ^4 0= C) 3 -4 r > ZI If __I

a I __________ exact result (eqn(5-3) - - - - - --— non -linear model ing (eqn(5 -1) -linear modeling (eqn(5-2) ) Hz MT CO) H 1 2 10 4.-Il -.% -._ - - - (0A. 08 - 01.06 - 0.04 -i.02 - C)~ 0 a a a i 0 20 40 60 80 i i i i i I Z 100 120 140 160 180 4 rrees). IP- z 0 (in deg FIGURE 5. 2: Imaginary Part of the Pattern Factor for a Soft 10:1 Prolate Spheroid (ka=l)

THE UNIVERSITY OF MICHIGAN 4405-1-F preferable when calculating the real part. Roughly speaking both (5-1) and (5-2) produce values of the pattern factor which approximate the exact values equally well and this approximation is reasonably good. The maximum deviation for the entire 0 -range of either real or imaginary parts is less than 25 percent of the exact values. It may be argued however, that the value of ka chosen (ka = 1) is too small for this agreement to be significant. Unfortunately a similar comparison for larger values of ka would involve a significant increase in computational difficulty. The fact that the non-linear modeling function does not appear significantly better than the linear one and that the linear modeling function can be reasonably limited to small ka values by the argument presented in Section IV make it appear doubtful that either modeling function has more than limited application. 20

THE UNIVERSITY OF MICHIGAN 4405-1-F APPENDIX A AN IDENTITY INVOLVING SPHEROIDAL FUNCTIONS A useful identity expressing products of spheroidal functions in terms of spherical functions of spherical arguments is derived here. Specifically, it is established that rn-n S (rO)he (m)=n i d (o,n)h (kp)P (cos0) (A-l) on on m m m m=o, 1 where the spheroidal function notation of Morse and Feshbach (1953) is employed (S is the angular function, he the radial function, g and T) are the radial and angular variables, respectively, dm(o,n) are expansion coefficients defined recursively in terms of the eigenvalues,hmis a spherical Hankel function of the first kind, P a Legendre polynomial, and p, 0, 0 are spherical coordinates). The relation (A-1) is given by Flammer (1953);however since his result apparently contains a misprint some details of the derivation are presented. We establish (A-l) by first considering the intermediate result 7r/2-i oo eikostPn (cos t)sint dt = iA (kp), (A-2) n n 0 To prove (A-2) let X- = cos t and then the left side, call it F(p), becomes 1 F(p)= [ ei P W(X dX. i OD 21

THE UNIVERSITY OF MICHIGAN 4405-1-F Now integrate by parts n times to obtain ikp n e,T (m) F(p) P (1) ikp n m=o Now kp n d n ( )m p(m) (n ) = m p (Y) n kp d n ' (nM) (n+m)! P (1 2m m(n-m)m n 2 m!(n-m)! so one has ikp n F(p) = e (n+m) ikp m(n-m)! 2kp m=o = in h (kp) n Thus (A-2) is established. We now show that if cosY = cos 0 cos 0 + s sin 0 sin cos 0, then I(p, 0) P (cosO)h (kp)= do0 e p c (cosO )sino d n d io n o o o (A-3) It can be shown that if 'Y = a+ if3 then the infinite integral in (A-3) exists if 0 < a < 7r -0, and 3 < 0. We give a formal derivation of (A-3) by showing that both sides are solutions of (V2 + k2)I(p, 0) = 0, that both sides agree for 0 = 0, and that the normal derivatives(at 0=0) of both sides are equal. That both sides agree for 0 = 0 is established by showing that 22

THE UNIVERSITY OF MICHIGAN 4405-1-F 2v w7r/2-i oo h (kp)= - n - d0 eikPcoo P (cosO )sinO dO n 27rin 0 n o o o O 0 and that this is valid follows from (A-2) upon performing the integration with respect to 0. The normal derivative of the left side is F [P(cosO)h(kp)l p ao Ln n and this is clearly zero at 0 = 0. The corresponding derivative of the right side becomes 7r/2 -i oO 2 7r ikn eikpcoso p (cosO )sin20 S cos 0 d0, 27rin I n o 0 0 0 0 which is obviously zero. Finally the operator 72 + k2 annihilates the left side of (A-3) and when applied to the right side it gives 2r 7r/2 -i co ik d0 (V2+k2)eikps P (cosO )sinO dO 27rin d Jo n o o o 0 0 Now it is easy to show that (V2+k2 )eikpcosY k2sin2e sin2 + ik sin eocos0 ikpcosY L o 0 p sin 0 thus 2 2wr 2l 2wr (2 ik p0 cos 22kpos ik sini o ikpcosYT (2 +k2) e P zd0 =k2sin0 sin20 e d0 + p- \ cos0 e d0 O 0 (A Psin 4) ' 0 0 (A-4) I 23

