AFCRC -TN-59-394 THE UNIVERSITY OF MICHIGAN 2871-2 -T SCALAR DIFFRACTION BY AN ELLIPTIC CYLINDER by N. D. Kazarinoff and R. K. Ritt June 1959 Report No. 2871-2-T on Contract AF 19(604)-4993 Prepared for ELECTRONICS RESEARCH DIRECTORATE AIR FORCE CAMBRIDGE RESEARCH CENTER AIR RESEARCH AND DEVELOPMENT COMMAND UNITED STATES AIR FORCE BEDFORD, MASSACHUSETTS

THE UNIVERSITY OF MICHIGAN 2871-2-T Requests for additional copies by Agencies of the Department of Defense, their contractors, and other Government agencies should be directed to the: ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA Department of Defense contractors must be established for ASTIA services or have their "need-to-know" certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to the: U.S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 25, D.C. ii

THE UNIVERSITY OF MICHIGAN 2871 -2-T Abstract. The method given in Reference 1 is applied to the case of scalar scattering by a perfectly reflecting elliptic cylinder illuminated by waves from a line source parallel to the axis of the cylinder. The surface distribution in the shadow zone is calculated, and the "creeping wave" representation for the scattered field in the shadow zone is derived. It is shown that the results are applicable if and only if Row>> 1, where Ro is the smallest radius of curvature on the cylinder and w is the wave number. 1. In a recent paper we have developed a theory of scalar diffraction for bodies whose boundary surfaces are level surfaces in coordinate systems in which the scalar wave equation is separable. We applied the theory to the case of diffraction by a prolate spheroid and calculated the surface distribution. In this paper we make a similar application to the case of an elliptic cylinder; but, in addition, we consider the off-angle case and derive the "creeping wave" representation for the scattered field in the shadow zone. The surface distribution is discussed in 6, and the scattered field is described in ) 7. N. D. Kazarinoff and R. K. Ritt, On the Theory of Scalar Diffraction and its Application to the Prolate Spheroid, Annals of Phys. 6, 277-299 (1959). 1

THE UNIVERSITY OF MICHIGAN 2871 -2-T The asymptotic theory which we use to obtain our results gives us more terms of the series, in descending powers of w, for the exponents of the "creeping 2 wave" terms than does the elegant geometric theory of Keller These extra terms show that the condition Row>> 1 is essential for our asymptotic theory to give a meaningful result for the diffracted field. It is reasonable to believe that the same restriction also applies to Keller's geometric theory. More specifically, -213 we show that it is the coefficient C of the attenuation exponent C R ds which behaves in an unknown fashion when R — 0; the evaluations of C which have been made by Keller and the authors are performed under the hypothesis that R w >> 1. In an appendix, we compare the magnitude of the attenuation term 0 o which he obtains with the next term of the asymptotic series for the exponent in the case of two prolate spheroids. We also note that Levy has applied the geometric theory to the case of an elliptic cylinder and has given a mathematical derivation of the results thus obtained which is based upon the use of the Watson transform. Generally, the analysis below closely follows that in Reference 1. Where this is true we only elucidate the principal points in the argument and omit most computations. Elsewhere we give a more complete discussion. 2 J. B. Keller, Diffraction by a Convex Cylinder, Trans.: I. R. E., AP-4, 312-321 (1956). 3 B. Levy, Diffraction by an Elliptic Cylinder, New York University, Institute of Mathematical Sciences, Report EM-121, December 1958. 2

THE UNIVERSITY OF MICHIGAN 2871-2-T 2. We consider an elliptic cylinder with semifocal distance c, eccentricity e = sech g, and semi-axes a and b. We introduce the (~, r, z) coordinate system defined by x = c cosh ~ cos n y = c sinh ~ sin r) (2.1) z =z On the surface of the cylinder, ~ = Z. For this cylinder, the operators L 0 and L appearing in Reference 1, Equation (3.2) are defined by the formulas dZ2 2 2 -L u = + y sinh2 u ( ) d2 o and d2U 2 2 -L u -= -+ sin 2u (-r L r] r). r7 dr2 The constant y appearing above is c(w - is), where s is a small positive number. The boundary conditions are du = 0 and u(~, r)) = u(, rj + 2ir). (2.2) d 0 Both the operators L and L are of the type considered in Reference 1, Section 3; and for them, respectively, q- $( 4(2 sinh2 ). 2wsc2 sinh2 E > 0 3

