THE UNIVERSITY OF MICHIGAN 7030-9-T NEAR CAUSTIC SURFACE FIELD by R. F. Goodrich May 1966 Contract AF 04(694)-683 I s*;.. t^., UN. *STA', AW.~ FORCE N,,.',..'',.. ~,.I..,: Prepared ior.::.: NORTON AIR ";ORCE BASE, A,. ALIFORNIA s ^'^- ^~.^'`sii ~~;IVlbO " <^-.^

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THE UNIVERSITY OF MICHIGAN 7030-9-T TABLE OF CONTENTS Page ABSTRACT iv I. INTRODUCTION 1 II. SCALAR DIFFRACTION BY A PROLATE SPHEROID 2 2.1 The Integral Representation 2 2.2 The Asymptotic Analysis 6 III. REFERENCES 20 iii

THE UNIVERSITY OF MICHIGAN 7030-9-T ABSTRACT At high frequency the acoustic surface fields for plane wave illumination of a prolate spheroid are found for near symmetric illumination. The surface caustic, in contrast to the symmetric case, is found to be diffuse. The values of the surface fields are used to find the modification of the creeping wave contribution to back scattering. * ~~~~~~~~~~~~iv

THE UNIVERSITY OF MICHIGAN --- 7030-9-T INTRODUCTION We consider the use of the creeping wave formalism (Keller, 1953; Fock, 1946) for the prediction of surface fields induced by a plane wave on smooth convex scatterers. The application is straightforward away from regions of the surface in which the creeping waves converge due to the geometry of the body. Here the local geometry is taken to be cylindrical. If the creeping waves converge to a point, we speak of this point as a caustic and the creeping wave formalism can be modified in order to predict the fields in the region of a caustic. However, if the creeping waves do not quite converge to a point and yet cannot be treated in the cylinder approximation of the bodies in question by cylinders, a more subtle modification of the formalism is required. We shall consider this problem and designate the nearcaustic region as a diffuse caustic. We attack the problem of describing the field in the region of the diffuse caustic through a canonical problem. We will study the surface fields on a prolate spheroid for a plane wave which is near axially incident and then attempt a generalization to other shapes. In analyzing the prolate spheroid we shall extend slightly the work of Kazarinoff and Ritt (1959) using the asymptotic analysis of Langer (1935, 1949). The generalization of these results to general convex bodies will then follow from a consideration of the creeping wave paths, the geodesics, on a prolate spheroid. 1 ii ii__________________

THE UNIVERSITY OF MICHIGAN 7030-9-T SCALAR DIFFRACTION BY A PROLATE SPHEROID 2. 1 The Integral Representation We suppose a scalar plane wave be incident on a prolate spheroid and study the solutions of the Helmholtz equation satisfying either the Neumann or Dirichlet boundary conditions. We use the prolate spheroidal coordinates (, r, 0) where 2 21 rxc=c 1 l-r1 cos, y =c -1 - sin, (2.1):z c a, where c is the semifocal distance. The equation of the spheroid is 0 = ~ so the major and minor axes are given by a=c0, (2.2) b o The Helmholtz equation (V2 k2)= 0 (2.3) becomes, in the prolate spheroidal coordinates, 2 ____~

THE UNIVERSITY OF MICHIGAN 7030-9-T at (g21)] +a) + + 1)+(l —T P-=0, -i i-~r? a^ ^(2.4) where y =kc. We require f to be of the form = ei + A (2.5) s where s satisfies the radiation condition and i is to satisfy either of the boundary conditions',(~,)0 )-o, or =, (2.6) -(^ m0) =0 ik r and when e is the incident wave with k=k(sin0, 0, cosO ). (2.7) o 0 We make the expansion 00 =A eim0 (2. 8) -00 and consider the component equations ~ ~~~~~~~~~~~~3~

