THE UNIVERSITY OF MICHIGAN SAMSO-TR-68-35 8525-4-T SURFACE CURRENTS INDUCED BY A PLANE WAVE ON A PARABOLIC CYLINDER WITH A FOCAL LENGTH COMPARABLE TO THE INCIDENT WAVELENGTH Technical Report No. 4 F 04694-67-C-0055 Captain J. Wheatley, Contract Monitor By S ~. Houng and R. F. Goodrich November 1967 Prepared for Department of the Air Force Space and Missile Systems Organization (AFSC) Norton Air Force Base, California 92409 This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Space and Missile Systems Organization (SMSD ), Los Angeles Air Force Station, Los Angeles, California 90045.

THE UNIVERSITY OF MICHIGAN 8525-4-T ABSTRACT Expressions are obtained for surface currents excited by a plane wave on the surface of a perfectly conducting parabolic cylinder whose focal length is comparable to the incident wavelength. In the shadow region, surface currents are expressed by the residue series which represents creeping waves propagating along the surface. In the illuminated region, surface currents may be represented by the summation of a geometrical optic term and a residue series which may be defined as the reflected creeping waves. In the penumbra region, surface currents may be obtained by the series expansion of the integral representation about a point on the shadow boundary. 111 -

- THE UNIVERSITY OF MICHIGAN 8525-4-T FOREWORD This report (SAMSO-TR-68-35) was prepared by the Radiation Laboratory of the Department of Electrical Engineering, The University of Michigan. The work was performed under Contract F 04694-67-C-0055, "Investigation of Re-entry Vehicle Surface Fields (Backscatter) (SURF)". Dr. Raymond F. Goodrich is the Principal Investigator and Mr. Burton A. Harrison, Contract Manager. The work was administered under the direction of the Air Force Space and Missile Systems Organization (AFSC), Norton Air Force Base, California, 92409 by Captain James Wheatley and was monitored by H. J. Katzman of the Aerospace Corporation. This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of Space and Missile Systems Organization (SMSD ), Los Angeles Air Force Station, Los Angeles, California 90045. Information in this report is embargoed under the Department of State International Traffic in Arms Regulations. Private individuals or firms require a Department of State export license. The publication of this report does not constitute Air Force Approval of the report's findings or conclusions. It is published for the exchange and stimulation of ideas. SAMSO Approving Authority. WILLIAM J. SCHLERF CONTRACTING OFFICER ---------------—. iv

THE UNIVERSITY OF MICHIGAN 8525-4-T TABLE OF CONTENTS Page ABSTRACT iii FOREWORD iv LIST OF ILLUSTRATIONS vi I INTRODUCTION 1 II INTEGRAL REPRESENTATION FOR SURFACE CURRENTS 3 III SURFACE CURRENTS IN THE SHADOW REGION 7 3.1 Formulation 7 3.2 Zeros of Wn(zo) 11 3.3 Zeros of'Wn(zo) a 16 3.4 The Value of the Functions -W (z ) and -'W (z ) an n o an no Evaluated at Zeros 18 3.5 Creeping Waves 22 IV SURFACE CURRENTS IN THE ILLUMINATED REGION 33 4.1 Formulation 33 4. 2 The Method of Geometrical Optics 36 4.3 Reflected Creeping Waves 37 V THE SURFACE CURRENT IN THE REGION OF PENUMBRA 40 REFERENCES 42 DD 1473 DISTRIBUTION LIST i* — - - - - -- v li i i - - -— V

THE UNIVERSITY OF MICHIGAN 8525-4-T LIST OF ILLUSTRATIONS Figure Page 1-1 Geometrical Optics and Creeping Waves. 2 2-1 Geometrical Optics. 4 2-2 Paths of Integration for the Function U (z) and W (z) 6 n n 2-3 Paths of Integration in the Complex n-Plane. 6 3-1 Line of Zeros for the Function W (z) When z = p. 9 3-2 The w-Plane When z =; —i P; = 2kh 9 3-3 Graphical Solution for Zeros of W (z)When z =-p. 14 3-4 Locus of Zeros of Wn(z) in the m-Plane When z = -f p. 15 3-5 Graphical Solution for Zeros of'W (z) When z = F p. 19 3-6 Locus'of Zeros of'W (z) in the m-Plane When z = p. 20 3-7 The Surface Current Density at the Crest, 0 = 0. 27 3-8 a(g) Versus ~k. 28 3-9 a'({) Versus g. 29 3-10 Attenuation Factor Versus Arc Length. 30 3-11 Attenuation Versus Arc Length. 31 3-12 Attenuation Factors on Parabolic Cylinders. 32 4-1 Region in the Complex m-Plane Corresponding to Asymptotic Expressions When z =ik-.. 35 4-2 In IC(m)l versus p. 41 vi.

