THE UNIVERSITY OF MICHIGAN 7030-4-T Scattering by a Paraboloid of Revolution Due to an Interior Axial Point Source By Stephen E Stone Technical Report 7030-4-T November 1965 Contract AF04(694)-683 Prepared For USAF Ballistic Systems Division Norton AFB, California 92409

THE UNIVERSITY OF MICHIGAN 7030-4-T TABLE OF CONTENTS ABSTRACT v I INTRODUCTION 1 1.1 Preliminary Discussion 1 1.2 Mathematical Statement of the Problem 2 1.3 Coordinates of the Paraboloid of Revolution 3 1.4 Integral Representations of the Solution 5 1. 5 Discussion of Integral Representations with Respect to Asymptotic Expansions 16 II UNIFORM ASYMPTOTIC EXPANSIONS 18 2.1 Airy Function Representation 18 2.2 Bessel Function Representation 24 m1 LOW FREQUENCY (THIN PARABOLOID) SCATTERING 36 3. 1 Low Frequency (Thin Paraboloid) Poles 37 3.2 Residue Series for the Near Field 41 3.3 Dirichlet Potential Problem 49 3.4 Residue Series for the Far Field 53 IV HIGH FREQUENCY (FAT PARABOLOID) SCATTERING 60. 1 High Frequency (Fat Paraboloid) Poles 60 4.2 Equivalent Integral Representation 61 4.3 Source Located "Far" from Focus, Surface 64 4.4 Source Located "Near" Focus 65 4.5 Source Located "Near" Surface 65 V APPLICATION OF RESULTS TO CONTINUING INVESTIGATION 66 REFERENCES 67 APPENDIX A. 1: NORMALIZATION (POINT SOURCE NORMALIZATION) 69 APPENDIX A. 2: ANALYTICITY OF RESOLVENT GREEN'S FUNCTION RA 71 APPENDIX A. 3: CLOSING THE CONTOUR (CONVERGENCE OF RESIDUE SERIES) 72 iii

THE UNIVERSITY OF MICHIGAN 7030-4-T ABSTRACT If one considers a paraboloid of revolution of focal length or with an interior point source located anywhere on the axis, the exact solution to the Dirichlet or Neumann problem (Green's function of the first or second kind) may be written in the form of an integral representation. In this report we consider the asymptotic evaluation of these integrals for both low (k ~<< 1) and high (k 0 >> 1) frequencies. The low frequency results are obtainable from an infinite series over the zeros of a particular Whittaker function, corresponding to a Mie series found in the scattering by closed convex bodies. For high frequencies, we find multiple reflections and caustics arising from saddle point evaluations as well as "whispering gallery" waves, which arise from the nature of the behavior of the above zeros at high frequencies. The work at high frequencies is only briefly discussed since it covers research that is actually in progress at the present time. The aim of this discussion is to introduce possible approaches to the solution. This work on the paraboloid is preliminary to a description of scattering by general concave surfaces. Having integrated the formalism in terms of the physical phenomena such as the whispering gallery waves, multiple reflections and caustics, we now are in a position to search for generalizations of these to other concave surfaces. The scheme we propose is to determine the dependence of the physical effects on the local geometry of the paraboloid and then to make the essentially physical argument that this geometric dependence is the same for other concave shapes. This approach is similar to that used in determining creeping waves on general convex shapes and is an application of the physical arguments used in Keller's genmetric theory of diffraction. v

THE UNIVERSITY OF MICHIGAN 7030-4-T I INTRODUCTION 1.1 Preliminary Discussion For the most part the solutions of scattering problems have been confined to convex surfaces; relatively little has been done in the case of concave surfaces. The latter usually give rise, in the short wavelength limit, to such effects as caustics, multiple reflections and whispering gallery waves (a form of traveling waves). Some of the early considerations of these effects are found in the book by Rayleigh (1945); modern investigations are illustrated by the papers of Kimber (196la, b) which treat the circular cylinder and sphere respectively. In this paper we will consider the paraboloid of revolution (Dirichlet or Neumann boundary condition) with an interior point source on the axis, but not necessarily at the focal point. The scattering by a paraboloid of revolution differs from the circular cylinder and sphere mentioned above (both closed bodies). Although it is found that multiple reflections, caustics and whispering gallery waves occur when the point source is not at the focal point, the point source at the focal point gives a concave s rface scattering problem which does not exhibit these effects. In the short wavelength limit it shows only a single reflection. The case of a dipole, with moment perpendicular to the axis, at the focal point of a perfectly conducting paraboloid of revolution has been investigated by Fock (1957) and Skalskaya (1955). Pinney (1946a, b) considered the moment both perpendicular and parallel to the axis. Although there is a double reflection, it is natural to consider the plane wave problem in this category. The scattering of a high frequency plane wave by the interior of a parabolic cylinder (Dirichlet boundary condition) was studied by Lamb (1906), who indicated that the method could be extended to the paraboloid of revolution. Perhaps the best starting point for a study of scattering problems pertaining to the paraboloid of revolution is the book by Buchholz (1953) which inoludes a complete bibliography. Akowrtyof mention is the earlier written paper by Buchholz (1942/3). 1

THE UNIVERSITY OF MICHIGAN 7030-4-T 1.2 Mathematical Statement of the Problem Let D be the closed interior domain of the paraboloid of revolution and let p(r) (r is the usual position vector) denote the point source distribution. The precise form of p(r) depends on the definition of a point source; it will be specified later. The arguments of Ritt and Kazarinoff (1959, 1960) can be applied to characterize the solution to the problems stated above as the ergodic limit T lira I if (r, t) dt, T —- o T where v(r,t) = u(r, t)e and u(r, t) is the twice continuously differentiable in D for fixed time t, twice continuously differentiable in time, function in the space time domain {D x <t < ooa satisfying V2 1 a _,, iUt u -. - = p(r)e c at" u or u = 0 on the boundary of D an some prescribed initial conditions. Ritt and Kazarinoff (Ibid) also show that the above limit, which we call v(r), is a twice continuously differentiable function in D satisfying V v+kv = p(r) av v or = on the boundary of D rlim r + ikv (Sommerfeld radiation condition). This, of course, is the more familiar formulation of the scattering problem. Ritt and a inff Ii f he limt Kazarinoff (Ibid) further show that v(r) is the limit v( s) where v(r s) is the s + v(r, s) where v(r, s) is the

THE UNIVERSITY OF MICHIGAN 7030-4-T function, differentiable as above, in the domain {Dx 0 < a < ao satisfying V2v+y v = p(r) (y = (w - is)) av v or n = 0 on the boundary of D an Iv(r, s)2 dV <oo; SDI dV is the usual volume element and D' is the closed exterior to D. 1.3 Coordinates of the Paraboloid of Revolution The natural system of coordinates, i. e. the system for which the wave equation separates and the boundary of D is a level surface, is defined in the following manner: ito families of confocal paraboloids of revolution e = g, rl = o with focal point at the o origin, given by the equations 2 2 2 22 p2 4( - z), p = 4rn(r+ z) (p2 x+y2) together with the usual azimuth angle p. If we make the natural choice for the domains of these variables, 0, I, r< < oo, 0, 0 < 2r, they can be related to the rectangular (x, y, z), cylindrical (p, B, z) and spherical (r, 0, I) coordinates by the following equations x = pcos0 = rsinecos0 = 2 r cos y = psind = rsinOsin 2 = 2i sinf z = z = rcosO = e-fl. This system of coordinates is called the coordinates of the paraboloid of revolution and is illustrated in Fig. 1. Thus if a physical problem is shown in Fig. 2, we see that the domain D under consideration may be represented by 0< 7 l r1 0 0 4 e < ao, 0 0 < 27r, the boundary of D by 1 = 7r. The point source is shown lying to the right of the focal point, since this is the case that will occupy most of our attention. The case of the point source lying to the left of the focal point Ivill be introduced, but no calculations performed. 3

THE UNIVERSITY OF MICHIGAN 7030-4-T x aX ra = n -n =0 n=o y FIG. 1: Coordinates of the Infinite Paraboloid of Revolution x!/A ~ro'(^'') - z (_,0) y FIG. 2: Point Source at (_,0), Field Point at (, n) 4

THE UNIVERSITY OF MICHIGAN 7030-4-T 1.4 Integral Representations of the Solution Integral representations for the solutions v(e, r), p) can be derived from the second and third formulations of (1.2). We first show how the method of Ritt and Kazarinoff (1959, 1960) can be applied to derive an integral representation from the third formulation. Thus we begin with the inhomogeneous wave equation in which the wave number has an imaginary part. V2v + 2v = p(r) (7= ( -is) (1.1) In the coordinates of the paraboloid of revolution V2v has the representation given by (Buchholz, 1953) 2v= i a (2 av)a f a(2Z)I+i} Lv aiari [r ) o2] But since p(r) represents a point source on the axis of the paraboloid of revolution, the problems have axial symmetry; hence the 0 dependence can be removed. Therefore equation (1.1) becomes U aV+ a ( +av)+2(+4) = (+ n)p(,r1). (1.2) For the point source at (E 0), ( +r) p(g, nl) = c&(~ - )6(rY) where c is a constant which depends on the precise form of p(r), i.e. on the definition of a point source. We now make the stipulation (or normalization) that our definition of a point source is such that c = 1. (Ti limplie that p(r) = 4r 6(r- r ), r the position vector of the point source. The'- - -o — o normalization is discussed further in Appendix A. 1.) Substitution of this choice in (1.2) yields a v + a( a- + 2(+ (+ ) = ( —6(7) which can be written as -L v-L-v = 6(e-E)6(q) (1.3) T 5 5

we THE UNIVERSITY OF MICHIGAN 7030-4-T Lxsy = - (p(x) +q(x) with p(x) = x, q(x) = -x. In order to proceed we must consider the operators L, L{ defined by the relations _ dy _ L y = -drl 2 dO C d9 d9/ Y 0 inasmuch as they do not correspond exactly with the ones studied in Ritt and Kazarinoff,2 2ws (1959,1960). However, since p(x) = x, q(x) = -Y x implying Imq(x) = x is'sodme q, c the conditions on p and Imq correspond. For L the homogeneous differential equation to be studied is L y-Xy = 0. It can be written as -1/2 di72 n dn (t 1) Y (1.4) The substitution y = url/2 results in the equation d2u+2 +X+ 1) drn 4i or d2. i 1 du + d(2i )2 4 + 4(+2it7))) 2i 0 which is Whittaker's equation. It has the two Whittaker functions M+ X/2iz 0(2iyr), W+ /i 0(+f 2irm) as linearly independent solutions. A complete discussion of the X/2Fy, o equation together with these functions is found in Buchholz (1953). The solutions MX/2iya (~ 2iqy) are regular at n = 0 and E 4(O0, n) where.4(a, b) is the class of all square integrable functions on (a, b). Except for certain values of X the solutions W+.Xiy. 0(~ 2iYY) are not regular at zero, but are EZ4(0, 1 ) for all values of X. 6

THE UNIVERSITY OF MICHIGAN To see whioh solutiona should be oered for te definition of the rolv.'rt GraSe a function we make use oltheoondition v(r, s)[2dV < oo. D' For large n + x/2i(y, 2im) ~ e = e = thus the solution W /2 (-2i) cannot be used to build a linear coipponent of v(, a) and we must consider the solution W /2. (21m). Since the solutions M+/i 0(+ 2iyr) are linearly dependent, the solution M./2iy 0(2iTr) can be considered. Therefore y, >.) = n,1/2 M/2, (2r1) and Y2(r, X) = -1/2 W/2 (21) are two linearly independent solutions of (1.4) such that (1) Y1(i A.) is regular at r = 0 (2) y2(r), A) is not regular at r = 0 except for the value 2 = n+ 2 or 2 2kY2 2 = i7(2n+ 1). This leads at once to the properties of L, 0< rl < rl (a) = 0 is a regular singular point of L y-Xy = 0, p(O) = 0 (b) p(r ) i 0 (c) For ImX <q = 0 (hence X cannot equal i'y(2n+1)), the homogeneous equation L y- AXy = 0 has exactly one linearly independent solution regular at n = 0. Now to find the resolvant Green's function of the operator L, 0 g rl r lo, we need a solution 0l(rl, X) of L y-Xy = C which satisfies the boundary condition at r10, together with a solution 02(7 ) of L y- Xy = 0 which is regular at = O. Considering first the Neumann problem, this is accomplished by the choice ( 1(Xr ) (( (l) 2 ( ( )\ P(VIA " S =2 M dyi \ dT17A 7

THE UNIVERSITY OF MICHIGAN 7030-4-T (Hereafter we shall denote the derivative ( dr 1) by F'(ro,X).) With these definitions the Wronskian W [l(n, ). 02(n X)J of l(r. X) and 2( X) becomes w [pn. X), 02(n X)] = (2iT)yi(j X) W[Y2( 1 ) X)1 y1 (, l)] which reduces to (Buchholz, 1953) w[01l(nrl)., 2( I)= (i )( It should be noted that.01(n0X) = 2irw[y2(n ).y 1(nX] = T.v 2iThe resolvent Green's function G(r, rl', \) thus has the value 0 1(IMX)(nI'MX) rI>n' nr, rl, X) = - 2 i,'x)2(n),<. the resolvent operator the representation Rxy = \ G(n n', )y(r') d'. Jo For the Dirichlet problem we can choose the two-functions 1 (n, X) = Y2(r, X)y1(o, X) - Y1(n, X)y2(o, X) 2(r1' X) = Y1(r,X) Thus W[(,. x).,2(rl, )]= (2i7)y1(.X) W[,Y2(r a), YI(n, x) 8

THE UNIVERSITY OF MICHIGAN 7030-4-T which reduces to (Buchholz, 1953) (21ty) Yl(nO X) w [P1(n.,L (n)'2 = 1 A) The resolvent Green's function then has the value 1 -l1(u X)2(a'X) n>n (2iy)yl.(rod) 0 1(,.,O(,.x) the resolvent operator the same representation as above. The properties of the operator L, can be written down at once. (a) f = O is a regular singular point for Ly-Xy = 0, jO) = 0. (b) For ImX< q = 0 the homogeneous equation L-y-Xy = 0 has exactly one linearly independent solution ( ) = -1/20 y(, X) = e M.X/2k, 02i7 K) which is regular at e = 0 and e.C(0, g ) 0 < o < o, plus exactly one linearly independent solution y2(, X) = 1/2 W/21 o(2i') X/2 2y,.0 which is regular at infinity and et (,oo) 2 0 The resolvent Green's function G(, S', X) thus has the value pfl J(9j, )y2(( y') k<)' G(C,',A)= - 2iX) 2iy 7Y1(9.)Y2(Y' (" y ) e>', the resolvent operator Rk the representation 9

THE UNIVERSITY OF MICHIGAN 7030-4-T RXy = G(g, e')y(g') de' 0 It should be observed that in this case R is not only analytic in ImX < q = 0, but also in the larger domain Im X < k. As in Ritt and Kazarinoff (1959, 1960) the papers by Sims (1957) and Phillips (1952) are the basis for the work on resolvents and resolvent Greeen's functions. We can now proceed with the method of Ritt and Kazarinoff to find the integral representations for the solutions. Consider the Neumann problem and let r be a path in the complex X-plane defined by the straight line running from -o - ia to o- ia 0 < a < k. Then applying the resolvents R, R -t successively to equation (1.3), using the resolvent relation R(Lxy - Xy) = y and integration along r, noting that the singular points of Rx lie above r while those of R, lie below P, we arrive at the integral representation VNt(, Tr s) = 2 r i G(,*-X)G(7n, )dX or where vN(E, ri, s) will denote the solution to the Neumann problem. Substituting for G(, -X) we obtain lima 1 d ______________ N 70+ 2-ri - dX2 )(Y (2 ( Y1(1,-X)y2(2, -X)yl(0, X)l (rl X) N" S(2iT)2 y'(i X) where 1 = min( ), 2 =max(g,_). But *See Appendix A. 2. 10

