THE UNIVERSITY OF MICHIGAN 7741-2 -T Asymptotic Theory of Diffraction by Smooth Convex Surfaces of Variable Curvature Technical Report No. 2 AF 04(694)-834 by Soonsung Hong August 1966 Prepared for Ballistic Systems Division, USAF, AFSC Norton Air Force Base, California

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THE UNIVERSITY OF MICHIGAN 7741-2-T FOREWORD This report was prepared by the Radiation Laboratory of the Department of Electrical Engineering of The University of Michigan under the direction of Dr. Raymond F. Goodrich, Principal Investigator, and Burton A. Harrison, Contract Manager. The work was performed under Contract AF04(694)-834 "Investigation of Re-entry Vehicle Surface Fields (SURF)". The work was administered under the direction of the Air Force Ballistic Systems Division, Norton Air Force Base, California 92409 by Major A. Aharonian BSYDF and was monitored by Mr. Henry J. Katzman of the Aerospace Corporation. Publication of this report does not constitute Air Force approval of the report's findings or conclusions. It is published only for the exchange and stimulation of ideas. BSD Approving Authority George N. Kent, Major USAF Director, Office of Information BSD/BSE iii

THE UNIVERSITY OF MICHIGAN 7741-2 -T ABSTRACT A general method is presented for obtaining successive terms in shortwavelength asymptotic expansions of the diffracted field produced by plane acoustic and electromagnetic waves incident on an arbitrary smooth convex surface. By introducing the geodesic coordinate system on arbitrary surfaces of non-constant curvature, both scalar and vector integral equations governing the surface fields are solved directly. The expressions for leading and second order terms in the asymptotic expansion of the diffracted fields are obtained explicitly and the differences between acoustic and electromagnetic creeping waves are shown. iv

THE UNIVERSITY OF MICHIGAN 7741-2 -T TABLE OF CONTENTS FOREWORD iii ABSTRACT iv I INTRODUCTION 1 II THE GEODESIC COORDINATE SYSTEM 3 mI DIFFRACTION OF A PLANE ACOUSTIC (SCALAR) WAVE 6 3.1 Integral Equation Governing the Surface Field 6 3.2 The Surface Field in the Penumbra Region 9 3. 3 The Surface Field in the Shadow Region 17 IV DIFFRACTION OF A PLANE ELECTROMAGNETIC (VECTOR) WAVE 25 4. 1 Integral Equation Governing the Induced Currents on the Conducting Surface 25 4.2 The Induced Currents in the Penumbra and the Shadow Regions 29 V DISCUSSION 35 VI ACKNOWLEDGMENTS 37 VII REFERENCES 38 v

THE UNIVERSITY OF MICHIGAN 7741-2 -T I INTRODUCTION A shadow is formed when a wave is incident upon a smooth convex body which is large compared to the incident wavelength. In the neighborhood of the shadow boundary the surface field does not abruptly vanish and a penumbra region exists. Some waves penetrate into the shadow region and account for the non-zero fields there. These phenomena are due to diffraction of the incident wave by the object. The mathematical problem of analyzing diffraction of waves involves finding the short-wavelength asymptotic form of a solution of the wave equation satisfying an appropriate boundary condition of the diffracting surface and the radiation condition at infinity. Detailed studies of the surface field on a circular cylinder and a sphere (Franz, 1954) for which the exact solutions are available, indicate that the incident wave is diffracted near the shadow boundary and the diffracted waves proceed along the geodesic into the shadow region, spilling off energy as they travel. Their phases are determined primarily by the distance traveled from the shadow boundary. The waves diffracted by a smooth convex surface are frequently called creeping waves. In obtaining a description of the waves diffracted by an arbitrary smooth convex surface of variable curvature, two techniques can be used: (1) finding the asymptotic form of an exact solution for a canonical body and generalizing the results; (2) solving the boundary value problem directly by an asymptotic method for a general surface but in restricted regions. The difficulty with the first method is that very few canonical problems can be solved exactly. Thus, in the well-known geometrical theory of diffraction (Levy and Keller, 1959), a locally cylindrical body is chosen as the canonical body in 1

THE UNIVERSITY OF MICHIGAN 7741-2-T analyzing diffraction of waves by arbitrary smooth surfaces. While this theory gives the correct leading term in the asymptotic expansion of the diffracted fields, it does not yield higher-order terms. The purpose of this paper is to discuss an integral equation approach based on the second technique. It can yield not only the leading term but also higher-order terms in the asymptotic expansion of the fields diffracted by an arbitrary shape with a smooth convex surface. The method to be used is the following. The geodesic coordinate system is introduced to describe the geometry of the diffracting surface (Sect. 2). In terms of this coordinate system, the short-wavelength asymptotic form of the integral equation governing the surface fields is derived (Sect. 3. 1) for the acoustic case, and its solutions are derived for the penumbra (Sect. 3.2) and shadow (Sect. 3.3) regions. The same procedure is repeated for the electromagnetic case (Chap. 4). 2