THE UNIVERSITY OF MICHIGAN 8525-4-T INTRODUCTION The first theoretical work on the diffraction of plane electromagnetic waves by a parabolic cylinder was done by P.S. Epstein (1914). His work makes use of a series of parabolic cylinder functions. When the radius of curvature at the vertex of the cylinder is large compared to the wavelength of the incident wave many terms are required for computation. V. Fock (1946) used an entirely different approach. He sketches the derivation of an integral for the current density on a large paraboloid of revolution. His result gives the change in current density on a large and perfect conducting parabolic cylinder as we go from the illuminated region into the shadow. In 1954 S. 0. Rice by starting with Epstein's series investigated the diffraction of plane electromagnetic waves by a parabolic cylinder. The series is converted into an integral and then the path of integration is deformed. He studied the behavior of parabolic cylinder functions of complex order in great detail. V.I. Ivanov (1960, 1963) by following Rice's procedure derives asymptotic formulae for a field which are uniformly true for regions of the umbra and penumbra behind a large parabolic cylinder and are connected with the formulae of geometrical optics in the illuminated region. In the shadow region he interpretates the results in terms of the "geometric theory of diffraction" [Keller, 1956]. In the past work, no one has considered the solution for the small parabolic cylinder which we mean a short focal length comparable to the incident wavelength. Therefore in this report we derive the asymptotic currents excited by a plane wave on the surface of a perfect conducting small parabolic cylinder by using the residue series representation. The graphical method is applied to obtaine the location of the pole and Rice's results are used in all asymptotic expressions. Our results are sketched graphically in Fig. 1-1.'-1 i1 —------

THE UNIVERSITY OF MICHIGAN 8525-4-T y Reflected Wave Plane, Wave ~~/ ~/~~~~" ",.^ Shadow Reflected r Transmitted Creeping Creeping Waves Waves FIG. 1-1: GEOMETRICAL OPTICS AND CREEPING WAVES. 2

THE UNIVERSITY OF MICHIGAN 8525-4-T II INTEGRAL REPRESENTATION FOR SURFACE CURRENTS 2 Let us consider a perfectly conducting parabolic cylinder x = 4 h (h - y) with the focal length h and the focus at the origin of coordinates. In parabolic coor122 dinates, x = 7 r/ and y = - (r2 - j2), the given parabolic cylinder is a coordinate surface r7 = j2h > 0. When r = 0 the cylinder reduces to the half-plane x = 0, y < 0. Let there be a plane wave U = e-ik (x sin - y cos) with the time icut0 factor e impinged upon a parabolic cylinder at an angle i as shown in Fig. 2-1. From Rice's (1954) results, surface currents are obtained in the form of a series Iik -ikr o o Un(z') JD = e sec2 ^tan ) W ( ) (2.1) D jf2ir 2 e2 W(z _n= n o co U (z') i 1 ikr is -i ain n n (2.2) N = I- e see (-itan )'W() (2.2) n 0 n o where z' =, z = F r = -i2kh = fp JD and JN indicate surface currents for Dirichlet and Neumann Problem respectively. The functions U (z) and W (z) are defined by contour integrals of the form W (z) = _ 1 (2.3) n ~ i27r^i exp [f(t)]dt (2.) (z) = exp [f (t)] d t (2.4) n 2ri -U exp [f-t)]dt(24

THE UNIVERSITY OF MICHIGAN 8525-4-T y x < 0 Reflected x > 0 < 0 Wave > 0 Incident Plane Wave s - -. ^ -2__ i, c= cot 8\~_~ 0 Focus / \ Shadow 2 n = r =2h' x =4h(h-y) o Parabolic Cylinder FIG. 2-1: GEOMETRICAL OPTICS 4

THE UNIVERSITY OF MICHIGAN 8525-4-T where f(t) = - t +2zt - (n+1) nt. (2.5) The pathes of integration for W (z) and U (z) are indicated by W and U respectively in Fig. 2-2. The function'W (z) is defined by W (z) = -zW (z) + W (z) (2.6) n n az n By Watson's transformation, the series can be converted into contour integrals with n as the complex variable of integration. Thus expressions (2. 1) and (2. 2) are transformed respectively into = - 1G e-ikr (itan ) U(z') sec -dn (2 7) ^D =127r ri 2 2 sinr n W (z dn C1 no and _1 -ikr a (itan-) U (z) J e) s2se(z dn (2.8) JN 4C 2 s 2 lC sin7rn'W (z). no with the path of integration C1 shown in Fig. 2-3. 5 -

THE UNIVERSITY OF MICHIGAN 8525-4-T t. 1 t - plane U _. U cut t r w FIG. 2-2: PATHS OF INTEGRATION FOR THE FUNCTION U (z) AND W (z). n: n - plane C2 C1 -3 -2 -1 _*"T-ir0 L - _ __ *_ _ —_ Zeros of Wn(z ) 1 I O- -.- i kh C3 FIG. 2-3: PATHS OF INTEGRATION IN THE COMPLEX n-PLANE. 6

THE UNIVERSITY OF MICHIGAN 8525-4-T III SURFACE CURRENTS IN THE SHADOW REGION 3.1 Formulation It has been shown that all zeros of both function W (z ) and'W (z ) are n o n o located in the third quadrant of the n-plane, while the points n = -1, -2, -3,.. are not singular [Rice, 1954]. Therefore the contour C1 may be deformed into C3 contained all zeros, and the asymptote-of the integral is defined by the poles of the integrand, i. e. surface currents may be expressed by the sum of the residues at the poles T- eI e see.- [: (3.1) s 1 sinwT - n-W (z) an n o n=n (i tan )nU (z' JN = i e-ikr see (3. 2) JN7 -2 a s=O sin7rn'W (z) where n and n' are zeros of functions W (z ) and'W (z ) respectively. s s n o n o In order to locate zeros, the saddle-point method of approximate integration is used to obtain asymptotic expressions for the function W (z ). Two saddle points n o in complex t-plane are obtained from (2. 5) by setting f'(t) = 0, i. e. 1 2. t -- -= z -2 7 — (34.3) tl_ 2 o Al7_2 e%