THE UNIVERSITY OF MICHIGAN 7030-4-T y1(O,X) = (21-)1/2 (1' ), = y X)y1 (o, ) - Y l (' X)y(,( 1.X) and therefore.3/2 ~ + v ) li (2iy)-3/2 d 2 2i 2 VN(' ) a 0+ 27ri jyd(X.X ) -- 0+ 2xi2' 0 J.F 1 0 [* (n X)yi(no X) - yl(t, X)y2(no 2)] (1. 5) When n = 7o, 0l(1o'X) = (22 ) thus the field on the surface is given by the simpler formula )i-1/2 X lim (2!)- 2 VN(, no sO 0+2 rl(ri) dX yl(n,X Y1(1 -X)y2(2,-X) (1.6) The above procedure may be repeated exactly to obtain vD(g, r7), the solution to the Dirichlet problem. Let r be a path in the complex X-plane as before, then l (27)3/2 r P -.- +,,, lim (2iPy) (, - \2'~7 ["(~ + \2 2i'. VD( =) s —0+ 2i j y(n ) 2l2X) [y2(, X)y1(r0, X) )- Y1(r, )y2(o, X> (1.7) The surface field for the Dirichlet problem is given by the normal derivative (vD(, n)/8n). This derivative is governed by the relation 11

THE UNIVERSITY OF MICHIGAN 7030-4-T avD _1/2 avD an ( )1/2 arn thus the equation for the surface field is given by D(, rl) ( 2 t+ -1/2 D u _ \_8 -0 \rY/_ ci 2 a7n ) =o 12 2ri dX y((n x) Y1(1-X)y2( - 2 0 1- o(o+iO)1 2r (1.8) Consider now equations (1. 5) through (1.8). We note that there is an essential difference between these equations and the similar equations in Ritt and Kazarinoff (1959, 1960). For these equations r is independent of s, thus it may be possible to take the limit as s-> 0+ inside the integral. According to Buchholz (1953) the functions y1(x X), y,(x, X) are entire functions of X/2iy; the path P is defined so that the functions ri +') r +- + are analytic functions of X/2iy on r. Therefore we can take the limit as s -- 0+ inside the integral and equations (1.5) through (1.8) are valid without the 0+ condition when the parameter y is replaced by the parameter k. Buchholz (1942/3,1953) derives an integral representation from the second formulation of section 1.2. For p(r) = 4r6(r- r ) (where r denotes the vector to the point - -O -O -ikR source at (, 0)) the free space Green's function has the form - e /R (time dependence e, R=r-r |). Thus he first derives (Buchholz, 1953) an integral representation -ikR for -e /R as -o'+ io -ikR e -cT+icD ek 2ik 1 1 (0) (0) (0) R. - - i dsrs+ )F(-s+-)m (2ik)w (2ik2)w (2iknr) t 27 i 2 2s i s 2-s -o'- ioo where la'I < 1/2, i, > 0), m(O)(2ikx) = (2ikx) l/2Ms (2ikx) and w()(2ikx) = (2ikx) 1/2W (2ikx). Then he assumes v(,, n) has the form 8 8, 0 / -ikR\ v(,7) = v'(_e_)+ (+ e with 12

THE UNIVERSITY OF MICHIGAN 7030-4-T -a'-l+ ioD v'(~..) = 2 i- I0)(2il)W (0)(2 ) m(0)(k I((r dsP(s+-)P(-s+-)m 2wYi ci 2 2 + i))w (sik Am Is E2 s -) where again a'J <1/2 and A is an unknown function of s. Thus v(F,r ) formally satisfies the inhomogeneous wave equation while the boundary condition may be satisfied by a suitable choice of A. For 5 (0)'2i vn =0 on the boundary A i 2 (k A8 m 2ik..). m-8(2ikro) w() (2ik1) v 0 O on the boundary A =21k 0 where again F'(2iko1) = (d F(2ikl)) We continue by considering explicitly only the Neumann problem. The obvious modifications can be made for the Dirichlet problem. Substituting for A we have 2 )1 1 (2ik1 0 2) 2i * i a-at i OD In (0) (2 k (0)2 vNM n) = 2,. di ds r(s+ )r(-s+ 2) 1 vN 2w 1 ia s+ mn s(2ikr o) -a'- iOD -is 0 * W( (2iklo)m(, S(2ik{) - m(O)' (21kyoo)V (2ikn)i (1.9) as a formal solution to the inhomogeneous wave equation which satisfies the boundary (0) condition. From the asymptotic behavior of w (2ik 2) at infinity is seen that vN(e, n) satisfies the radiation condition. Thus it remains to show that the integral exists. Let us return to the previous Y1(x, X), y2(x, X) notation and substltute -s for a in equation (1.9). Then we obtain 13

THE UNIVERSITY OF MICHIGAN 7030-4-T 3 k-1/2 Sa'-L oo (21ik /2 a'1 1 O y l(,-21ks)y2(2a -2iks) VN(9) 2ri Ps rs+-s).(-s+.).2.N= 2r ri 2 2 y 6 2 (iks) a'+ ioo * Yl(n,' 2iks)y(no, 2iks) - y2(n, 2iks)y;(n, 2iks) (1.10) It is shown in the Appendix A.3*and Buchholz (1942/3, 1953) that r(-s+) [y (n. 2iks)y'(n0, 2iks) - y2(n, 2iks)y (n, 2iks) is analytic in the complex s-plane, y' (n, 2iks) has simple zeros which lie along the imaginary axis and the Integrand of equation (1. 10) vanishes exponentially on a large semi-circle in the right half s-plane. Thus the integral for 0 < a' < 1/2 represents the zero solution which can be omitted by the further -estriteton -1/2 <a' 4 0. However, for a' = 0 the integral is not defined and so we arrive finally at the restriction -1/2 < a' < 0. In addition it is seen in the appendix A. 3 that along this path the integral converges. Thus vN' n) given by equation (1.10) together with the restriction -1/2 <a' < 0 is a solution of the Neumann problem given by the second formulation of section 1.2. We write it as a'- iD 12ik)1/2 1 1 yl(1, -2iks)y2(g25 -2iks) -VNf r{) = 2\7 +\ ds'(s+ ) r(-s+-) y 2 NNjf'= 27 i 22 y (6o, 2iks) (-1/2 <a' <0) * [y2(n, 2iks)yi(ro 2iks) - Y (n, 2iks)y(n, 2iks) (1.11) If in (1.11) the substitution s = X/2ik is made, vN(, n) becomes V, R =) c=a (2ik)_32 ar 2 -X)' 2r \i\2ik ik~ -y + 2ak+ * [Y2(n X)y (no X) - y1(n, X)Y2 (' X) * In the Appendix A. 3 we again consider only the Neumann problem. The computations can be easily modified for the Dirichlet problem; the results are essentially the same. 14

THE UNIVERSITY OF MICHIGAN 7030-4-T with -1/2 < a' < 0 which implies 2ik)-3/2 rio — ~ i) (-.i) L ~~-3/2 ~ X \ 1(Y1, -X)y2(% -X) VN(dn)=r ~ r y+ -... VN( ) f 2)i 2i2k dX 42r-2 17 +() -oAD -ia (0 <a <k) [Y2(rl X)yi((. A) - y1(r, )y2(rlo0, X) (1.12) As is expected, equation (1. 12) agrees with (1.5). Substituting r= nr in (1.12) yields the eqution for the surface field (2ik) -/2 - io yi_______ VN(,D n~) = rio(2iri) dx( P1 X)Y2(S2, -X) *YN(' (E ) = (2( \" dA r +, ) (1.13) VN(~ ro) = (27i),eikk_/ y-o d X)Y) 0-cO-ia (0< a <k) which of course agrees with equation (1.6). Repeating the above arguments we obtain for the Dirichlet problem (+. + y(1 -X)y2(2 -X)) ( -3/2 (2i) = 1 yi (x 23, + -_oo - ia 2i 3 - \2 2 %2ilk 2/ - r ) (0 <a <k) * 2(n, X)Yl(no0 ) - y1 (. X)y2(o, X)] (1. 14) while for the surface field(avD({ n) = -1 (2ik~-/2 d P +Y ()I -X)Y2(' -X) n 12= ", [nOe+ ]1/2 2r i 1/ - ia kr 2) 0 0 (8+77 0 -OD - iW (0 <a<k) (1.15) These integral representations (equations (1. 12) through (1. 15)) will be used throughout the paper in preference to the corresponding representations obtained by the substitution s = X/2ik. 15

THE UNIVERSITY OF MICHIGAN 7030-4-T As indicated previously the above derivations correspond to the point source of (I, 0). The integral representations corresponding to the same point source at (0, H) can be derived in a like manner without any difficulty. For r11 = min(r), H), r2 = max(rl, H) they are -3/2 So - ia M IA.)y VN(~,DT) = (2ik)32 " Yia N'7 i N 27ik ( X) — oo- ia (0 <a <k) [Y2(2. )y(. ) - y(2n )y((no, x)] (1.16) for the Neumann problem, and -3/2 O(D -a i\ / y -X)Yl (rl X) V (I ) = (2i'-k) r + - f r (. -- 1 1' D 7 2iri J - (2ik 21' 2ik Y (,.X) (0 < a < k) rY2,(n2o.)y (, ( ) - l (T2g a)y2(r7o X]) 1 (1.17) for the Dirichlet Problem 1.5 Discussion of Integral Representations with Respect to Asymptotic Expansions For the present discussion we consider the integral representation given by equation (1. 12). The statements made can easily be changed to apply to the other representations. We have already mentioned that part of the integrand of (1.12), r (- 2k2 ) [Y2(,IX)yi(o, )- Yl (rI, )y(r is analytic in the complex X-plane, y' (ro, ) has simple zeros which lie along the real axis, and the integrand of equation (1. 12) vanishes exponentially on a large semi-circle in the upper half X-plane. Therefore the singularities of the integrand of (1.12) are of two types: 16

THE UNIVERSITY OF MICHIGAN 7030-4-T (1) simple poles at the zeros of yl(n10 X) which le on the real axis, (2) simple poles at the poles of the function (- 2 +- which lie at the points X = -ik(2n+1) n 0, 1,2,... on the itlative imaginary axis. Since the contour of integration runs between the two sets of poles and the integrand vanishes as described above, the contour can be closed around the poles corresponding to the zeros of y l(rn, ). In this manner, by use of the residue theorem, a series expansion for the solution can be obtained (Buchholz, 1953). The same asymptotic expansions used to demonstrate the vanishing of the integrand show that the series converges (Appendix A. 3). However, the series dmst lend itself to asymptotic analysis when k >> 1 and thus corresponds to the situation found in the scattering by closed convex bodies (Ritt and Cazarinoff; 1959, 1960). If the correspondence is complete the series should readily yield the first term of the asymptotic expansion when krj << 1. This is found to be true; the calculations are performed in Chapter 3. There are two cases of interest when ki) << 1. One is kE1<< 1, k2 >> 1 which corresponds to the point source (field point) in the near field and the field point (point source) in the far field. These conditions correspond to the most likely physical situations; a possible application is the use of the surface field to consider scattering from a body whose surface has a concave portion. The other case of interest is kg << 1, k << 1 1 2 and corresponds to both source and field points in the near field. This case is of interest because of the relationship with the potential problem. In Chapter 3 it is shown that the solution to the Dirichlet potential problem agrees with the first term of the asymptotic expansion of the solution to the Dirichlet problem when kr)o < 1, ki << 1 and kg < 1. While these low frequency results are complete, further low frequency nvestigations are in progress. In particular the Neumann potential problem as well as a physical interpretation of the far field results are being considered. The work for k r>> 1 is being carried out at the present time. Thus, Chapter IV is only a summary of a possible approach to the analysis. 17

THE UNIVERSITY OF MICHIGAN 7030-4-T II UNIFORM ASYMPTOTIC EXPANSIONS In this section we will consider separately the two asymptotic representations of the Whittaker functions necessary for later use. The cases studied all correspond to those used in the problems investigated in future sections. The results obtained are based on the work of Langer (1935, 1949). More detailed results are obtained in the memoir of Erdelyi and Swanson (1957) which discusses the necessity for the two representations, and the paper by Taylor (1939). 2.1 Airy Function Representation For this section the Whittaker equation will be written as 2 [+ + u =; (2.1) dz 4z ] hence the two Whittaker functions of interest are M O(z) and W (z). We make the substitution s = z/41, thus equation (2.1) becomes d 4 + )+ u= O ds L 4s 2s and upon defining = -2Ui we obtain d2u 2t(-)+ u = O (2 2) ds 4s Considering p to be a complex parameter such that Ip I~1 and s belonging to a simply connected, closed domain of the complex plane which includes the point s = 1 but excludes the point s = 0, we find equation (2.2) can be of the type studied in Langer (1949) and his results may be applicable. Let 0 (s) s= where 0(s) is the root of p2(s) such that 18

THE UNIVERSITY OF MICHIGAN 7030-4-T We also define 1(s) = \ ()d7,? = p~ i8) (2.3) and ~(s) = [(s)s1/6[()}-1/2 with 1) = ); (2.4) these are the functions to be used in the theory. For the applications we must consider the region args = [-6, 6] where 6 is a small positive number. Then, provided Is >> 13 Langer's results are applicable (Taylor, 1939). For [s >1 we can write )(s) = s(s - 1) - log (s - 1 + Vij) (2.5) while for | 1 we use -(s) - ti 12 d (2.6) 1 where the upper (lower) sign corresponds to the argument of negative real s being r(-ir). When s w 1 s- l 1 = O(1/p-2/3),(Taylor, 1939; Erdelyi and Swanson, 1957; Buchholz, 1953) we employ the expansions (s) = (s- l)2 [1- (s-1)+ (s-1)2] as s- 1 (2.7) and (s) =3 ~(s-1)3/2 ( l)5/2+0((-1)7/2) as s- 1. (2.8) In order to derive the desired asymptotic representations we need the behavior of the 1/2 above functions as s -* o. Thus expding (1 - 1/s)1/ about s = o leads directly to 1 2 (s) = 1- 2 + 0(1/s2) as so (2.9) 19

THE UNIVERSITY OF MICHIGAN 7030-4-T and using (2. 5) and (2. 3) respectively yields (s) = s- - -log2Vi + 0(1/s) as s —oo (2.10) and = p - 2 -log2V'+0(1/s) as s->oo (2.10a) We wish to find the asymptotic representations of the functions Ml O(z) = MI 0(41s) asymptotic to -4 21s ( -21s -2rs -i(I- -) (41s) e +(41s) e 2 + 1 e r(l - ) r(l + ) 2 2 (4s) - 2s for large Is, -7r/2 <argls<3r/2, and W 0(4es) asymptotic to (s e for (- I) large Is, jargIs | < 3r/2. We shall accomplish this by finding the asymptotic representations for WI 0(41s) and WI 0(-41s) and using the relation (Buchholz, 1953) e=7W i 0(-41s) -Iri 7ri/2 M (4k, ) - 0 + e 0 e (41s). (2.11) 1,0O 1 1 10 0 0r( -I) r( + ) 2 2 (In choosing the particular form of (2. 11) we make use of the value argls = 7r/2 which occurs in the applications.) To find these asymptotic representations we must compare WI 0(4)s) and W 0(4s) with the functions V)(s) = (7r/21/2 e 57i/12 s))1/3 H(j) () j = 1, 2 "1/3' which are solutions of the differential equation d+ LP2 ( s -) 1 y =, ds2 Ps) called the related equation of (2. 2). These functions have the asymptotic behavior for large? 20