THE UNIVERSITY OF MICHIGAN 7741-2 -T THE GEODESIC COORDINATE SYSTEM From the analysis of the sphere solution (Franz, 1954), it is observed that the creeping waves propagate along the geodesic. Thus, we propose to use the geodesic coordinate system to describe the diffracting surface. An important advantage of this coordinate system is that it can be defined on any smooth surface. For the sake of simplicity, it is assumed that the diffracting surface is symmetric with respect to the shadow boundary and that the torsion of the geodesic is zero. \^^ —-~ ~ Geodesic b n \vO tt uO 0o t shadow boundary - J u = constant FIG. 2-1: GEODESIC COORDINATE SYSTEM 3

THE UNIVERSITY OF MICHIGAN 7741-2 -T Let us define the geodesic coordinate system as follows. The curve u=0 is taken to represent the shadow boundary with v denoting arc length along it. At each point of u=O the incident wave is tangent in a given direction, and thia defines a geodesic through each point of u-0; these geodesics are taken as the coordinate curves vxcinast; with u taken as arc length along the geodesic measured poaitivety from the shadow boundary. The geodesic coordinate system is orthogonal and the linear element is given as: ds2 = du2 +G dv2 with G(u=O) 1. (2.1) Because of the assumption that the geodesics are planar, G is independent of v. The diffracting surface may be described by the Gauss-Weingarten equations (Struik, -1950):: at au g at ab'-= -K b av au tt (2.2) abc 2 2 __ -GKt+K ] With K K I Kt av G tt tn tt= tttn an - n - au K t av Ktnb where ai au, a =s b with F = the position vector. (2 3) au 8v Here i, t and -M are unit normal, tangent and binormal vectors along the geodesic, respectively. Kg is the curvature of the geodesic. Ktt and Ktn are 4

THE UNIVERSITY OF MICHIGAN 7741-2 -T respectively, the tangential and the normal components of the curvature of the u= const. curves. Thus, the two principal curvatures are K and K and their g tn product is 1 2 u2 KK, d - (2.4) g tn F' au2 while- Ktt is related to the function G by aG u 5 K = (2.5) tt 2G In addition to Eq. (2.2), the Codazzi equation must be satisfied: aKt tn (K -_ au tt g tn (2. 6) aK - = 0 av A more detailed analysis of the geodesic coordinate system can be found in most books on differential geometry (e. g. Struik, 1950). 5

THE UNIVERSITY OF MICHIGAN 7741-2 -T II DIFFRACTION OF A PLANE ACOUSTIC (SCALAR) WAVE The first problem to be considered is the diffraction of a plane acoustic (scalar) wave by an acoustically hard surface, i. e a Neumann boundary condition is imposed. 3.1 Integral Equation Governing the Surface Field. We suppose that a plane acoustic wave is incident upon a smooth convex surface and that the normal derivative of the total field on the surface vanishes. Then the integral equation governing the surface field can he easily de — rived by Green's theorem (Hbni et al, 1961): U()2a_1 1-ikR f U(7) 2U inc) -./da. U(') 1 (r). Re e (3.1) inc. 22- R 3 where R= r'-r, and U. is the incident field. Without loss of generality, inc. we will consider the surface field on a geodesic which will be called the curve vL0Q Inlerms of the geodesic coordinate system, the incident wate on the geodesic v=O is U i (u, vO) eild(u=0, v=0) -(u,v0) (3.2) In the above two equations the time dependence factor e is omitted. As observed in the study of a circular cylinder and a sphere (Franz, 1954), the phase of the diffracted (creeping) wave is determined mainly by the distance traveled from the shadow boundary, thus we shall set U(r)-e ik I(r) (3.3) 6

THE UNIVERSITY OF MICHIGAN 7741-2-T 2r and for a large k (=;, the wave number) I(r) is assumed to be slowly varyiku ing in comparison with e. Substitution of this expression into Eq. (3.1) gives: I(u, 0)= 2 exp. {ik (0, 0) (u, 0)- iku } (u' du'dv' I(u1, v') R {(u', v') R} exp. {ikR-ik(u-u'}. (3.4) Since we are interested in the short-wavelength behavior of the solution, we will replace the second term in Eq. (3.4) by its asymptotic form. For large k, the integrand has a saddle point where the derivative of the function R-(u-t1) vanishes. The Taylor series expansion of the vector R near r=r' is easily derived by means of Eq. (2.2) and is given by: - l Ri-r'(u', v')-i(u, O) (u'-u) (u)+ v' b(u)- [(ut-u) vK ^()b(u) -2 L g tt + vG K G (u)(u) K +(u-u)3 (U)(U) + K...) ~ F tt,,- ] (3t6) Above,andinfollowingpages, thecurvatures (K etc. b and without the argument for the v coordinate represents their values at vO. The dots denote the derivative with respect to the argument of the function. Using the above expression, the solution of the equation av' L -(u-ut)] = G(u) [(u-u')Ktt(u) + -0 (3 6) 7

THE UNIVERSITY OF MICHIGAN 7741-2 -T yields the saddle point at v'= 0 for the v' integration. Applying the method of steepest descents (Brekhovskikh, 1960) to the v' integration in Eq. 3>. 4) we obtain an asymptotic expression of the integral equation for large k. e f ikR -iku uI) G(u'). I(u. 0)2exp.{ik(0.0,0).(u,0)-ik — du' e ~, r 4 (I's.R 4R(v 0) n(u', 0) R (v 0) - _' "_ _ _}v' IK, 2 Iuh2 2( )} I(u',0- oi (u+ ) Io2u') + (u), 0 +0 (k 3/2) (3. 7) where R I' (u', 0)-'r (u, 0) (3.8) K2(U) K() (u) (u) R (u-u') —u (u-u')+ -- ) (u-') o 24 24 4'2 K() K (U (u) K() (U) 5 ( + _- —. (u-u)5 (3.98 L1920 90 80 j 8