THE UNIVERSITY OF MICHIGAN 4405-1-F From the known result /.27 e1tc0 d0 =27r J (t), O O 0 one finds, without difficulty, that S27r sin20e itcos0 d0 = 2[J (t)+J"(t)1 o0 and [ 27 O cos0 itcos0o d0 = -27ri J' (t) 0 0 Thus, if one lets t = kp sin0 sin0, it follows from (A-4) and the definition of cos T that S27r (72+k2)e 0 ikpcosY i ikpcos0cos0O d0 =2rk2 e 0 sin20 [J (t)+1 J (t)+J (t 0. O O 0 Thus (A-3) is formally established. Next, as stated by Flammer (1957) ^2 7w /2 -i 00 S (rh)he (E)= in do eikpcosY on on 27rin 0 n and if we substitute the known expansion and if we substitute the known expansion S (cos0 )sin0 dO on o o o (A-5) 00, S (cose )= 2 on o -, m=o, 1 d (o,n)P (cos ) m m o 24

THE UNIVERSITY OF MICHIGAN 4405-1-F into (A-5) and utilize (A-3) there results 00 Son ( r)he on()= m=o, 1 *m-n i d (o,n)P (cos0)h (kp), m m m which is the expansion (A-l). The misprint mentioned above is the omission of the factor i in Flammer's result. 25

. THE UNIVERSITY OF MICHIGAN.. 4405-1-F APPENDIX B SOME COMMENTS ON THE "FAR FIELD" An item of much concern in three-dimensional scalar diffraction problems is the behavior of the scattered field in the far zone. This concept is more meaningful when the scatterer is finite in extent since then the scatterer can be considered as being concentrated at the center of a sphere for observation points exterior to the sphere provided its radius is sufficiently large. Much is known about the behavior of scattered fields in the far zone for a fairly general class of scatterers, at least for all finite convex bodies. For these scatterers in the far field, a diffracted field, 0s, can be represented mathematically in spherical coordinates as ikr-iwt s = e f(0, ). (B-l) s r Actually (B-l) results from neglecting all but the first term in an asymptotic expansion of 0. Nevertheless if f(O, 0), the pattern factor, is known in any particular case then the field 0, is determined everywhere, not only in the far zone, and an explicit representation can be given in terms of f(0, 0). The proof of this statement is implicit in all standard treatments of scalar wave functions, e. g. Sommerfeld (1949), Stratton (1941), but a direct statement of the fact is curiously missing. At the risk of belabouring the obvious, we present here an outline of the proof and derive the representation alluded to. What we shall do is essentially reverse the steps by which the far field is obtained from an expansion of any three-dimensional scattered field in spherical waves. We know (Sommerfeld, 1949)that in any scalar scattering problem (Wilcox, 1956, also treats the vector case ) the scattered field, 0, exterior to a sphere containing all scattering bodies (here assumed to be finite in extent) can be 26

THE UNIVERSITY OF MICHIGAN 4405-1-F written in the absolutely, uniformly convergent series ikr O f (0,0) e n 0 =- n (B-2) s r rn n=o -iw~t ikr The time dependence e is assumed so that e /r represents an outgoing spherical wave. Furthermore, since (V2+k2)0= 0 (B-3) s we have the following differential recursion relation between the f's: 1 1 a a 1 a2 f 0 (0,0)= sin0 )+ +n(n-) f(,0) n 2ikn sin0 O- a6 sinr aP0 n 1 (B-4) Using the abbreviation 1 a a 1 a2 L = - (sin0 a)+ - +m(m-l) (B-5) m sin0 aO ao sin20 aO we can iterate (B-4) to obtain f in terms of f, as n n o L f (O,0) f (0,0) = - (B-6) n (2ik)n nW With (B-6) and (B-2) we see that if we have an explicit analytic representation of the far zone field, f (0, 0), we can write down all the higher order terms in a convergent representation. Wilcox (1956) establishes the fact that (B-2) is absolutely and uniformly convergent. A spherical wave expansion of the scattered field can be constructed from f (0, 0) as follows. We expand the pattern factor in spherical harmonics, 27