THE UNIVERSITY OF MICHIGAN 7030-4-T (j) =/6 + =1, - < arg< 2ir V (s) N(sK)r /6i +r/ j=2, -2r <argC<wr Now if arg s = a e-, 61 then due to the nature of the applications argl must have the value argl I=/2-a and thus as -coo arg — 0. Therefore either of the above expan(2) sions is valid as s -oo; from (2. lOa) we see that V (s) has the correct exponential dependence in s for WL 0(41s), V (s) the same for W -1 0(-41s) and hence according to Langer (1949) W_,1 (-41s) = [v(l)(+ 0(1/] N (2. 12a) -W, 0 (-41s) = C V (8)0 ] |I|>N (2.12b) W2, (41s) = C2 [(2() + O(l/P)] IC N (2. 13a) W, o 41) = C 2 (s)( + ()-/ e ~l ] ll>N (2. 13b) where N is a large positive number and the C. j = 1, 2 are determined by the relations lim W 1 0 ( —41) 1 e -1, e (1)~, 1/6 e - 11 s —oo - W (4s) 1 1s)' lim W 0(41s (4) (4s) e 2 s -*a V-(2)() - s)-1/6 Using the definition (2.4) and the expansions (2.9) and (2. 10) we obtain C, = (-2lU)1/6 e-log-1/e C= (2tU)1/6 elogl/e Thus (2. 12) and (2. 13) become for C = eog /e W 1 0(-418) = (_21)1/6 C0_[V(1)(s) + O(1/)] 1 1 >N (2. 14a) 21

THE UNIVERSITY OF MICHIGAN 7030-4-T 1/6 - 1/ 6 i? OM W 0(-41s)= (-2i)1 C "(1)(s) + S) e' O i >N (2.14b) w-1,0-s) - P W (41s) = (-2iU)1/6 C[V(2)(s)+ O(1/p) I N (2.15a) I, 0 = (41s) = (-2i)l/6c v2)(s)+ (s) N (2.15b) W,0 0 { 14s) j Using (2.11) and the relations (Erdelyi et al, 1953) 1(+ )= eI log +e O(1/1e valid for |I>> 1, we obtain ( _i) 1/6- 7r if Iv(1i e.ri/2-(2)(,M o(41s)= - V (s) + s)+ eO(1/p 1.1 N (2. 16a) _> e1/6 - l [' (2) )+16 - (li M (41s)- ( - L (s) + e (s) + [e (1) + e |1|> N (2.16b) For purposes of calculation it is often more convenient to represent the Hankel functions H13 (?) in terms of the Airy function Ai(g) defined by the integral representation 1/3 1 \ 13 Ai(z) = - cos(3s + zs)ds IT o 3+ Jo We can use the relations (Abramowitz and Stegun, 1964) (1) -ri/6 =3- 2/3 H1/3() = e / Ai(-)- - = ( )2/3 1/3 2 [Ai(-)- Bi(-.] = 2 e) i/3 - /3 (2) - i/6 Ai( iBi(-a] ()2/3 222 22

THE UNIVERSITY OF MICHIGAN l - 7030_ —T [Ai-a)+ i Bi(-i = 2eiA3 Ai(-Ue'27i/3) (Bi(g) is an Airy function linearly independent of Ai(g) which does not enter in the final result and so it is not defined here) to find H(1) -ri1/2 (2)4/3 (3)1/6 21i/3 H(1/3) = e12 i /3 A(-ae <(2).- _ i/2 (2)4/3 (3)1/6 A 27i/3) H(1)(M) = e i( A-e-2X 1/3 A1/3 Thus equation (2.14) and (2.15) become Wt (_41s) = 2(3)1/6 1/2 ()1/6 -7ri/12C[ )Ai(-2i/3)+0( W~1'00(-41s) = 2(3) 7r e C s)Ai(-ae + 0(1/ I I<N (2.17a) rW Ile (- s) = 2(3)1/ 7 (i)1/6e- ri/2C [(s)Ai(-ae /3)+ p L (4) =2(3)1/6 1/2( )11/6 e- i/ 20 i3s)Ai(-27 e /) +. O(1/ lil(N (2.18a) w^(4s) = 2(3)1/6 1/2 ( 1-l/6 7ri/12C[)Ai ri/3) (S) F/' 6 i > N (2.18b) with [3]2/3,, 2r/. r/.~-2i/3 23

THE UNIVERSITY OF MICHIGAN 7030-4-T M 0(41s) = (2)1/3 (6)l/6 eri6 (1)1/6 -[e- (s)Ai(-&)+ O(1/p) IC| N (2.19a) Ml 0(41e) = (2) /6(6)16ei'/ /6()l16e (s)Ai(-a)+,s)r 1/6fe i0(1)+-i O(1l M, 0{L. P I1I>N (2. 19b) 2.2 Bessel Function Representation If after the substitution s = z/41 in (2. 1) we define p = 21, then equation (2.1) becomes d2u 2 1-s 1 2 + p(S)+ u1 = 0. (2.20) ds 4s j Suppose then we consider p a complex parameter such that I p |, 1 and s belonging to a simply connected, closed domain R of the complex plane which includes the point s = 0, but excludes the point s = 1, then (2.20) is of the type studied in Langer (1935) and his results may be applied providing R has all the properties he requires. We will s assume that R has these required properties since it is straightforward to show that 8 the domains used in the applications have them. Let 0 (s) = - (1 - s) where p(s) is to be the root of p (s) determined by the relation 8 {slimo alp( =- 1. We also define 8 (s) = \ (T)dr,? = p (s) (2.21) JO -1/2 lim ab(s) = [(s)(s)- with =(0) = s (s) (2.22) 24

THE UNIVERSITY OF MICHIGAN 7030-4-T For the applications we must consider three regions of s, defined for 6 small, positive as (i) argse[r-6r,r, (ii) argse [-r,-r+6], and(iii) argse[-6,6], Isl<l. When argse [- 6, r], equation (2.21) can be written as h(s) = -i - 1)/2 (2. 23a) while for arg s e [-.r,-r+ 6] we can write 8 _ -o(s) = i 1)/dr. (2.23b) In either case (region (i) or (ii)) i( /2dr = -^s(s-1+ 1og(F- -i) (2.24) 0 where again the square root is such that im Vls =1. For region (iii) Langer's results are applicable provided I11-s8 1 (Taylor, 1939) 1P2/3 In order to derive the asymtotic representations, the behavior of these functions as s-40 and s —^a must be known. As s-40 we expand (1-s)1/2 about s=0 resulting in () 1 1 81/2 + /2>s3/2) as - 0 (2.25) Therefore equations (2.21) and (2. 22) yield 25

THE UNIVERSITY OF MICHIGAN 7030-4-T (s) = 281/2 - 83/2 + O(s5/2) as s- 0 (2.26) = pu(s) = 2ps1/2[11- I + o as s-+0 (2,26a) (s) = 12 [1+ s+ O(s2] as s- O (2.27) and () = 2s 1+ s + 0(s2 as s-O (2.28) 0(8) 1 In examining the behavior at infinity we will only consider regions (i) and (ii) since in region (iii) we are concerned only with I s < 1. Thus expanding (1- 1 1/2 about s = o in region (i) yields Ad = -i[i- 21 +0(1/s2] as s-4-x (2.29a) while expanding about s = i) in region (ii) yields (s) = i - - + (l/s2)] as s-4oo. (2.29b) Equations (2.21) and (2.24) then given for region (i) r = p~(s) = -ipLs- -log(2 s) + 0/s) as s —o (2.30a) and for region (ii) = pl(s) = ip[- ] - log(2 V) +0(1/s)] as s- oo. (2.3Ob) We wish to find the asymptotic representations of the functions Ml (z) = M (41s) I -2ts regular at the origin and W (z) = W (41s) asymptotic to (41s) es for large Is, 1,0 1,0 jargls | < 37/2. Then for V~(s) = f(s))Jo(0) the theory in Langer (1935) asserts 26

THE UNIVERSITY OF MICHIGAN 7030-4-T M O(4^) = C [V (a) + -)~ I,<N (2.31a) P M, 0,(41s) = C [v~(. + 1/2[lO() + e-Ol) |5|>N (2.31b) where N is i large positive number and C a function of I determined by a comparison of the behavior of V (s) and Ml 0(41s) as s- 0. V~(s) is a solution of the differential equation + _)+...(. S() Y = 0O ds2 4s2 which is called the related equation of (2.20). For the value of C we can writf lim Ml 0(41s) — 0 VO( ) and thus (4f) 1/2 1 - _ _ = 2 1/2P 1/2 Therefore, equations (2.31a), (2.31b) become (41) 1/= 2 (f(&/2 ((P)+ m) IrI<N (2.32a) N 0(41s) L' [-J~J + 9/2 1/2 1/2 [BC 0(1) C Ml 0(41s) = pl/2 + e 4 >N (2.33/2b) |?|>N (2.32b) In uThtbtr 3 we will be primarily concerned with the case IpaI < 1. Hence s -0 and we can use expansions (2.26) through (2.28) to find for I| | O N 27

THE UNIVERSITY OF MICHIGAN 7030-4-T M f(41s) = (2ps)l/2J (i ) [1+0(s) 0 +O (ps) /2s2)'t O 0 Upon retaining only the order term of the lowest order in s this becomes M,0 0 o except at a zero of J (r), where then the additional term is the required estimate. For r|> N and? not a zero of J (r) we have Ml. (41s) = (2ps) 1/2 Jo()0l+0(s)+O(i/p) while at a zero of J (I) the estimate 0 Ml 0(41s) = ~ /4 is valid. Upon comparing order terms the above equation reduces to M, 0(41s) = (2ps)1/2 J () 1+ (l/p)]. (2.33b) We note that the above results for Ml 0(41s) are valid for a in any of the regions (i), (ii) or (iii). However, in deriving the asymptotic representations for WI 0(41s) we must be careful to distinguish between regions. In any case we need the following solutions of the related equation V()(s) = (r/2) /2ti ir/4 g)H(J)(H ) =1 2 with asymptotic expansions for large I | given by (j) 1/2 e+ir + E =1, -7 < arg <27r V(s) = S (s) IeiL +O (/p1 ] j = 2, -2r < arg C < 2r j=2, -28<arg<r. 28

THE UNIVERSITY OF MICHIGAN 7030-4-T Case 1: arg sEr- 6, 7] Let args = 7r -ae [r- 6, ]. Due to the nature of the applications argp = -r/2 + a and thus as s -ao, arg r — 0. Hence, either of the above asymptotic expansions are valid as s-B o and from equation (2. 30a) we see that V (s) has the correct exponential dependence in s for W 0(4s), V)(s) the.same for WI (-41s). Therefore according to Langer (1935) W- 0(41s) = CLv(1)(s) + iMs)logC0( 1N (2.34a) W!i, 0(4rs C[v<() Cg 12 ep OM W- (41s) = C1 [V(2)(S) + >/ 1 I|>N (2.354b) (2)(s) + ~(s)S1/,2 i~ot1 wI. o(41s) = C2 V )irO ] l >N (2.35b) where N is a large positive number and the C. j = 1, 2 are determined by the relations w (-41s) -A 21s lim W-, 0(4s) (-4s)- e2 1 s —co V(1) (s) 1/2 i (41s) -2(s) /2 lim V. o(48) (4/s) e2 8 -0o V(2)( 1/2 iC Using the definitions (2.21) and the expansions (2.26) and (2.27) we obtain (-i) 1/2 -log/e and (-i)1/2 log -/e Cl ~ 1/2 e and C 1/2 P P Thus with D = e Ilog -/e equations (2.34) and (2.35) become 29

THE UNIVERSITY OF MICHIGAN 7030-4-T W I0(-41) = (- )l/2 (1) + L(s)r log O(1) | N (2. 36a) 0 (-41) = 1/2 DF )(s) +' IC 1< N (2.36a) 1,0 1/2 ( P1 W (41s) = D [v(2)+ ()r ( ] 1i>N (2.37b) 1,0 0 1/2 1 I I (236b) 1,0(41s) = -iz D l[v(2)(s)+ CS):lp I < N (2.37a) Case 2: args [-6, 6] s <l1, args = r for s negative real. In this region we cannot directly derive an asymptotic representation for W 0(4es). However, we have the representation (2.32) for Ml 0 (41s) and can show (Erdelyi and Swanson, 1957) that the function W-V() (-) 1/ D (2)(s) + E(s)logCO(1] 1I N {1/2 3 P is a solution of the differential equation (2.20) in this region. From the definition of (2) V (s) we see that these two solutions are linearly independent and thus W( (4s) = AM (4s) + BW(s) or W 1(4)s) = A- (s)CJ() + O(termj ])1/2 + B 1_2)(s)+ /(s)1/ 2 + B -)1/2 D1 [( /2s12 ei/4?s) H( o(() + O(termj 30

THE UNIVERSITY OF MICHIGAN 7030-4-T where we have used O(term) to denote the two order terms for I| | N or |I |> N. Now we have the relation H(1)(C) + H(2) O O J (~) = o 2 thus the above equation may be written as ) H(1) H (2)'I —-+ - o + O(termj o(4s) = A 2 + 0 + Wit0 P1/2 L 2 2 + B (i)/2 1/)2 -i()4 H (2) O(term) +B 1/2 DiL(7r/2) e + () H (+O(term) ~~~p~~~~ ~ (~~'2.38) Since the representation (2.32) holds in region (iii), jl - s j > l p so does equation (2.38). Now from section 2.1 we know that equations (2.15) hold in region (iii) provided Is >> 1/Ip /3 Then we can substitute this representation in (2.38) provided 1s I|~> /IP /3 and since s - 1 >> 1/[ p 2 /3 the asymptotic expansions of the Hankel functions may be used in equation (2.38) provided -ir < arg r < 7r, -27r < arg < 7. Consider then args = 0, from the nature of the applications argp = argl = 7r/2 implying argr = 0. But then argp = 0 implying arg = -Xr/2. Thus for args =0, (-2)1/6 ellogl/e -8) -1/6 e-i 1 1/2 e_ 2 i 01/2 (as) 2 /7r) 2 + (2/7 )1/2 e + B 1/2 - e is)c e 1/2 This then immediately implies the values A = 2( r/2)1/2 (_i)1/2 ei /4 elog-1/e B = +1 31

THE UNIVERSITY OF MICHIGAN 7030-4-T Therefore equation (2.38) becomes We o ) = + (- i)1/2 e-i/4 /2 H(1 )(D) H + ) log C O(1)] 1,0 +1/2 L o P 1 rl N (2.39a),r 0(411)= (1) 1/2 i( WIN, O(4s + - (r/2) /2Di/ ) I ) () (+ P/2eO I||>N (2.39b) The question of sign can be resolved by again considering args = 0. Then we have argl = r/2, thus we can compare equations (2.39) with the result in Erdelyi and Swanson (1957). Therefore we see that the negative sign must be used. Since (i)1/2 e-ir/4 = (i)1/2 ei/4 we obtain WIN 0(41s) = *1/2 V(1 ) + (s)lg )] 11 (2.40a 1,0) D 1 1/2 p 1 W, (41s) = i()l /2 D (21s) /2 Hl() + (s) + (/ V(1)(s0 1 except at a zero of H (r) (ther aee no zeros of H (exp) on the principal branch, 0 0 Erdelyi et al, 1953) when the addition term in (2.40a) is the required estimate. Upon comparing order terms this becomes W (41s) = i(r)l/2D(21s) /2H(1)() + O(l/p (2.41a) Similarly for > N (with the above exception for zeros of H 1)(g)) 32

THE UNIVERSITY OF MICHIGAN 7030-4-T W (Q4Us) = i) /)1/2 DH(2s)1/2 H1) l+(1/p (2.41b) It 0 o Case 3: argse [-, -lr+ 6] For this case, we consider p to have been defined by p = -21. Then equation (2.20) remains the same together with the resulting definitions and expansions. We let args = -r +ae [wr, -r+6]. From the nature of the applications argp = r/2-a, and thus as s-oo arg —-0. The asymptotic expansions for H( )(s) are valid as s- oo and from equation (2.30b) we see that V(2)(s) has the correct exponential dependence in s for W (4rs), V(1)(s) the same for W o(-41s). Therefore according to Langer (1935) W, 0(-41s) = E1 [V()(s) + >(s)r Og C0(1)] Ir6 N (2.42a) W_ 0'41s) - E 1(8)+ )L/ e 1 i>N (2.42b) Wo(4(s) = E2 [V( )(s) + (s)lg 2 | r| 6N (2.43a) Wt(4) E2 V (s)+ r>N (2.43b) where N is a large positive number and the E j = 1, 2 are determined by the relations lim -1(' (-41) -esa "1 s —a V(1) el=2ir V (s) 94)11 e lim W, (4 (41s)e2 s -)oo V(2)( 1/2 i V (s) p(s)n e Using the definitions (2.21) and the expansions (2.26) and (2.27) we obtain E = ge e E. e - 1 (_2)1/2'2 (_ g1//2 But for args = -r+a, argl = -r/2-a, hence (-I) = e 1r. Therefore, 33