THE UNIVERSITY OF MICHIGAN 8525-4-T The path of steepest descent which passes through t is that branch of the curv Im [f(t) -f(t)] = 0 and Re [f(t)- f(t)] < 0 for which t is the highest point. The path of steepest descent has been shown by Rice (1954) to have the following properties: 1) If z is regarded as fixed and to, t are functions of m defined by (3. 3) and (3.4) the equation Im[f(to) - f(t)] =0 (3 5) defines a critical boundary in the complex m-plane. On this boundary the steepest descent contour passes through two saddle points, t and t in the complex t-plane. In this case both saddle points will contribute to the asymptotic expression of the function W (z ). In general this critical boundary defines a region in the complex n o m-plane within which a function is approximately evaluated from two saddle points (Fig. 3-1). 2) If m is such that the path of integration W must be deformed along the steepest descent contour to pass two saddle points, each one will contribute to the value of W (z ), Furthermore, if m is such that n o Re [f(t) - f(t)] = 0 (3.6) t and tl have the same height and the two contributions have a chance of cancelling each other and giving a value of zero for W (z ). Thus (3. 6) defines the line in the complex m-plane along which zeros of W (i ) are asymptotically distributed (Fig. 3-1). I —------— 8 —----------

THE UNIVERSITY OF MICHIGAN 8525-4-T m m - plane m = n+ \ / Boundary of \ -a / Imff(to-f(tl)]=0 II \ / II \ ----- ^ --- 7T --- ~f —----- m \ 0 r Im l p2 II Zeros of W(z) m = -2 Re[f(t )-f(t )]=0 FIG. 3-1: LINE OF ZEROS FOR THE FUNCTION W (z) WHEN.i,1~ n z=f-ip. 3 m=oo Arg m = 2 3~ > ~- w- ~ -w -plane \I^.~~~~~~ ^ ~w = u + iv \ w=u+iv 2 < * \sArg m = -r 1 I /.' - _- Argm = 2 o 0 — -- 2 —-- m — 0 [-7/2 x ~ w 0; m -ip/2 > -1. \ Boundary! \ Arg m = O Im[f(t) -f(t)]= 0 -2" ] -' —-...... I — I // o - Lines of Zeros -3 - _Argm=7r/2 Re[f(t)-f(t =0 m = oo 0 1 2 3 4 5 u- axis FIG. 3-2: THE w-PLANE WHEN z = %7; p =2kh. 9

THE UNIVERSITY OF MICHIGAN 8525-4-T 3) The lines in the complex m-plane defined by (3.5) and (3. 6) may be obtained by the following transformation W = n( = u+ iv (3.7) t\ From this transformation we obtain 2 z 0 m = n+ 1 = (3.8) coshw + 1 f(t ) - f(t ) = m(sinhw - w) o 1 z (sinhw - w) 0 (3.9) coshw + 1 Since |to| > |tl and jargto-argtl1 < 7 we have u > and v < - for mapping (Fig. 3-2). 4) For the special case z = =7-2ikh' = ~-p, (3.9) gives (cosh u + cos v - v sinv) sinh u = (coshu cosv + 1) u (3.10) (cosv + coshu +usinhu) sinv = (coshu cosv + 1) v (3.11) respectively for Im[f(t) - f(t )] = 0 and Re[f(t) - f(tl)] = 0 i —----- 3210.

THE UNIVERSITY OF MICHIGAN 8525-4-T 3.2 Zeros of Wn(za) The zeros of W (zo), regarded as function n, occur when the contribution from two saddle points cancel each other. From Rice's (1954) results, the asymptotic expression of W (z ) for the region III (Fig. 3-1) where the path of integration passes through two saddle points t and t1 is W n(z) A -A1 (3.12) n o o where f(t ) Ao = 2 /(3.13) 0 -2i %- (-ip2- 2m)/4 f(tl) t1 e A 2= -1 (3.14) 2 tg (-ip - 2m) t f(to) - ( n - - n + i pt (3.15) 3 2 2 1 f(t -n- - In + 4-i p tl (3.16) and 7r< arg m < 2 -3 <- arg (-ip - 2m) < 2- 2 53 < 3t 4< arg t < 4 - 1 4 5 7r 3 r - 4 < argtl < 4 - -- -- ^ - -- i^ ^ -- -- -- -.11

THE UNIVERSITY OF MICHIGAN 8525-4-T Therefore zeros of W (z ) are located at n o A -A = 0 o 1 i.e. exp f(t )- f(tl) = - (3.17) 1t /t1 Using the transformation (3. 7) and (3. 9), we obtain 2 sinhw -w w - coshw+ i 7 (1 -4s) (3.18) where s =1, 2, 3,.... By separating the real part and the imaginary part of (3.18), we obtain two simultaneous equations p (cos v + cosh sinhu)sin (cosh u cos v + 1)v] = - 2 u cosh u os v + 1) + (sinh u sin v)] (3.19) p cos v + cosh u - v sin v) sinh u - (cosh u cos v + 1) = 2 [v+ (1 - 4s) ] (coshucos v+ 1) + (sinh u sin v). (3.20) Let (3.19) be divided by (3. 20), we obtain (cos v + cosh u + u sinh u) sin v - (cosh u cos v + 1) v -u (cos v + coshu - v sinv) sinhu - (coshu cos v + l)u v - (1 -4s)7r (3.21) 12

