a - THE AFCRL-65-639 UNIVERSITY OF MICHIGAN 7133-4-T LOW FREQUENCY SOLUTION OF THREE-DIMENSIONAL SCATTERING PROBLEMS by R.E. Kleinman August 1965 Scientific Report No. 2 Contract AF;:,9(6:28)-'43 2 r.oJec.t:5835 Task 563.502 AR FORCE CAMBRqIGE'RE1.EARCH LABORATORIES OFFICE OF AV,'fC. IESEARCH UNITED STATES " I'FORCE BEDFORD, MASSACHUSETTS ~ I I

THE UNIVERSITY OF MICHIGAN 7133-4-T TABLE OF CONTENTS ABSTRACT iv I. Introduction 1 II. Scalar Scattering 3 II. Stevenson's Method 12 IV. Electromagnetic Scattering - An Alternate Approach 24 V. An Example - Scattering by a Sphere 43 5.1 Zeroth Order Terms 43 5.2 First Order Terms 46 REFERENCES 50 iii - - ~ -~

THE UNIVERSITY OF MICHIGAN 7133-4-T ABSTRACT A deficiency is pointed out in Stevenson's method of reducing the solution of electromagnetic scattering problems to a succession of standard potential problems whose solutions determine terms in the low frequency expansion of the scattered field. An alternate approach is presented, for perfectly conducting scatterers, which not only removes the difficulty but also is simpler and more explicit than Stevenson's method. The details of the analogous, though simpler, scalar scattering problems are also presented. iv

THE UNIVERSITY OF MICHIGAN 7133-4-T I INTRODUCTION The purpose of this report is to describe a method of reducing scattering problems to a series of potential problems. We deal with a general class of three dimensional scatterers, smooth, closed, bounded, in short those surfaces for which Green's theorem in any of its guises may be invoked. The solution of a scattering problem, for arbitrary excitation, is expressed as a series in ascending powers of wave number, k. This series is known by a variety of names, including Rayleigh series, quasi-static series, and low frequency expansion. That the first term in such a series could be found as the solution of a potential problem was observed by Rayleigh (1897) who determined this term explicitly for a variety of scatterers of both acoustic and electromagnetic waves. For scalar scattering, the determination of succeeding terms in this series as solutions of potential problems has been described, in varying detail, by Noble (1962), Morse and Feshbach (1953), and Darling and Senior (1965). (See Kleinman (1965a) for a more complete bibliography.) The derivation of successive terms in this series for electromagnetic scattering was described by Stevenson (1953a). Actually Stevenson described two methods, one for finding the general term in the series and a second special technique for finding the first three terms. All of his specific calculations (Stevenson, 1953b) were carried out using this special technique. No attempt to utilize the general method for obtaining higher order terms has, to this writer's knowledge, been reported, which indicates that if attempts were made, they were unsuccessful. More likely, there were none. This is due to the fact that the analysis is sufficiently involved to discourage most efforts to derive more than three terms in a low frequency expansion (that Stevenson treats the more general case of penetrable scatterers certainly doesn't help). For these, Stevenson's special simpler technique suffices. An attempt to clarify the Stevenson method was made by Senior and Sleator (1964) and the present report may be considered an outgrowth of their work. [ 1

I THE UNIVERSITY OF MICHIGAN 7133-4-T The present work demonstrates that the method proposed by Stevenson for finding the general term in the series needs clarification at best and at worst leads to incorrect results. An alternate method, preserving the spirit of Stevenson's approach and indeed largely based on it, is presented which hopefully embodies both clarity and correctness. Conciseness has been sacrificed in an attempt to minimize the chances of further obscuring the subject. The procedure in the electromagnetic (vector) case is a natural extension of the technique employed in the scalar case. For this reason, and also to introduce some notation as well as concepts in the simplest setting, the next section is devoted to a discussion of how scalar scattering problems may be reduced to the study of a succession of potential problems. In Section 3 we describe Stevenson's method for treating the analogous vector problem and show why it is unsatisfactory. Section 4 presents an alternative to Stevenson's method which eliminates its shortcomings. Section 5 is devoted to an illustrative example. 31 0 1 0 2

THE UNIVERSITY OF MICHIGAN 7133-4-T II SCALAR SCATTERING In this section we show how a scalar scattering problem with Dirichlet or Neumann boundary conditions may be reduced to a succession of "standard" potential problems. These terms will be precisely defined as they are introduced. Let B denote the boundary of a smooth, closed, bounded surface in Euclidian 3-space (or the union of a finite number of such surfaces provided they are disjoint), let n denote the outward drawn unit normal at any point of B and let V be the volume exterior to B. Erect a cartesian coordinate system with origin in B an let r denote a radius vector to a general point (x, y, z) and rB denote a point on B. Furthermore denote by R the distance between r and rB, i.e., R = jr-r = V (X-XB) +(Y-YB) +(Z-zB). (2.1) The geometry is illustrated in Fig. 1. V! ~Figure 1~ Figure 1 I I 3 I

I THE UNIVERSITY OF MICHIGAN 7133-4-T By a scalar scattering problem for the surface B is meant the determination of how the presence of the surface perturbs an incident field, nc, that is, finding a function ( Or) such that (V+k2) i 0 reV, (2.2) l r - ik- 0 (2.3) and either P(rB) r= nc B) (2.4a) or aQ( r) a^inc(~r ) an-. = -' an.r. r=rB r=rB Equation (2.3) is a statement of Sommerfeld's radiation condition which imeikr plies that outgoing waves look like ei f(0, 0) for large r. The boundary condir tions (2.4a) and (2.4b) are Dirichlet and Neumann conditions respectively. Specifying either one is sufficient to guarantee the existence of a unique function ( hence both the values of the function and its normal derivative may not be assigned arbitrarily. We will consider the Dirichlet and Neumann problems separately but the analysis is quite similar. The starting point is the Helmholtz integral representation of regular solutions of (2.2); viz, 1 rikR ikR a (2. 5) The intgrBn ir an The integration is carried out over the surface and the normal derivative is. V. Next we assume that the unknown function ( may be expressed as a conver I l 4

THE UNIVERSITY OF MICHIGAN 7133-4-T gent power series in k. Actually this need not be assumed, that is, it may be proven that there does exist such an expansion, convergent for k sufficiently small (see Werner, 1962 and Kleinman, 1965b). It should be noted that we are considering k real and positive though the results may be extended to include complex val - ues of k. We write the expansion 00 0(r = I I m()(ik)1m (2.6) m=0 where the factor i is included in the expansion parameter merely as a convenience. The functions 6m are independent of k and each of them may be determined as follows. ikR Since e is an entire function, the series 00 ikR= E (ikR) (2.7) 1=0 converges for all k. Substituting (2.6) and (2.7) in (2. 5), we obtain OD OD OD r m(r)(ik)m 1 EdB m )(ik) n m=0 dB Lm=0 ~ =0 O~ (ik 1 D /_ ~ (m~ik~lR1 % ~ t..m3&S) 4n E ( E an m9, } _=0 i0- m=0on (- (2.8) As long as Ik [ is strictly less than the radius of convergence we may interchange summation and integration and reorder terms in the double series obtaining E'(6 {.im _i ~ "N-(ik) \ " JR-m-1 R-m-1 a - m= m 47-: m= -m) m an R -R (2.9) I 1 5 I - -I