THE UNIVERSITY OF MICHIGAN 7030-4-T E D E D E1 1/2 D-1 2 1/2 D1 1 (2U) (2) and equations (2.42) and (2.43) become -1, DS) 1/2 D 1V(F)(1) + s8) logO(1) (2.1a) W s,(-42s)= - D 1V0^)0 +(2.44a) -,0 (21M 1/2 - p W (-41ss) + 1 S) ~ e O(1)](2.44b) "(U1A2 P WI 0(41s) = 11/2 DIv(2)(s). + (s)C log o 0(1] (2.45a) (2~)1/2 D P + W i(-ts = _1 (1 0s)l/2 e-i 0(1)t W 0(41s) = 1/ D[V(2)(s)+I (s)~ /eirQ(1)] (2.45b) ^.0 M /2 4 P Having done this case we note that the restriction of arg s = ir for s negative real can be lifted from the results of Case 2. To demonstrate this we consider I s I < 1, args e r-6, ], arg s = -r for s negative real, and show that the result for W 0(41s) is the same as in Case 2. We have the representation (2.32) for M 0(41s) and can show as before that the function w,(s) = 1 D Fv(2)()+ (s)iCog (l) i = 1 D [V(2)(s)+ (s)1/ le () ICI<N (2i1)1/2 1 P (2 ) (s)c1/2 -i OM is a solution of the differential equation (2. 20) in this region. But then we must have WI 0(41s) = A'Mi 0(41s) + B'W'(s) and arguing as previously, we find 34

THE UNIVERSITY OF MICHIGAN 7030-4-T -i A' =;2(ir/2)1 -— e D (1)1/2 I B' = +1 This implies — i (4 _ e 4. 1/2 (1) ] | WI 0(s) =0 1+ 2 7(/2)a)D CH0 (+B)+ 3~ /e OM) ICr N -i 1., 4-i 1/2 [ EV(1)(S, + l( ( /2e iC 0 (1)],I N W o0(41s) = 1 (/2)/2 D,[ s)H( H / ()+ i |I|>N (2i1)112 O)] N whereupon we see that as in Case 2 the negative sign is correct. But since 4 i_ e = ()1/2e 4 (i)1/2 we have /2 e 0(1)M, 0(4sV() s= )/2 I V (s )] I N,0' 1/2 pr 1>N (21) This agrees with equations (2.40) since there p = 21 by definition. 35

THE UNIVERSITY OF MICHIGAN 7030-4-T m LOW FREQUENCY (THIN PARABOLOID) SCATTERING As indicated in section 1.2 the paraboloid of revolution may be characterized by the focal length rio. Then for a given wave number k, the mathematical condition kn~< 1 corresponds either to small k (low frequency scattering) or small no (thin paraboloid). We wish to investigate the integral representations in this case. For convenience these representations are now given a slightly different form. Let us define vl(x,)) = m>ik(2ikx) = (2ikx) 1/2 Mx/2i (2ikx) = (2ik)-/2yl(x,X) (0}'2ik(2ix) = 1 v2(x, X) = w((/2ikx) = (2ikx)12Wx) = (2ik)1 /2 2(X,X) and i(Xo'X) =(d2 )v' (X), =) 0 (Ths notation is different from that of Chapter 1 and Appendix A. 3 where the primes simply refer to differentiation with respect to x.) Then equations (1.12) through (1.15) become oo - io VN/. 2 i dXr, x I\ - +,,,.-. VN 2V(oi j k2 2ik 2 V X) -co - i7 (0< a <k) V* (, viv;( ) - vl(, X)v(n1. x (3.1) 1 X00'1f. \vl(S 1 -X)v2(2-X) V dXr +.. (, (3.2) VN('ro) = 2 i(2ikr k ) v(n ) -oD - io (0 < a< k) 36

THE UNIVERSITY OF MICHIGAN 7030-4-T [v2(n, )v(n.X)- v x)v2(r1, (3.3) (VdZD(' =r]i 21 1 So ia rr( + v y' 22-) Dn 2'oiri=) LNW i 2ilk 2 vl(, 1)7X) (0<a(k) (3.4) with the conditions of the derivation! ~ _, r > 0 mentioned for completeness. 3.1 Low Frequency (Thin Paraboloid) Poles The low frequency poles are those of the integrands of equation (3.1) through (3.4) which lend themselves to asymptotic analysis only when krn << 1. Hence they correspond to the zeros of the functions v! (rc X) and vl(o, (X) when k 1. In order to analyze these zeros we consider M/2i~ 0(2&) when kl << 1. If A = 0 then (Buchholz, 1953) v(n,X). But (, 0) = O(kr); therefore for k < 1, =0 is a zero of v( ). Suppose now 0 <IX/ 2k 0:(1), then M,2*/ o(2ikr) has the power series expansion (Buchholz, 1953) MX/2ik ( ) = I(2ik)1/2[1 +0(i )] whichimplies vl (, a ) = 1+ O(k ) ad Thene v (o<! ) = 0(1) and Vl(noi ) t = (1) indicatig that the re are no z for z 0 m io /2k or com(1).pl e To investigate y pes are tos of t ra ean (1 thoe e f | /2(k or e an us h ory which lend themselves to asymptotic analysis only when k () O1. hence they correspond to the zeros of the fctions v O( ) and v1 (, X) when k=l <. In order to analyze developed in Chapter 2. For the Whittaker equation 37

THE UNIVERSITY OF MICHIGAN 7030-4-T d2 1 X 1 du ++ u 0 d(2iki)2 + 4 2ik(2ki) +4(2ikil)2) d(2ikn)2 ( I = X/2ik, z=2ikn and thus s = 2k = -k21/X = -kt(k/X) where Is = knlk/XI << 1. = ~/2ik Now for 6 and B small positive numbers the results of section 2.2 for M 0(41 s) can be applied in the domains defined by |IsI|3, argse[Xr-6, r] or argse [-6,6] Isl<,. argse[-7r.-r+6] or argseC-6,6]. But since the zeros of MX,2ik O(2ikr7) occur for real X, s = -k 2r/X lies in one of the above domains. In particular (i s| = kn/2 << 1 and we can use equations (2.33). Therefore, we have MX/2ik, (2ik = (2l)l/2J(r) [l+o((kn0)]] |J( N MX/2ik o(2in) = (2ikU)l/2 J ()1+ O(k/X] R > N where C is given by equation (2.26a). In order to use this equation we must specify argX = 7r for X real and negative. Then we can write 2X 1 6 - - Z (3.5) 2X1/2nh1/2L+ ~kA k +O kn)2 k2)] (35 It immediately follows that vl(1 X) = J() 1 + ((knr) ) | K | N (3.6a) v1(, X) = Jo(C) [ + o(k/\) C I>N (3.6b) from which 38

THE UNIVERSITY OF MICHIGAN 7030-4-T vi(ri, ) Jo( d(2iC ) [1+o((kn)) ]+J (r)O(k/x) I< N vIn, X) = J, () d(i ) [ +O(k/X] +J () O(k/X) II >N. The second term for | | > N is present since the order term 0 ((km) -) also occurs in equation (3.6b) but is not written explicitly since it is of lower order than O(k/X). Using equation (3.5) we obtain vi(rX) = x7/2 J' )l + (kY)i) 1 +Jo() O(k/X) |r N (3.7a) 2ikr 1/2 0 0 1/2 vI (UX) X= 2-;;ik J'(r) [l +0(k/X)] + JO () 0(k/X) I > N (3. 7b) Let Cr' r = 12, 3,..., r=-1,-2,-3,... denote respectively the positive and negative zeros of J' (). Then the zeros of the function v (r), X) are given by the equation = r + o (k)1/2 k3/) (3.8) for arbitrary C. Since we are explicitly interested in the zeros Xr in the complex X-plane, we must solve equation (3.8) for X. Substituting for C we obtain the equation 2X1/2 / [1+0k) k- r+(k)1/2 k3/2 l>o S+ r rl) X3/2 which has the solution r 4r 4L E2 (3.9) But since C r = we need only consider r = 1, 2, 3,... Then kr << 1 implies Btsne=-r -r3 39

THE UNIVERSITY OF MICHIGAN 7030-4-T X /k >1 since r A,2 2 r r 1 k 4kin 4kn because C 3. 832. This demonstrates consistency with the assumption of | /k ~>> 1 and therefore v' (r1, X) has positive zeros given by the equation 2 =r 0(kr) X 5 [1+ (0 - r = 1,2,3,... (3. 9a) r 4 - 2 r To find the zeros of v1(io, X) we need, in addition to equation (3. 6), the estimate 1/2 2 k2):(kt) (kr) 2) of vl(r, X) at a zero of J (r) (Chapter 2). Thus if 13 r=1, 2,3,..., r=-1, -2, -3,... denotes respectively the positive and negative zeros of J (C), the zeros of v (rI, X) are given by the equation r 3r+ 0 (kt))/2(kn)2 k) (3.10) for arbitrary C. Substituting for C we obtain the equation 1/2 1/2 i+o ((kn) = 3r+ (k1o)l/ (kl 2) which has the solution riL 2 =r =+ ~+0. (3.11) r Again since f3 = -, we need only consider r = 1, 2, 3.. Also kn << 1 implies 40

THE UNIVERSITY OF MICHIGAN 7030-4-T X /k 1 since X 2 2 r rr 1 k 4kn y 4kr) because 381- 2.405. Therefore we again have consistency with the assumption of k/kl >> 1 and v (ro, k) has positive zeros given by 2 = l+o((k )] r=1,2,3,.... (3. a) r 4i L \ r From cos (- 7) which governs the zeros of J (r) for large r we note that the 4 o zeros given by equation (3.9a) for large?r go into the zeros given in Aypendix A. 3. That this limiting relationship exists follows from the fact that s =- - -kr(k/) can go to zero for any ktr provided k/X is small enough (X/k is large enough). Then although - s = kr) is not small the asymptotic representations (equations (2.32)) and expansions (equations (2.26) through (2.28)) still apply. Since in this case r is large, cos ( - W) governs the zeros. 3.2 Residue Series for the Near Field Let X = 0 and r r=1, 2, 3,.. denote the positive zeros of v'(rlo, X). Then we have shown in appendix A. 3 that equation (3.1) can be written as o0 V (,t1) E f" (2ik+ r( 2 -i 2 ) -;-^ r [v2(n, Xr)vI( X) r - v(l, X)V (ro r] which upon separating the r = 0 term reduces to 41

THE UNIVERSITY OF MICHIGAN 7030-4-T P(1/2)r(l/2)v 1(1, O)v 2(2 o)r N (dV v(vp' 2x N ( d rl (V A)l )v 2(2. ~) rt(O 0 - v(_. ~)' +2 r(ik 2) r + r (3.12) 2ikveo 2 d' -xr r:i: -x 1 --'(o X) Vl rlo' We make this separation since although VI (n O) O, v2(),, ) and v((no, 0) oo and thus the ratio \dX1 o )/I (~dA vl(ri0,)) L2 0 0) - 0) P o] must be carefully evaluated. In order to compute the derivative in the r = 0 term we need the definition (Buchholz, 1953) -1/2z O() = e~F 1 X; ) and thus oo I r mo() =-Z1/2Mlo(X) = e/2 ()rr! Z, 0 s r=0 rr If we consider small z we can write 2 3[1' ^+ ( 1 + 21) 2+ m, o(-) = 1 +( -2)z + (2 2 + O 0'"' ji2 4 2 4 2z2 1 1 1 2 3 z 1 z2 1 z2 ( -7i-~ +l) z2 3 mx(ob) M 1 Z+ O(za) 0 1-2ii2 2 4 + 4 2 2- +Oz 4 42

THE UNIVERSITY OF MICHIGAN 7030-4-T Hence d z 1 1 3 z 2 dz m; 0(z) = - ( -X)z+ ( ) (- + o(z2) and d_ d m o(z) = -1 + z + (z2) Od dzm0 Using these forms with = 2k and z = 2ik we have d v(,) = 2k + Xro+ o((k2) and (d (n0)); 2i ~ + 0 ) ) =0 21k The other members of the r = term are given by (Buchholz, 1953) vlsl( 0) = J (kgl) vt(%2, 0) = - 2r H(2 (k ) 2,2 2 0 2 v(, 0) = J (k7) vl,~o) - 41 v2(, o) = H2(k) + 0() + ^ 2 o 1 2 2 2 T 47r k"o k17o VI (TIO, 0) + n In - + O(ko) 2ik r 4i7r1/2 2 oD Therefore equation (3.12) becomes 43

THE UNIVERSITY OF MICHIGAN 7030-4-T VN() i J (k )H(2)(k )[l+O(k 2) N 2r o 1 o 2 L o~ mo (& iv1(' Xr),,r ('V 1 r 2 2 r 2ik r= (21k v r) *. (3.13) r To compute the r) members of the rth (r ) 1) term we use the asymptotic representations obtained in Section 3.1. v!1(qDX) = J((r) )[+ ok_) (r)f2N (3. 6a) L )1+0 I-r vl('W,) = J(Cr)) [ +0( ] r(r)W >N (3. 6b) o. ) 0r2 r ~l,/2- O tn0) + O ((krio), X+ J o(,) O(k/A) C ol,<S N (3. 7a) 1/2 2iknh/o 0o VI(7o = -/2 Jo r1 + 0(k/A)]+J )0(k/A) | >N (3.7a) 0 where 1/2 1/2[l+o( k)] rro2A nr [1 + 0~) 0 o (r) = 2X1 l/2 1/2[1 + (( r) k Thus v(dA= 1 )J"(r ))o [ 0('(ko)2] (r) |< N r 1r 44

THE UNIVERSITY OF MICHIGAN 7030-4-T (- (o, )) A + ((r) > N r r and using Bessel's differential equatiun j,(,(r)) + ~r) J(r(r)) + J (r)) = 0 0 ^ (r) o o0 10 these derivatives become, ince (r) = the positive zeros of J' (c), rX r o ( v110, =' 21k o(r) + ( 2 r N (3. 14a) r r /d \ l r /(kn) 7 VI 60I%7 )a= 2Xic io( [I~ 2 lrl N (3.14b) r r In order to further evaluate (3.13) we must use the mathematical conditions of the near field. Since the near field corresponds to the physical problem of both the source and field points near (with respect to wavelength) the origin, these conditions are k<< 1, k << 1. Then we have J(kl) = 1 + ((k1 )2 (3.15a) (2) 1 k2 1 2 (3. 15b) 0 2 it 2 n1/2 K2 2 H(2) (k2) =I - 1'- n- +( - 2) (3.15b)+ where y is the Euler constant. We now find v1(51,-X ) and v2(E2, - ) for 3 1, k << 1, =1, 2. It suffices to obtain M_)x/2 D(21kl1) and W X/2ik, 0(21k2). These functions are solutions of the equation 45