THE UNIVERSITY OF MICHIGAN 7741-2 -T av' 0 a. (v' =O) [ {-RO tt(u') T(u')+ K(u')n(ul)}] (3.10) and a4R 3G(u') iaV4 R 2 tt(u (u)+Kt(u')n(u) 3v'4 o +,_- - a R (vtO} (3.11) In Eq. (3.7), the contribution from the u' integration between u and infinity is neglected. The reason is the following. From Eq. (3. 5) it can be shown that R near u=u' is o R (u-u') +... for u> u' o (u'- u) +.. for u < u', thus, exp. iktR -(u-u'}} in Eq. (3. 7) has a saddle point at u=u' only if u > u'; therefore, by integrating by parts, one can show that the contribution from the region u< u'<oo is asymptotically negligible for large 1k. Now the integral equation governing the surface field is thus reduced to a one dimensional Volterra equation. 3. 2 The Surface Field in the Penumbra Region. In this section, the asymptotic integral equation (3. 7) governing the surface field is solved for the penumbra region. It is assumed that the curvatures are slowly varying and that -- is of order one or less. In order to obtain an appropriate tn form of Eq, (3. 7) in the rieighborhood of the shadow boundary, we shall set 9

THE UNIVERSITY OF MICHIGAN 7741-2 -T M Lp (O.P(0) (p -) o g g Kg 2 ku M 2 k= 2 (3.12) ku' =M, o 1/3 and further assume that [kp (u, v)] > 1. Near the shadow boundary (u=0), the phase function, t(0, 0). (u, 0)-u, of the incident wave term in Eq. (3. 7) can be expanded in Taylor series by means of Eq. (2.2) 3 5 t(0, 0)- (, 0 0)-u - K2(,0) + 1 O) +4p (0,0) P(0, O) (3.13) 6 g 120g g g (p (0, 0)= 0 by assumption of symmetry of the diffracting surface with respect to the shadow boundary. Substitution of Eq. (3.12) into the above expression yields the asymptotic form of the incident wave: ikt (O, 0). (u, 0) -iku e 6 r p )P( )} 5 + (M 3) e — e Ll i f + 0(M ) 0 (3.14) Similarly, an appropriate asymptotic form of the second term in the right-hand side of Eq. (3. 7) can be easily derived by expanding the integrand near u=u' and by substituting the relationships of (3.12) along with 2 p g(u,0)P p (0, 0) +near u=0 10

THE UNIVERSITY OF MICHIGAN 7741-2 -T The high frequency asymptotic form of the integral equation governing the surface field in the penumbra region is then 3 -i 6T {l+4p (0,O)p (0, 0)} 5 I(e,0)=2e 1+ --- 120 M 0 -i ( )3 24 -i -T) 1/2 K2(~, T d- / dTI(, 0)e e' O ) J2 M2 o -0 0 (3.15) where K(, T) e (-r) [() +i ( -T) r 1 p(O, O) (0, 0) 224 1/2 96 20 192 8 + P(O, 0)0 (0, o) - +i 27(U.3 p +(0, 2 2) 3 pg(O., ) li6 2 t2(0,0)J + i] * (3.16) -1 Since there is no term of order M in the above equation, we shall take the 0 asymptotic expansion of I as I(,0)=I() -3 I( 0)-= Io(, 0)+ 2 +O(M ) (3.17) M 11

THE UNIVERSITY OF MICHIGAN 7741-2 -T Substitution of this expression into Eq. (3. 15)' simplifies the integral equation and the following equations for I and I are obtained: o0 4 o -iS 3.n / -i (. 0)- 2e - -X dr- I(.,O)(-r) 24 -oo (3.18) and {l +4p (O,0)Pg(0,0)} -i Iei o)~ O) -S 9__KV 5 e 6 1 dT Io(r, 0) K (, T)e -00 Jr 3 e -i1T1/232 e/ dTl.dOI1 (T,0)(-r) /2e 24 (3. 19) 4 1 124 -co -00 We observe, from the above two equations, that the kernel functions are the same, and that substitution of the solution for the leading term I yields the solution for the second order term. Similarly, integral equations governing higher order terms in the high frequency expansion of the field can be derived by including further terms in the asymptotic expansion of Eqs. (3. 7) and (3.15). Since Eq. (3. 18) is a Volterra type and its kernel is a function of — r only, the use of Fourier transform is suggested. We shall set I(t) = Io(0) ee dt. (3.20) -00o 12