THE UNIVERSITY OF MICHIGAN 7133-4-T Equating coefficients of like powers of k, with an obvious change in notation, yields r()= 4ir E (.- m)! { mm R m -R — l an mj dB, (2.10) 17' -i-R —)! m- an an 2 I = 0, 1,2,. In order to determine l we must employ the boundary conditions, hence we must distinguish between the two problems under consideration. Whether the incident field is a plane wave, point source, or linear combination of such sources it remains true that the representation of the incident field is analytic in k. Thus we may write oo in (rc) = in (r)(ik). (2.11) e =0 The boundary conditions (2.4a) and (2.4b) then imply that either S(rB) = rB- ( B(2. 12a) or Q — = a 8inc(.0r) (2. 12b) 8n I A.n 8n r=rB r=rB Consider first the Dirichlet problem, (2.12a). Inserting the boundary values in the integral representation (2.10) produces the system of equations' = — m aa - m 1 a' ] }dBa rIy) 4i- __ (2.-m)n + R n n mjB 13) = ~ - 1 SB ~-m-1 8(2.13) I, 6

THE UNIVERSITY OF MICHIGAN 7133-4-T We treat first the case when I = 0. Equation (2.13) becomes simply _ 1 Binc 1 I S 1 a0 -o(.) - - - dB 1 - dB (2.14) SB B The unknown term on the right is clearly an exterior potential function or in the language of potential theory (e.g. Kellog, 1929) the field of a single layer distribution of density ao/an. That is, if we designate by 00 the unknown function, -(r) = 4 25dB (2.15) r R n then 0 satisfies the equation V2(r) = 0, reV, (2.16) and 0 is regular at infinity in the sense of Kellog, viz. lim lim 2 a0o r r0 < o r o r r < oo. (2.17) r -eoo o r —-eoD or Furthermore, with the boundary condition (2. 12a) and the expression (2. 14), the values of 0 on B are specified, i.e.,._B - inc( lim 1 \ a 0o(rB) =f (.B)+ lim 41 S inc a (.)dB (2.18) 0 B 0 o(rB) r —rB 4 -0 an B ^B Note that the integration in (2.18) must be carried out before the limit is taken so the integrand is always defined. With this proviso the right hand side of (2.18) is well behaved and completely specified in terms of the incident field. Equations (2. 16)-(2. 18) constitute a standard exterior Dirichlet potential problem which has a unique solution. Next we show that succeeding terms ( may be written in terms of solutions of similar problems. To this end assume that o, y1'...- 1 are all I I 7

THE UNIVERSITY OF MICHIGAN 7133-4-T known. Then (2.13) may be written (r) = F,(r)+((r) (2.19) where 1 1 C Flr) 4 -= 0 ( -m)! inc a R-m-A dB Fr = 472-.7irm) m 0n dB -1 1 m -1B -m- m ndB 47r =(l- m)! an and 1 ak1 PI(,) 47T R an dB B With the assumption that all l's are know up to, but not including F, Fr ) is a known function. Clearly 0,O(r) is again a single layer distribution, satisfies (2.16) and (2.17), and is uniquely determined with the boundary condition rB)= (-Q gcrB )-F(rB) (2.20) Again, care must be used in letting r -b rB in one term of F but there is no intrinsic difficulty. Thus l is determined in terms of a known function FQ and a solution of a standard exterior Dirichlet potential problem, P0. We have shown that is true for I =0, and also for i >0 provided ( 1, - 2'.. o have previously been found. The solution of the Dirichlet scattering problem is then given by (2.6). An exactly analogous procedure may be followed for the Neumann boundary condition, (2.12b). Corresponding to (2.13) we have i 8 I

THE (r) = - 1 4m Ub 1I =0 Furthermore this may be qIVERSITY OF MICHIGAN 7133-4-T m) m c + a R } dB B written ( r) = Gf()+rQQf ) - -m) -m-1 a inc dB B A-1 1 \ 1 B r a R- m- dB, 4m=~O jmaRman -^^ (ir^ da, (2.21) (2.22) where IA m=C the second sum is identically zero if 0 = 0, and -. ) - 1 Ac(c) -47r aSB dB - an RR U GA is completely determined if -1' "..o are known and i (i) is a double layer distribution. That is, pb is the solution of a standard exterior Neumann potential problem, namely, v2() r o A reV Vi regular in the sense of Kellog, (2.17), and r=rB an an r=rB 8G - Ir an A r=rB 9

I THE UNIVERSITY OF MICHIGAN 7133-4-T We have thus demonstrated that for either Dirichlet or Neumann scattering problems successive terms in the low frequency expansion may be determined by solving a succession of standard potential problems. That is, the first term is the solution of such a potential problem the second term is expressed in terms of the first and the solution of a potential problem, the third is given in terms of the first two and a potential solution, etc. Before closing this section, a word should be said about low frequency expansions of the far field. The Rayleigh series, (2.6), may be considered as an expansion of the near field which, if all terms are included, is also valid in the far field. If only a finite number of terms are known, then the truncated series does not in itself give much useful information about the far field. Such information is available if we again make use of the integral representation (2.5). To this end note that for large r r* Xri ikR ikr-ik ikr-ikr. r e e e. ^ e e = e, r=l|r|l, r= r/r (2.23a) and....a ikR ikr- ikr r r ikR B Ve, ik - - r (2.23b) If (r, 0, ) and (r B 0B, ) are spherical coordinates of points r and tB respectively then r.rB = rB os0cosB sinsinB (0-B). (2.24) Substituting (2.23a) and (2.23b) in (2.5) we obtain, for large r, () i-d47rer Bani e - B d. (2.25) *r 47rr a n B ] 10 I,_ I I

THE UNIVERSITY OF MICHIGAN 7133-4-T A. -ikr rB Now if we substitute expansions of J and e in the right hand side of (2.25) and rearrange terms we obtain ikr ir0, - -.8 Tyv e (-1) A i-m / A A am \_. ^ pr)'= -rr' mik 0 ( -m) (rB) r 2m- - an dB 10- m-0 (2.26) where 0.1 0 Examination of equation (2.26) reveals that knowledge of a finite number of terms in the low frequency expansion of the near field (the's) provides similar information about the low frequency expansion of the far field coefficient, i. e. the coefficient of eikr/r. More specifically, in the Dirichlet case, when the boundary conditions specify ~m on B for all m, then knowledge of the first I I''s d/ai 1 c Q_\ -,... - will provide, with equation (2.26), the first I terms of a\n an an n the far field expansion. In the Neumann case, ( a- on B given for all m) the an first I m's will apparently give 1+ 1 terms in the far field. However, it may be shown that, whether inc is a plane wave or a point source, ainc 0 dB = 0 (2.27) an hence only S terms in the far field are specified. Or another way of saying this is that, in the Neumann problem, S near field terms produce ++ 1 far field terms but the first term, i.e., the coefficient of (ik), is always zero. 11

I THE UNIVERSITY OF MICHIGAN 7133-4-T III STEVENSON'S HOD STEVENSON'S METHOD In this section we shall describe Stevenson's attempt to generalize the approach of Section 2 to electromagnetic scattering and pay particular attention to the shortcomings, rather than the strong points (which are numerous) of Stevenson's work. To effect some simplification, we shall treat only the case of scattering by a perfectly conducting surface whereas Stevenson considered more general scatterers. It seems clear, however, that both the criticism in this section and the correction in the following section may be applied in the more general case. The surface geometry and notation are the same as introduced in Section 2 and depicted in Fig. 1, which is here reproduced for convenience. Iz r V y II Figure 1 x By an electromagnetic scattering problem for the perfectly conducting surface B is meant the problem of determining how the presence of the surface per- inc hinc turbs an incident electromagnetic field, (E, H ). That is, we seek a solution of Maxwell's equations VxE = ikH, VxH = -ikE, V-E=O, V H = 0 rEV (3.1) 12