THE UNIVERSITY OF MICHIGAN 7030-9-T a (2 2 2 2 m2 2 --—,Vr/+'y (1-ri )V/=O - (2.9)' -1 1-r7 With X denoting a separation constant, (2. 9) is seen to be equivalent to the separated equations 2 a 2 a m 2 22 [a(~2 1) -d m2 +d(t2-1)-X u= 0 (2.10a) a 2 a ^m2 2 2 (2. b) 0Lar (1-2) a_ 2+'y (1-)+ 0 (2.10b) 1-r7 From the solutions'of (2. 10a) which are asymptotically in ~ of the form -iT/ e Y2(,X) and (2. 11) ei't y (X) e we form either (Mx) =X yMIX) y2 (o )-y (Q y QX)y ) ~2 (1,) Y l 2 o' or (2. 12a) N' P2 (C, ) y1(, X) Y2 (,o') -2( X)Y1 (O) and 0(1 (X) = Y1(, X) (2. 13)

THE UNIVERSITY OF MICHIGAN 7030-9-T With the solutions 01 and 02 we then form the resolvent Green's function -2('<) 01( > ) G -'(2.14) (2 -1)W (01, 02) where W (01 02) is the Wronskian of the solutions 01 and 02. This becomes for the Dirichlet and Neuman boundary conditions GD= 2i1(eo, D)2 (I>, X) (2. 15a) N 1 N G 02 (e< x)p1(> X) (2.15b) 2i42 ( 1,X) 1 o,0 Similarly, we take the solutions of (2. lOb) which are regular at r=l,1, ( -X) and at rl=l, i(, -X), where -1'( -X)=-)'~ ~(2.16) and form the resolvent Green's function G = 1 l(r?<, -XA)(r>, -X) (2.17) (1- )W(-1,'l) The solution P can then be represented by the plane wave limit of m m 2= i |(rl, t7, -X) G(I, X) d (2. 18) rm 2;i 5

THE UNIVERSITY OF MICHIGAN 7030 -9-T 2.2 The Asymptotic Analysis We wish to evaluate (2. 18) for -y large. We will parallel the work of Kazarinoff and Ritt (1959) making a turning point analysis of (2. 6a) (Langer, 1949) and a Stokes phenomenon analysis of (2. 6b) (Langer, 1935). On puting u - v, (2.19) Eq. (2.6a) becomes 1)2 ~,~ g _ 3g m2 ], (2.20) C~-1 (c -1) We define r by 2 2 i1-m x ( -1)(y+ 2 2) (2.21) r 2 2 ( -1) r and rewrite (2.20), 2_ 2 2 2 r 1 v"+^ 2 1+ v 2 =0 (2.22) 2-1 (2.2 2'- 1)( ) - 1 Now for m=0 or for m<<y the asymptotic forms of two solutions of (2.22) for y -- oo with C> 1 are, after Kazarinoff and Ritt vj) 1/6 e +if(rv) (j=l 2) (2.23),~ 6 ~6

THE UNIVERSITY OF MICHIGAN 7030-9-T + 5i/ 12 1/3 (j) VJ()= e- 1(E) H13() = u (T)dT r 2 2 2 r U (Q) 2 2 -1 1/6 1 2 -1/2 r 1-t f~j "2 dt 0 As m grows large, each solution retains the same form with 2_ T, — =m=1+ 2 2 (2.24) (~ -1) provided C is near r. We note that for e increasing without bound, the asymptotic solutions have leading terms independent of m. Since we shall wish to evaluate (2.18) as a residue series (where this is possible) we look for the roots of ~ ~~~~~~~~~~~~7 __________________

THE UNIVERSITY OF MICHIGAN 7030-9-T 01 (o' ) =0 or (2.25) 0( X)=0 02 O The zeros of these solutions occur at the zeros of (1) H( (h )=O 1/3 (hr 0 or (2.26) (1)', 1/3 (hr) = From the definition of ~ these are the values of s such that r 2 2 r = m 2r horh' 2 d. (2.27) Moreover, if ~ is near to % 1/2 r r m3 2 r o 0 We now rewrite (2.23) using the Airy integral W)= e 2ri/3( _t1/2 H(1) 2 (-t)3I2 3/(2. 29) hcwith r 23 (-t)1/3 3 2 3/2 hence, with ~ = (-t)