THE UNIVERSITY OF MICHIGAN 8525-4-T This equation is independent of the parameter p. Setting s = 1, we calculate the first zero as the following by graphical means. Equation (3. 21) may be approximated by a circle in the w-plane as [u - (r - a)] + [v 2 = r (3.22) where a + (/2)2 2a a = uIv _ = ~0.575, s=1 v = 7r/2 <u < a < 1 For u < 1, (3. 20) can be evaluated approximately by 2 1r22 -p uv sinv = (v - 3r)[(cosv + 1)2 + (usinv)] (3.23) If we plot (3. 22) and (3. 23) on the w-plane, the points of intersection between the two curves determine the zeros of Wn(zo). A typical plot is given in Fig. 3-3. Mapping the zeros of W (z ) from the auxilliary w-plane with the help of n m=-ip /(coshw+1) gives the location of zeros on the m-plane. If we consider p as the variable parameter, the locus of the first zero in the m-plane is expressed approximately by m =- 2 [(P +2.8 + p + 1.4)] (3.24) where we limit the range of p as 0 < p < 10. Rem and Im m are plotted in P P Fig. 3-4. Similarly, loci for s = 2, 3, 4,... may be obtained by the graphical method. ---------, ~~~~~~~~~13

THE UNIVERSITY OF MICHIGAN 8525-4-T v Q7r m,-_ _ w - plane / \ " w - u+ iv 2,,,,~- Im [f(t)-f(t1)] =O // \ \P's~/ // =^ =1 - i - Tn I 2 2 p= 7 20 /" / a2 +(x)2 a =0.55 0 1 2 FIG. 3-3: GRAPHICAL SOLUTION FOR ZEROS OF Wn(z) WHEN z =lp. 14 \~~~~1

-Re m (p + 2.8) 7 I / * 6 Z 1 2 -, -Im m - (p + p +1.4) ~~0.0 0 00-.10e Mo* -m 2 [iP+2.8)+i(p +p+ 1.4)] ~ O 2 4 6 8 10 12 14 4 -Im / 1- - p+ 1 4 1.,_y -m ~ ~ +2.8)+i(p+l.4;] / / -Re m 0 2 4 6 8 10 12 14 P z FIG. 3-4: LOCUS OF ZEROS OF W (z) IN THE m-PLANE WHEN z =-ip. n

THE UNIVERSITY OF MICHIGAN 8525-4-T 3.3 Zeros of'Wn(zo) In Neumann's problem we define the function'Wn(Z) = -z W () + W() (3 25) no on z no 0 Here W n(z ) = -a Wn(z ) no 0 az n o 0 has the asymptotic expression W' (z ) - 2t contribution of t to W (z) + no o o oj + 2tl [contribution of t to W (z)l (3.26) 1 L I n oJ from the saddle points t and tl. If the path of integration does not pass through a particular saddle point, its contribution to (3. 26) is zero. Upon replacing t an t1 by their expressions and subtracting the corresponding expression for z W (z ) O n o we obtain W (z ) (z ) 2m ft contributionto W (z n o 0L\o n o (t contribution to Wn(z). (3.27) W1hn z -ih -oWhen z = f-2ikh = St p, the asymptotic expression of TW (z ) for the case that the path of integration passes through two saddle points t and t is 2 12~~~ ~ 1'W (z) = -ip - [A +A12, (3.28) no 0r 0 o where A and A are expressed by (3.13) and (3.14) respectively. 0.. 16

THE UNIVERSITY OF MICHIGAN 8525-4-T Therefore, the zeros of'W (z ) are located at n o A +A = 0 o 1 or exp [f(to)- f(t)] = i (3.29) Using the transformation w = in (to/tl) = u + iv, we obtain 2 sinh w - w. w -i oshw+ = i~ (1 + 4s)- (3.30) cosh w + 1 2 where s = 0, ati 2 3 By separating the real part and the imaginary part of (3. 30) we obtain two simultaneous equations p (cos v + coshu + u sinhu) sinv - (coshu cos v +1)v =- u (coshucosv+ 1) +(sinhusinv)2] (3.31) 2 (cosv + coshu - v sinv) sinhu - (cosh ucos v + 1) u] = - - (1 +4s)] [coshu cos v + 1) + (sinhu sinv)]. (3.32) Dividing (3.31) by (3.32) we have (cos v + cosh u + sinhu) sinv - (coshu cos v + 1) v -u (cosv + coshu - vsinv) sinhu - (coshu cosv + 1) u v - 7r(1+4s) Setting s = 0, (3.33) mnay be approximated by a circle in the w-plane 1 —--------------------— ~~~~~~ 17 --------------

THE UNIVERSITY OF MICHIGAN 8525-4-T 2 [u-(r-a)] 2 +[v = r2 (3.34) where 2, 2 a + (f) r 2a 2a a = u| 2,- 0.48 \V = 7r/2 0< u a < 1. For u <, (3.32) may be approximated by -2uvsinv = (v - 7r) (cosv+1) + (usinv)2]. (3.35) The location of zeros are determined by the graphical method from (3. 34) and (3. 35). A typical plot is shown in Fig. 3-5. Mapping the zeros from the w-plane to m-plane gives approximately the locus of the first zero as mp = - p+ ) - i p(p+l) (3.36) where p is limited in the range 0 < p < 10. Re m' and Im m' are plotted in P P Fig. 3-6. Similarly, loci for s = 1, 2, 3,... may be obtained by the graphical method. a a 3.4 The Value of the Functions - W (z ) and -'W (z ) Evaluated at Zeros a.n n on an n o The function W (z ) is defined by no W (z) 2i --- e dt (3.37) n o 27ri 18.....