THE UNIVERSITY OF MICHIGAN 7741-2-T Application of the Fourier transform to both sides of Eq. (3.18) and use of the convolution theorem yields oD 3 -igt -i C 4 - igt- i - d d~e 6 e1/2 24 The denaminator and other integrals for I can be evaluated by means of the functions o oo F S - dx x e 173^ 3 (3.23) 0 The numerator of the above equation is an Airy function (Miller, 1946) 3/22 41i(t21/3) for various n (Weston, 1960). I particular, F = r3/2 223 3-1/6 e 4 Ai(p)[Ai(p)-iBi(p)] (3.24) F^3/2 3-1/2 e 4 [2Ai(p){Ai(p)-ii(p+ (3.25) -5r 2, 3 /2 3-1/2 (12)- 1/3 i 4 [{Ai(p)} P {A(p } -i{pAi(p)Bi(p) +Ai(pBi(pd}] x (3.26).3~13

THE UNIVERSITY OF MICHIGAN 7741-2 -T The remaining F may be expressed in terms of the above three functions by n the relation, n i - (3i(n- 2) F (p) C (12) i F(p)+D (12) 3 i( )F(p)+E(12) 3 2) F2() n n o n n 2(p) (3.27) where Cn, D and E are given in Table I. n n n C D E __ n n n 3 2 4p 0 4 0 6 4p 5 8p 16p2 10 6 28 80p 16p2 7 32p2 108+64p3 112p 8 288p 672p2 220 +64p3 9 2912 +512p3 10048p+1024p4 3456p2 TABLE I: THE FUNCTION F n Using the F1 function, Eq. (3.21) reduces to 24/3 /r I( t) t213/328) where wl- i [Ai- iBi] 14

THE UNIVERSITY OF MICHIGAN 7741-2-T The inverse Fourier transform of Eq. (3.28) yields the solution for the leading term: ep-1/3 1 eip^2 1/J3 I(ev, 0)- / dP () (p2 t 2 (3.29) -00 Similarly, the application of the Fourier transform to the integral equation governing the second order term (3.19) gives N 1/3 Il(t) N ---- N -(p t21/3 (3.30) 2; ^Ai (p)w (p) where 00 3 6 1. -, -r) -it{ 1 / 6 1 2 Ne d i e [t g e 6- I d ( 0)r2(, )e 24 60 0 2 -00 -oo (3.31) Using the convolution theorem, the functions F of Table I and Eq. (3.28), one can show that: rpw 2 w7'1( ) 1 ( 6 -3 4 ]F) (oo) o}(00)2() p[- (. Po)i g (o,go ) + 0)o Wj' 00.(p) 5 30 5 5 [ 3 (0, )g, O)+p7(o, ) ] Pg( pg( (3 3 32 (3. 32) Substitution of this expression into Eq. (3. 30) and inverse Fourier transformation gives the solution for Il. Combining I1 with Eqs. (3.3) and (3.29), we obtain 15

THE UNIVERSITY OF MICHIGAN 7741-2 -T the desired expression for the surface field in the penumbra region: i0~iku U as ikul d eiku / d pe dU r 2 U (u,O). e.dp + )p/3 e * l 15 di~ 2.iSW-~P +P 17 P(0,O) 1 w_(p) 5 +^ 0)^ (0,0 (0,0) + 2 2 rpw1(p)1r7 d p p (0,0)1 4 p pw1(p)o3 (o,0)-( (, 0 + g2 1 tndu (3.33) 3 {vd(p)} 30 p {P dl( g(O', O)5 ( P pds (.P). and aX = the incident wavelength the shadow region) (3 33) can be expressed as a rapidly convergent series in terms gudu When U is positive and sufficiently large (far away from the shadow boundary into of the residues at the poles wl(p)O. This residue series represents the creeping 16

THE UNIVERSITY OF MICHIGAN 7741-2 -T waves. When a goes to negative infinity (illuminated region), Eq. (3.33) reduces ik (0, 0). - (u, 0) to 2 eik' ) (Logan, 1959) which is the geometrical optics term. The width of the penumbra region is of order d. 3.3 The Surface Field in the Shadow Region. The incident plane wave cannot reach the shadow region directly (otherwise the shadow does not exist), and only the waves diffracted near the shadow boundary proceed into the shadow region. AnLexpression for the surface field in this region may be obtained by following two steps (H6nl et al, 1961): 1) obtain the initial values of the diffracted (creeping) waves from the solution for the penumbra region at the shadow boundary; and 2) solve the homogeneous integral equation (without the plane wave term in Eq. 3. 7 and the limit of the integration only over the surface in the shadow region) and match the initial values at the shadow boundary. The initial values of the diffracted waves may be obtained from Eq. (3. 33). When u is positive, the integrals of this equation can be expressed in terms of the residues at the poles wl(p) 0, and each residue represents a creeping wave (Goodrich, 1959). The values of these residues at the shadow boundary (u0=) yield She necessary initial values of the creeping waves. The residue series of Eq. (3. 33) at the shadow boundary is 00 P (0, 0) U (0,0) 2'2 \ 1 + 1 1 2 d^^ 0 ^T (f(Of 0) d2 0,0 + (O, o) - g (o, o) + - (o' (~. o) 2 (o, o 90 Pg() (2 5 2 (0, 0) 30 Pg( ~) 2 du Pi tn 9 du (3.34) where pt is the lth root of wl(p1)=0. In the above expression, each term in the series represents the initial value (birth weight) of the Ith mode of the creeping waves. 17