THE UNIVERSITY OF MICHIGAN 7133-4-T subject to the boundary conditions AnxE n -n xE |. n H | -n- H |I (3.2) =-nxE, n.H1 k~r~ nr =rB r=rB r=rB B and the radiation condition lim Ek lira m rlm rx(VxE)+ikrE = rm rx(VxH)+ikrH =0 (3.3) r —OO r —um uniformly in i (The divergence conditions and boundary condition on H are redundant, i.e. may be deduced from the other conditions.) In attempting to show how to reduce this problem to that of solving a series of potential problems, the procedure parallels that followed in the scalar problem. Corresponding to the Helmholtz integral representation (2. 2) we employ the expression derived by Stratton and Chu (see Stratton, 1941) which expresses the field at any exterior point in terms of its values on the surface B. [Wilcox (1956) also derives these formulas but strangely omits any reference to the Stratton-Chu work] ikR ikR ikR 1 e^ikRA - ik e A A 1 eikR ^ E() = Vx - nxEdB+ nxHdB- EdB -- R 47r R 47 R B B (3.4a) ikR - ikR ikR H(r) = Vx Vx Bie HE dBEV - n HdB. 47r \ R 47n R 47r R (3.4b) Recall that R is a function of the coordinates of two points r and rB, everything else in the integrands on the right hand sides is a function of the integration variables (coordinates of?B) and V operates on r. For future use, we denote by VB the operator on r B and note that I 13

THE UNIVERSITY OF MICHIGAN 7133-4-T ikR ikR V e e (3.5) R B R Now following Stevenson as well as the procedure in the scalar case we assume that E and H may be expanded in series of powers of k, i.e., E(') = ( rJ)(ik)m, H() = H (r)(ik)m (3.6) m=0 m=0 As before, this assumption has been proven (Werner, 1963), that is, it is no longer an assumption but a consequence of (3.1), (3.2) and (3.3). It is perhaps worthy of note that the reason this entire discussion concerns three-dimensional scattering problems is that convergent expansions of the form (3. 6) do not exist for two-dimen sional scattered fields. ikR Next we expand the free space Green's function, e /R, in a series, viz, ikR CO ik,. Re (ik) R R = Ew- ( (3.7) R 1=0 then substitute (3.6) and (3.7) in (3.4a), (3.4b). After interchanging summation and integration, reordering terms and equating like powers of (ik), we obtain 1 ii nE(r) 4 Vx E x R dB ~( _4V f m! E I-m 1-1 1 M- 1 r-iS + snxH R dB- 4 E: V\^E1 mR dB 4 m =-m-1 - m=0 = 0,1,2,... (3.8a) 14

THE UNIVERSITY OF MICHIGAN 7133-4-T -A.~ 1 1 ^ i mr n 4!' V n n Hm R dB B (3. 8b) Q-1 where 0 when I = 0. Furthermore, substituting the series (3.6) in Maxwell's equations (3. 1) yields VxE = 0 o VxE = H V- E= 0 VxH = 0 o VxH = -E_V He = 0 = 1,2,3... = = 0,1,2,3 (3. 9) (3.10) (3.11) and the boundary conditions (3.2) become A &x nXEI. - _A binc = -nfxE I r-r. r-rB n HA r=rB 0. inc = -n r H r=r (3.12) These last equations result from the fact that, as with scalar sources, representations of electromagnetic plane waves or point sources are analytic in k. There is one more condition of importance. With Maxwell's equations and Stokes' theorem it is a simple matter to show that E dB = 0, B n i HdB = JB It follows then from the series expansions (3.6) that S n- E dB = 0, ni HdB = 0 JB JB I = 0, 1, 2, 3... (3.13) (3.14) 15

THE UNIVERSITY OF MICHIGAN 7133-4-T Stevenson then proceeds to show how the zeroth order terms, E and Ho, o o may be determined as solutions of potential problems. For the perfectly conducting case this reduction to potential problems for the zeroth order terms will be included in the general treatment of the following section, and, since we have no quarrel with Stevenson's results for these terms, the details will be omitted here. To calculate higher order terms, Stevenson proposes the following procedure: Suppose E, E,..".EI 1' H0, H1..'.H 1i are known. To find E2 or HV, determine first a particular solution of (3.10), that is, find functions F and G~ such that VxF = H and VxG = -E 1 The differences between these particular solutions and the true coefficients, -^. f.U h -* E[ - FE and H- Gl, are gradients of unknown potential functions (not necessarily regular at infinity) i.e., E = F+Vo (3.16a) H- = GI+VI (3.16b) Substitute (3.16a) and (3.16b) into the integral expressions (3.8a) and (3.8b) respectively, also introduce the boundary conditions, (3.12). There results equations for El and HI which contain some known terms and some unknown. It is then possible to show that the unknown terms are now exterior potential functions (regular at infinity) which may be determined as solutions of standard potential problems. The process, once begun, appears to be both correct and, in the details of its execution, ingenious. The source of trouble, however, is right at the beginning; namely how does one determine particular solutions of the equations VxE H =x H 1-E? I 1-1, I 1-1~~~~~~~~~c i I I I Ed 16

I THE UNIVERSITY OF MICHIGAN 7133-4-T In a separate paper, Stevenson (1954) points out that necessary and sufficient conditions for the equation Vx F(r) = f(r), reV (3.17) to have a solution are V-f = 0 (3.18) and. n fds = 0 B (3.19) where, if B consists of a number of disjoint surfaces, B1,...Bi, then (3.19) must hold for each separately, as well as the sum. With this we have no quarrel. Stevenson then goes on to assert that an explicit solution of the problem is given by 1 f(r ),v dv V (3.20) space provided that f satisfies (3. 18) and (3.19). Since the integration is over all space, not merely V, the exterior of B, this expression requires some explanation. In the first place, f(A) is originally defined only exterior to B. To extend the definition to the interior, Stevenson proposes to choose f so that (3.18) remains true and that ni. f is continuous at B. This he accomplishes by choosing f(r) = Vu(C), r interior to B (3.21) where V2u = 0 r in B (3.22) 17

THE UNIVERSITY OF MICHIGAN 7133-4-T and n.Vu au n.f (3.23) r=rB r=r r=r This is a standard interior Neumann potential problem for u and has a unique solution provided that \ dB = O. (3.24) an B That (3.24) holds is guaranteed by (3.19). Thus the extension to the interior is carried out, once this potential problem is solved. Equation (3.20) then is the required solution provided the integral exists, that is, provided [ ~3 f = O(1/r3) as r —eoo. (3.25) Stevenson describes the proof and we shall demonstrate it in detail in the following section where we again make use of this device. Now however, we accept it and finally get to the heart of the matter, namely, what do we do if f is defined originally in the infinite region V, but does not satisfy the necessary order condition at infinity, equation (3.25)? This in fact is exactly what happens since E. and He 3 vanish as 1/r only for i = 0 which allows us, using the method described, to determine E1 and H1 but apparently no higher order terms. (Actually we may go one term further since the 1/r terms don't contribute to the integral.) Stevenson was aware of this and proposed the following procedure: If V is the unbounded region exterior to B and if f does not vanish at infinity to the required order, first surround B by a surface B. Then redefine f exterior to B in terms of the solution of an exterior potential problem, namely, let f(A) = Vu, r exterior to B (3.26) 0 I 18