THE UNIVERSITY OF MICHIGAN 8525-4-T v w - plane Y \ 1/, / - = 2a p= I FIG. 3-5: GRAPHICAL SOLUTION FOR ZEROS OF'W (z) WHEN z =p. 19 a=0.48 " 2 i)2 0 1 2 9a FIG. 3-5: GRAPHICAL SOLUTION FOR ZEROS OF'W (z) WHEN z = — p.

p4 / _ 7 - R m-Re m' p + 7 p iT 10 2 6 -- ~-Imm' 12 ~ ~s p 2l 3-mm' (pO + O p) IN( +- 3-// ^^-m'"""' ~2 —.)+ i 10 -Re m'n P 0 0 2 4 6 8 10 12 14 -Im mm P FIG. 3-6: LOCUS OF ZEROS OF'W (z) IN THE m-PLANE WHEN z = i —i p. n

THE UNIVERSITY OF MICHIGAN 8525-4-T where f(t) = -t2 + 2z t- (n+1) nt.(3.38) o Differentiating (3.37) we obtain the function - W (z ) as an n o - W (z ) = (tn t) e dt.(3.39) an n o 27ri W 4 If we assume the path of integration W does not pass through the point t = 0, the function In (t) may be considered as a slowly varying function in comparison with the integrand and put outside the integration sign at the saddle point. Therefore the saddle-point method may be applied to evaluate the asymptotic expression as the following -aW (z) - n t ) [contribution of t to W (z) an n o o n - (nt) [contribution of tl to W (z) = -[n to)A - (In t) A] (3.40) where A and A are expressed by (3. 13) and (3. 14) respectively, and o 1 _ z = I-2ikh = --. At the zero, A - A, =, the asymptotic expression gives - W (z ) = In (t /t ) A (3.41) an n o/ 1 ) n=n s where n is the sth zero of W (z ) in the n-plane. s n o 21

THE UNIVERSITY OF MICHIGAN 8525-4-T Next let us consider the definition'W (z ) = -z W (z + W (z ).(3.42) n o o n o 3z n o 0 By a similar consideration as before, we obtain the asymptotic expression for -'W (z ) as an n o -'W (z ) - z I- (nt )A + (nt )A ] + an n o o o o 1 + 2[- (nt )t A + (nt )t1 A + L oo 001 1 =- (z 2 2m [(nt )A + (int )A (3.43) At the zero, A + A 0, we obtain o 1 -a W (z (z) 2 2m- I [n(t/t)] A0 (3.44) Tn n o n P n=n' s where n' is the sth zero of'W (z ) in the n-plane. s no 3.5 Creeping Waves After zeros are obtained, the asymptotic expression of the function U (z') in n the region III was given by Rice (1954) as U (z') = (1 -i) A'1 (3.45) n 1 where t'1exp f(t'1) A' 2'r (z - 2m)/4 22

THE UNIVERSITY OF MICHIGAN 8525-4-T Z, = V-k f(t'l) = -t21 + 2z't' - (n+l) nt' t = k - 2m] (3.46) m = n+l = m for Dirichlet's problem P m' for Neumann's problem P Now we can evaluate (3. 1) and (3. 2). If we assume ~ = 2 and consider the leading terms of the residue series, the surface currents in the shadow region become =2 d4 - r 1 n (t /t ] 2 1\ qk k+k +2im - - ^ p + p - 2im xp 2 - n2imP (3.47)2 + (m - ~) fn P x-2 +ip ip-_2im 1 (3.47) v 2T P, _ J~~~~~~~~~~ ---------------------- 2 -------------

THE UNIVERSITY OF MICHIGAN 8525-4-T 2 427' i3/2 N n(t. /t) [(ike2 -2m' )(-ip - 2m' P P {(m p4) -' + k2++2im' + ----- P /* 1~ p+p- 2im' k + 2im' p2 2 p + ({ m p [ i- 1 + i p - 2im p}. (3.48) The asymptotic expressions obtained may be interpreted in terms of the "geometric theory of diffraction " (Keller, 1956). Let the length of the arc of the parabola between the points e = 0 and ~ = e be [ 1 S SO l,2 ~ 2 + + +2h S = J 2+ 2h d - + 2h + h In 2h J, (3.49) and the radius of curvature of the parabola at the point with coordinates (E, 2h) is R() ==.( + - (3.50) 12h Finally let us express the integral over the arc of the parabola as D s ds 3 1 ds O [Rs]213 (- +2h)h = (2h)V13 n +2h. (3.51) 24

THE UNIVERSITY OF MICHIGAN 8525-4-T Comparing these expressions with formulae (3.47) and (3. 48), we obtainthe asymptotic surface current in the forms 1~ k1/3 k JD/k^A(p) [R() ] exp -ikS 2 (p+3.8)+i(p+L4)] 2/3 (3.52) JN v-A'I (p)[R () exp -ikS k +1.2 +ip __ (3.53) where A(p) and A'(p) express amplitude functions which are a function of p only. Here p = 1ikh1. Formulae (3. 52) and (3. 53) have the same expression as the results of Keller and Levy (1959). Therefore it is clear that the creeping wave theory may be extended into the region where the radius of curvature is comparable to the incident wavelength. The only place needing modification is the coefficient of D in the exponent of (3. 52) and (3. 53), where the coefficient is expressed as a function of the focal length p = 12kh. For large cylinders these coefficients are equal to 2.338 ei /6 and 1.0188ei /6 for Dirichlet's and Neumann's problems respectively (Ivanov, 1960). In our case these coefficients are [(p + 3.8) + i (p + 1.4)] /2p 3 and [(' + 1.2) +ip] /2p3 in Dirichlet's and Neumann's problems respectively. When the focal length of the parabolic cylinder is large compared to an incident wavelength, the asymptote of the function W (z) may be expressed by Fock type formulae, i.e. the function can be expressed in terms of the Airy function (Rice, 1954; Ivanov, 1960). This is due to the fact that the asymptotic expressions given by the saddle-point method fail when m and m' are near -ikh, i.e. zeros of P P W (z) are very close to - (ikh + 1). In this case two saddle points t and t coincide, and f"(t ) vaniches in Taylor expansion of the function f(t). Therefore the unvanished terms will start from the third derivative of the function f (t), and the 25