THE UNIVERSITY OF MICHIGAN 7741-2 -T The short-wavelength asymptotic form of the homogeneous integral equation governing the surface field in the shadow region is the same as Eq. (3. 7), except for the incident field term which vanishes now. Before attempting to solve the integral equation, we can observe that the common factor 2 -1/2 G (u') - ain the kernel of Eq. (3..7) behaves near the saddle point u=u' as follows FJ (u R (v=o)] J G(U' l-Ro.tK(u')t(u') V Lov,2 Lo ti +Kt(u)(u)}]l1/2 [_;]1/4 (3.35) (Refer to Eqs. (2.2), (2. 4), (2.5) and (3. 10.) The above relation indicates that the solution of the homogeneous integral equation has a factor G (u] -1/4 In view of the phase factor for the solution in the penumbra region, Eq. (3. 33), we shall set the solution of the homogeneous integral equation in the form rI(u, 0 [ -1/4 2rAexp [ / ls2 o(S+Y (s)'+ (s) I(u, 0)A[ A (u exp 22'y(s)+ + —-+.. d I [GIikjds *1]/32 ^o(s) M (s) M (s) 0 (3.36) where the constant A is the initial value, 18

THE UNIVERSITY OF MICHIGAN 7741-2 -T and M(u)- [kpg(uO) 1/3 The propagation factors yo 1, and 2 are yet to be determined. On setting k(u-u) M (u)T (3.37) we obtain the following expression under the assumption that the cur'vatures are slowly varying: u k 2 L(3 2 ( J k M () Y () (s) 2.. 12(U) 2 6.T g u' + 24 [ T 172( p0) - (u'- - ( (u, 0) 0)L - 3 g g o - pgu. o0)pg(u, O)o(u) + 0 pg (u, 2 ) yo(u) 23-3 -T pg(u, 0)l(u)-p (u, 0) u ] +0(M ). (3.38) Now combine Eq. (a. 7). without the incident plane wave term, with Eq. (3. 36) and expand the integrand near the saddle point u-u' by Taylor series (asing Eq. 2. a). 19

THE UNIVERSITY OF MICHIGAN 7741-2 -T After these algebraic manipulations and making use of Eq. (3. 37) and (3. 38), we can obtain the following asymptotic homogeneous equation; for the propagation factors 1 ~ PI — - 1/2 d 7' 2 -- - 3; - T 2T [1- (U,) {3 t + - 190 360 Pg g ( 5 r 1 2 p (u,O)p (u,O 8 r3L - 1 23 _ _2_ _ T 2 g + 1 - P (u) + 2( ) + g o) (U) 192 36 p0) P 3 pu) 4 12g l8 P(u~ol2,luoo),o 0)(g1/3 3 +e + 1-2 (u) 2 / — + 2M( (u)T - (g(U 0)Vo(u, O) + 0(M-3 (U)- go 0) u))}] (3(39) 20

THE UNIVERSITY OF MICHIGAN 7741-2 -T Upon comparing coefficients of the leading term in the expansion in powers of M we obtain t00 1 - - e2 d1T /2 exp {i -- i o(U) 2/3} (3.40) 0 Comparing with Eq.(3. 23), it can be shown that the right-hand side is related to the function F of Eq. (3.25). Substitution of Eq. (3.23) and (3.25) into Eq. (3.40) yields the following: oAi )1 ( ) = 0 (3.41) The solution of this equation determines -Q. To be consistent with the initial values (Eq. 3.34) of the creeping waves, the roots of v ('y )=Q must be chosen. In terms of the definition'o le e 3 the various roots are given by Table II at the end of this section. Since yo is constant, comparison of the coefficients of M in Eq. (3. 39) 0 M yields 21

THE UNIVERSITY OF MICHIGAN 7741-2 -T ed 4 71/2 F2 (uO) 24) ~0 -W dT L1 TP'1'P 0) T 2 L j/2d4L3g 0 -iT {fV(u)+-p (U.*u0)T} 21/3] exp. {-ly2/ r-i } (3.42) Various integrals in Eq. (3.42) can be identified with Eq. (3.28). Thus, after substitution of F from Table I, we can evaluate 71 from Eq. (3.42) and the result is 1/3 p (u, 0) ly(u) = i 2 /3 g (3.43) 1 6 From the coefficients of M- in Eq. (3. 39), one finds: T2(u) 2-2/3 F() (-2 /3 Fo po' +2P(Uo () P 0(uO) 2& 2~ 0F (2 00 5 2p (u, 0) 30 2 tn (U,45 o 2(" O pg(U, O)pg(U O)+ 4 p2(U, O4] u(o)u r 0( +-45 2LOc 45 go (u0)+. 4% (u (3,44) and upon substituting the values of F and F2 given by Eqa. (3.24) and (3,26), 0 2 we obtain Y2, namely 22

THE UNIVERSITY OF MICHIGAN 7741-2 -T -2/3 1 p (U, (O) p (U,0 ) ) p (u0)'V24' (o L1 ^4p(0) 60 90 2 1 2 4 2 L+ 60 [ 4(Us O)p (U, O) + 15 p(u o) (3 45) Combining Eqs. (3. 36), (3. 43) and (3.45) and matching the initial values given by Eq. (3.34) by letting u=O, we obtain the desired solution for the surface field in the shadow region: r/,o,.,L"',~ OD, i - U (U. O) L'1p iku 1 G[~ " E - i (,0 A0) d2p 1. e P (0,0) 2 -2/3 0' 0 4Ptn(00) 180g (0du 2, (0,0) d2 60 o,2 tn du u u r 6' ds M) 2 1/3 6i/ ds 2 13 exp, -e M) ^/ Ol 2 "e - Mlsf g 21/3 ~exp. e2 (s,0)M(s) J0 -o 23