THE UNIVERSITY OF MICHIGAN 2871-2-T and J q=j(y2 sin2 ) 0. The radial operator L is to be considered on the interval [|, oo), t > 0. For Le, p-1, and hence p(%o)# 0. In order to construct the resolvent Green's function for Lt, we consider the homogeneous equation L y-Xy =, (2.3) where iX<2csc sinh2.(2.4) 0 It has linearly independent solutions w., j = 1, 2, with the asymptotic forms -1/2 i Yicosh r{ (1) w(g) = (sinh e) e + - in which &(1) denotes a function which is bounded for ~ > N, Il > N, and |X < N. In this and in succeeding formulas, the upper sign is to be used when j = 1, the lower one when j = 2. Since ~6 (i > ) > 0, the only solutions of (2. 3) in L (~, oD) are multiples of w2; therefore, L falls into Case I of Reference 1. We next single out the solution 5l of (2.3) which satisfies the 2 boundary condition (2.2) and a solution P2 in L ( c, oo): P ( ) = w ) Pi (, x) =w1(E, X) w2 (%, X)-w2(EX)w1 (E, X) P2 (~,;) -= W2 (, A). 4

THE UNIVERSITY OF MICHIGAN 2871-2 -T At e, p1 = -2iJ'. Thus, the resolvent Green's function is i ()02 (') ( <') G(~,', A) = - 2i/w2(5, k) 2 ( (0') ('< t) The operator L is to be considered on [-Ir, ir]. To construct its resolvent Green's function, we consider the homogeneous equation L y - (-Xy) = 0, where X satisfies the condition (2. 4). Using the notation of Meixner and Schafke, we let yI(r, -X) and YI (r, -X) be the solutions of the homogeneous equation for which YI(O) = 1, y'(O) 0, y (0) =, y0 (0) =1. It is then a routine computation to show that the resolvent Green's function for the periodic problem is _ yf I -r yi (T) yi(7r - r) G(rZ', -) - 1..I + 1- II (2.5) I II 4 J. Meixner and F. Schlafke, Mathieusche Funktionen und Spheroidfunktionen, pp. 98-100, Springer-Verlag, Berlin, (1954). 5

THE UNIVERSITY OF MICHIGAN 2871 -2-T for T < r). The relation G(,, -X) = G(r,, -X) (2.6) then serves to define G for Ur < T. 3. We are now in a position to write down the contour integral representation, guaranteed by the theory in Reference 1 for the solution v(~, rT) of [V2 + (W-iS)2] v=p( t) (s >0) which we seek. The function p ( -, ) is a distribution corresponding to a line source at (, T). The representation is v(g, nr, 3,tC) =- 1 G(r, T, -X) G(j,2, A) dX, 27ri r where 7is a path in the X-plane defined by the conditions X = + i S, 0 < < < 2w sc sinh2, 0 in which ~ and f are real. The integration path r is oriented in the direction of increasingi. When s->0, v(, r,, t ) reduces to the Green's function for the elliptic cylinder relative to the line source at (,, " ). We shall consider the distribution on the surface of the cylinder and off the surface separately. On the cylinder, 6

THE UNIVERSITY OF MICHIGAN 2871-2-T 1V(=,/w2(, )G(),,- dX. (3.1) v( ~o2,J'U) - 1 0 2ri wr W2 (o x As in Reference 1, we shall evaluate this integral by residues, the residues contributed by the zeros of w. ( o, A). If > f > v,,,z - _ [l(g w1(, X)w4(o h0-wo )W2(g,;0wl( oJ w2(, ) v( ) = 1 J w2(,)- G(7,, -A) dX. (3.2) In the case where it is practical to evaluate this integral by the residues contributed by the zeros of w.(o, A), the representation (3.2) reduces to vg S') — )1 d w2(S wl(f,' )w2(o ) G ( ~-X d X * (3 3) 4rY W2(SA) 4. Our first objective is to determine the zeros of w( o, A). We need only sketch the analysis in view of its similarity to that in Reference 1, Sections 6 and 7. The differential equation satisfied by the wj is d2y + (2 sinh2 + A) y = 0. d52 If we let X= -Y sinh2 1, (4.1) then this equation takes the form 2 2 2 2 dy + (sinh2 - sinh ) Y = 0. d 2 We define #(,51) = sinh2 5 -sinh2, 1 )( 5, I) =' 0(t, ) dt, (, )= (, 7),