THE UNIVERSITY OF MICHIGAN 7030-9-T 2 1/4 (1)()= e-i/4e-( t))122 ) w(t) (2.30) r We now apaproximate v )( o) and v ( ) ) with r assumed to be near C. From (2.28) we find 1/3 1 ( - 1) yr z ) tr/t (2.31) 2 V o r where t is a root of r w(t) = 0, (Dirichlet problem) or w'(t) = 0 (Neumann problem) Substituting in (2. 30) 2 1/6 (1 -i/4 -i'f() m( -1) 1 v(1) )e' re ) w(t) (2.32) On noting that -1/3 _a_'_m___ _) ~l= 9o

THE UNIVERSITY OF MICHIGAN 7030-9-T we find i2 — 1/6 (1)el ( — i- ) 0'(t) (2.33) v (o) e- e e o For the case y>> m, we put and define 2 1/3.'2 - 1) 0 1/3 (2.34) kb Hence (2. 32) and (2. 33) become (1)() = e-i'/4 e-iT f( 1) 1/2 V )0 e e M W(t) (2. 35) (1)' -ir/4 -i'y f() -1/2 1 v ()=e e M ywl(t). (2. 36) The angular Eq. (2. 10b) is transformed by letting co =' z (2.37) 2 and becomes 2z +(12 + } + z =0 (2.38) 10

THE UNIVERSITY OF MICHIGAN 7030-9-T As before, we put 2 2 1-m xY ( -1)(1+ r ) (2.39) r 22 y2(2 -1) r so that (2.26) becomes 22 2 2 1-m 1-m v+ r r - + v v+O (2.40) 1 12 I % u- 1) 2 1-Tri 1. ry (~ ~1)} (1- V 0 or, by defining 2- 2 1-m r r 22 r 2(2 1) r 2 2 -r2 2 1-m v + y (-2 2)v+ v=0 (2. 41) r? 2 2' 1-nr 2 (1-r ) Using the Stokes phenomenon analysis of Langer (1935) we may approximate the solution of (2. 41) that is regular at r= 1, as follows: v J(Cy) (2. 42) 1 where ar =- y P (t) dt, 2 r 1 ~11

THE UNIVERSITY OF MICHIGAN 7030-9-T The solutions of (2. l1b) are, 2-X)^ 2 p[(l- )( ]) J (/T) (2.43a) l,-1 -~2)2-~2 m 1Cl(7 -X) =^C1(- 1X ) (2.43b) and (1-r)W(_,o)-2 cos(2 (o)) (). (2.44) The angular Green's function is from (2. 44) and (2. 17) 0 (n< -x) (n>, -X) G(rn,' -X) 2..1 1 (2.45) Icos (2(o)) (-)m We now form the integral representations for the surface fields using Eqs. (2.45) and (2. 35) or (2. 36). In the case of the Dirichlet boundary condition the normal derivative of the field on the surface is given by D e o 1 11 im ( ) +l) an - Mr 2y 2 22 Z o o m ei / 1/4 /4 -i (o) 1 /4 jm ()rT)) jm(( O r )) | dt m d2 cos(2a(o)) 2 2 2 " (t) () r r 0 r (2.46) where we have noted that'yf(C )= -c (o) and that dX =2^y~ Mdt. r o 12