THE UNIVERSITY OF MICHIGAN 7741-2 -T 2 1- (1 P p(s, 0) g(s0) (80) (8, 0) 131 1O4ptn(s, 0) 60 90 2 1 2 4 2 + 1 6 - P4g(.,) (O80)+ (35 Pg (SO)}] (f,46 where M(u)= kp(u, 0)] 1/3 and various values of j3 and Ai(-1) are given in Table II. L 3, i Ai(i3) 1 1.01879 + 0.53566 2 3.24820 -0.41902 3 4.82010 + 0.38041 TABLE I: THE VALUES OF 31 and Ai(-31) In deriving the Eq. (3.46) the following relationships are used: ir E 2r i i- iY ol =e 1 and wl(p)= e 2 fAi(p (3.47) 24

THE UNIVERSITY OF MICHIGAN 7741-2 -T IV DIFFRACTION OF A PLANE ELECTROMAGNETIC (VECTOR) WAVE The second problem to be investigated is the diffraction of a plane electromagnetic wave by a perfectly conducting smooth convex surface of nonconstant curvature. Since much of the analysis is similar to that which we have already discussed for the acoustic case, the details will be omitted wherever possible. 4.1 Integral Equation Governing the Induced Currents an the Conducting Surface. If a plane electromagnetic wave is incident upon a smooth convex conducting surface, the integral equation governing the induced currents on the conductor is ( inlet al, 1961) J () 2n Cr) xHin ( )- n)x da' -()xR e (4.1) R 3 -inc -i.t Here H is the incident field and the time dependence factor e is omitted. Again, without loss of generality, we shall consider the induced current along the geodesic v=O. The expression for the incident field is H nc.(u,0 [-cosO b(0)+sin (0, 0)] e t(o0 r (u) (4.2) Hhor 0~ -n(0,0 O)X o where 8 0-sin 1 i- - is the polarization angle of the incident wave. With the substitution J (), [~t(r) It(r) +b (r)!)] e (4. 3) 25

THE UNIVERSITY OF MICHIGAN 7741-2-T the vector integral Eq. (4.1) is reduced to two coupled scalar equations: R T(u). n(u)x [It(u,, v') t(u', v')xR+Ib(u', v)b(u, v')x-R] and inc 1 1 -ikR K(u, 0)o= 2 1b (u, 0) -- u'dv JG(u R e'",0 6 (u,o) 3 b(u)- *Cn( x Iu, v') (u'vx, v')xR+ V)u'')'v')xR}] ikR- ik (u-u') (4.5) e * The above two equations are similar to that of acoustic case, Eq. (3. 4). They also have saddle-points at v'-O for the v' integration and at uxu' for the a' integration along the v'O0 curve. Performing the v' integration by the method of steepest descents, we obtain 26

THE UNIVERSITY OF MICHIGAN 7741-2-T u O R(v O -D, Lav2 0' -It(U'O)2 G (u'. t (U) (u) (x (u) * xR I ( t( u ) ~~}2 a R2 + (u'') T(u') to(u) ) xiu') )Nu)} (u' O) -2RR{ ('-0 J 0 - alb +.t G(u')N 7'+u')-tt(u').Ro (t(u).N(u) N {(u') 27

THE UNIVERSITY OF MICHIGAN 7741-2-T and u uo~)=2Ibc. (upv=O)- d, duJ' 21 1 -a o J'"2 (v':0)l COL [ i{n(u)R }{b(u).b(u)} [ (vt ok n(u) R {b(u) b(u)} LR 2 av'2 12 G(u') I (u's 0) (u',0)- f2 ( {b(u). b(u')}{(K(u')ni(u)- R l 2a(VI =o) 2R a2I 2 + t(y'=,u') 3- {Kt (u')n(u).R -n(u) t(u')} + 0 (k3/2) (4.7) where Kt(u)N (u) Ktt(u) t (u) + K(- U) (u). 28

THE UNIVERSITY OF MICHIGAN 7741-2 -T 4. 2 The Induced Currents in the Penumbra and the Shadow Regions. In this section, Eqs. (4.6) and (4.7) are solved by the same technique used in the acoustic case. In the penumbra region, substitution of Eq. (3.12) into Eq. (4.2) gives the asymptotic form of the incident field: It. (uvO0)T- (u). [(u)xrn e 3 -1- ii p (0)D 5fl - - cos0 e 26..... S 5 +o(M 3 ) L 120 M2 J o (4.8) and I. (u,v=0) b(u)* [(u)xH in e inccb e -i sin e + 0 (M ). (4.9) o M o 0o Combining Eqs. (3. 12), (4.6) and (4. 8), we have: 3 t 3 29'f 2 0 -it- r l+4p 4 Op() ngI ) I 0)= 2cos 0e + i -M 120 M o It(,. 0) e e 2 2.... _ ^o ~0 (4.10) 29