THE UNIVERSITY OF MICHIGAN 2871-2-T and t(5 )= /( > 1/6/ (gi) (5+ 1), with j(1,~1) = lim u (5, ) In terms of the above notation, the solutions wj have the asymptotic forms w. = /6 e +iY f( 1) v + B(5 ) w y~,1/6 e+iaf(gj) (j) w.= H e V () + B(5,Y) when i 4 N, and the forms Wj ^ (g ) e L )(1[I +B( )] when i > N. In these formulas B is used generically for a function which is uniformly bounded for the range of. in question and for o| > N, ir22 f( 1) = - \|sinh2 1+sin2& de, 0 and () 1/ 5i _ e1/3 v(J)( ) = e (S) () 2 1/3 (j) where H is a Hankel function. 1/3 2 When X < ~ |Y, it can be seen from these formulas that w2'(, ) has 0 no zeros. Provided (~,] ) is not too close to the shadow boundary, the zeros of 2 w( o, k) corresponding to values of X with XI >> 12 have large imaginary parts, and the terms which they contribute to the residue series may be neglected 8

THE UNIVERSITY OF MICHIGAN 2871-2-T (see, for example, Franz5 and Levy3). When Xj is comparable to il2 w2(~, A) vanishes only if 1/3 1 / 3 (2)\ / dS { H1 3(V ) + ('(Y) = 0. If g =, 5 fixed, and. is considered as variable, the value i of O O r which corresponds to the rth zero of w.(, A) may be thought of as the value of 0 ~' J ^(t, i(X, ) ) dt O1(Xya ) which is attained when X = Xsince.1 and. are related by (4. 1). Thus 9 = ^(^, vn *1 r 0 Because the zeros h of r dt H1/3(t) are simple and because this function is analytic in a neighborhood of each of its zeros, - =h + (y~( ) r r The zeros hr and the values of related functions such as t H(123(t) at these zeros 1/3 6 are known. The relation -1 -2 LI X(t,i( i(,y)) dt h Y + =(/ ) 5 W. Franz, Ueber die Greenschen Funktionen des Zylinders und der Kugel, Z. Naturforsh, 9a, 705-716 (1954). British Association Mathematical Tables, "The Airy Integral", Cambridge University Press, London and New York, 1946. 9

THE UNIVERSITY OF MICHIGAN 2871 -2-T may now be used to compute 5r = 1(Xr, X) by expanding the integrand on the left hand side in powers of (t- 51) or (t - S). It is vital to note that both of these expansions are slowly convergent as 5 -- 0. Therefore, the approximation 0 o for r which we obtain by neglecting all but the first two terms is not useful when;-4 0 and w is fixed. We henceforward assume that S is bounded away from O O zero. We find, under this hypothesis, that 7_ nr~ ~ ~ ~~~~-i 3 27/3 _ - 32/3 1hr 2/3. 7e 3 (sinh: o+osh o) h r 2(sinhcoshS )l/3 60(sinh coshg )4/3 + ( y -53) (4.2) The specific value of X will not be needed. Computation of aw2(go, A) ci a a x Ar now leads to an approximation for the residue contribution of w' at Xr: ( -3)1/r 01/2 iqf(g )+ 5 (2) -1/3 - wY(g X) =ej81sinh f e r 4 hH (hr) 1 + ( 1)J ( X r 8 ( ( sinh g cosh r 1/3 r 10 (4.3) In subsequent work, we shall also need an approximation for w{(~o, s)' An easy calculation yields the formula.1/2 y5 1/6 -if( )+ 3-ri wI( r (-1 (3 Ysinh cosh ) e r 4 hr ff H( 1) 10