THE UNIVERSITY OF MICHIGAN 8525-4-T asymptotic formulae may be expressed in terms of Airy integrals. In the case of the short focal length compared to the incident wavelength, the locations of zeros of the function W (z) are not closed to - (ikh + 1). Therefore the saddle-point metho n may be applied to evaluate the asymptotic expressions. This is what we have used to obtain the asymptotic formulae for surface currents in (3. 47) and (3. 48). Let us define the surface currents as J/k = Ae (3. 54) = A' e (3.55) where A and A' express the surface current density at the crest = = 0; a(g) and a' (0) are the attenuation factor as a function of the parabolic coordinate g. Then the surface current density at the crest is plotted approximately as a function of p in Fig. 3-7. From Fig. 3-8 to Fig. 3-12, they show attenuation factors as a function of the parabolic coordinate 5 and of the arc length along the parabolic cylinder. Fig. 3-12 shows the constant attenuation contour on parabolic cylinders. The wavelength X of the incident plane wave is plotted against parabolic cylinders for scaling. From these figures one can see that the attenuation factor for a large cylinder increases more rapidly than for a small one in the deep shadow region, i. e. the larger the cylinder the darker it is. In general it is much darker behind a parabolic cylinder than behind a half-plane. For the same cylinder, it is much darker for Dirichlet problem than for Neumann problem. 26

IIA A A' 2.0 \.8 \ I.4 \\j. 6 4~~~~~~~~~~~~~~ -z1 ~ ~ ~ - cm) 0.2 N 0 ~o 08 p o 0 FIG. 3-7: THE SURFACE CURREN ^ o~ 0.4 -\' 0~2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~z 0r,,l-T c --- -. —--- ~

a () (db) p = 1, kh = 1/2 Dirichlet' s Problem r)70 - ~2kh C 70 p=2kh / p = 3, kh = 4.5 p=5, kh = 12. 5 z 60 p = 7, kh =24.5 a -,x Ox _ 0 2 4 6, kh =10 12 50 o 20 FIG. 3-8: f(?) VERSUS:.k7:. FIG. 3-8: oi(~) VERSUS*/-~.

(db) e 30 Neumann's Problem p=1 t Fi, A / j'p=3 C 25 x z <u.^A^/^)=~9 P 0 Half Plane 20 ~ -.^^000 / 0 ~o#, Y^/^/ 7:p44.7 kh 1000 o0 0 15 2 x Large Cylinder Q 3lR 4 10- 0 — ~ 0 0.0-:0 0 2 4 6 8 10 12 y'~ j z FIG. 3-9: a'(~) VERSUS ~k

oam) (cb) 80 Dirichlet's Problem p =42-kh' phs1/ t 70 X = wavelength 3 -7 50 9rF- 3 I o I I I I. I I I I I E o - 30 VS THE ARC LENGTH ALONG THE PARABOLA.; 0.1.2.4.6.810 24 6 8 1 2040 0 10 C VS THE ARC LENGTH ALONG THE PARABOLA.D

25 Neumann's Problem (db) 20 -0-1 010 15 I / /^^~~~~~ / / / /S i-^~~~~~~~~Crl. - ~10,-..- p 3 p =0. Half Plane..00. Rdlnaoe 0 - 0.1 1.0 2 4 6 8 10 20 40 60 80 100 0 O. 1X O. 2X 0. 4X 0.6X 0.8X X 2 4X 6). 0X oZ FIG. 3-11: ATTENUATION FACTOR FOR THE SURFACE CURRENT J VS THE ARC LENGTH ALONG THE PARABOLA.

ky Neumann's Problem Dirichlet's Problem 5db 5db db 0. 2 3X: 15 db I~ / 25db ~3d 20 db 35pO I z Haf / $od b P FIG. / iO CID >. I\ i,0 \ 40.0dble db a Z 00^ kxk ~ FG 3-12 ATENATO kx O: 0, le db C/J Half Plane FIG. 3-12' ATTENUATION FACTORS ON PARABOLIC CYLINDERS.

THE UNIVERSITY OF MICHIGAN 8525-4-T IV SURFACE CURRENTS IN THE ILLUMINATED REGION 4.1 Formulation If we consider the region x < 0 where ~ is negative, we have the following relations 2n -2n U (-z) =-i (z) - i W (z) (4.1) n n n U (z) + V (z) + W (z) 0 (4.2) n n n Following Rice's (1954) derivation, the leading terms in the asymptotic expansion for U (-z') along the contour C2 is obtained as follows: n 2 (-2ni2n) A 1 U (-z') - 2 in A' + i A' (4.3) n o 1 f(t' ) A' =' ~______t 2ir (ik2 -2m) /4 f (t') -.e A' (4. 5) 2 2 (ik2 - 2m) /4 m f(t') = zt' + - - m n t' (4.6) 2 t? = -1 [^Ti + ik2 - 2m] (4.7) t' = 1 - lik2 - 2m] (4.8) ----------- ~~~33 -