THE UNIVERSITY OF MICHIGAN 7030-4-T 2 1 X 1 u + - +) u=O 1=1,2 --- "~d( 2 14 k(2)2ikg ) d(2ik )2 4 2D is) 4(21k ) = which upon the substitutions 2ikF k2 X 2Egi k {i 1 Xr ~ X ~ \' ~ 2ik -4 r r 2ik reduces to d. u-12 ds2 L 4s 2 Since equation (3.9a) implies s positive and k2i 4(kg) 3 (- ) <<21 r O r we can apply the results of section 2.2 (equations (2.33) and (2.41)) to obtain M / (2iktl) = (2ikF )1/2J ((r)) (+ O1(r) N -X3/2ik, 0 1 2 ) M (2 ikg) =2 lr) I >N M A z/iko(2V= (~Eik1)2 /JO*r)) 2+ ~( 2)] | 1> > N Xr/2iko 0 1 1, 2 r W XA2ik, (2ik2) = (2 2 D X k/2 (2ik)/ H (r))1+ 1 2o 2 r As before, D = ep log 2 DXr/2ik -ikex-J 46

THE UNIVERSITY OF MICHIGAN 7030-4-T and from equation (2.26a) we have r(r) = 2X1f/2ti I+ O1 ( 0 Hence v( (kgr) = Jo(H)[1+ ~(X)) (3.16a) Vl(S1 -x ) = J ~(r) 1 +O0 2 r vi(lr = ~((r)[+O r)] (3. 16b) r 2 Now Xr/2ik1 ~ /8ky0 >> 1, thus from Sterling's formula (Erdelyi, et al, 1953) r((r +-X r 1 = Jeg lk[(r7og oi r Substituting this and equations (3.6), (3.14), (3.15), (3.16) and (3.17) into (3.13) we kg 2 ) o f2 O N'w 1~~21k\"~ 2- 20 % r=l1 0 Ho((btain( (r) + O whe~re 2n o 2to l re Eg0 where only the largest order term ha been retained. Expanding) and ( 47

THE UNIVERSITY OF MICHIGAN 7030-4-T about r(0) = 2i XA/2(0) /2 1=1,2, X (0) = 2/4n; expanding J ((r)) about r i r r o l r)(0) = 2X1/2(0) r1/ and substituting into equation (3.18) we find 7r r ) = 1/2 i1/2 ( Eo 0 0 2 V -~ nk2 _In kg - - 2oi-Z-I2 2rlo 2 2ro Jo r=l' o r E+o( — (3.19) where again only the largest order term has been retained. The solution for the Dirichlet problem is found in much the same fashion as above beginning with equation (3.3). If X, X2,...,... denote the positive zeros of vl(rn, X) (given by equation (3.1 la)), then equation (3.3) may be written as \^ /x A\ r 1 V1 /1 A-r)v2(E2-X r v D(en) =2 r(~ +2) r( 2-ik +2) V( X) r)[v. (vT(t, r)] X=X r and on using the Wronskian relation this becomes vD(, D) = (2ik~_ 2 1 r'(f. + 1 1' 1 r 2 2 2' r Lvi(frlf) 1D2rV-r2) 2- v2 (2,k d v. 0 rv 1(Y,rX) r (3.20) In order to write equation (3.20) when k << ~1, i= 1, 2 we need only find (- vl(ro, X); the remainder of the asymptotic representations have been given r previously (equations (3.6), (3.7), (3.16) and (3.17)). From equations (3.6) 48

THE UNIVERSITY OF MICHIGAN 703y7i-T d 0 -v- )+o0 krq)- |< N Vl(r1,) = J(1o ) yi[1+0 (n A K0 1/2 d > dA —l-o 0) = J0(> VI)1 [+lO(k/A i N.] Thus 2n L (ki I2 r r whrd (1, 2, 2/o J'2 (0l/ V( 7l1i^ = 3 - [1+ O ()]r >N (3.21b) r r O J X(r)(Oi) H ( O)) J( 8(r)(o ) k( g V*. Q n) = - E+~(-2)] (322) D 1 0\'1 o r J1 2 2*r the largest order term has been retained. 3.3 Dirichlet Potential Problem 2 2 It can be shown that the term in equation (3.22) which is independent of k agrees x= q cos, y= *sing, z= (X-w2) (r= i(r +Q ), Morse and Feshbach (1953, chapter 10) give an integral representation for 1/R. For the source at the point (X = X, o= 0) this reduces to 49

THE UNIVERSITY OF MICHIGAN 7030-4-T OD i Jo(tX) o(tU ) H (t1 ) tdt >O. Then if the substitutions X = V2, * =, X = \ are made, the above coordinate O system reduces to the one which we are oonsidering and the above integral representation for 1/R becomes R Br I) (1)1 - = if tJti( JtV'2~!tZ tdt n >C (3.23) 0 where we have used f l min(S,), 2 = max(E,2). The potential problem corresponding to the Dirichlet problem pobtulated in Chapter 1 is the solution LD(F, n) of the equation d0 = 4r6(.r r) (o is the vector to the point (, 0)) together with the boundary condition = 0 on the boundary, and the condition at infinity?b-=O as The free space Green's function for the above problem has the form -1/R where 1/R is given by (3.28). Then we can assume that D(9, r1) has the form 0 D(, ") =R + 0 (, rl) where (E,ri) is given by (8.) JJo(t i ) J ot ) A (it )tJdt with At an unknown function of t. The boundary condition implies 50

THE UNIVERSITY OF MICHIGAN 7030-4-T (1)it2o A - t J (itVo 1) 0 0 and hence OD J (tV2~l )Jo(tV 2~) l(t )J~t 2~)-H(1)(it2 (t-).D('n) =-ir 0\ tdt J (it ) oit)Jo(it2)-Ho)(it2;)Joit Jo 0 0 ~ - J is a solution of the inhomogeneous potential equation which satisfies the boundary condition. That D( n) has the proper behavior at infinity follows from the fact that 1/R has the desired behavior and the form of s (g, l). Thus we have a solution, in the form of an integral representation, to the potential problem posed above. If in this integral representation we substitute t = 2v, we obtain OD00 J(2vVi~ )J(2v)g r-) (e, ) = -2iir vdv J) H (2iv.)Jo(2ivV) OD E$Jo (21Vo ) -1 -0 70 D0 0 ~ -H(1) (Zl\Viro)J (2iv' ) (3.24) In order to analyze equation (3.24) we consider the function;t(v) = J (2 iv ) H(o) (2iv\')J (2ivV ) - H(l)(2iv )Jo(2ivvjil J (2 iv ) o o O whereupon we note (ve-i) = J (2i 2H(1) (2i +(2)(2iv i} J (2iv 1o) Jo(iV H (2iv')+ H )(2iv - {2H'(2ivJiV ) + H(2)(2iv ol) Jo(2ivfi)] which reduces to ^eI i) = rFmn%1 F (1) (1) ve ) (2ie) )(2iv')J (2ivi o )-) J)(2ivo) -J (2iv - o v%~O 0 0 ivO O /o or 51

THE UNIVERSITY OF MICHIGAN 7030-4-T 0(ve -) = (v) Therefore if we write equation (3.24) as o0 OD)S 0D(~'v) = -iDr vdvJ (2v1~;pH'H (2v)zWiv) (r) - ir vdvJ (2v V~)H( (2)v( )(v), 0 -7ri we can consider the substitution v = we in the second integrand to obtain w t7) - - =^ vdv Jo(2v\v) Ho (2v 2) ((v) 0 r- + ir wdw Jo(2w ) H(1)(2w (w). -0 This reduces to (? {) s= -7ir ~ vdv J (2v\ )H(l)(2v ) W(v) and thus the equation for the potential becomes ( 52 d^ n), s, -1A vdv 0 i- VH(1) (2iv 7)J (2 iv <J-oo - H(1)(2ivVoo) J (2v-. (3.25) 52

THE UNIVERSITY OF MICHIGAN 7030-4-T For large Iv I in the upper half plane the usual asymptotic representations for Jo(2v ), H 1)(2vV), H 1)(2ivVi) and J (2ivyi) hold, therefore it is straightforward to show that the integrand of equation (3. 25) is vanishing exponentially for large v | in the upper half plane provided 1 < 2. Suppose then 1 < 2 ( -) and let r = 1,2,... r=-1 -2,... again denote the positive and negative zeros of J (B). Thus v = ir/2Vif r = 12, 3,.. denotes the zeros of J (2iv fF) along the positive imaginary axis and hence also the poles of the integrand of equation (3. 2) in the upper half plane. Using the residue theorem we obtain J2l)(2V r 2) 1) D(, = 2) - 2, i(-ilr) Vr j(2vrf ) E )(2iv ) r=l J (2v \ yo \dv /=v r which upon using the Wronskian relation for the Bessel functions J (2ivVo), H )(2iv Vii) reduces to = ^ o J (2v riv [ )H (2v ) (3.26) n ~[o o (r)]21 Equation (3. 26) yields the Dirichlet potential subject to the conditions e f _, n > 0. These conditions were also assumed in the derivation of equation (3.22). Upon comparing the k independent term in equation (3.22) with the Dirichlet potential of (3.29) we see that they agree. 3.4 Residue Series for the Far Field Suppose we consider equation (3.13) without any assumptions on kg 1 =1, 2. Then using the formulas developed in section 3.2 we can write Lo r(2(3.27) 53

THE UNIVERSITY OF MICHIGAN 7030-4-T where the error terms have been omitted for the sake of simplicity. We will continue to omit them throughout most of the section. The physical situation of interest is the source (field point) in the near field and the field point (source) in the far field. This corresponds to the mathematical conditions kgl<< 1, kg2 >> 1, thus the behavior of v1 (1-X r) is at once determined by equations (3.16). It also immediately follows that Jo(k)' 1 (2) (k\1/2 -ik92 /4 Hii (Ilkg2 (e 2 e. o 2 \7rkS2/ However, we will write equation (3. 27) as 12,/ W f2 -ikc J 0 o C ( v( i r7) vJ (kg [ (2 e 2 —2- r ) 2 ro r k e [Jo( r]12 exp i log 2ikej V2(2' r) (3.28) so as to retain the dependence on kg1 in the first term. To find v2( 2, -X ) we will make use of the methods of Chapter 2 and thus need to determine the order of magnitude of k2 k 4k^o = = (k2 (k2) g2 xr r 2 Since k2 >> 1 there are two different possibilities for kg2; they are kg >> 1/kr)o and kg2 = 0(1/kTrl). For kg2 >> 1/kyro we can write the series in equation (3.28) as 54

THE UNIVERSITY OF MICHIGAN 7030-4-T M J (rr) j ((r)(0)) _ ~i~ ^ - y 9 1 Xr r - r[lo 2rr - exp2 2p log 2 2(2' - r here M is such that rM mples ) >> 1) N is su t r >N imples -' 7 y [7 - - 2 -exp log 23 exp lo 2ik lv 2' ("2 r r=M+1r (C )2 L lo r=N+1 2to r og v2(2 2 ) 0O rl (3. 29) where M is such that r4M implies C = 0(1) (s >>~1), N is such-that r>N implies C2 >0(kEg) (sa ~1), and re[M+l. N] implies C2 O((k)(kC2) (s 0(1)). We will refer to the three sums as,. Thus the problem of finding v2(2. -X) is reduced to finding it for the three sums Z2. Z, 2' In order to evaluate v2,(2 -X ) for 1 we need the expansions (2.9) and (2.10). 2Ar!5 Then p=-2il =-2i(-X /2ik) = Xr/k 2~;/4kn, thus f is large with s, large and so equations (2.15) yield -1/2 -"21k log(2ik2) -ik2 V'2%J -X), (2ik2) e e e (3.30) 2 r 2 Hence ~1 becomes 1 C. (4o) exp ^log exp - og(2ik%.} ~e Evaluating the two exponentials we have 55

THE UNIVERSITY OF MICHIGAN 7030-4-T,1 M l (J ~(r)(o Jo( r)(0)) 2 1 0 o 7 ( Cr Or o(____ )__ -ikg2 7~ \ r=1 i[ 2oriIP 8k1 ex C 2 2 exp 8i log 16(kg The expansions (2.7) and (2.8) apply to v2(2, -r) for. From equations (2.15) we have v2(2,-xr) - (2ikg )-l/2(r /k)I/6 exp 2k log - 2i (so(0)) 2(0) 3 H2(0) (3.31) with 4klr S (0) = - (k%2) and iC(tO) = - (s (O)) Therefore. becomes N 2 7/6 J ((r)(0)) J (r) (0)) 2 nJo o2 = l n2 8k1 -()H(2),(r) (Qi' ( <(0>) C (0,) H, (O)'(2 92 1/3''2 (3.32) In finding v2(g2, -X ) for ~ we need only use the results of section 3.2, in particular equation (317). Then we find for equation (3. 17). Then we find for 56

THE UNIVERSITY OF MICHIGAN 7030-4-T 0 I ( (r) ()) J (r)(o) l) H (r)( 3 no rQ=+ [Jo 2r Since (r)0) = 1/2 we may write 3 as ( -) io E= - nr2 exp r(l/ro) exp r(2/lo Thus we can show that 3 is vanishing exponentially along with E and 2 by considering the product of the two exponentials. We write it as er e( 1/lo)/2 C_ (f92/ro 1/2 e e which of course reduces to (w1/ro)1/2 _(e2/Xo)1/2 e e e But 51/n = 0(1) and so the latter product is -(g h )1/2 -(E2/tlo)i/ 0(1)e 2 2 2 However, kg ~>>1/kr~ implying 2/r >> 1/k 1. Therefore -( / )1/2 -1/k 0(1) e << O(l)e and 3 is vanishing as asserted. We have now shown that the contribution from the residue series in equation (3.27) is exponentially vanishing and this is much smaller than the order terms involved in the approximations for the r =0 term. Therefore we should write equation (3.27) as 57

THE UNIVERSITY OF MICHIGAN 7030-4-T v( (2) VN(kg)= ) J ( k)H( )(k )+ 0 ko)) Upon using the conditions k1 << 1, k2 >>1, and k2>> 1/kro this becomes VN(ln) = 1 \)1 e k 1+ O(kgx )+ O(/kg2)+0(ko )] (3.33) oN ik2 In the case of kg2 = O(1/k o) we can write the series in equation (3.28) as M (J ()Jo (( ) o E opn o e l { Alog }X lrl _ r D Jo W ( (0 fr(r(> X A } -0(r)()J(fr(Oexp log log 2j v2( V -2 r o -r r=M+[' ri 21 e j22 ro where M is such that r~<M implies = 0(1) ( =01) and for r>M r >1 abov for to 2 r (s << 1). Thenthe discussion above for applies to the first sum here, a similar discussion to the one above for - applies to the second sum (here we have -1 /n 1/2 -/k 0(l)e = 0(1)e. Thus both sums are vanishing exponentially, again the contribution from the residue series in equation (3.27) is much smaller than the order terms involved in the approximation from the r =0 term. This time equation (3.27) becomes ~ 1/2 -iko vN( ) (2 e 1/2 -t O(k21 )+O(1/kg2) (3 34) 2 since l/kg >> (kn )2. &2~ 058 58

THE UNIVERSITY OF MICHIGAN 7030-4-T For the Dirichlet problem we must consider equation (3. 20) without any restrictions on ki, i= 1, 2. Using the formulas developed in section 3.2 we have?lo rp,l 10.Llog r o ) ViE)k lo 2ike V) r1'r)( rx 00~~~ rr~~~~~ L(3.3 5 ) However, the preceding arguments for the Neumann problem show that this residue series is vanishing exponentially, thus we have the result that VD(, f) ~ 0. This result is also true for the field on the surface. Writing equation (3.4) as a residue series we find which upon using the results of section 3.2 becomes ivD V 2 \ _ 1 (p 9 r, A v(9 ~r)v-r2(2, r) 2 1 exponentially and therefore /Iv \ ane (,r)) -.r 590 59