THE UNIVERSITY OF MICHIGAN 2871-2-T 2 2 5. We next approximate G(,t -Xr). For sinh, the Liouville r r asymptotic representations for YI and yII are Kr (0)l 1/2 In YI Kr() cos [ K(t) dt (5.1) and sin (j Kr(t) dt) [Kr(0) Kr (1)] 1/2 where 1/2 Kr(t = sin2t + sinh2 S (5.2) r These results and the relations (2. 5), (2.6), and (5.1) combine to yield the formula I r l -cos Kr(t) dt - I Kr(t) dt 3 G(1nV, - - 1/ * (5.3) 2 Ysin( r Kr(t) dt) [Kr( ) Kr(17)] 0 6. In this section we consider the surface distribution, and we derive the residue series for the integral in (3.1). The residue series we seek is a sum of terms of the form W2( 2? Xr) G(-7, o-Xr) dra rWai(cN1 X=X r These can be approximated by using the results of 4 and 5. And, in fact, we need such approximations in order to investigate the convergence of the residue series and to see if the boundary of the region of convergence coincides with the geometric shadow boundary. We perform further approximations by expanding 11

THE UNIVERSITY OF MICHIGAN 2871-2-T in powers of ( 5 - 0 ) and neglecting the terms which cannot be specifically computed using the estimate (4.2) for -. Henceforward, we shall assume r o that the parameter s involved in Y is zero. We first consider w2(, ) Since is large, W2(Xr) = exp -iy H(t) dt + Kr(0) dO + e(Ie )1 1+('( 3/ e )] where 1/2 Hr(t) = (sinht - sinh r) (6.1) Performing an integration by parts, we find that W2( ) = exp -i / [coth H K (t) dt (6.2) with -1 sinh ) (t) =sin r - r. (6.3) r t sinh t J The exponential factors in the remainder of the r residue term are -iS Kr(t) dt cos S Kr(t) dt - Kr(t) dt e Jo r --, sin [ K (t) dt (or i r/2 7/2 -i |J K (t) dt - Kr(t) dte -id X3 Krdt K(t) dt | -4i +/ KP.t) dtH 1 - e S r (6.4) 12

THE UNIVERSITY OF MICHIGAN 2871-2-T It is important to note that because sinhS cosh dt K(t) dt (t) dt= + () - o o +o Jo ^o o... 0 0 0 K0(t) and because 9(8 - ) < o< r /2 (O (i \ Kr(t) dt) > 0. (6.5) 0 Since. (Y) is large, it follows that the dominant term in (6.4) is / r r'1 lr/2 exp - Kt) dt - K (t) dt. Therefore, by the relations (3.1), (6.2), (5.3), (4.3), and (6.4), the surface distribution 37ri/4 2 y sinh,cosh 5 o v(5,1,2t), e 2 si, o 2 Y 37r r exp -i c c oth Hr(Z) + K (t) dt + K r (t) dt - Kr(t) dt r (2 r 11/2 hr1/3() LKr() Kr(Z) Hr() (6.6) where only the dominant part of the first creeping wave term has been included. In this formula the functions Hr, Kr and -r are defined by the relations (6.1), (5.2), and (6.3), respectively. Let us assume that T > Z. Then if we expand Kr in powers of (g - ), we find that the exponential terms in the above summation take the form 13

THE UNIVERSITY OF MICHIGAN 2871-2-T Ai 07 tF\+ X - K (t) dt+ coth Ho() ( )o (0(Z) r/2) 2u 3 3u 0 NT 0 0 () H0(() )/Z exp(-ia +(sinh5 cosh ) and (- ). U g the trso r o'0 \ inh4 e + (cosh 22 + sinh2 = ) sin tanh b cosh i' 1 —-------------------- + —-----— l+ d t Ko (t) H() Each of the above integrals is a real quantity. The only terms with an imaginary represents an integral / R / ds where R is the local radius of curvature on the ellipse. This tells us that to a first approximation, the attenuation of the "creeping waves" is as predicted by Keller. Now by (4. 2), 2/3/ o o r o c r 1 where F 1 = (-s9wh)1 /3 7e 3 (3hr 2/3(2/3 R 1\/3 14 where~~~~~~~~~~~