THE UNIVERSITY OF MICHIGAN 8525-4-T m = n+l z' = lik' e, ~ > 0. In fact, the asymptotic expressions of the function U (- z') in the various regions n 1 of the m-plane are listed in the following table when z' = i1/2 -f, e > 0. Region in m-plane m =n+ 1 n 2n -2n 2n II -i A' + i At 0o 1 Ib (-2n -i2n)(T +A' 1) LIII |_(?-2n -i 2n)AT The contour C2 passes through regions I and II in which U (-z') has different asymptotic forms (Fig. 4-1). Because the stationary phase point 2 is found within the region I, i.e. a -i 2 < < i2 We may use the asymptotic form in I for U (-z') along the entire contour C2 When (4. 3) is substituted into (2. 7) and (2. 8) we obtain k I -ikr 2 _ { f n A' D k e 2 ^sec -2if (tan dn D 2wri 2 2 2 W(z) dn 2 n o n Js, ( i — ) sinnW(z ) (9) C2^ \ n o34 34

THE UNIVERSITY OF MICHIGAN 8525-4-T m. D j a -2/ -1 0 1 2 r III / a \ II rl/. \ Im[f(t'o )-f(t) - 0 C2 / m = -i - - \ Lo 1J FIG. 4-1: REGION IN THE COMPLEX m-PLANE CORRESPONDING TO ASYMPTOTIC EXPRESSIONS WHEN z =. 35

THE UNIVERSITY OF MICHIGAN 8525-4-T n A'l dn ) s ii )t sin 7rnW (z ) 4. i2'W Mzc 4.2 The nethod of Geometrical Optic o geometrical optic term. The second term may be expressed by a residue series which represents creeping waves launched from the shadow boundary and traveling along the surface of the parabolic cylinder into the illuminated region. It may be called the reflected creeping waves (Fig. 1-1). 4.2 The Method of Geometrical Optics The first term of (4. 9) and (4. 10) may be calculated by the stationary phase method when k-> oo. The asymptotic forms of the functions W (z ) and'W (z ) no n o along the contour C2 were given by Rice (1954) as W (z ) = A (4.11) n o o and'W (z ) = (z) - 2m A (4.12) n o o o Introducing the new variable of integration a = i (-), we obtain the first term of (4. 9) and (4. 10) as follows: ___i 2 2 r72 - 1/ 2 1/4 osn2 +"J2o J- L / +2aj |T s ik n + 2 i -ik ~ C2d a (4.13) 36

THE UNIVERSITY OF MICHIGAN 8525-4-T 12 Iike -ik (a) J. = __ - + f'j__ ^2 e' dL 1 ik + + 2a e da C [r r2~J] ]2/4 (4. 14) No sin + 2 _-2 2' 4. N 2 2 where ~(a) = 1. +2a + 2 r -2a +2~ r2 + i-2 r)-2a 2 w = tan - The stationary point of the phase; (a) is obtained /+ 2+ 2a (a)= In [w7Y] = 0 as 2 2 (> + 1r7 - 2a) = a + +2a (4.16) 0 0 2 The equation (4.16) has a real root if r < K y + 2 a +. On solving this inequality, we find > - ra cot n 0 The point e = -r)0cot / is the boundary of the shadow and the points g > - /'cot are located in the illuminated region g. 2-1). Thus the stationary point of the phase ( (ar) exists only if the point of observation is situated in the illuminated region. Solving (4.16), the stationary point is found as 37 -

THE UNIVERSITY OF MICHIGAN 8525-4-T 2 2 a = sin sin - o cos. (4.17) o 2 Substituting a into the phase function ~ (a) we have 0 2 2 r7 - (a)= i sin b - -2 cos / (4.18) x sin r - y cos and also h2 + 2a = Tr sinb - ~ cos b 0 2 r( - 2a = r lcos + 5 sin b o sin (or cos ~ + sin) (r7 sinV - cox b) r =o + Now the asymptotic expressions of the surface current in the illuminated region are obtained as follows: rT cosr + esin J = -2ik exp -ik (xsin- y cos (4.19) o 2 2 J = 2 exp -ik (x sinb - y cos). (4.20) 38

THE UNIVERSITY OF MICHIGAN 8525-4-T It has been shown that the quantity (r~ cos b + i sin b) /' + ro in (4. 19) is the cosine of the angle of incidence 0 (Fig. 2-1), and the exponential factor is the incident plane wave U (Ivanov, 1963). Thus JD = -2ikcos0U (4.21) D o 0 J = 2U (4.22) N o 0 i. e. the distribution of current in the illuminated region is described asymptotically by geometrical optics. 4.3 Reflected Creeping Waves The second term of (4. 9) and (4. 10) may be calculated by the sum of the residues at poles given by W (z ) = and'W (z) = 0 Thus we obtain,-I~ F ( o 00 aan0/n A 1 J = JJi ek'' i sec 2 1 (423 c sinn W (z ) an n o D V~r L a n =n no ~s, o'( Ai)~" A' JN = i1 i e sec- 1 Wz nn (4.24) c o sin7rn tw (z ) n=n5 (4.24) Comparing above expression with (3. 1) and (3. 2), one can see that the reflected creeping waves in the illuminated region are exactly the same with the transmitted creeping waves in the shadow region except for a constant factor -im7T e i2m7r (1 -e ) 39

THE UNIVERSITY OF MICHIGAN 8525-4-T Both are launched from the shadow boundary. One propagates into the illuminated region and the other into the shadow region. Let us define this constant factor as the ratio of reflection to transmission, then we have -i m C (m) - e (4.25) i 2 m 7T where m = m for Dirichlet problem P = m' for Neumann problem P In general, C (m)i is negligibly small for p > 1 (Fig. 4-2). Therefore, reflected creeping waves may be neglected in the case of large parabolic cylinders. 40