THE UNIVERSITY OF MICHIGAN - 7133-4-T where V 2u = 0 aulN I — B == nf(3.27) an -= -A. r=rB r=rB 0 0 u regular at infinity. This problem has a unique solution u and, since \ n fdB = a -dB = 0 (3.28) B B o o 2 _ it follows that u = O(l/r ) hence f will satisfy (3.25). With f thus redefined, the solution (3.20) exists and is valid in the portion of V interior to B where B can o o be taken arbitrarily large. With that, Stevenson apparently considers the subject closed. The implication is that since B may be taken arbitrarily large we may take it as a sphere whose radius becomes infinite and then (3.20) will represent the solution we seek throughout V. But, unfortunately, if f were a function whose original behavior at infinity was insufficient to guarantee existence of the integral in (3. 20), then the limit of the integral with f redefined may not exist as the radius of B becomes infinite. This argument by which the unpleasant behavior at infinity is avoided (that is, confining attention to a finite. volume, carrying out the calculation, and then letting the volume become infinite) is not only employed by Stevenson but others as well, e.g. Morse and Feshbach (1953, I, p. 53). It does produce the desired results in many cases. For example, the process is valid whenever f is the gradient of a potential function, regardless of its behavior at infinity (which includes the example *If B is any surface entirely containing B and equations (3.18) and (3.19) hold, then (3.28) follows from Gauss' theorem relating volume and surface integrals. I ],, LI I -- LI I-II 19 - I I -r-

THE UNIVERSITY OF MICHIGAN 7133-4-T used by Morse and Feshbach). That it may also yield unacceptable results is illustrated in the following example, where f is not the gradient of a potential function though still satisfies (3.18) and (3.19). This indeed is representative of the behavior one would encounter in actually attempting to find particular solutions of (3.10). Let (r) = Vxr i = mr (z -y ) (3.29) r = +x +y +z2 Clearly a particular solution of VxF = f is F r i (3.30) X However, let us attempt to determine a particular solution using equation (3.20). First of all it is a trivial calculation to observe that (3.18) and (3.19) are satisfied with this particular f. For this simple example we have no scattering surface B, but with Gauss' theorem it is clear that for any closed surface B, n. fdB = 0. Furthermore, the function f clearly misbehaves at infinity so that to use (3.20) we must employ the redefinition of f. Thus choose B to be a large sphere of radius r. Next define O o f = r rr (3.31) =Vu, r r 0 where V2u = 0 r r O n Vu = f r=r r=r o o u regular at infinity. I mm 20

I THE UNIVERSITY OF MICHIGAN 7133-4-T Then 1 rV) ~47TrVX5 __ dv. all space (3.32) Note that with our choice of B and f, O nr=r r o = Vx mr (sin+coscos) = 0. = r. Vxr i = r',mr ( sin 0+ aeos0Cos ) = 0. X O (3.33) Hence u is a solution of the homogeneous Laplace equation, regular at infinity, satisfying homogeneous boundary conditions on B which means that O u - 0 (3.34) Thus all spac f() C r-rv = I r -r V v 0 e f(r ) v r rI (3.35) This integration is easily performed yielding A. ^/ \4r 27r 73r m f(r ) O V xr }\._ dv = \ dr \ d \ d r sinO -a |r-r \ v \ v V V V r-r x r:r 0 0 JO v ro- m+3 y(z-yl'), rr 3 m r y z 0' m+3 47T m4 o 3 m+3 3 (y z) rz ) r Now we form F using (3.32) and find (3.36) 21

THE UNIVERSITY OF MICHIGAN 7133-4-T rn- 1 A> \ 2 m' m+2 mi mxr M A F r -r 1 - r i - r r<r 3 o x m+3 x m+3 o (3.37) m+3 mr 0= ~ (i -3x X) r>r 3(m+3)r3 x r It is a simple calculation to show that this F is indeed a solution of Vx F = f when r < r. However, it is also clear that F as defined in (3.37) does not exist as r >ao. Furthermore, if r remains finite, then the function F not only exhibits an unwanted dependence on an arbitrary parameter ( the radius r of B ) but also o o is discontinuous on B. This violates the tacit requirement that F be a differentiable solution of VxF = f for all points in V. How then do we proceed in those cases when Stevenson's scheme for finding particular solutions apparently fails? One method would be to attempt to show that the undesirable part of F is the gradient of a scalar function and can therefore be neglected; the remaining part of F would still be a solution of Vx F = f. In the example above it is easily seen that F may be written -% m+2 m'> mxr A1 A - 2 m o F= rm- rxi. r+V(-V-r x (3.38) m+3 x m+3 3 (3.38) hence a particular solution of Vx F = f, valid everywhere in V may be obtained merely by deleting the term V (- r x). In general, however, the process of identifying the unwanted terms with the gradient of some functions may not be so easily accomplished and in any event adds yet another complication to an already involved procedure. Rather than attempt to prove that this procedure can be made correct in the manner indicated, we shall end this section having demonstrated that, as it stands, Stevenson's procedure is ambiguous. In the next section we shall show that this I 22

THE UNIVERSITY OF MICHIGAN 7133-4-T problem of finding particular solutions of (3.10) may be avoided entirely and the process of determining successive terms in the low frequency expansion may be made more straightforward. 23

THE UNIVERSITY OF MICHIGAN 7133-4-T IV ELECTROMAGNETIC SCATTERING - AN ALTERNATE APPROACH In this section we again treat the problem of extending to electromagnetic scattering the method of Section 2 whereby scalar scattering problems are reduced to a series of standard potential problems. Though the method described here departs from Stevenson's approach, the debt to his work, both in ideas and technique, is large. We formulate the problem exactly as in Section 3 and the details will not be repeated. The starting point for this analysis is the integral representation of the coefficients in the low frequency expansions of the scattered field, equations (3.8a) and (3.8b). That is, we write the field scattered from the perfectly conducting body B, as E( ) = E( r)(ik), H( r) = H( )(ik), (4.1) Q=0 o=0 then the boundary conditions at the surface B are nxE =-fixE a n H A-n Hinc = 0, 1,2... (4.2) and, furthermore, \no EdB = 0 n- HdB = 0 = 0,1,2,... (4.3) B B The Stratton-Chu integral representation, after expanding in powers of k, equating coefficients and using the boundary conditions may be written E()- A dB (4.4a) )= F()- n47r R dB (4.4a) 1 24

THE UNIVERSITY OF MICHIGAN 7133-4-T nxH B where 1~ 1 ~A 1 inc Rm-1 F(r) = - rVx nxE R dB (4.5a) +1 -1 1 1 1 +HdB El_ n\H R dB 47r m 1-m-l 47r m' I Bm=0 m=lB and -1 7i 1 G ) =! Vx xH R dB (4.5b) SI 47r -m m=l |B 1 ( 1 \ A inc m- 1 \ 1 A in m-1 47+ E~ m,- ncxE. R dB + 1 V n H R dB. 47Tr m1 m' -m-1 47r -' - n -m mrn=0 m=0 Equations (4. 4a, b) and (4. 5a, b) hold for all I = 0, 1, 2,..., however, the terms Q-1 S Z and i are identically zero when I = 0. Observe that F and G are exm=0 m=l pressed in terms of the incident field and preceding terms in the series for E and H, i.e., E, E1,...E H, H,...H Thus if we consider the problem of 0 1'*E-1' k 0' 1 i J. k-I finding E and H, assuming that the preceding terms have already been determined, then F and G. are known functions. The approach, ours as well as Stevenson's, is to show that the unknown terms in (4. 4a) and (4. 4b) are gradients of exterior potential functions which may be determined as solutions of standard potential problems. Stevenson went to considerable effort and complication to formulate these problems. The method of this section, though still complicated, hopefully represents a simplification. In any j 25