THE UNIVERSITY OF MICHIGAN 2871-2-T 2 Ro =b /a is the radius of curvature at the ends of the semi-major axis of the ellipse. Thus the second term in the above exponent becomes s(j) s(o(3)) s(7r/2) / +i C, + - R ds. s(t) s(O) s(0) The formula for C1 reveals an essential limitation upon our theory and perhaps that of Keller; namely, the above expansion of the creeping wave exponents in descending powers of aL is meaningful only if Row >> 1. In particular, for a fixed L, Ro cannot be taken too small. Thus, we have derived mathematically the expected physical restriction upon theories of this kind. 2 The ( - S ) term in the exponent is of less interest. We have carried r o out its computation only in the case of the prolate spheroid; see the Appendix. As the discussion there would indicate, when 5 - co and the cylinder becomes 0 2 circular the ( 5 - o) and ( g - g ) terms cancel in such a way as to produce the r r 0 expected exponent. It remains to investigate the convergence of the residue series when summed in the "creeping wave" form. The condition (6.5) shows-that the convergence will not be rapid unless o() - /2 -f(-g5,) -- + \ \Sr Ko (t) dt > 0 z 0 0 In fact for a given', the condition f(7, 7) = 0 (6.7) 15

THE UNIVERSITY OF MICHIGAN 2871-2-T determines the boundary of the region of convergence. We shall show that this is indeed the optical shadow boundary. To do this, it is convenient to put the elliptic integrals of the first kind involved in f(,r) into Legendre form. We then find sinh2 o o cos -cos sinh cosZ -cos + jo as Q0 0 O where cosh cossincos7sin (cosh2 os2o) -cossin coshg0coshV) - os 0 0 A tangent to the o-ellipse drawn from the i an point ( Z) touches the ellipse o o at the point( ) such that cosh sinh cosCos+sinh32cosh - sinhcos sinh sin =sh coshosh (6.8) When Z = 0, it is a trivial matter to verify that (6.7) and (6.8) are equivalent. For nonzero V, the verification is easy but tedious. 7. Lastly, we discuss the far field in the shadow zone. In particular, we derive the residue series for the integral in (3.3) for large f and > S > i, or O.~. Ther rside Ris recsel0 o for large and ~ _ > >. The rth residue Rr is precisely 16

THE UNIVERSITY OF MICHIGAN 2871 -2 -T (-2iY) w2(5, )wl (5 ) times the rth residue in the case of the surface distribution. It therefore follows from the relations (6.2), (4.4), and (6.6) that Rr FrEr Rr r' where o (sinhsrcosh ) / 3 l/3 H(1)] =h 1/3 r 2 3 1 /3 h 2r)/3(2) r 23 (gh r)2H 1/ 3(hr) [Hr Hr(6) (I r ) and cos{ Y[ J Kr(t) dt - | Kr(t) dt j } r sin ( ^' \ K (t) dt) 0 * exp -i Y coth )Hr(K)+coth( )Hr(3)+ + Kr(t) dt. In these formulas the functions Hr, Kr and~r are defined by the relations (6.1), (5.2), and (6.3), respectively. We can also write E as a sum of creeping wave terms: r E =i iY - (2n-1) )Kr(t) dt -iY (2n+l) Kr(t) dtl 0 Jo0L Er = i e7 1 e n=0 I exp -i / cothsH (S) coth, Hr() + + Kr(t) dt]. 17

THE UNIVERSITY OF MICHIGAN 2 871 -2 -T Let I (f) = - i (2n- ) + + f(t) dt r 0 Then a typical exponent is 2/3 -1 -iX cothSHo(,) - icothS H o() -iY I(K ) +i C1(I ( K )+... u - 0 0 0 - C 0 0 The description of the terms involving I in terms of physical parameters is 0 essentially the same as that given at the end of: 6; and, of course, the remarks made there upon the region of validity of the expansion also apply. It is a considerably more tedious matter to verify that the creeping wave expansion converges in the geometric shadow zone. 18