I - THE UNIVERSITY OF MICHIGAN 4405-1-F 00 f (0, 0) =21 J=o s=-j A. Ps (cosO)ei is j (B-7) where 2j+1 A. =4 Js 47r (j-s) ( (j+s)! dO d fo(0, 0) P (cos9)eis0 sinO -To (B-8) Substituting (B-7) in (B-6) we find that n TT L m f (0,0) = mi (2ik)n n! O0 J=o S s= -j A. Ps (cosO)eiso js j (B-9) Since isO U i Lm P (cos0)e m(m-)-j(j+i)i P(cos0)e equation (B-9) can be rewritten as (B-10) n 0 (2ik)nn. 00=o j=o s=-j A is n s.is T [m(m-1)-j (j+1) PS(cos0)e m= (B-1l) However m(m-l) -j(j+i)=(m+j )(m-j -1) from which we find that 28

THE UNIVERSITY OF 4405-1-F MICHIGAN i(h [m-I)-j j+J = nI iT(m+j)(m-j-l) m1l I I m1l =(-1)n (j+n)! = (ji-n)! = (-j) (j+l) n n IV (B-12) hence (B-li) becomes f (O ) = 1 j =0 i A. (-j) (0+1) PsicsOe S=i s n n j (B-13) Since (-j)n = 0 if j < n (B-13) can also be written f (O 0)= 1 n (2ik)nn! ID j=n i F.5= -J A. (-j)n(j+l)nPj(cosO)e15 (B-14) Substituting (B-14) in (B-2) we obtain ikr e 0 = 5 r OD 21 n~o OD j n j 27A 5=- -j (-) (j+l) P.(Cos 0) n n i iso n! (2ikr) n ~ e (B-15) Rearranging the terms, (B-15) can also be written ikr 0= s r n~o 7 i=o = -n (-n).(n+l).P (cos 0) J j n - eiso A ns (B-1 6) j! (2ikr)3 But ikr e r i=o (-n).(n+l) r+ () I 4 =kin~ h~ (kr) j! (2ikr)j n. (B-17) 29

THE UNIVERSITY OF MICHIGAN 4405-1-F where h(1) is a spherical Hankel function, e. g. Magnus and Oberhettinger (1954). n Making use of(B-17) in (B-16) we have o0 n 0 kA in+l h() (kr)P (cos0)eiis (B-18) s ns n n n=o s=-n which is the usual expansion in spherical wave functions convergent for r > 0. The coefficients A are defined in terms of the far field in (B-8). ns We have thus explicitly demonstrated that it is always possible to recover the near zone from the far zone. This inverted scattering problem is not without interest since in many practical situations, it is possible to measure the far zone field even though the precise nature of the scatterer is not known. Theoretically it would be possible to then compute the coefficients A and even calculate level ns surfaces of 0 + incident' thus determining the configuration of the scatterer assuming Dirichlet boundary conditions. More complicated boundary conditions ar not so simply handled. Even if it is known that the scatterer (or scatterers) could be characterized by Dirichlet conditions there is a question as to the practicality of this scheme since accurate measurements of the far field are often not available over the full range of 0 and 0. Still this procedure could certainly be used as an approximation, by interpolating between measured values of the far field. 30

THE UNIVERSITY OF MICHIGAN 4405-1-F APPENDIX C MODELING IN QUANTUM THEORY Following the general treatment of Ritt (1956) we present another example of the existence of models of a given system which evolves in some variable t, according to v(Q,t)= exp A t vo(Q), (C-i) where A is an operator independent of t, vo(Q) is the given initial condition, and exp At is to be interpreted as v n n exp At = tA (C-2) n=o Q symbolizes the remaining variables required to describe the system. In the Heisenberg picture of time-dependent quantum theory, the wave function describing a quantum system evolves according to o(r, t)= exp ( "^ )O (). (C-3) Here, H is the Hamiltonian operator of the system, and the exponential operator is to be interpreted as in (C-2). 0 is the wave function describing the initial state of the system. Thus for quantum systems, the operator A of (C-1) is - iH/-fi. It is possible to say a great deal about this operator without specifying the system more completely than we have done already. Quantum systems are generally discussed (by physicists, at any rate) in the manifold of L composed of "finite" functions, i. e. 2 functions non-zero only in some finite domain. + In this manifold, H is always selfadjoint. As a result, the eigenvalues of H are all real, and its eigenvectors span This has the effect of limiting consideration to the point spectrum. I 31