THE UNIVERSITY OF MICHIGAN 7133-4-T event, the present procedure for finding E and Hi, or at least defining them in terms of solutions of potential problems is based on the integral relation (4. 4a) and (4.4b) and does not require, as Stevenson does, first finding particular solutions of Maxwell's equations. Consider first the task of determining E ). We observe, and this is the essence of the approach, that the unknown term on the right hand side of equation (4.4a) is itself the gradient of an exterior potential function, a single layer distribution of density n * EQ. It is possible to formulate a boundary value problem for this term as follows. Let 0 denote the unknown potential, i.e. 13\ R dB (4.6) B Then E = F+ Vp (4.7) where F is known and V =0 r in V 0 regular at infinity in the sense of Kellog ^ A.~inc - fnXV)0 -nx(E F r=rB r=rB This is not quite a standard Dirichlet potential problem in that the boundary condition as given may be shown to specify the function 0 on the boundary to within a constant. That is, specifying nxV0 on B is equivalent to specifying 0~+c on B where c is constant but unknown. This constant is evaluated by solving the potential problem with the ambiguous boundary condition, constructing the corresponding E with equation (4.7) and then imposing the requirement! j 26

THE UNIVERSITY OF MICHIGAN 7133-4-T n- EdB = 0 (4.3) B The procedure for finding E is thus seen to be reasonably straightforward once we observe that the unknown part of E is the gradient of an exterior potential. This observation spares us much of the complication of Stevenson's approach. The determination of H requires more work since it is not obvious that the unknown term on the right hand side of (4. 4b) is the gradient of an exterior potential, except when I = 0. In fact it may be shown that when I j 0, this term is definitely not the gradient of an exterior potential function. Nevertheless it is possible to retain some of the simplicity inherent in the determination of El by adding a known function to the unknown term such that the sum is the gradient of an exterior potential function. The determination of the function we must add again requires the solution of a potential problem. Thus we introduce a function g2 as yet unspecified, into equation (4.4b), obtaining 1xH H((r) =GI(r) - g( r)+ 4 Vx R dB +g ). (4.9) vB It is well known and/or easily verified that a condition sufficient to guarantee that a vector be the gradient of a scalar is that the curl of the vector vanish, i.e. VxA = 0 = A = V/. (4.10) Thus a condition sufficient to guarantee that we may write 1 fnxHI 4 Vx\ R dB+g(r) = V (4.11) ~is~ ~B is 27

THE UNIVERSITY OF MICHIGAN 7133-4-T A - 1 nxH VxVx dB + Vx = (4.12) B Since g is as yet unspecified, we use (4.12) as an equation for g and seek a particular solution in terms of known functions, i.e., terms not involving Hl or EI. Since Hi appears, we must first put (4.12) in suitable form. First we use the vector identity VxV x =V(V -V2 together with the fact that for r in V, V2 1 = V R2i^ i~) -0 to rewrite (4.12) as 1 _ Vx'g(r) = - VV dB. (4.13) ~ 41 1 Recall that R is symmetric in r and rB (eqn. 2.1) and V- = -V where V B R BR' operates on r and VB on ri, hence (4.13) may be written Vxgr) = V VBR nxHdB, (4.14) or, on employing the properties of the scalar triple product, Vxg(r) =- 4 V n * VB R xH dB. (4.15) This we rewrite as C H xg(r) = -47r V n (VBxR R VBXH)dB (4.16) I m I 28

I - THE UNIVERSITY OF MICHIGAN 7133-4-T and, since Stokes' theorem implies n.VBX - dB = 0, (4.17) PU B we have 1 n A Vxg,(r) = 4 V VBXH dB. (4.18) PB But Maxwell's equations (3.10) imply that VBXH(rB)i -E (-1 ) L >0 (4.19) =0 = 0 hence we have, finally,. 1 c'Eid- 1 Vxg()= - V V SL nE dB, I >0 (4.20) wB =0, =0. We have thus succeeded in rewriting (4.12) in terms of known functions since we have assumed that E is known. Now we want a particular solution of (4.20). Clearly when I = 0, go= 0 is a solution. When I > 0, we employ Stevenson's.~~~~~~0~ ~ method or producing particular solutions of the equation Vx F = f. First of all note that the right hand side of (4.20) is the gradient of an exterior potential function (single layer distribution). Thus, introducing the notation e( _ 1 -, (4.21)1 u,(") =f -dB (4.21) BRdB 1 29

THE UNIVERSITY OF MICHIGAN 7133-4-T (4.20) may be written Vxg(r) = Vu,. reV (4.22) Stevenson has shown that necessary and sufficient conditions for (4.22) to have a solution are [3.18), (3.19) V. Vue = 0 reV (4.23) aue B an dB = 0 (4.24) on B e The first condition, (4.23), is clearly satisfied since, as noted, u is a potential function. To show that (4.24) is also satisfied we use Gauss' theorem to write e \ u ^ A e 2 e e dBan dB= V d rVu dB= - (4.25) a\ n \ JL \ B B B B aD where BaO denotes a large sphere whose radius approaches infinity. The volume integral term vanishes by virtue of (4.23) and the surface integral over B will also vanish if Vue = o(l/r2) as r -+o. (4.26) e That (4.26) is satisfied may be seen by examining the structure of ue exhibited in the defining equation (4.21). Thus (r1 r i r. r dB (4.27) oB B or, for r > max rB, I 30 I I I I

r THE UNIVERSITY OF MICHIGAN 7133-4-T Vue(r) = - 4v E m+1 -l - P cos) dB m= r (4.28) where cos y = cosOcosB+ sinO sin0Bcos( - 0B) and Pm is a Legendre polynomial of order m. The m = 0 term vanishes by virtue of (4.3), S A nS'E dB = 0 B hence V ( r) = 4r1 V 1mV1m+l - rPm(cos dB m=l r (4.29) From (4.29) it is clear that Vue(r) = O(1/r3) or o(1/r2) as r —oo (4.30) hence (4.26) holds which in turn means that (4.24) is valid. Thus we have established that equation (4.22) has a solution. Furthermore we have shown, in the process, that the right hand side of (4.22) is O(1/r ) at infinity. Now we use Stevenson's solution to this problem. i We define an interiorlpotential function, u, (r), t interior to B, as follows. V2u( r) = 0 n Vu( r) r=rB r interior to B = nI Vu(r ) I r=rB (4.31) (4.32) This is a standard interior Neumann problem for u (r) and has a solution provided that 31

THE UNIVERSITY OF MICHIGAN 7133-4-T Ac n\ -Vu dB = 0 B (4.33) but this is satisfied by virtue of (4.24) and the boundary condition (4.32). Recall that n above is always directed from B into V, the exterior of B. Now, according to Stevenson a particular solution of the equation (4.22) is given by frC ~ v Vu (r v V / 1 7 \ V i V \ g( r) = 4ir Vx{ vR dv+ 1 V ul ) ( vi dv R (4.34) where V. is the interior of B, and V the exterior. i To demonstrate that (4.34) is indeed a solution of (4.22) is a relatively simple calculation. Again using the identity Vx Vx = V(V -V2 we have e 4 J Vu (rv) Vxg()) = 4v v -dv+V. V.V 1 i V u (r ) R dv) 1 2( \T v vu(rv) - V dv+ \v V. v u(rv R dv) R, reV (4.35) 1 but 2 (= 6(fr^-r1) 47T RR(r- r ) v \V therefore, for r in V, e.b ~ & 1 V W u, (&v) V udrv dv +V' Vxg(-r ) = Vu('r)+ V(V. vI v dv + V- V V. 1 (4.36) V ur ) v -v dv. (4.37) R I 32