THE UNIVERSITY OF MICHIGAN 2871-2-T APPENDIX In our paper we developed an expression for the surface distribution induced by a plane wave whose plane is perpendicular to the axis of the spheroid. This expression is in the form of the well-known "creeping wave" representation, namely ei27 (n iVrd(r(7)+nLr]+ X i (+e-1 A e iY dr ( ) + nL ] r r r 4 -- e We should like to point out the restriction upon our result and that given by the geometrical optics theory of Keller. The restriction stems from the terms (dr() + nLr) and (d* (1) + nLr). We have shown that i /d (") =i \a2 -b2t 2 r dt where a and b are the semi-major and minor axes of the spheroid, w is the wave number, and g is related to the rth zero of the Airy function (Ref. 1, Sect. 10). If one expands the integral above in a series of ascending powers of 5 -, where r 0 -1 5 = e is the eccentricity, one obtains the result (taking into account only the 0 first 3 terms of the expansion) i/Z 1/3 -1/3 v -2/3 i-vrdr () iw ds + i(c1w + c2w ) R ds Arc cArc cost +ic 3 -1/3? dt _ +c [(1 -et)3(1 t2)l 1/2 19

THE UNIVERSITY OF MICHIGAN 2871-2-T where R is the local radius of curvature, s is arc length, -ri/3 2/3 =e (3h e -i/(3h 4/3 (8- 7Ro/a) R 1/3 C3 -ri/ 3(h 4/3R2/3 (3hr) R c= - 8a R = b2/a is the radius of curvature at the tip of the spheroid, and h is the rth zero of [t /3 H /(t)]. In the geometrical optics theory of Keller only the terms ds and -2/3 cl R ds are present. Two observations of interest can be made from these formulas. Firstly, we note that in the case of the sphere (e = 0 and b = a) the term -27ri/3 4/3 1/3 ie (3hr) 43R -2/3 w —- 1/3a - R ds L a is the negative of the term -1/3 7 dt J o [(l-e2t2)3(lt2)]1/2 This is consistent with the known results for the sphere. Secondly, we observe that if R -- 0, that is e ->1, and if w is fixed, the c2 and c3 terms completely 4c2 dominate the attenuation. Let us estimate 4 2 in the case n = r = 0, the v2/3, c 20

THE UNIVERSITY OF MICHIGAN 2871-2-T case of the most significant term in the creeping wave representation. One finds ~C2 __3ho - (8-7RO/a) R01/3 2/3 cl w2/3 4 1 5R2/3 a where ho, 3 (1.0188) 2 If a = 6 in., b =.6 in., andX = 1.25 in., d /2 3 3/8; while if a = 1 in., b =.1 in., and X = 1.25 in., 4c2 4 2/3 c * L / cl Therefore, for such spheroids and such a X it appears that our theory and that of Keller will not give a significant result for the diffracted field. That is to say, the condition Row >> 1 is essential for the expansion of the creeping wave exponents in descending powers of w to be meaningful. This condition is a consequence of the fact that in order for the expansion of 2_t2 112 ( r dt J o \1 - t2 in powers of 5 - S to be useful, we must know E -; whereas S - r o r o r o has been estimated under the hypothesis that S is bounded away from 1. Further even if Rw > 1, since our theory is only an asymptotic one, we have no a priori way of predicting for a particular choice of parameters whether or not the additional terms in this series, which we have found, give a more or less 21

THE UNIVERSITY OF MICHIGAN 2871-2-T accurate result than that obtained by consideration of only the first term. The examples a = 6 in., 1 in. have been chosen for discussion because of recent 7 measurements of Olte and Silver on such spheroids. 7 A. Olte and S. Silver, New Results on Backscattering from Cones and Spheroids, URSI-Toronto Symposium, June 1959. 22

THE UNIVERSITY OF MICHIGAN 2781-2-T A CORRECTION TO A PREVIOUS REPORT In "Studies in Radar Cross Sections XXX, The University of Michigan, Radiation Laboratory Report 2591-4-T, (August, 1958)", two errors are made which lead to erroneous conclusions. Equation (2. 5) is incorrect. It should read: [, o r x2 _ x2 wll + {p2 r - + = 0. (Xr - 1) (1 - X ) (1 - X2)2 O In its original form, the equation led to an incorrect evaluation of the creeping wave exponents. This was observed in Reference 1 of the present report, and the corrected exponents appear in the appendix of the present report. al y'Also, the formula for on page 39, should now read: ay' (X2 (1) ( 3 1/2 (2 H() (r 4 r This results in a corrected formula (8.27): lim Rn = 2.3 i hn H ) (h-2)(n ( s-n0 1/3 if nd(-)+ i iVnd*()- 4 | e +e +e23 n 23