11764-517-M = RL-2227 011764-517-M 12 December 1973 Memo to: File From: T.B.A. Senior Subject: The "split" functions of Senior (1952). The problem of the diffraction of a plane electromagnetic wave by a half plane at which an impedance boundary conditions is imposed was first solved by Senior (1952). Subsequently, Senior (1959) and Maliuzhinets (1955, 1959) obtained the analogous solution for an impedance wedge, but since the functions in terms of which Maliuzhinets expressed his results are, in the particular case of a half plane, much simpler than the "split" functions of Senior (1952), it is of interest to determine the connection between them. For an E-polarized plane wave, Senior's solution is given in terms of the split functions K+ (), K (0); however, K () - + k( h ) K+() (1) K +()K (-e) = 1 (2) and it is therefore sufficient to consider only K+(e). The expression for this is given in eq. (22) of Senior (1952). If, for simplicity, we write = k cos0, implying sin - = - -0 and = secx, implying - = itan2 ]= secx, implying -ri itanx K (+) becomes

011764-517-M K () = cosxc(1cose) 1/2 + L cosX+sinO (20 - 1)/4 -o- cosX-sin cos sin X +cos 0 cos X +sin O exp L. 7 7 0 c2 log \cos X cos s ds 0os X s (3) But I7 lo cosx-cos s ds \ cosx+cos s J0 -) \cosX-sin0 - 2 j)'~p~~~Xi~ne/I 7T 7r 2 O. sin s cos X s ds 2 2 s cos X-cos s (4) on integration by parts, and when this is substituted into (3), the first term on the right hand side of (4) cancels the third factor on the right hand side of (3). Moreover, 2 sins cos 2 2 cos X-cos s 1 1 sin(s - X) sin (s ) sin(s-X) sin (s+X) and 7IT 0 -- 2 sds sin(s -X) 0 71 -.-0-x 2 -\ xdu sinu JX 7r- -0x 2 x udu + log 1-sin(X + 0) i sin(x + - 2 ) 0 sn 2 sinX y 1 -cosx Since x udu sinu 0 =-Sc udu sin u 2

011764-517-M it follows that (j-0 / -0 X r+-0+X sins co s \ X udu ds2 2 + Jo cos X-coSs o J ++ log- + sin(X - ) sin(x+ 2 and when this is substituted into (4) and then (3), the second term on the right hand side of (5) cancels the second factor on the right hand side of (3), leaving K+(kcos) = ( c +os ) exp + udu K kcosxo+1 f e sinu(6) which is certainly a great deal simpler than the expression we started with. Maliuzhinets (1958) defines a meromorphic function ~ (/3) one of whose 7T representations is (Bowman, 1967) 2b 1/2 - yCosj 2 2+2b \/ r, O1 2 ud exp udu (7) - +sin e 4 — sinu 2 where b =- K with K = 0. 9159656... (Catalan's constant). In particular 7f (/2) = 2 ) ( (8) and thus 3

011764-517-M b (13) 2'(2> f S cos2 +1 2 1/2 ) ( 2 1/83 J \l+sin J exp { 7 2 - udu } sinu J. (9) Comparison of (6) and (9) now shows that K +(kos ) = COS X(l +cos ) 1l/2 l cosX+sin J 4 (cos + 2 ( c + 1) 1+sin( +X) 1 +sin(0e -X) 1/4f 7r_( +X) 2 2 2 y) (7r/2) ( (0-X) 2 1(7/2) J and since {1 +sin(e+X)} 1 +sin( - X)} = (cos + sin 0)2 it follows that K (k cos 0) = 2 {2 cos X(l +cos e)1/2 (cos + cos + 1 r (o +x) I0 (7r/2) b (O-x) ( (1r/2) (10) A special case of this is f 1/2 r 4 K+(k) ( o. 1)2 W,,(11) (,B) being an even function of,3. 7T Equation (10) can be used to determine the behavior of K +() for large | |. For this purpose, let 0 = i7 where y is real and will be allowed to approach +oo. From one of the alternative representations of IT (3), we have 4