THE UNIVERSITY OF MICHIGAN 7133-4-T 1 1 e i Now using the facts that V- = -V - and u and u are both potential functions we R VR ci obtain (7 e.. Vir e 1 u dv- V V dv. V x r) = Vur 4T \ V. * dv 47 v. v vR. R l ~ (4.38) Now we use Gauss' theorem, taking care of the signs of the normals (n on B is always directed into V, the exterior) to obtain.~ e 1e. Vxg r) = Vu(r) + V n Bu (rB)- VBu(r )BdB (4.39) wB Actually there is another surface integral term over a large sphere at infinity but this vanishes by virtue of (4.30). The integral in (4.39) vanishes because of the boundary condition (4.32) thus verifying that Vxg(?r) = Vu,(r), rV. We may cast g( ) in slightly more convenient form as follows. Again 1 1 using the fact that V- = -V -R (4.34) becomes 1 1 e1 i vR v;ulv v ~ r)= v4r -xVvu(r. )dv- V -xVv u(rv)dv, (4.40) 47r vv 47r vv v V. or since curl of the gradient is identically zero, e i -,\ V Or\ V uW (,) d* /\ 17 \ n dv (4 41) = - _ \ V _x vRv dv - 7 V x v v dv. (4.41) V47 V. 1 33

THE UNIVERSITY OF MICHIGAN 7133-4-T Now employing a famous, but apparently nameless theorem of vector analysis, C xAA VxAdv n= nxAds (S encloses V and n is out of S), V S equation (4.41) becomes r i) -u 1\ nx V ue' dB 0 (4. 42) B An alternate form of (4.42) is found to be 1 \_ __11 _ _ _ g(r) = r Vx dB (4.43) Again the behavior of Vu at infinity, (4.30), causes a similar integral over a large sphere to vanish. e i In this form it is clear that the tangential components of Vu, and Vu, on B must be unequal if g is to be different from zero. In fact they are necessarily dise i continuous. Since u is a potential function regular exterior to B, u is a potential function regular interior to B and their normal derivatives were defined to be continuous at B, then the tangential derivatives cannot also be continuous. If so, i e u, would be a continuation into the interior of B of u. The resulting function would be a potential function regular everywhere in space and therefore would nece essarily be zero. But u (see eqn. 4.21) is not identically zero. We have thus determined a particular g such that (4.12) is satisfied. This in turn guarantees that equation (4.11) holds, that is, with the gp we have found we may write I 34

- THE UNIVERSITY OF MICHIGAN - 7133-4-T 1 1 Vx nxH 47r Vx R dB + g = VV. B (4.11) With equation (4.34) or (4.43) which expresses gl as a curl, it follows upon taking the divergence of (4.11) that V2i = 0. On expanding 1/R in (4.43) it follows that I g (r) = O(1/r2) as r —o Also S nxH1 2 Vx R dB = 0(l/r ) as r — co hence p is regular in the sense of Kellog. (Actually VOq = O(l/r ) does not imply completely that ip is regular. There may be an additive constant which would imply rp1t is not bounded. Since we are interested in Vt, which removes this constant anyway, we may choose it as zero to begin with and take ip to be regular.) With equation (4.9) and the boundary conditions (4.2) we may formulate a standard exterior Neumann potential problem for 01, namely V2p = o rEV 0l regular at infinity in the sense of Kellog (4.44) A - r=rB rnI A "inc A - an -n H n G + B r=rB nxV +n- \ - -'J! i R The solution of this problem then determines H (r). I I, 35

a THE UNIVERSITY 7133-4-T OF MICHIGAN To summarize the procedure we have established: If an electromagnetic field, OD 00 Einc() = E ( )(ik)Q Q=O OD'inc. Hinc( r) = i i(r )(ik)" Q=0 (4.45) is incident on a smooth finite perfectly conducting surface B in three space then the coefficients in the low frequency expansion of the scattered field { D00 E()=E,0r)(ik 1=0 00 1=0 (4.46) are given by EQ(r) = F(r) + V0 HI( r ) = G( )g r )+ VI (4.47) (4.48) where I/?) I 1 7'l 1 =- 4 Vx - -- 47r m=. 4nP~~C~~~~s..-Lvry L A -^inc Rm-1 nxE - B 1-1 dB +m=1 1 4rm m BE I xHm- dB 1 nxH -M-1R dB- 4= m' 1V S A m-1 n- E R dB, 3. >0 3 (4.49) -F r. 1 F (i) = - -Vx o 47r I SB A - inc nxE -. dB R 36

THE UNIVERSITY OF MICHIGAN --- 7133-4-T 1 I 1 Gl(.r) = 47 Vx >m' G(") =-Vym~ l ml=l S nxH R dB +Q-m 43r $ 1-1 1 m=O 5 A 1inc Rm-lnxE -m-1 dB 1-n + 1 47r I 1 m=O M rV 5. Rm-dB, B -m 1>0 G () =-V o 4r B A -binc n.H - dB R (4.50) " b nxV (ue - u1 &. B ^R dB = - -1Vx R 47 SB A n e i (u -u)dB, R Ii 1>0 g ( ) = 0 (4.51) 1 ea 1 B A E R dB R I >0 (4.52) i and ui, (, and p1 are all solutions of standard potential problems. u,(r) is an interior Neumann potential:,v2 = 0 n. i A^ e n -Vu/ = n.Vue r interior to B on B IT (4.53) 0 is an exterior Dirichlet potential: V20 = 0 r exterior to B l reg at o nx V A _ - inc + - ) nxV 1=- X(E +Fl) I 1 I (4.54) on B 37

THE UNIVERSITY OF MICHIGAN - 7133-4-T (we must use n. E dB = 0 to determine an arbitrary constant arising from this form oa the boundary condition), and 0L is an exterior Neumann potential: V20 = 0 I r exterior to B 0l reg at oo (4.55) A A - inc A A n- Vd =,HC-n - n G+n-g on B J n is the unit normal on B always pointing into V, the exterior of B, and R is the distance I r-i from a point r" on the surface (the integration variables) to a field point r. We complete this section with a brief discussion of the low frequency expansion of the far field. Here we proceed exactly as in the scalar case. We incorporate the facts that, for large r, ikR ik(r- r- rB) e e R r ikR ik(r - r r ) VeRikR — B e R A e v — ikr.. R r in the Stratton-Chu integral representations of the scattered field, equations (3.4a) and (3.4b), also employing the boundary conditions on the surface (3.12), obtaining ikrik 47rr SB ikrHr,) H(r) - - e 47r r JB -ikA r -.nx Einc r rB _A Ain A A A.-'d -rxCnxE )+ XH- r n- E-dB (4.56) ikrB [ A r x(n x H)+nxE +rn+ H dB (4.57) Now we expand the field quantities, equations (4.45) and (4.46) and the factor 38

THE UNIVERSITY OF MICHIGAN 7133-4-T ~ A. -ikr* rB e and reorder the terms to find E(r) 4 1=(ik) m0 (i-m)! SB [rxxEm ) ikr -m +nxH -rnE -idB (4.58) +nxE + rnH dB (4.59) m m The = 0 term in (4.58) and. 59) always vanishes the integrals * E dB and \n * H are zero for all m, (equation 4.3) and, since E and H may to see that fnxH dB and \ nxE dB also vanish.. Therefore we may 4 2rrm (2+ +-m rewx+itxH - rn. 5 d (4.)E dB (r, e -v (k)+ 2. (-k) e [( B' B L=0m m.[xf -+ ~ ~dinc (4.60) and and I 39