THE UNIVERSITY OF MICHIGAN 7030-4-T IV HIGH FREQUENCY (FAT PARABOLOID) SCATTERING For this chapter we will consider only the surface field for the Neumann problem. In addition we will consider the point source to be located anywhere on the axis. Consequently the pertaining integral representations are 1 (~,~l V ('-)v(1 -x) VN('o) 27ri(21ikr ok ) vI(7 -,) (4.1) -o- ia -OD- i (0 < < k) with e i _, for the point source located at (-, 0) to the right of the focal point, and 1 -) icr v 2(, e -X)v1 (H, X) N(' - 27r i(\ikno) -dX +2ik ) v+(no, 2) (4.2) 0 -o -.ia (0 < a < k) with qr 0 H, for the point source located at (0, H) to the left of the focal point. 4.1 High Frequency (Fat Paraboloid) Poles As shown in Chapter III the poles of the above integrands which lead to the residue series readily summable for kr << 1 correspond to the zeros o0 the function vI(ro, X). 0 1 0 If we wish to consider an analogy with the scattering by closed convex bodies we must now seek an alternative set of poles which, when the geometric term (terms) is (are) removed, lead to a residue series readily summable for kr) ~ 1. The other set of poles of the above integrands are the poles of the r-function r(2k + at X = -ik(2n+1), n= 0, 1, 2,... in the lower half plane. However, as seen in Appendix A. 3 these poles lend to a residue series only provided ~ + / J+ < o~ (or [/+ Jl< i in the case of equation 4.2). Although this residue series may be analyzed for kr >> 1, the inequality sharply limits the range of source and field points which may be considered. Therefore we derive an integral representation which can be analyzed for a wider range of source and field points. In performing this analysis we find the poles of ri +.2) play only a minor role, while once again the poles of l/v(ro, X) are of paramount importance. Thus we must consider in detail the zeros of v{(r1], X) for ko >> 1. 60

THE UNIVERSITY OF MICHIGAN 7030-4-T According to Buchholz (1953) the zeros of v ( X) all e on the real axis to the 2 1o right of the value -k ro. In Appendix A. 3 we found the positive zeros of v(r, A) for X > krio. Thus it remains to consider the th ree regions: 2 (i) A negative, A Ov0(k z) ) (ii) |A/k|4 0(l) (iii) X positive, X v O(k2in,) In region (i), v l(r, X) is governed by the Airy function representation while in region (iii) the Bessel functionrepresentation determines the behaviorof v1(rp, X). The explicit representations in these regions are derived from the results of Chapter II. The zeros of V, (n, X) in region (i) give rise to the "whispering gallery" waves and thus can be called the "whispering gallery" poles. In region (ii) we can write the asymptotic representation (Buchholz, 1953) X X X in 1/2 ik) 2ik kW7 2ik -ik 2k 2 v1(ln,X) - 1(2ikrn) / + (2ikn) (e -. e (4.3) i 2 2ik 2 In addition to the zeros of vl' (7 X), we have need for the zeros of the function v' (oe, -X) in the region -r < argX < 0. Again an explicit representation of tbheseaj zos 2 o can be found using the results of Chapter II. They are found to arise from the Airy function representation of v'(rt_ e, -A). 2 o 4.2 Equivalent Integral Representation Consider the integral representation (4.1) together with the following representation (Buchholz, 1953) of v&(ro, X) x x i2k VI (wi 2k 2 e v(rl e, -X) e e V(, X) v(rX) = 1 + 2 F2 21k 2 2 ik Then we can write 61

THE UNIVERSITY OF MICHIGAN 7030-4-T 12k [r ii i r 1. e 2 v(Uo eri4 _-X) F-i ( vl(n\)= 2o l+e 1 2 2o'^^ -iik; _ II ^\ r/i-,(__ _ v 2 and if we define F(E 1 2o, k, X) = 2( " x - -)v2) X,2k 7ri vi(.. e VI'( e2 2 i r 2: X(,k,X) = e' n o r + v(n2 2 i-) -o- ia to oo- ia (O < a < k) by c, equation (4.1) becomes _ F(1_ _2" no' k, X) F VN(f,.o) = VI e X (4.4) = 1 (no.. o,) = \ dA -X) But then we have the decomposition VN(, rl) - S dXF d FX N 09g g(1 - X) c c (*o'F(1' 2 r' k, X) ='V(ro, X) X(o, k, ) c If we now assume 62

THE UNIVERSITY OF MICHIGAN 7030-4-T N-1 VN(f )Z5 — dx x dg g(1 - l-X) then N 1-X _ o xF. N+1 (1 - X) C k=O g g(l -X) X - F(eil 2,k.a F).M+1 k=O = r V' dXox Ac v,(n X) [((ok, X) (4.5) v1 c If we examine the integrand of the remainder term in (4. 5) we see that it is of the form of the original integrand of equation (4.1) multiplied by the factor [X(o, k, )]M+l. Therefore its behavior on a large semi-circle in the lower half plane may be investigated by considering the behavior of [X(n, k, kM+1 together with the already known (Appendix F(1. 2'r o k, X) A. 3) behavior of v-2'( ). We find that the contour c may be closed in the lower half plane provided the inequality g1 + 2 - o - 2(M+) 0 < 0 (4.6) is satisfied. But for any triplet ( 7,Z, n ) there must be some M such that the inequality (4. 6) is satisfied. For this M we can determine the value of the remainder by the sum of its residues in the lower half plane. Now 63

THE UNIVERSITY OF MICHIGAN 7030-4-T k x M+1 X 1 F(l 2 0o kA X) * It r )v2(9 k (E2' -X) t(M+l) 2 2- 2 v(o, X) M+Il " A+ VI r1 ei and hence the only poles of the remainder integrand are ones of order (M +1) at the 7Ti zeros of v.(rn e, -X) which lie in the quadrant -Tr < argX < -r/2 of the lower half plane. 0 0 M c or to be more precise M vN(if, ) = dl X +O(e 0) (4.7) from the tip of the paraboloid (k 1), we can aluate th integrals occurring in (47) by ~ k=0 O 4.3 Source Located "Far" from Focus. Surface Since the representation (4. 7) was derived from (4. 1) for the source to the right of the focal point we already have that the source is "far" from the surface. If in addition we consider the source to be "far" from the focus (k` ~ 1) and the field point to be "far" from the tip of the paraboloid (kg ~ 1), we can evaluate the integrals occurring in (4.7) by saddle point integrations. The first term corresponds to the usual geometric term (twice the incident field) while the higher order terms correspond to the multiple reflections. Since the number of higher order terms is governed by the inequality (4.6), we can for a fixed source identify regions of the surface with the number of multiple reflections received. 64

THE UNIVERSITY OF MICHIGAN 7030-4-T 4.4 Source Located "Near" Focus In the representation (4.7) we can consider k.= 0(1) (source "near" the focus) or kg = 0(1) (field "near" the tip of the paraboloid) but not both. The effect obtained is that the saddle points of the various terms of (4.7) disappear. In addition we can estimate the neighborhoods of the focal point or tip where this effect occurs. As the source approaches the focus this neighborhood corresponds to the rays leaving the surface almost parallel so that no multiple reflections occur. In the region of the tip this neighborhood corresponds to the region where no multiple reflected ray is received. In order to explicitly evaluate the terms of (4.7) we note that the portion of the contour contributing most to the integrals corresponds to IX/k | O(1) where the functions vi(x, X) are gverned by asymptotic representations of the type (4.3). The functions can then be replaced by these asymptotic representations and the resulting integrals evaluated. In either case only the term corresponsing to twice the incident field remains. 4.5 Source Located "Near'LSurface If we wish to consider the source to the left of the focal point we must derive a representation corresponding to (4.7) starting with equation (4.2). In this case we must nay careful attention to the "whispering gallery" poles, since when H is close to ro they -kno are no longer of O(e ) as in the case of (4. l),but are of order comparable with the other terms. Thus the "whispering gallery" wave which travels along the surface of the paraboloid becomes more evident as H approaches rl. While this is true for kr >> 1 it is felt that this "whispering gallery" wave is the key to the behavior for krl not so large, even when the source is not close to the focal point. 65

THE UNIVERSITY OF MICHIGAN 7030-4-T V APPLICATION OF RESULTS TO CONTINUING INVESTIGATION This work on the paraboloid is preliminary to a description of scattering by general concave surfaces. Having integrated the formalism in terms of the physical phenomena such as the whispering gallery waves, multiple reflections and caustics we now are in a position to search for generalizations of these to other concave surfaces. The scheme we propose is to determine the dependance of the physical effects on the local geometry of the paraboloid and then to make the essentially physical argument that this geometric dependance is the same for other concave shapes. This approach is similar to that used in determining creeping waves on general convex shapes and is an application of the physical arguments used in Keller's geometric theory of diffraction. 66

THE UNIVERSITY OF MICHIGAN 7030-4-T REFERENCES Abramowitz, M. and I. A. Stegun (1964) Handbook of Mathematical Functions, NBS Applied Mathematics Series 55. Buchholz, H. (1942/43) "Die Ausbreitung der Schallwellen in einen Horn von der Gestalt eines Rotationsparaboloides bei Anregung durch eine in Brennpunkt befindliche puntformige Schallquelle", Ann. der Phys., 42, 423-460. Buchholz, H. (1953) Die Konfluente Hypergeometrische Funktion (Springer Verlag, Berlin). Erdelyi, A., W. Magnus, F. Oberhettinger and F.G. Tricomi (1953) Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York). Erdelyi, A. and C.A. Swanson (1957) "Asymptotic Forms of Whittaker's Confluent Hypergeometric Functions", Memor of the Amer. Math. Soc. No. 25. Fock, V.A. (1957) "The Theory of Diffraction of a Paraboloid of Revolution", from Diffraction of Electromagnetic Waves on Certain Bodies of Revolution (Moscow). Kazarinoff, N. D. and R. K. Ritt (1959) "On the Theory of Scalar Diffraction and its Application to the Prolate Spheroid", Annals of Physics 6, 277-299. Kazarinoff, N.D. and R.K. Ritt (1960) "Scalar Diffraction Theory and Turning Point Problems", Arch. Rational Mech. Anal. 5, 177-186. Kinber, B. Y. (1961a) "Diffraction of Electromagnetic Waves by Concave Surface of Circular Cylinder", Radio Eng. and Elec. Phys. 8, 1130-1139. Kinber, B.Y. (1961b) "Diffraction of Electromagnetic Waves on a Concave Spherical Surface", Radio Eng. and Elec. Phys. 10, 1474-1475. Lamb, H. (1906) "On Sommerfelds' Diffraction Problem; And on Reflection by a Parabolic Mirror", Proc. London Math. Soc., Second Series, 4, 190-203. Langer, R. E. (1935) "On the Asymptotic Solutions of Ordinary Differential Equations with Reference to the Stokes Phenomenon About a Singular Point", Trans. Amer. Math. Soc. 37, 397-416. Langer, R.E. (1949) "The Asymptotic Solutions of Ordinary Linear Differential Equations of the Second Order with Special Reference to a Turning Point", Trans. Amer. Math. Soc. 67, 461-490. Phillips, R.S. (1952) "Linear Ordinary Differential Operators of the Second Order", New York University Report No. EM-42. Pinney, E. (1946) "Laguerre Functions in the Mathematical Foundations of the Electromagnetic Theory of the Paraboloidal Reflector", J. Math. Phys. 25, 49-79. Pinney, E. (1947) "Electromagnetic Fields in a Paraboloidal Reflector", J. Math. Phys. 26, 42-55. Rayleigh, J. W. S. (1945) The Theory of Sound (Dover Publications, New York). 67

THE UNIVERSITY OF MICHIGAN 7030-4-T Sims, A.R. (1957) "Boundary Conditions for Linear Differential Operators of the Second Order", J. Math. Mech. 6, 247-285. Skalskaya, I. P. (1955) "The Electromagnetic Field of a Dipole Located in the Interior of a Parabolic Reflector", J. Tech. Phys. USSR 25, translated in New York University Report No. EM-103. Taylor, W.C. (1939) "A Complete Set of Asymptotic Formulas for the Whittaker Function and the Laguerre Polynomials", J. Math. Phys. 18, 34-48. 68

THE UNIVERSITY OF MICHIGAN 7030-4-T APPENDIX A. 1 NORMALIZATION (POINT SOURCE NORMALIZATION) In the text we first assumed the point sburoe in question to be represented by p(r). Let J(g, r, 0) denote the volume Jacobian in the coordinates of the paraboloid of revolution. Then for the point source at (, 0) 1 C p(, rl) = 6 i ('9 -3 "- 6 (6C( ^6(F -6(n). and so prne~~ r r2 oD o I\\\ 5 p(g, r)dV = \ 2C6(f -)6(n7)dF dr1d = 4rC J JJ Jo 0 0 0 = 4r for C = 1. Therefore for C = 1 we must have \\ Cp(rr)dV = \ C6(r-r )dV 4 7 implying C"=4r and p(r) =47r6(r- r ). Since the free space Green's function for this -ikR p(r) is -R (R = |r-ro ), then in order to have consistency (already demonstrated by the agreement of equations (1.5) and (1. 12)) we must show the solution to -L v- L v = 6(9 -z )6 (r;) Ti? r (~ Iv(, n s)2dV < co all space -ikR has e- (R= jr-r ), for the limit as s- 0-. The solution to (*) can be represented as v(t ns) = - \ G(.,.-X) G(r, 0, X)dX

THE UNIVERSITY OF MICHIGAN 7030-4-T where r is a straight line contour between the poles of G(_,X, -X) and G(n, 0, X). But 1 X r 2 2i G(r, O, ) = 2i 1(0, X)y2(r ) o < analytic in Im X < k, and.rll + ) yl, [1( -X)Y2(i -X) < ^.^ 2 2' G 2i' y (- -X)Y2(, -X) > analytic in Im X > -k. ^ Therefore if r is a path defined by -o - ic < X < Oo - ia or < k, v(n, s) = 2r (27r i) y2(, -X)y2(, -)2(. Arguing as in section (1.4) we see that the limit as s -0+ may be taken inside the integral by replacing the parameter y with the parameter k. But then the integral representation -ikR for -- in Buchholz (1953) shows that R-L li. e-ikR -- 0+ v(, r, s) = - R which demonstrates consistency. See Appendix A. 2. 70

THE UNIVERSITY OF MICHIGAN 7030-4-T APPENDIX A. 2 ANALYTICITY OF RESOLVENT GREEN'S FUNCTION R. The analyticity of RA follows that of G(,', X) which was represented as ( j'~ ) = 1(, )Y2(' X)? <' ov 2 2 7 2i-y1(' y )y( A) 9> 9' According to Buchholz (1953) the functions y1 and y2 are entire functions of X, thus the singularities of G(c,'I, X) are those of r ) which are simple poles at the >1 1 1 points 2 = n+ n = 0, 1, 2,... G(2, 0', X) will be analytic in any domain which 21ry 2' excludes these points. Consider then the expression A/2iy with X =x+iy and y = - (w- is). We have C jx = -iA = -i(x+ iy) A(c is)) 217 27 r21'y 21yy Thus aX + Y + i+ Xi c c c c 2iy 2|yr and if this is to be real - =- implying x= y. Then C C 2 2 s + Y X _WC WC _ JL (2 +2 A 1(2+ 212 217 2271 2wc 2 Now 1y1 y y= 2 (w +s ) and so for real A/2iy we have 2 2 c 2w To exclude the pbles of the (-function we need for real X/21-y x- 1 2 < implying < or y < = k. 217 2 2w 2 c But since X = x+ iy, y =ImX and therefore we arrive at Im X < k. 71

THE UNIVERSITY OF MICHIGAN 7030-4-T APPENDIX A. 3 CLOSING THE CONTOUR (CONVERGENCE OF RESIDUE SERIES) In this section we consider the integral representation given by equation (1.12) O- io (2ik)3/2 r00" /X 2\ 2 x i\ Yi.r-^o2o'-^ N 27r'7ri \2i'ik+2 Y\ + ) y ( X) (0 <a < k) *y2(n, X)y(ro, X) - y1(nr, k)y(ro, X)] We first show that k- + Y) [Y2(r, X)y (o X) - yl(nr, X)y2(o X) (A.3.1) is analytic in the complex X-plane, and for X] — cao (i) the zeros of y (rno, X) lie along the real axis, (in the text we discuss the zeros of y;(no' X) (when ko << 1 and kr >> 1) for other ranges of J X, a discussion for arbitrary kr? appears in Buchholz (1942/3, 1953)) (ii) the integrand vanishes exponentially in the upper half plane (Im X ~ -a). Thus the contour may be closed and the residue series obtained. We also show that the residue series converges. The only possible poles of (A. 3. 1) occur at the poles of r- + ) which lie on the positive imaginary axis at X = ik(2n+ 1), n = 0, 1, 2,.... But at these points y (, X) = y[I. ik(2n+1k) = M2 0(2ik) (2ik) /2e ikr L ()(2ikn) y2(n,X) = y2[n. ik(2n+lj1 = 1/2W +/ (2ikn) = (-1) nn (2ik) /2 e-ikL(0)(2ikn) 2 2^ -1 n+ /2-, n where L) (2ikr) is the corresponding Laguerre polynomial. Thus n Y2 [n ik(2n+ 1) Y1 ik(2n+ 1 - y1, ik(2n+l y [, ik(2n+ 1) equals 72