THE UNIVERSITY OF MICHIGAN 7030-9-T If we take the incident direction to be near axial so that 0 is small, we can o truncate the m series at some mom so that -y can be chosen so large that 0 0 ~>> m. In this case we put 0 0 = r>> 1 r r and, except in oscillatory functions, put r o Equation (2. 43) then becomes 2 1/4 1/4 alrD,e-i/4 1 0o' e-ia (o) 12 2 )(1- /4 0 0o r w(t) Z eim (0'J( (r(T)) J (a(T)) (2. 47) We note that the summation in (2. 47) can be done explicitly to give D 2i/4 r -21/(o) e _____ 1 dt e anD M 2 24 ~,1/4 Nt2 cos(2a(o)) c - M (1-T2) t 0 0 (1 r 1 2, i 2~~ J a(12(rm)+T ()+2a (T)() (acosJ). (2.48) (Magnus and Oberhettinger, 1949). 13

THE UNIVERSITY OF MICHIGAN 7030-9-T With a similar argument the surface field satisfying the Neumann boundary conditions can be written as 2 1/4 ri/4 IO, e-io (o)' 0 \2fy 2 2 1/4 (o)) $(o)=,e,, 172' 2,,,2 1 _22 cos (2a (o) 0, tt) ^C (2. 49) A/(t) o2 where we have put 2 Z 2 (2. 50 E =a(f17)+o~2(c )+2c(y(rl)(rl )cosp (2.50) From the definition of o(r ) we write 2 2 2 2 C-x -x o (r)=o(o)- (r)Yg 1 -x- 3y r dx (2.51) \\-x2 \ - 0 0 and expand about r = r o 22 2' | idxka.1 dx+M t. (2. 52) I-x V1 -x Jl-e x 2)(l-x ) ^ ~ ~0 The first term on the right of (2. 52) is the wave number k multiplied by the path length measured from r) = O to r? along a curve of constant; the second term is the reduced distance given by Fock (1946) multiplied by t. Rewriting 14

THE UNIVERSITY OF MICHIGAN 7030-9-T a (r1)= kS(r) + (rn)t (2.53) where S(rl) a ai dx (2. 54) 0 S) kRs s) ds ()f (2 1/3 () (s) R( ds (2. 55) 2 R(s) 0 dx - M (1-e -2x2)(1-x2 0 with R(s) the radius of curvature at S(r1) along a curve of constant 0. Since the path of integration F' is such that Imt> 0 we have from (2. 52) that Imr (o) > 0 and we make the convergent expansion -i (o) o = ei ( )o () e) i(o). (2.56) 2 cos (20 (o)): e =0 We now consider the forms (2.48) and (2. 49) for r) near enough zero so that lo(r)~>> 1. In this case, we may use the asymptotic form of the Bessel function and approximate Z by 1.co(r7)+o-(+ ) cos0 =- (o)- (n) +u (n?) cos 0 (2.57) ~~ 15 ____

THE UNIVERSITY OF MICHIGAN 7030-9-T in oscillatory functions and X -o (r7) (2.58) in non-oscillatory functions. On substituting the asymptotic form of the Bessel function and the expansion (2. 56) in (2.48) we have 2 1/4 rDC 1 0 1 2 dt an o 2M~ 3/2 (2 2) 1/4 ( w(t) 0 -.^ _(JL~) I v= / L. "(1-2):o e e e -ie. f ei(e () (%)cos0) i (2(o)-(r1)+cT(r)cos0)} (2.59) Using the forms (2. 51) and (2.53) and noting that for 7N near to one 0 oC(r?)~ kb + Mt (2. 60) 0 o b o and rewriting L o((o) = k -+ t, where 1 22 4L 1- x A 4 a 2 1l-x 0 is one quarter of the distance around, the spheroid for = constant mod r and 16

THE UNIVERSITY OF MICHIGAN 7030-9-T _- M,' dx 0 (1- e2x2)(1-x2).is the corresponding reduced distance, we have 2 1/4 31 o 1 )_)~ dt ilkL+4il-t o (1 I =0 r i kS() -kb cos] i[() b sin0]t i[k 2 kS( r) + kb 0cos 0 * e e -ie i[ - - (()+ M + sin ] t (2.61) 4 b o0 e If we now evaluate (2. 61) on the shadow boundary and include only the first term in the back scattered direction we have an expression of the form 2M 3/2 M2 2 1/4 j 1 1 2 1 cos10 1.j de [ot / (t)+ 0sn]t(2.62) r 17