THE UNIVERSITY OF MICHIGAN 7133-4-T ikr2 i +! (1) +1-m 47r r 1m: = ( + 1 - m): B 4n B X (A / A inc A A ^nc* rx(nxH )+ nxE inc rn H dB m m m J (4.61) This illustrates a famous result of Rayleigh: the leading term in a low frequency 2 expansion of the far field is proportional to k. Stevenson criticized this form as being inefficient since one apparently needs to determine i+ 1 non-vanishing near field terms in order to obtain I non-vanishing far field terms. Actually this is not completely true, as a close examination of the "extra" near field terms reveals. These are the m = 1+ 1 terms in (4.60) and (4.61), namely A inc A A A r [-rx(nxE_ )+nxH -rnA. E ]dB (4.62a) B and [ x(a x H )+nxE + rn.H +dB (4.62b) S + inc +x~.lH~++dx B which we rewrite as -r ix nxEl dBxH+ dB - r xn Eii dB (4.63a) Q^B JB JB and C x n xxH d+ nxdB + r nxEndB (4.63b) J B +JB B The terms involving the incident field are effectively known since the incident field is given. Also, from equation (3.14), S.n E+l dB = 0 and the only B I I 40

THE UNIVERSITY OF MICHIGAN 7133-4-T unknown part of these "extra" terms involves A A nxHL dB. (4.64) B With (4.48), however, it follows that A A A fxH+1 = (xG1 nxg+ + V 1)dB (4.65) JB B But I nxV l dB = 0, using a vector identify we have employed before, and G,+1 is given, (4.50), in terms of the first I near field terms. The only unknown part of this "extra" term involves gl which does require the solution of an interior Neumann problem see equations (4. 51)-(4.53). This is considerably less than requiring complete determination of EN 1 and Hf1', but is still unsatisfactory. Repeated attempts to determine this "unknown" part without solving for g, have so far been fruitless. The alternatives are also less than overwhelmingly desirable Stevenson provides a generalization of Rayleigh's continuation method whereby the near field terms for large r are matched with multipoles for small k (thus defining the multipole moments) then using the far fields of the multipoles. This of course involves expanding the near field terms in spherical harmonics which may involve as much labor as solving the required interior Neumann problem. Still, in principle, Stevenson's method of continuing into the far field is preferable since it does not require the solution of another, albeit simple, problem in order to obtain the same number of terms in a low frequency expansion of the far field as are available in a low frequency expansion of the near field. The price is apparently requiring both to be represented as expansions in spherical harmonics.! 41

THE UNIVERSITY OF MICHIGAN 7133-4-T V AN EXAMPLE-SCATTERING BY A SPHERE To illustrate the procedure derived in the previous section, we consider the problem of scattering of a linearly polarized plane wave by a sphere. The incident -iinc field is taken to propagate down the z-axis, with E along the x-direction (see Fig. 2), i.e. oi 1-inc i -ikz \'e Einc y =0 oo -inc A -ikz 7 -.inc H = i e(iik) H y =0 -inc (-z) ^ El f'. J! x -inc (-z) I II ~! y (5.1) binc H _ y Figure 2 We shall proceed to calculate the first two terms in the series for the scattered field, 42

THE UNIVERSITY OF MICHIGAN 7133-4-T 00 00 i EI E = (ikE, H = (ik) H, (5.2) Q=0 O=0 by straightforward application of equations (4.46) - (4.55). 5.1 Zeroth Order Terms From (5.1) we see that Ec =. (5.3) o x The scattered electric field to this order is (4.47) E0 = F +V0 (5.4) and (Eq. 4.49) 1 ^ &xi B F Vx dB. (5. 5) Equation (5.5), written in its entirety is, 27rT 7 F 4 Vx dB deBa sinOB J0 ~0 (5.6) A A r xi rB x r +a -2arososB+sinOsinB cos(1- | where the unit normal is rB = sin cos0+ sinsini 0 +i cos0. Be in is c o z B The integration is carried out using the well known expansion of 1/R in spherical harmonics and we find (using a mixture of rectangular and spherical unit vectors) 43

THE UNIVERSITY OF MICHIGAN 7133-4-T 3 F o( r=a 3 - i (5.7) o 3 3 3 x r r Now we use (4.54) to find 0. The boundary condition A~ A inc -- M nxV -nx(E +F), r=a (5.8) 0 0 0 is seen to imply that 0 = - sin0cos0+c. (5.9) 0 3 r=a The exterior potential function taking on this boundary value is found to be (write 0 Do a as a series expansion n- Pn(cosy) whose unknown coefficients are detern+l n n= r mined using the boundary condition), 3 0 = -2 a7 sine cos0+ ac (5.10) o 3 2 r r Substituting (5.7) and (5.10) in (5.4) we find 3 3 < a x a ac A E 3 r — - r (5.11) o 4 3x 2 r r r The auxilliary condition \S B.E =0 (5.12) implies, with(5.11), that c=0. (5.13) To find H we see (Eqs. 4.48, 4.51) that o { 44

I --- THE UNIVERSITY 7133-4-T OF MICHIGAN H = G +V. 0 0 o (5.14) From (4.50) "-I 1 G = -V o 47r.A - iinc B dB 1t (5.15) and inc - -A H = o y (5.16) from which G = 33 o -3 r r (5.17) io (Eq. 4.55) is an 0 exterior potential function with boundary values aoi 0o ^ rHinc A G - = -r ~ H -r. G ar o o r=a A O 1 a = r.l + 3 — y 3 ar (.3) Vr (5.18) 1 = - sinOsino. 3 Such a function is easily seen to be 1 a o = — - sinOsino o 6 2 r which, with (5.14) and (5.17) leads to (5.19) H — V a o 2 3 r 3#. a 1 = _2 y 2r3 3 3 ay^ +- r 2 4 r (5.20) The zeroth order results may be rewritten entirely in terms of spherical unit vec tors as 45

- THE UNIVERSITY OF MICHIGAN 7133-4-T o 22 3 ^ 3 o = -3- sin0 cos - 08- 3 cos0cos0+0 a3- sin0 (5.21) r r r A3 3 3 ra A a H 3 sin sin0-0 a cos sin0-0 cos. (5.22) r 2r 2r 5.2 First Order Terms The next terms are found using these results, again following the procedure of the preceding section. It is to be noted that even at this stage, the calculations become tedious. With (4.49) we see that if' A-x'inc.A 1 1 1 inc 1 n 0 F - Vx\ - dB -- Vx nxE dB+ dB 1,JB JB jB 1' - - V - E dB (5.23) 41r o All terms are well defined, E and H in (5.21)and (5.22)above and E and -zinc 0 E in (5.1), namely -inc A inc A E = E = -zi. (5.24) o x 1 x Carrying out the indicated integrations we find 5 3 5 -* a xz A a A A. a' q + A\ F 6 r- r2 a y + (zi +x (5.25) r 2r 5r Now we use (4.54) to find 1. The boundary condition nxV01 = -nx(E +F1) r=a is seen to imply I 46

THE UNIVERSITY OF MICHIGAN 7133-4-T 3a2 2 0 = -0 -cos sin20+c = - P2(cos0)cos +c (5.26) r=a Here we employ the definition of the associated Legendre functions given by Magnus and Oberhettinger (1949). The exterior potential function taking on these boundary values is found to be 5 a P12csco+~ 0, = 1_ 3 P2(cos0)cos +. r(5.27) 10r " r Forming E1 with (4.47), and applying the auxilliary condition in E1 dB = 0 which implies c = 0, we find E -- rxi + (V 5. 2r2 y + 1 Proceeding to the determination of H1 we see that (Eq. 4.48) H1 G -g9 +V0 (5.29) With (4.50) ^A -Minc A ^ inc 1 A 1 B 0dB+ G - 1x \ xH dB+ - --- dB + R d 47 S B 47r R 4 gr dR OB J Bi 4+ V Br 0H dB. (5.30) 4B All the quantities involved have already been defined in (5. 1) and (5.22). Carrying out the integration yields. 3 35 A a A A G = - rx +V a5 yz (5.31) 3r \5r/ I 47