THE UNIVERSITY OF MICHIGAN 7030-4-T (-1)n! (2ik) eik nL02j d - L(0)(2 [ ek L(0)( r) ( n1).; (2ik) e-^ L~ikr,) ^ (e(lkr)] - ()nn. (2ik)e L 2ik (e d Lik 0( L()ikr) = 0 i0) 70 and therefore cancels the simple pole at X = ik(2n+ 1), n = 0, 1, 2,... This implies the analyticity of (A. 3. 1). In order to investigate the zeros of yI(rto, X) for I X -oD we first note that the X-plane will be considered to be cut at X = Xr or X = -r. Thus the upper half plane (ItnX -a) for |L-4So can be characterized by |XI-4 oo, -6 argX <, arg= r, a rr or -r, -r < argX< -,r+6, where 6 is a small positive angle which decreases as IiX increases. Similarly the lower half plane (ImX A -a) is characterized by I| X —a>, -ir+ 6 argX -6, where 6 is as above. Therefore we can investigate the zeros of y'(r,X) in these regions. For -r, argX <, y\(ro,X) has the representation (Buchholz, 1953, p. 98, pertainng to equation 17a) o 4 yJ(rn e) e e (A. 3.2) and thus there are no zeros in this region. For 0 < argX < r, Buchholz (1953, p. 98, eq. 17a) can be used for Yl(ro,X). Thus Y(nX) 1/2 (4ktl cos 2iAx -4- (A.3.3) which implies y(nX)) — ( - ) sin[2 —. (A. 3.4) Therefore for |X A| - the zeros of y'(nr,X) are the zeros of sin [2 \ 4 - which obey the equation 73

THE UNIVERSITY OF MICHIGAN 7030-4-T X 1 r + NNr 2 - N 2 8 -2 Neo rio and consequently lie on the positive real axis. To demonstrate the exponential vanishing of the integrand we will examine the factors 2 ik+ 2) y1,, (A.3. 5) and r(- k + 1) [y2, X)Y(, ) -y(n X)y(n, X (A. 3.1) separately. We further divide the upper half plane into the intervals -6 K argX < 0, 0 argA < 7r/2, argXA= r/2, 7r/2 <argX Air, -r < argX. -r+6. (We make this choice so that either arg X = ir or argX = -ir may be considered.) Then in order to estimate (A. 3. 5) we need to investigate Y1 (1, -X) and y2(%2, -X) on these intervals. For o < argX. 7r, Buchholz (1953, p. 98, eq.l7b) applies to y1(l1,-X). Therefore as -if 4] 2V'f — 1\ -2 1k i4 y -)cos[ 2 e 2 - 7r e e +e e (A. 3.6) When -Tr < argX ~ 0, we can use the same equation with the opposite sign to assert as I X|-o i -2\ -i) 2 V|A i y ( -A.) cos [2 X e -2 i] e e + e e (A.3.7) The function y2(Q2, -X) is not so simple. In order to examine its behavior we must consider the intervals defined above. The first two intervals may be combined, the rest must be considered separately. Upon obtaining the behavior of y2( 2' -4) in each interval, the factor (A. 3. 5) can then be examined. 74

THE UNIVERSITY OF MICHIGAN (i) -6 argX< r/2 = RpX > 030-4-T In this region we cannot obtain the behavior of y2(g2, -X) directly. Instead we must make use of equation (21b), p. 19 of Buchholz (1953) which asserts X ) 2 X )eX X X 2ik 22ik ri 2k -Wi.J y^2 (2 2 -r e y2(2e,i)-e y2(2 e 2 Now -6 argX < r/2 = -r/2 - 6<arg 2 < 0 =Im- <0. In addition arg2k 2ikg2e7 = argO(e i) Tr-6. 3r/2) X -iri a nr i - arg k 2ik2 e = arg(,e )e[-r-6,-r/2) therefore Buchholz, (1953, p..90, eq. 19a) applies to both y2(2e 1,X) and y2(g2e X0i). But since RpX > 0 we have y? ({ o-kX "/1 -X 2k 7" -yi 9y2g,-A)~ r( lk r (2 e y2(e ). Using the above-mentioned equation for yf,(2e, X) x 2 r iX 2-ri 2g2e A e pn2ik o 2ike} e e 2kr i. thus from Erdelyi et al -22(1953) y2(2 2ik -) e - 2 log 2ke (A. 3.8) thus from Erdlyi et al (1953)

THE UNIVERSITY OF MICHIGAN 7030-4-T and so y2(e2. -A) becomes u X ~(, rr~l 1 1 A. 2k -2~2 9 y2( 2 - ik)r'( ) + e 2(2 e 2e However, from Erdelyl et al (1953) we also have P\2 2=- c2:r + = X (A.3.9) 2ik and using the exponential representation of the cosine y(2, A-X) r(, - 2k e e (A.3.10) Therefore using (A. 3.2), (A. 3.7), (A. 3.9) and (A. 3.10) we find l, ^-xR-\) 2Rpj 2Rp\ki`j (1 ),X2 21k 7mF ) -2<Im on -6 ~ arg X < 0 = -6/2 < arg > > 0 RpT\ > 0, ImnF < 0, and using (A. 3.3), (A.3.6), (A. 3.9) and (A. 3.10) we find r, (+- 2_o e e. \ 2ikJ 7. ) ~ 2Impj~ e on 0 < argA < vr/2 0 0 arg\T < r/4 = Rpfr > 0, ImT > 0. In either case (A. 3.5) is vanishing exponentially since f2 > f 1 (ii) arg = 2r/2 = = X l[Xk argr = xr/4 =' RpT' >0, ImT > 0. For argX = r/2 we note -. Thus Buchholz (1953, p. 100, eq. 20) 21k 2' 2ik 2k can be used to give 76

THE UNIVERSITY OF MICHIGAN 7030-4-T - X 1xxl i -2iW 2 y2 -X ) exp - 2 k log - je e exp ~i2 log 2'kee 2 e (A. 3.11) But (A. 3.8) applies for r(i+ -) and using this together with (A. 3.3), (A. 3.6) and (A. 3.11) we have / ik x -Xy ^ ^ 2 -X) ~2 2 ~~COer W 1-2 e Cos 4 2XR1 cos 4 yi'' 2 22 e e rfA_____ __ ONe (o2 2ik) Y 2 cos Ir/4 e which vanishes exponentially since,2 > 1 (iii) 7T/2 < argX <~ r = 7/4<arg rg 7r/2 ==Rp\Vt>0, Im P' >0. In this region we note that 0 < arg 2ik ( r/2 == Im k > 0 But in addition -- = e 2'....21k 21k 21k' arg'i 2ik52 r argAe (-7r, 3r)., Hence Buchholz (1953, p. 100, eq. 20) applies to y2(2. -) giving -2,y2(i2 -X) - exp- 2 log 2ik e 2 (A. 3.12) Now equation (A. 3.8) also applies, hence, using it together with (A. 3.3), (A. 3.6) and (A. 3.12) we find /ix\' Y1, A)Y2 ) (-2Rp{ X2 2Rp X2 r'_ ^U* M [2 2ik y{ (o, X) 2 Im iXmj e which vanishes exponentially since 2 > 51 and ImmrT > 0. (iv) -r, argX< -7r+6 RpX < 0 ir X ir X iX In this region -X = e X, thus --- = e 2ik - In addition iX ix -r - argAX -7r+6 = I-r/2 < arg- 2 -7r/2 +6 =: Im 2k <0. We also have 77

THE UNIVERSITY OF MICHIGAN 7030-4-T ar i arg2-Xe CO, 6]C [o, 2). Then we can use eq. (19a), p. 99 of Buchholz (1953) to assert -xr -2i. Y2(2, -X) exp k log 2kj e e r 2 x - -2ye exp gik ike e But for this region Stirling's formula for the F-function rF - is applicable (Erdelyi et al, 1953). Thus 2k) e 2ik i} 2i(A. 3.13) and so y2(2, -X) 1 C -2 2 ig Y2 -2 r - )e 2ie (A.3.14) Then using (A. 3.2), (A. 3.7), (A. 3.9) and (A. 3.14) we have 2Rp\f v 2Rp X9~ [l+ A _) Yl(!-X)Y2(v2'- ) e e _'2 2ik y\(\^A -2Im^(X j e since -+/2 6 arg-< - - = RpfX 0, ImX < 0. But as IXI —co Rp < k while ImBrT decreases without bound, therefore 2RpVX~2 2Rpf\(X,_ 2kVg2 2k (1 &e ___ e e e -2Im JWo -2Im /X30 e e and (A. 3.5) is vanishing exponentially. We now examine the factor (A. 3.1). We consider first the interval -7I < arg X < -r+6. Thus from Buchholz (1953, p. 98, eq. 17a) we can derive the representations valid in -wT argXk O 78

THE UNIVERSITY OF MICHIGAN 7030-4-T y l(tX) - e 2 (A.3.15a) y' (r1.) e21i (A. 3.15b) In order to derive y2(r, X), y(l~, X) in this interval we recall equation (A. 3.10) valid in 0 < argX < 7r/2. If we then consider the substitution v = Xee and also recall that in 0 argX < r/2, -X = e -A, (A.3.10) implies for -r <argv < -/2 2v 2'k Y2(v2,Dvhr(i) " + )e e = r(2+2) - -2e i But since v, 2 are dummy variables we can write for -r,< arg < -/2 and in particular for -r < argAX -7+6 A y X(, ) "- r + e 2 e 2 i e f (A. 3.16a) Ii - -i e 2-k-e )y,(I r(2+2e) e+e. (A.3.16b) Therefore 2ik) r 2ik [ e 21XT and using (A. 3.9) together with the exponential representation of the cosine, as wllas RpX<0, Im <0 and ri<,o7 we find A -2i 2i 0o (A.3.1) e2 e Hence the integrand (product of A. 3.1 and A. 3.5) behaves as 2RpV2 2Rp\ XT 2Im Integrand ^ e e e 79

THE UNIVERSITY OF MICHIGAN 7030-4-T As previously 2Rp /2 2RpyI 2Im % e2k e2k 72Im 7e e e' e e e which vanishes exponentially since ImVT < 0. Consider the interval -6 ~ arg X ( 0. Equations (A. 3. 15a) and (A. 3. 15b) for Yl(tr, X) and y,(r, X) apply; thus it remains to find y2(n, X) and y(ri, X). But now we recall equation (A. 3.12) valid in r/2 <arg X r. If we again make the substitution -ri -Wri v = X e and recall that in r/2 < argX < r, -X = e X, equation (A. 3.12) implies for -f/2 <argv <0 Wi -2 T ~9' v -2i y2(2',v) = exp~ log e = P kllog - e 2kk e Again v and E2 are dummy variables, thus for -7r/2 < argX,< 0 and in particular for -68 argX <0 y (n )-exp xlog e 2 e-2i (A. 3.17a) y'2rL X) ~ exp - k log 2i yI(rn, ) ev exp log X e 2k e r' (A.3.17b) 2r 2ike Now for -r /2 < argX: 0, -r < arg 2i ~ -7r/2, hence equation (A. 3.8) applies for r(2 + -) and equations (A. 3.17) become ^y r + i 2k e) 2ie e (A. 3.18a) Y2( 2 +?r2-k yW(titXr)e2kee -2e (A.3.18b) We proceed exactly as before except that in this case RpX > 0; therefore we find 80

THE UNIVERSITY OF MICHIGAN 7030-4-T,-2i" 21 (A. 3.1) v e ~ e and the integrand behaves as -2Rpi 2RpJ 2Imai Integrand e e e which vanishes exponentially since InS < 0, RiVT > 0 and g > E. We can now examine the behavior of (A. 3.1) on the interval 0 $ argX $ w. Equations (A. 3.3) and (A. 3.4) apply for yl(n, X) and y(r), X); we must derive equations for y 2(,Xa) and y2(,X). For 0 argX<Ir, -r/2 <arg 2ik<r/2, and arg 2i (2ikl) = arge [0, r], thus Buchholz (1953, p. 99, eq. 19) yields 2ie (2 /2 (4k 1/4 r A A + x 1r (2)1/2 ~ -2 1/4 Y- log2 oo" - + y(nA) -, _ (-4k20X) exp{ log2 } sin [2 - r 2Ak +. Now equation (A. 3.8) for the r-function ^( - + ) is valid, hence 1/2 4k2r1/4 / i y,(rln,) 2 ) 4 k r)1) cos[2~VX- ikr~- + ] (A.3.19a) yI(nX) - 2 (-4k 2 r()1i/4n 2i) -[Ain- 2ik + T (A.3.19b) Yl X ) ik -2- ( - 2i24/ Therefore (.2ik 2~ ik 2 A+io8C2O X2k+i 4n[2 i - cos [2 -]s- sin [2 - which upon using the exponential representations of the sine and the cosine becomes 81

THE UNIVERSITY OF MICHIGAN 7030-4-T (A.3. 1) r +) + e -2 e2 +e 21 e2i + r(-+ +)r( +4) e-2k e2i 2ie +e 2 e-2ii] (A.3.20) Let us now divide 0 K argX < r into the sub-intervals 0,< argX < r/2, argX = r/2, r/2 < argX,< 7r. (i) 0 argX < r/2 = RpX > 0, 0 argV < 7r/4 ==Rp > 0, ImX >0 Using (A. 3.9) and (A. 3.20) together with the exponential representation of the cosine and rl rlo we have (A.3.1).- eIm V 2 ImeXf Then the integrand obeys -2 RpX2 2Rpk -2 Im. \ Integrand ^ e e e which vanishes exponentially since >2 > 1 iii) argX = 7r/2 RpX = 0, argX= r/4 =- RlpW >0, Im\/ > 0 Thus using (A. 3.9) and (A. 3.20) as above -2 J v Cos 2 1Xo os1 (A. 3.1) v e e while the integrand obeys 2X cosT CosT -2 4 cos 4 2 l 4 -2 i cos 4 Integrand * e e e which vanishes exponentially since t>2 > 1. (iii) 7r/2 <arg X < r == RpX <0, ir/4 <argf < 7r/2 = Rp' O0, ImV > 0 82

THE UNIVERSITY OF MICHIGAN 7030-4-T The arguments here are the same as above. We find -2Rp \C 2Rpk -2 Im Integrand v e e e which vanishes exponentially since f2 > 1 and Im\' > 0. If we now consider equation (1.12), the above argument implies that the path of integration may be closed by an infinitely large semi-circle in the upper half plane (ImX >-a). Then by use of the residue theorem vN(L, tr) can be evaluated as a sum,. of residues. Let X1 < 2 < X3... denote the zeros of yl(rlO X) along the real axis. (For finite X we appeal to Buchholz (1942/3, 1953) for the location of the zeros.) Thus 3/2 n n1A t r 1 ni92 n=1l + / Y /noA n l-y (7 X,)YH(Z10 XI But from the Wronskian relation we have -y rlo Xn)Y2(ro Xn) = - 1 0i n n 1) therefore vN(, rn) becomes (2ik) 1/2' n +1 Yl(AL' -n)Y2( 1 2 -n) Y1 ( )n_ o 1i \ ik n2ld [4 yj y,(<o, ]n X=X n In this series argXA =0 for n sufficiently large. Hence the previously developed n asymptotic forms can be used to investigate the convergence. By arguing as above, it is seen without difficulty that the series converges. We should remember, however, the conditions 51 = min(F,Z), - 2 = max(E,-), iE_, rl >0 under which the integral representation was developed; they still apply to the series. 83