THE UNIVERSITY OF MICHIGAN 4405-1-F the manifold. Furthermore, the physical interpretation of the spectrum of H is as the set of allowed values of the energy of the system. An essential point to be gleaned from the above is that the spectrum of A is purely imaginary. This introduces certain complications into the existence proof, as was noted by Ritt (1956). For, if we consider a model quantum system governed by a different Hamiltonian j%#, then a necessary and sufficient condition for the existence of a modeling function relating it to the original (H) system is that -iHt exp ( -i ) ( =) o() (C-4) implies exp ( 0 0 (r) (C-5) 0 0 The fact that the spectrum of H (andSF) is real guarantees that (C-4) has a nonzero solution a: the smallest number such that o En is an integer multiple of h for all n. Then a modeling function exists if a En is also an integer multiple of h for all n and the same o. Thus, whether or not a system may be used to model another depends on the allowed energy values of both. This seems very reasonable from a physical standpoint. Actually constructing a modeling function, once we know it exists, is in general a complicated problem. However, when both model and prototype are in stationary states of the energy, it becomes quite simple. For, in that case, -iHt -iErt i (r, t)=exp (-,-) = exp( ) while -~ ) 0(r,t)=exp( lt ) 0(r)=exp ( -1 ) 0o' The existence condition exp(-iE (/f) = exp(-id scr/) = 1 r s 32

THE UNIVERSITY OF MICHIGAN 4405-1-F is satisfied as long as Er/(s is the ratio of two integers. The modeling function relating the two systems is then clearly Er/&s [ r, t) = ]0 (C-6) Suppose that both model and prototype systems are in mixed states of energy. Let the representatives of bO in the H-representation be {an} and the representation of 00 in the p-representation be bn}. These are presumed to be given. Then w(ri t)= a exp ( -iE while n 0(r t) =bn exp (-i nt) vn. n Since {vn} is an orthonormal set, t= + v v() 0(r, t) dr (5 for all s. By substitution, then (C, t) = an (r) 0(r,t)df fn(r). (C-7) n n 33

THE UNIVERSITY OF MICHIGAN 4405-1-F REFERENCES Belyea, J. E., R. D. Low and K. M. Siegel,(1959), University of Michigan Radiation Laboratory Report 2871-4-T. Belyea, J.E., J.W.Crispin, Jr., R.D.Low, D.M. Raybin, R.K.Ritt, O.G.Ruehr and F.B. Sleator, (1960), University of Michigan Radiation Laboratory Report 2871-6-F. Darling, D. A., (1960), University of Michigan Radiation Laboratory Report 2871-5-T. Flammer, C., (1957), Spheroidal Wave Functions, Stanford University Press. Gumprecht, R.O. and C.M.Sliepcevich, (1951), Tables of Riccati Bessel Functions for Large Arguments and Orders, University of Michigan Engineering Research Institute. Magnus, W. and F.Oberhetting, (1954), Formulas and Theorems for the Functions of Mathematical Physics, Chelsea Press, New York, p. 22. Morse, P.M. and H.Feshbach, (1953), Methods of Theoretical Physics, McGraw-Hill New York, p. 1502. Ritt, R.K., (1956), Trans. IRE-PGAP, AP-4, 216-218. Ruehr,O.G., (1962), Non-linear Modeling Functions of a Special Type, University of Michigan Radiation Laboratory Report (to be published). Senior, T.B.A., (1961), University of Michigan Radiation Laboratory Report 3648-4-T. Senior, T.B.A., D.A.Darling and R.E.Hiatt, (1962), Low Frequency Expansion for Scattering by Separable and Non-separable Bodies. To be presented at Symposium on Electromagnetic Theory and Antennas, Technical University of Denmark, June 25-30, 1962. Siegel,K.M., (1958), Appl. Sci. Res., Sec.B, 7, 293-328. Siegel,K.M., R.F.Goodrich and V.H.Weston, (1959), Appl. Sci. Res., Sec. B, 8, 8-12. Siegel, K.M., (1962), The Quasi-static Radar Cross Sections of Complex Bodies of Revolution, Electromagnetic Waves, (ed. R.E.Langer), University of Wisconsin Press, pp.181-197. Stratton, J.A., (1941), Electromagnetic Theory, McGraw-Hill, New York. Wilcox, C.H., (1956), Comm. Pure and Appl.Math., IX, 115-134. - - -- mmw 34