THE UNIVERSITY OF MICHIGAN 7133-4-T e i To determine g we must first find ul and u1. From (4.52) nE e 1 0 eU=- \ R dB (5.32) 1- 4 Br R which, with (5.21), may be evaluated as 3 e = 2 a Pl(co18)cos1 (5.33) r Following (4. 53) we determine the interior potential function whose normal derivae tive matches that of ue on the boundary. Here we assume a series of the form 00 anr Pn(cosy) and determine the an using the boundary condition. In the present case, this is easily seen to be i 4 1 u1= - rP (cos) cos p. (5.34) 1 3 1 With (4.51) we see that A _ e i 1 n B^ 1- 4 1 _ \ __ 1)dB (5.35) g, 4= r SB and using the expressions (5.33) and (5.34), this may be found explicitly as 3 2 a 3 g 3 2 rxi. (5.36) =-3 2 x r Now we proceed using (4. 55) to find an exterior potential function, a1, satisfying the boundary condition *1 n A +n*g (5inC 37)A A =-n-H -n-G+n-. (5.37) an 1 1 r =a I 48

THE UNIVERSITY OF MICHIGAN 7133-4-T With (5.1), (5.31) and (5.36), this boundary condition becomes 1 2 2a 1 - = - asinOcos sin = -1 P2(cosS)sino. (5.38) Or r=a5 2 r=a and the solution is found to be 2a 1 2a___z - 3 P (cose)sin = 2a(5.39) 1 3 2 5 45r 15 r Substituting (5.31), (5.35) and (5.39) in (5.29) we obtain 3 /5 H1 a A e 1 "ay H rx + V (5.40) r r The first order terms may be written entirely in spherical coordinates as 5 A /5 3. 3a 3a a ^a E - - a- sinOcos0cos0 r+ a cos20+ -a2 os 1 2 4 \WC~; ( 24cos2r+ 2r 2r 2r4 a 5 ^ Hcosin (5.4 + sin) 5 / 5 3 \ 1 = - 4 sOossin r4CO2n + r \3r r / +cosecos0( + -a 4)0 (5.42) These results for the first two terms in the low frequency expansion may be shown to be in complete agreement with comparable expressions derived from the standard Mie series. 49

-THE UNIVERSITY OF MICHIGAN 7133-4-T REFERENCES Darling, D.A. and T.B.A. Senior (1965) "Low Frequency Expansions for Scattering by Separable and Non-Separable Bodies", J. Acoust. Soc. Amer. 37, 228-234. Kellog, O.D. (1953) Foundations of Potential Theory (Dover Publications, New York). (Springer, Berlin, 1929). Kleinman, R.E. (1965a)"The Rayleigh Region", Proc. IEEE Special Issue on Radar Reflectivity. Kleinman, R. E. (1965b) "The Dirichlet Problem for the Helmholtz Equation", Arch. Rat. Mech. Anal. 18, 205-229. Magnus, W. and F. Oberhettinger (1949) Formulas and Theorems for the Special Functions of Mathematical Physics (Chelsea Publishing Co., New York). (Springer, Berlin, 1943). Morse, P.M. and H. Feshbach (1953) Methods of Theoretical Physics (McGrawHill Book Co., New York). Noble, B. (1962) "Integral Equation Perturbation Methods in Low Frequency Diffraction", appearing in Electromagnetic Waves, ed. R.E. Langer (University of Wisconsin Press, Madison). Lord Rayleigh (1897) "On the Incidence of Aerial and Electric Waves upon Small Obstacles in the Form of Ellipsoids or Elliptic Cylinders and on the Passage of Electric Waves through a Circular Aperture in a Conducting Screen", Philos. Mag. XLIV, 28-52. Senior, T. B.A. and F.B. Sleator (1964) "Notes on Stevenson's Solution for Low Frequency Scattering", The University of Michigan Radiation Laboratory Internal Memorandum No. 667T-504-M. Stevenson, A. F. (1953a) "Solution of Electromagnetic Scattering Problems as Power Series in the Ratio (Dimension of Scatterer)/Wavelength", J. Appl. Phys. 24, 1134-1142. Stevenson, A. F. (1953b) "Electromagnetic Scattering by an Ellipsoid in the Third Approximation", J. Appl. Phys. 24, 1143-1151. Stevenson, A. F. (1954) "Note on the Existence and Determination of a Vector Potential", Quart. Appl. Math. XII, 194-197. Stratton, J.A. (1941) Electromagnetic Theory (McGraw-Hill Book Co., New York) p. 466. Werner, P. (1962) "Randwertprobleme der Mathematischen Akustik", Arch. Rat. Mech. Anal. 10, 29-66. - -------- -— 50

I THE UNIVERSITY OF MICHIGAN --- I I 7133-4-T Werner, P. (1963) "On the Exterior Boundary Value Problem of Perfect Reflection for Stationary Electromagnetic Wave Fields", J. Math. Anal. Appl. 7, 348-396. Wilcox, C. (1956) "An Expansion Theorem for Electromagnetic Fields", Comm. Pure Appl. Math. IX, 115-134. 51

TTn -hi1 ciffiPw. 1 I Security Classification DOCUMENT CONTROL DATA - R&D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. ORIGINATIN G ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan Unclassified 2b. GROUP 3. REPORT TITLE LOW FREQUENCY SOLUTION OF THREE-DIMENSIONAL SCATTERING PROBLEMS 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Technical Report 5. AUTHOR(S) (Last name, first name, initial) Kleinman, Ralph E. 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS August 1965 51 16 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) AF 19(628)-4328 7133-4-T b. PROJECT NO. 5635 C.~563 502| 9 b. OTHER REPORT NO(S) (Any other numbers that may be assigned 563502 this report) d. AFCRL- 65-639 10. A VA IL ABILITY/LIMITATION NOTICES Qualified requestors may obtain copies of this report directly from DDC. Other qualified users shall request through CFSTI. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Hq., AFCRL, OAR (CRD) United States Air Force L.G. Hanscom Field, Bedford, Mass. 13. ABSTRACT A deficiency is pointed out in Stevenson's method of reducing the solution of electromagnetic scattering problems to a succession of standard potential problems whose solutions determine terms in the low frequency expansion of the scattered field. An alternate approach is presented, for perfectly conducting scatterers, which not only removes the difficulty but also is simpler and more explicit than Stevenson's method. The details of the analogous, though simpler, scalar scattering problems are also presented. I I --- It ~ I -I I I ~ I I I DD I JANR64 1473 T Tr1 rI o c i fi -'r Security Classification

Unclassified Security Classification I 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Scalar Scattering Stevenson's Method _i I i INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. 3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5.-AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 6. REPORT DATE Enter the date of the report as day, month, year; or month, year. If more than one date appears on the report, use date of publication. 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, i.e., enter the number of pages containing information. 7b. NUMBER OF REFERENCES: Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report. 9b. OTHER REPORT NUMBER(S): If the report has been assigned any other report numbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those I imposed by security classification, using standard statements such as: (1) "Qualified requesters may obtain copies of this report from DDC.' (2) "Foreign announcement and dissemination of this report by DDC is not authorized. " (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through,, (4) "U. S. military agencies may obtain copies of this report directly from DDC. Other qualified users shall request through (5) "All distribution of this report is controlled. Qualified DDC users shall request through If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 11. SUPPLEMENTARY NOTES: Use for additional explanatory notes. i2. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S), (C), or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. 0 Unclassified Security Classification

UNIVERSITY OF MICHIGAN 3 9015 03023 8359