DIFFKACTIOX OF 1{.\1)I() WAVKS HY A I'A K \ l<( )1,IC ( "» 1,1 NDKK 457 



// is parallel to the crest of the parabolic cylinder. Since (he ( yliiidcr 

 is a perfect conductor, the cui-rent ilensily ./,. on the surface v — rjo 

 is (Hjual in magnitude to H and its dii-ection is that of inci-easing ^. 

 Thus settings' = ^,i in (expression (5.17) for // and usin.u; the U'ronskian 

 (9.9) j>ives 



-/. = X f -^ {nc)"UMnV,Xz:^, 

 jl., sni Trn 



/ ^\ / ^^^-^^^ 



N = e-"- (^sec ^j/ 2x> ^ r = {y^l + ^)/2, 



where 'Wn{z'^) is defined by (4.19). 



Closing Lo on the right and left leads to the analogues of (0.4) and 

 (6.5): 



00 



J. = 2iXT. {-iwrUn{z)/'W{z',). < (/• < 1, (G.20) 



7i=0 



{iwYVniz) 



J, = -2l7rXYl 



1 < w < ^, (6.21) 



sin Trnd'Wn(zo)/dn_ 



where n's is the sth zero of 'W„(z'o). The zeros of both 'Wn(zo) and W„(zo) 

 are enclosed b\' the path of integration L4 shown in Fig. 5.1. 



The current density on a half-plane is obtained by setting ^o = in 

 (6.20), using 'ir„(0) = Wn'(O) = -i"-'/T(n/2 + 1^), from (9.4), 

 and the generating function series (9.22) for Un(z): 



/„ = 2{i/Ty"e-"-''"'' f c-''' dl (6.22) 



h sin (9/2) 



where r has the same significance as in (().6). This agrees with the ex- 

 pression obtained from Sommerfeld's result for the half-plane. 



When w = 1 and h is large, the path of steepest descent for (6.19) 

 becomes the same as that for the case of horizontal polarization, namely 

 ACD of Fig. 6.2. This follows from the fact that, as may be seen from 

 (12.2), the controlling exponential functions foi- '1T'„(^|') and Wniz'o) 

 are the same. The analogue of (6.16) is obtained \)\ using the approxi- 

 mation (1.3.24) for 'Wn{z',): 



J, ^ (1/2x0 exp [-/^77o - V/3] 



1 (-i2ir/3) 



exp(i''V) doi/Ai'{a) 



L 



=oexp(-i2W3) ^ ^^^ (G.23) 



'« exp ( i2ir/,'?) 



where At {a) denotes dAi{a.)/da and 7 is given by (6.15). 