THE UNIVERSITY OF MICHIGAN 7030-4-T For the most obvious attempt to represent the total field as the sum of the incident ikR plus scattered fields, we use the integral representation for -e kR/R (Appendix A. 1) to write equation (1.12) as = e -ikcR 3/2 C-iao (J-o- i o N R + 2ri / dik 2- 21k 2) Y (n,X) (0 < a < k) [ Yl((.n. )y(o, -I] The Integrand of the remaining integral now vanishes exponentially in the upper half plane if and only if \ 1- 2+ fr < 0. But this integrand posses, in the upper half plane, poles not only at the zeros of y'( o, X), but also at the poles of the r-function r - 2i + ) which lie along the positive imaginary axis at the points X = ik(2n+1), n = 0, 1, 2,... The contribution of these latter poles to the total residue series (conW-1cR vergent when 1 - 2+ < 0) simply cancels -e /R. This follows immediately from the existence of a residue series for e-ikR when - 2 +'T < 0, and the _iacultion which shows (A. 3.1) analytic in the upper half plane. Thus we arrive at the previously obtained residue series. Evaluating the remaining integral by residues in the lower half plane is discussed below. The other set of poles of the integral representation of (1. 12) are the poles of the r-function r[7 + 2) and lie along the negative imaginary axis at the points A = -ik(2n+ 1), n = 0,1,2,.... We now investigate the behavior of the integrand as IXI- -coo in the lower half plane (ImX -a) which we characterized by the relation I X O, -r+6 a arg X -6 where 6 is a small positive angle that decreases as \x I increases. It will still be convenient to examine the factors yl(rlo, X) n -+1) -z — y(A 31 2(92 2ik 2 y(l. (, X) (A. 3. 5) and rf i + f) [y2(n, ))yi(o, X) yR- y&n, X)yI(0o, X] (A.. 31) separately. 84

THE UNIVERSITY OF MICHIGAN 7030-4-T Let us first consider the factor (A. 3. 5). Equations (A. 3.15) apply for y1(rl, X) and yi(rl, X) while equation (A. 3.7) is valid for Yl(1. -X). Thus it remains to find Yo2(2, -X). We can derive an expression for y2( 2, -X) which is valid for -Xr < argX < 0. i7r X irX IX In this region -X = e X and so -2i =e 2ik = 2 Then -r < argX < 0 -r/2 < arg - < r/2 while arg2ik 2(- -k ) = arg(e X) = r+argXE(0, v). -7rf2<arg- <ir/2 whil a22ik i Hence Buchholz (1953, p. 99, eq. 19) may be used for y2(52, -X) giving y -X) -- exp log cos[2 eriX 2-ri +. We write it as X X X + T y2(,20 -X) exp 2ik log - ik cos i 2 4 -.] Since in this region equation (A. 3 13) applies for r- 2i + ) we can also write X 1 y2 -2 \2 ik 2 CO 2 2k 4(A.321) In order to calculate the behavior of (A. 3.5) as 1XI- c o, the interval -r+6. argX, -6 is divided into the sub-intervals -7r/2 < argXue -6, arg = -r/2, — r+6 b argX < -r/2 and (A. 3.5) examined on them separately. (i) -7/2 <argX< -6 = RpX >0, -7r/4 <argX-<. -6/2 = Rp\fL >0, Im < <0 From (A.3.21) we find X \ A 2tk -2 g2 2 A) r( 4 + e)e e (A.3.22) and using (A. 3.7), (A. 3.9) and (A. 3.15b) ( A + X-)y ( g-X) 2Rp X9~ -2Ip X9 2ik 2/ y (rno, ) — 2 Im ~ p which vanishes exponentially since 52 > 1. 85

THE UNIVERSITY OF MICHIGAN 7030-4-T (ii) argX = -r/2 => = -ilx|, arg, = -i/4 = RpF O > 0, Imti < O From (A. 3.21) we find y 2()L2I+x)- 5 r(%+-2)e 2 coo (A. 3.23) and using (A. 3.7), (A. 3.9) and (A. 3.15b) 2IXi2 coso 2 FijiM Cos0 (A.3.5).....5). 2e x-o2T cos 4/4 e which vanishes exponentially If and only if g + g < 7. (ii) -,+6 argX < -r/2 = Rp X<. 0, 2 arg < -r/4 > Rpf/ >O, Im f< From (A.3.21) we find y2(t2) r(-2ik+-2)e e (A. 3.24) andusng (A. 3.7), (A. 3.9) and (A. 3. 15b) 2Rp/X91 2Rp% (A. 3.5) e -2 Im e But in this region Rp V < -ImrVT, thus 2Rpm\|5 2Rp i 2 -2Im S -2Imni -2 Im (X -2 Im -X) e e which vanishes exponentially if ~ + 2 < r70. But as arg'' approaches -r/4, -Im fT approaches RpY6; hence (A. 3.5) vanishes exponentially only if [ + < 86

THE UNIVERSITY OF MICHIGAN 7030-4-T To obtain the behavior of the integrand the factor (A. 3. 1) is now investigated. As above, yl(rl, X) and yi(rn, X) are governed by equations (A. 3. 15a) and (A. 3.15b). Let us consider the previously defined intervals. (i) -r/2 < arg X -6 = RpX >0, -r/4 <argX<: -6/2 = Rp' > 0, Im \/ < 0. Here Y X(r, ) and yI(n, X) are governed by equations (A. 3. 18a) and (A. 3. 18b). Therefore (A.. 2ik 2 \2ik + -2) e e e e e e -2 Arguing as previously, O2Tm X I -2m F ir- 2Im Tyr (A.3.1) 2 eIm e VVo+e e Thus for the integrand -2Rp 2 2BRp-1K 2Im + Integrand,v e e e -2p Rp 2 2 Rp g 2 I -2R Im Rp -2 ImXo e e which vanishes exponentially since g2 > 1 and r }. (ii) argX = -r/2 X = = -iIXI, arg\V= r/4 = RprX >0, Im < 0. Here we can use Buchholz (1953, p. 100, eq. 20) to assert Y(, X) v exp - log 1XeJ e (A. 3.25a) 1-2VJBTTcos Y2(r, X) - exp{ - 2 log2k e (A.3.25b) (A. 3.2 5b) But now equation (A. 3.13) holds for the F-function r(- +), hence 87

THE UNIVERSITY OF MICHIGAN 7030-4-T -2Ix h coo 2 X coso (A.3.1) -e 4o 4 Therefore the integrand obeys 2,I cos8o 2fj COS -2coi 0oos Integrand -v e e e which vanishes exponentially if and only if ~ + ~ < Af. (iii) -Ir+6, argX < -r/2 == RpX < 6:arg\< 4 R0, -r+6 ImYX<O0, Rp <<-Im f Here y2(i7) and yo(i7, X) are governed by equations (A. 3.16a) and (A. 3.16b). Arguing as previously <.,..sl e.2Im^ e 2ImV \l +e-2m; 2Im X1 (A. 3. 1) Im e7e m 2me Then for the integrand 2 RpV l 2 Rp / +e e e mX2 hterandF t Km-2Im e e By arguing as above we see that the first term vanishes exponentially if and only if l + g2 < nf the second term vanishes exponentially if and only if + l + < 2 o. But the first condition implies the second, therefore the integrand vanishes exponentially if and only if it holds. Suppose then 6 + ( < fri and we again refer to equation (1.12); the above argument shows that the path of integration may be closed by an infinitely large semi-circle in the lowe r half plane. Then by the use of the residue theorem, vN(, rl) can be evaluated as a sum of residues. The residues of P(x + at the poles X = -ik(2n+1), n = 0, 1, 2,... 2k (-1)n G% I ()are i)-,! thus equation (1.12) yields 88

THE UNIVERSITY OF MICHIGAN 7030-4-T -1/2 y ik(2n+l y ik(2n+l) -) VN(9 r) = (2ik)-1/2 ( 1n Y[1' 2 2'2 * [ y2[.-i(2n+ 1l N W yl.-ik(2n+l L2 * y; [. -k(2n+ 1] -y [, -lk(2n+ 1 [r. - Lk(2n + 1)] From the form of the residue series it is seen that the asymptotic forms and arguments given above imply convergence for ~ + ~ < F<. The restriction 1 + < j' may be slightly eased. Using the integral representation of -e- IR we again write (1.12) as -ikR -3/2 (D 2, - X) ~e = _ (21k) t -3/2 2 1,l ( 2x) y2 vQ, ) d. + r + __.. - R 27r i Ve ik 2 2ik 2 Y1 2(ot -oxo-ia (0 <a <k) y(n, M)yI(o, ). The exact arguments used above show that the path of integration of the integral can be closed in the lower half plane when, + i<2 ~J — -. We obtain then -kR y1[ ik(2n+) ln Y2[e2' ik(2n+ 11 ~V,n) = -R + (2ik) n=0 [n LJ — - - yl[=-ik(2n+ Dy -ik(2n+l and < 2+ 2 < 2 no - F implies the series converges. It is instructive to consider the behavior when 1 f + = iri r/t ~~ = 2 i -' ) Then the exponential amplitude factors are equal to 1 and the behavior of the integrand as I XI |- is governed by the powers of X which appear. The arguments above show that we need only consider the behavior for -r+6,< argX < -r/2; the pertaining asymptotic forms show that the integrand -1/|l x /4 Thus the path of integration can be closed in the lower half plane. But at the poles |IX= k(2n+ ), consequently the residue series, whose terms behave like Ix 11/4 for large X, does not converge. 89

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA - R& (Security classification of title, body of abstract and indexing annotation must be entered when the overall report is clasailied) 1. ORIGINATIN G ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION The University of Michigan, Radiation Laboratory UNCLASSIFIED Department of Electrical Engineering zb GROUP 3. REPORT TITLE The Scattering by a Paraboloid of Revolution Due to an Interior Axial Point Source 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Technical Report 5. AUTHOR(S) (Liet name, firt name, initial) Stone, Stephen E. 6. REPORT DATE'7. TOTAL NO. OF PAGES 7b. NO. OF REFS November 1965 89 20 8a. CONTRACT OR GRANT NO. 4.. ORIGINATOR'S REPORT NUMBERfS) AF 04(694)-683 7030-4-T b. PROJECT NO. c. S b. OTHERR R EPORT N(S) (A ny other nmbers that may be aelssned this report) 10. AVA IL ABILITY/L!MMITATION NOTICES Qualified requestors may obtain copies of this report from DDC. 11. SUPPI EMENTARY NOTES 19. $PONSORING MILITAMY ACTIVITY USAF Ballistic Systems Division Norton AFB, California 92409 13. ABSTRACT If one considers a paraboloid of revolution of focal length rQ with an interior oint source located anywhere on the axis, the exact solution to the Dirichlet or Neumann pro blem (Green's function of the first or second kind) may be written in the form of an integral representation. In this report we consider the asymptotic evaluation of these integrals for both low (krbo<<) and high (ko>>1l) frequencies. The low frequency results are obtainable from an infinite series over the zeros of a particular Whittaker function, corresponding to a Mie series found in the scattering by closed convex bodies. For high frequencies, we find multiple reflections and caustics arising from saddle point evaluations as well as'whisperin gallery' waves, which arise from the nature of the behavior of the above zeros at high frequencies. The work at high frequencies is only briefly discussed since it covers research actually in rogress. The aim of this discussion is to introduce possible approaches to the problem. This work on the paraboloid is preliminary to a description of scattering by general concave surfaces. Having integrated the formalism in terms of the physical phenomena such as the whispering gallery waves, multiple reflections and caustics, we are now in a position to search for generalizations of these to other concave surfaces. The scheme we propose is to determine the dependence of the physical effects on the local geometry of the paraboloid and then to make the essentially physical argument that this geometric dependence is the same for other concave shapes. This approach is similar to that used in determining the creeping waves on general convex shapes andis an application of the physical arguments used in Keller's geometric theory of diffraction. D DJAN4 1473 UNCLASSIFIED Security Classification

UNCLASSIFIED Security Classification 14. KEYWORDS LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE 1 WT RADAR CROSS SECTIONS PARABOLOID OF REVOLUTION CONCAVE SURFACES WHISPERING GALLERY WAVES INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address imposed by security classification, using standard statements of the contractor, subcontractor, grantee, Department of De- such as: fense activity or other organization (corporate author) issuing (1) "Qualified requesters may obtain copies of this the report. report from DDC." 2a. REPORT SECURITY CLASSIFICATION: Enter the over2a. REPORT SECUIRTY CLASSIFICATION: Enter the over- (2) "Foreign announcement and dissemination of this all security classification of the report. Indicate whether ( announc ntn "Restricted Data" is included. Marking is to be in accord- report by DDC s not authorized." ance with appropriate security regulations. (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC 2b. GROUP: Automatic downgrading is specified in DoD Di- ths rer et from DDC ther quied DDC rective 5200. 10 and Armed Forces Industrial Manual. Enter users sha request through the group number. Also, when applicable, show that optional., markings have been used for Group 3 and Group 4 as author- (4) "U. S. military agencies may obtain copies of this ized. report directly from IDC Other qualified users 3. REPORT TITLE: Enter the complete report title in all shall request through capital letters. Titles in all cases should be unclassified... If a meaningful title cannot be selected without classific — tion, show title classification in all capitals in parenthesis (5) "All distribution of this report is controlled. Qualimmediately following the title. ified DDC users shall request through 4. DESCRIPTIVE NOTES: If appropriate, enter the type of.. report, e.g., interim, progress, summary, annual, or final. If the report has been furnished to the Office of Technical Give the inclusive dates when a specific reporting period is Services, Department of Commerce, for sale to the public, indicovered. cate this fact and enter the price, if known. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on 1. SUPPLEMENTARY NOTES: Use for additional explanaor in the report. Enter last name, first name, middle initial. tory notes. If military, show rank and branch of service. The name of the principal iauthor is an absolute minimum requirement. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (pay6. REPORT DATE; Enter the date of the report as day, in for) the research and development. Include address. month, year; or month, year. If more than one date appears on the report, use date of publication. 13. ABSTRACT: Enter an abstract giving a brief and factual. T L N R OF P: T t p c summary of the document indicative of the report, even though 7a. TOTAL NUMBER OF PAGES: The total page count it may also appear elsewhere in the body of the technical reshould follow normal pagination procedures, e., enter the port. If additional space is required, a continuation sheet shall number of pages containing information, be attached. 7b. NUMBER OF REFERENCES: Enter the total number of It is highly desirable that the abstract of classified reports references cited in the report. be unclassified. Each paragraph of the abstract shall end with 8a. CONTRACT OR GRANT NUMBER: If appropriate, enter an indication of the military security classification of the inthe applicable number of the contract or grant under which formation in the paragraph, represented as (TS), (S), (C), or (U). the report was written. There is no limitation on the length of the abstract. How8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate ever, the suggested length is from 150 to 225 words. military department identification, such as project number, 14. KEY WORDS: Key words are technically meaningful terms subproject number, system numbers, task number, etc..surenmryenb numb. ec. or short phrases that characterize a report and may be used as 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the offi- index entries for cataloging the report. Key words must be cial report number by which the document will be identified selected so that no security classification is required. Identiand controlled by the originating activity. This number must fiers, such as equipment model designation, trade name, military be unique to this report. project code name, geographic location, may be used as key 9b. OTHER REPORT NUMBER(S): If the report has been words but will be followed by an indication of technical conassigned any other repcrt numbers (either by the originator text. The assignment of links, rules, and weights is optional. 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 UNCLASSIFIED Security Classification

UNIVERSITY OF MICHIGAN II3 9015 0326 8666i 3 9015 03526 8666