THE UNIVERSITY OF MICHIGAN 7741-2-T where 2 F 2 5 P (u=0-)(0) i 1/2.( +5 ((-T') 1 g K t(, T)- e (i —r) l 96 z2t L 96 + 20 192 2 2 1-' -3 8 48 +T e 3- P ) + 1 (0) 12 2 Ptn3 (4.11) 8ptn Now the above asymptotic form of the integral equation governing It is independent of I6b and thus the original coupled vector integral Eq. (4.1) is decoupled in the asymptotic sense for large k. Upon comparing Eq. (4.10) with the acoustic Eq. (3.15), we can easily observe that the only difference between the two equations is the sign of the term p (0, 0) p (0 0)( -) in K 2t and K2. Thus, we can immediately obtain the solution 2p, (0,)(-T) 2t 2 tn for It from the acoustic solution given by Eq. (3. 33). The asymptotic form of the integral equation governing b in the penumbra region is 3 -i d _ (-i r)3 Ib(, 0)= -2sin 0 e 6 + dT I (r,0)e 24 o M i/ -wo e 4 (-')12 + 0(M 2) + 0 (M ) (4.12) 0 OO30 e A~-~ +o(o

THE UNIVERSITY OF MICHIGAN 7741-2 -T On setting (s,0)-= M l + 0(M - (4.13) o application of the Fourier transform and the F function (Eq. 3.23) yields: n ib= -isin 4 22/ (p)+ [-l +i 2rAi(p) A i(p)-ii (p)}]. (4.14) Substtituti-the Wronskian relation *. 1 AiBi - BiAi into Eq. (4, 14), the inverse Fourier transform gives the solution for I 0 sino in21/3 0)'2 0 1/3\ M - dp w (p (4.15) 21/3Mo I w1(p) v00 Thus, combining solutions for It and 1, we obtain the expression for the induced currents in the penumbra region as 31

THE UNIVERSITY OF MICHIGAN 7741-2-T o c"OGO -co 0 iku 1 d g t(O) cos 8 e 2 -23 m2 X (P 1 l5 g du2, ) l5 ( J 5 (} {d u2 2 t.p(o, o) d2 4ip ~pw6(p4 p (oo} df (o,0) (P) -3 uP P ) 2 A-L X, J, 32

THE UNIVERSITY OF MICHIGAN 7741-2-T Due to the similarity of asymptotic formsa of eiedtromagnelic and acoustic integral equations, solutions for the shadow region can be obtained by the same method used in the acoustic case. The induced current in the shadow region is: (u, o)- (u, o)cos eo I ]L- --- L ~'orI e19 ~ ro l ~ [1 e + o 0) PG(U) [ (pU, U0) Ai( L3) 2-2/3M2(0) p. (0,1 p 1 s1 P0 (O) 2 9 +1 S p 4(o' C)6 (1 P\ L 1 pg(30 (g(OO,O) du pg(OO2g d( /o1,o] 3^ ^10 4p,(e,0) 60 90 uU 6a ds -1/3 d6 exp 2 M (s) + 0(M. (4.117)/ I Pg O, p) (s,0)2 M(so0) 2 2 +b'P0)iO( O) dp e- 1 p[(sO) Pg(s O) 2 M)dsj+ 0) ( 1d ts. 0)) 60 60 45 g ds 2 135 ds 0(O 1/ (0, 0 e ~6 ds 13 -3 exp.,-e a~ pg(S:() — 1/ M a2S + O(M (4.17)

THE UNIVERSITY OF MICHIGAN 7741-2-T Here 31 and Ai(-l) are given in Table 2, and at is the Ith root of Ai(-a,)=0: I at Ai(-a,) 1 12.33811 +0.70121 2 4.08795 - 0.80311 3 5.52056 + 0.86520 TABLE 3: THE VALUES OF a1 AND Ai(-a) 34

THE UNIVERSITY OF MICHIGAN 7741-2-T V DISCUSSION In both the acoustic and the electromagnetic diffraction problems considered, the short-wavelength asymptotic expressions for the surface fields have been obtained for the penumbra and the shadow regions. The second order terms in the asymptotic expansion of the surface fields are new results. The leading terms are the same as those of Fock (1946) and Levy and Keller (1959). In the solutions for the shadow region, the factor G(u is of interest. By definition of the function G (Eq. 2.1),' dv represents the width between the two adjacent geodesics. Thus, referring to the geometrical theory of diffraction ( a e, Y 1/4 (Levy and Keller, 1959), (u) represents the so-called ray convergence factor for the creeping waves. In the geometrical theory of diffraction, this factor was obtained by physical reasoning (conservation of energy), and in the present paper, this factor is justified mathematically. The leading term for the acoustic and electromagnetic creeping waves is the same as that predicted by the geometrical theory of diffraction. This leading term, except the factor [G(0)/G(u) 1/4 is independent of curvature in the direction transverse to the geodesic. In the solutions of electromagnetic diffraction problems, it is shown that up to the terms of order [kpg] 2/3 in the asymptotic expansion, there is no coupling between the tangential and binormal components of the creeping waves. However, identity between the acoustic creeping waves under Neumann boundary condition and the tangential component of the electromagnetic creeping waves is true only in the leading term. The transverse curvature appears in the second order term. The effect of transverse curvature on the electromagnetic creeping waves differs from that on the acoustic creeping waves. This is one of the new results of the present investigation. 35