THE UNIVERSITY OF MICHIGAN 8525-4-T 0 1 2 3 4 5 p _______ Neumann -20 -- x x Dirichlet -40 -' -60 -- x n |C(m)l -80 \ -100 - x -120 - -140 \ -160 FIG. 4-2: In |C (m)| VERSUS p. 41

THE UNIVERSITY OF MICHIGAN 8525-4-T V THE SURFACE CURRENT IN THE REGION OF PENUMBRA The function U (z') in (2.7) and (2. 8) may be expanded into a series about a point on the shadow boundary. If we expand exp (2 z' t) in the following integral U (z ) = f exp -t + 2z't - (n + 1) nt dt (5.1) and integrate termwise, then the function U (z') becomes n s in - n n- n'\/ (5 2) n' =-sinn — 0 (-2z z')5 Therefore, we obtain the surface current in the following forms N 2T. J C *(z(. n n (i.)n r(n) where -Mi k r le N =,-5 — e - 2 sec - Mo 27 2 2 U = tan - 2+ —----- 42r

THE UNIVERSITY OF MICHIGAN 8525-4-T Now (5.3) and (5.4) may be evaluated by the residue series. If only leading terms are considered, we obtain 2r9(-ip2 -2m)/4 (iw) s (-2z) r( s 2 M4 27n ip2-2m )/ (i)ns ^-iM - o t( f-(t t 2 p exp( m I27n (-1Pto} 0 (' ( ) (5.5)| 2OA2 (1/4 n -- t P where z' = ik ^, n and n' are zeros of W (z ) and tW (z ), respectively 5 5 no no given by (3. 24) and (3. 36). - o —-- 43 en ( ~ 2 mIn -r) Pt0 14'=

THE UNIVERSITY OF MICHIGAN 8525-4-T When w = 1, we have > 0 for the shadow region g < 0 for the illuminated region Equations (5. 5) and (5. 6) converge absolutely for all finite values of. 44

. THE UNIVERSITY OF MICHIGAN 8525-4-T REFERENCES Epstein, P.S. (1914), Dissertation, Munich, see also the Encyklopaedie der Math. Wiss., Part 3 (1909-1926) Phys., p. 511. Fock, V. (1946), "The Distribution of Currents Induced by a Plane Wave on the Surface of a Conductor," J. Phys. (USSR), 10, pp. 130-136. Fock, V. (1946), "The Field of a Plane Wave Near the Surface of a Conducting Body," J. Phys. (USSR), 10, pp. 399-409. Ivanov, V. I. (1960), "Shortwave Asymptotic Diffraction Field in the Shadow of an Ideal Parabolic Cylinder," Radiotekhnika; elektronika, 5, No. 3, pp. 393-402. Ivanov, V. I. (1963), "Diffraction of Short Plane Waves on a Parabolic Cylinder" Computational Mathematics and Mathematical Physics, No. 2, pp. 255-271. Keller J. B. (1956), "Diffraction by a Convex Cylinder, " IRE Transactions on Antennas and Propagation, AP-4, pp. 312-321. Keller, J. B. and Levy, B.R. (1959), "Decay Exponents and Diffraction Coefficients for Surface Waves on Surfaces of Nonconstant Curvature, " IRE Transactions on Antennas and Propagation, AP-7, pp. S52-S61. Rice, S. 0. (1954), "Diffraction of Plane Radio Waves by a Parabolic Cylinder," Bell System Technical Jounal, 33, pp. 417-504. ________________________ 45 _____..

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA R & D (Security classification of title, body of abstract tind ilidexlinrd nnotntion mntl. be entered when the overall report 1. classified) 1. ORIGINATING ACTIVITY (Corporate author) 2a0. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 3. REPORT TITLE SURFACE CURRENTS INDUCED BY A PLANE WAVE ON A PARABOLIC CYLINDER WITH A FOCAL LENGTH COMPARABLE TO THE INCIDENT WAVELENGTH. 4. DESCRIPTIVE NOTES (Type of report and Inclusive dates) Technical Report No. 4 5. AUTHOR(S) (First name, middle initial, last name) Shun-Jen Houng, Raymond F. Goodrich 6. REPORT DATE 7,t. TOTAL NO. OF PAGES 7b. NO. OF REFS November 1967 43 8 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) F 04694-67-C-0055 b. PROJECT NO. 8525-4-T c..)h. OTHER REPORT NO(S) (Any other numbers thot maty he as.slned this report) d. |SAMSO-TR-68-35 10. DISTRIBUTION STATEMENT This document is subject to Special export controls. Transmittal to foreign governments or nationals may be made only with prior approval of SAMSO (SMSD), Los Angeles AF Station, Los Angeles, California 90045. I1. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Space and Missile Systems Organization (AFSC) Norton AFB, California 92409 13. ABSTRACT Expressions are obtained for surface currents excited by a plane wave on the surface of a perfectly conducting parabolic cylinder whose focal length is comparable to the incident wavelength. In the shadow region, surface currents are expressed by the residue series which represents creeping waves propagating along the surface. In the illuminated region, surface currents may be represented by the summation of a geometrical optic term and a residue series which may be defined as the reflected creeping waves. In the penumbra region, surface currents may be obtained by the series expansion of the integral representation about a point on the shadow boundary. D D FORM 14.73 UNCLASSIFIED Seacuri l (I.i;ssitci.ltion

UNCLASSIFIED Security Classification LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT R OLE WT SURFACE CURRENTS CREEPING WAVES PARABOLIC CYLINDER INTEGRAL REPRESENTATION UNCLASSIFIED Se- urilx (Cl.I siftri;" bic