THE UNIVERSITY OF MICHIGAN 7030-9-T where (17V 01) are the coordinates of the shadow boundary and a1, b0 ocos0. (2.63) I bo 1 Since f1 is small we make the approximation S(r1) at71 2 a = b - os01 (2.64) We will neglect the terms of order 0 in the t-integration and include 0 terms of this order only in the phase. This gives the approximation 2 ik-0 cos0 IB(r1 01)- IB(o) e (2.65) where I (o) is the value taken on the shadow boundary for symmetric illumination. B The back scattered field will be proportional to the integral of (2. 65) over the azimuthal angle 1 from 0 to 2r, that is 2a 2 ik - 0 cos u)- B(ojhe (2. 66) 0 But the zeroth order Bessel function has the integral representation 18

THE UNIVERSITY OF MICHIGAN 7030-9-T 21 to'Z' "^ I izcost J (z) e dt o 0 so that the leading creeping wave term in the back scattered field is given 2 B( ) $B(O)Jo(k b). Bo B o bo There is an analogous expression for the Neumann boundary condition. We have then, that for near axial illumination of the spheroid that the leading creeping wave term in the back scattered field is given by the field arising c2 from symmetric illumination multiplied by the shape factor J(k- ). We note o b o that this factor takes the value of one for either symmetric illumination, 0 = 0, or 0 for the sphere limit, c = 0. 19 _____

THE UNIVERSITY OF MICHIGAN 7030-9-T REFERENCES Fock, V. A. (1946), J. Phys. USSR, 10, 1. Kazarinoff, N.D. and R.K. Ritt (1959), Ann. of Phys., 6, 277. Keller, J.B. (1953), Symposium on Microwave Optics, McGill University. Langer, R.E. (1935), Trans. Amer. Math. Soc., 37, 397. Langer, R.E. (1949), Trans. Amer. Math. Soc., 67, 461. Magnus, W. and F. Oberhettinger (1949), Special Functions of Mathematical Physics, Chelsea, New York. 20

Unclassified Security Classification DOCUMENT CONTROL DATA- R&D (Security classielfication of title, body of abstract and indexing annotation must be entered when the overall report is clas~ilied) 1. ORIGINATING ACTIVITY (Corporate author) s. REPORT SECURITY C LASSIFICATION. The University of Michigan Unclassified College of Engineering, Department of Electrical Engineeri] 6ROUP Radiation Laborator Ann Arbor, Michigan M 3. REPORT TITLE Near Caustic Surface Field 4. DESCRIPTIVE NOTES (Type of report and inclusive datee) Technical Report 5. AUTHOR(S) (Lst nme. first name, inlittl) R. F. Goodrich 6- REPORT QAT.I ~'''~'~' 7a. TOTAL NO. OF PAGES 7b. NO. OF'REFI May 1966 20 6 i8. CONTRACT OR GRANT NO. AF 04(694)-683 94.ORIGINATOR'S REPORT NUMBER(S) 7030-9-T b. PROJECT NO. c. fib. OTH'.R REtPORT NQ('i.) (anly othaer n wbR thet mthtmay e asgned this faportJ d. 10. AVA iLABILITY/!.IMITAT!ON NOTICES Qualified Requestors May Obtain Copies of This Report From DDC 1. SUPPI.EMENTARY NOTES l.'l.-POiSRING MI.ITAY ACTIVITY Ballistic Systems Division Air Force Systems Command United States Air Force, NortonAir Force Base, 13. ABSTRACT California At high frequency the acoustic surface fields for plane wave illumination of a prolate spheroid are found for near symmetric illumination. The surface caustic, in contrast to the symmetric case, is found to be diffuse. The values of the surface fields are used to find the modification of the creeping wave contribution to back scattering. 1DD ^sJ4AN 1473 Unclassified Security Classification

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