011764-517-M 1/2 (3) =7r (cos ) 1/8 1 C vdv ex 4 r cos v (Bowman, 1967, eq. A. 3), and since v dv cos V 0 I cos 2 +( 1/2 ) ^ = 2- 1 i 4'~2+ 1 (cos3) 1/8 exp { -'"-~4 1 4+ io vdv cos v ( Hence, from (8), ~ (2) %( (/2) fcos +1 2 1/2 cosp/ i0o 1o v dv exp -\ -s V I1Jo8 (12) and for >> 1, vi (+x) (.r/2) iT exp { - } { 2 18 4 implying (e+x) % (T/2) ff J I.( X) 1 e^/4 V (7r/ 2) 2 When this is inserted into (10) and the remaining trigonometrical factors approximated for y > 1, we have K (e) (cos X)1/2 -1/2 >>1, (13) which is the result used in Memorandum 011764-515-M to specify the current at the edge of the half plane. 5

011764-517-M The edge diffraction coefficient P(r, 0) for edge-on incidence has been examined by Knott and Senior (1973), starting from Bowman's (1967) expression for P(r, 0) according to Maliuzhinets (1958). If, instead, we use Senior's (1952) results, i 20 K (k) P(r, 0) - cosec2 4 2K (kcos 0) and hence, from (2), i 26 P(ir,r-6) - sec K+(k)K (kcos6) (14) where, for convenience, we have written 0 = r- 6. In particular, for backscattering (6 = 0), i 2 P(7r, 7) = - K(k)} (15) ^ —(i^e. (16) i.e. P(, ) = i 4cosx r(x (16) (Vcos 2 +) L (2) on using (11). If we now insert the expression for V4 (X)/V (7T/2) obtained from (12), the expression for P(7r, ) becomes ioo P (7, r) = exp 2 vC, (17) in agreement with eq. (3. 8) of Knott and Senior (1973). For bistatic scattering, P(7,r -6) 26 K+(kcos6) =6 =se - - P(T,, a) 2 K+(k) and hence, from (10) and (11), 6

0 117 64-5 17-M P -6) 6 c2 + oss X+1 r x+6+ ) o(x- 6)1 P(S, C - 6) = 2 P( 7r) 2 Cos +1 2Cos +1 '(X) (X) J 2 2 (18) which is in agreement with eq. (3.20) of Knott and Senior (1973). There is one consequence of (18) which is rather interesting and markedly simplifies the task of computing the bistatic scattering. If we denote by Q(X) the expression given in eq. (16) for the backscattering coefficient for a surface impedance specified by the angle X, we have 21/4 = 2) os X (2cos+2 1) Q(x}1/4 If we now replace X by X t 6 and insert the resulting formulae into (18), it follows that P(7r -6) _ sec 6 cos QX( +6). Q(X-6) 1/4 ( PrQ() Q(x) cos X-sin26 W W which enables us to deduce the bistatic scattering from the backscattering coefficients for different face impedances. For small surface impedances (I|) n 1), x -i log and a small change 2 in -6 will have little effect; the bistatic scattering is therefore a slowly varying function of 6 for small 6. If the impedance is large ( In>>~ 1), however, 7i 1 X - -, and it is possible that even a small change in 6 can change the scat2 Y ' tering significantly. References Bowman, J. J. (1967), "High-frequency backscattering from an absorbing infinite strip with arbitrary face impedances", Can. J. Phys. 45, 2409-2430. Knott, E. F. (1973), "Non-specular radar cross section study", University of Michigan Radiation Laboratory Report No. 011062-1-T. Maliuzhinets, G.D. (1957), "The radiation of sound by the vibrating boundaries of an arbitrary wedge, Parts I and II", Sov. Phys. - Acoust. 1, 152-174 and 240-248. 7

W 011764-517-M Maliuzhinets, G. D. (1959), "Excitation, reflection and emission of surface waves from a wedge with given face impedances", Sov. Phys. Dokl. 3, 752-755. Senior, T. B.A. (1952), "Diffraction by a semi-infinite metallic sheet", Proc. Roy. Soc. (London) 213A, 436-458. Senior, T. B.A. (1959), "Diffraction by an imperfectly conducting wedge", Comm. Pure Appl. Math. 12, 337-372. 8