THE UNIVERSITY OF MICHIGAN 7741-2 -T When the radius of curvature pt in the transverse direction is infinite, the diffracting surface becomes cylindrical. In this case, the propagation factors of the creeping waves in the shadow region agree with those obtained by Franz and Klante (1959), and by Keller and Levy (1959). When the principal radii of curvature (p and tn) are the same and constant, the diffracting surface is spherical. In this case, the solutions of the creeping waves reduce to the results of Senior (1966), who obtained the creeping wave solution (including the second order terms) for the sphere by means of a Watson transformation of the Mie series (exact) solution. The solutions for the shadow regions are not valid near a caustic where the radius of curvature (pt) in the direction transverse to the geodesic is no longer large compared to the incident wavelength. The author feels that the integral equation method used here will be still applicable in investigating the surface fields near the caustic, provided that the saddle point integration for the v' coordinate (Sect. 3.1 and 4.1) is modified by some suitable means. 36

THE UNIVERSITY OF MICHIGAN 7741-2-T VI ACKNOWLEDGMENTS The author is grateful to Dr. V. H. Weston and Dr. R. F. Goodrich for their many and valuable suggestions and to Mr. L. P. Zukowski for checking the algebra. He is also indebted to many members of The University of Michigan Radiation Laboratory for comments which were of value in preparing the manuscript. 37

THE UNIVERSITY OF MICHIGAN 7741-2-T VI REFERENCES Brekhovskikh, L.M. (1960), Waves in Layered Media, (Academic Press), 245-250. Fock, V. (1946), "The Distribution of Currents Induced by a Plane Wave on the Surface of a Conductor, " J. Phys., 10, 130-136. Franz, W. (1954), "On the Green't Function of the Cylinder and Sphere," Z. fuir Naturforschung, 9A, 705-716. Franz, W. and K. Klante (1959), "Diffraction by Surfaces of Variable Curvature, " Trans. IRE-PGAP, AP-7, 568-570. Goodrich, R. F. (1959), "Fock Theory-An Appraisal and Exposition, " Trans. IREPGAP, AP-7, 528-536. H8nl, H., A.W. Maue and K. Westpfahl, (1961), "Theorie der Bengung," Handbuch der Physik, (Springer-Verlag), Band XXV/1, 218- 544. Keller, J.B. and B.R. Levy (1959), "Decay Exponents and Diffraction Coefficients for Surface Waves on Surfaces of Nonconstant Curvature," Trans. IRE-PGAP, AP-7 552-561. Levy, B.R. and J.B. Keller (1959), "Diffraction by a Smooth Object," Commun. Pure Appl. Math., 12 159-209. Logan, N.A. (1959), "General Research in Diffraction Theory Vol. 1," Lockheed Missiles and Space Division Tech. Report No. LMSD-288087. Miller, J. C.P. (1946), The Airy Integral, (Cambridge University Press). Senior, T.B.A. (1966), an unpublished note. The University of Michigan Radiation Laboratory. Struik, D. J. (1950), Differential Geometry (Addison Wesley Press). Weston, V.H. (1960), "Pressure Pulse Received Due to an Explosion in the Atmosphere at an Arbitrary Altitude, Part I, " The University of Michigan Radiation Laboratory Tech. Report 2886-1-T, C. 1-C. 4. 38

UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA - R&P (Security classification of title, body of abstract ancd ndexing annotation must be entered when the overall report is classified) 1. ORIGINATING ACTIVITY (Corporate author).a. REPORT SECURITY C LASSIFICATION The University of Michigan UNCLASSIFIED Radiation Laboratory 2b. SROUP Department of Electrical Engineering 4 3. REPORT TITLE Asymptotic Theory of Diffraction by Smooth Convex Surfaces of Nonconstant Curvatures 4. DESCRIPTIVE NOTES (Type of report and inclusive dstee) Technical Report No. 2 5. AUTHOR()' (Lest name, first naome, initl).. Hong, Soonsung 6. REPORT kAT'T E 7. TOTAL NO. OF PAGES 7fr. O. OF REFS August 1966 38 13 8a. CONTRACT OR GRANT NO. ". ORIGINATOR'S REPORT NUMBER(S) AF 04(694)-834 7741-2-T 7741-2-T b. PROJECT NO. c.? Sb. OTH R R R PORT NQ(S) (Any othernptbera that may be *aasined thise rport) d. 10. AVA IL ABILITY/.!IMITATION NOTICgS Qualified requestors may obtain copies of this report from DDC. 11. SUPP EMENTARY NOTES 1..IPQPONSORIN6 MI.ITARY ACTIVITY Ballistic Systems Division, USAF, AFSC Norton A FB, California 92409 13. ABSTRACT A general method is presented for obtaining successive terms in short-wavelength asymptotic expansions of the diffracted field produced by plane acoustic and electromagnetic waves incident on an arbitrary smooth convex surface. By introducing the geodesic coordinate system on arbitrary surfaces of non-corstant curvature, both scalar and vector integral equations governing the surface fields are solved directly. The expressions for leading and second order terms in the asymptotic expansion of the diffracted fields are obtained explicitly and the differences between acoustic and electromagnetic creeping waves are shown. DD iJAN64 1473 UNCLASSIFIED Security Classification

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