DIFFRACTION OF RADIO WAVES BY A rARAHOLIC f'VLINDKH 153 



when the integral for r(l + n/2) is inserted and the sum (9.22) [i.e., 

 the sum for the generating function of U„(z)] used. Tntogratiug part of 

 the result gi\'es 



a 



fo-/ = {2/iirry - cos - 



((3.G) 

 - 2/t sin ^ c"'''""^ / exp (-//') dt . 



■i ^5 sin («/2) 



In (G.6) /• is the distance along the half-plane as measuied from the 

 edge: r = ^~/2 = \ y \. Positive values of ^ correspond to the shadowed 

 side of the half- plane and negative values to the illuminated side. With 

 this interpretation (6.6) agrees with the current density obtained from 

 Sommerfeld's expression for the field. 



Although (6.6) has been derived from (6.2) on the assumption that 

 < u? < 1 it also holds for < w < oo as may be shown by analytic 

 continuation. Again, (6.6) may be obtained from (6.5). 



Since the factor r~^'" comes from the multiplier Mo in (6.4), it is pos- 

 sible that (6.6) may give one an idea of how the current density behaves 

 near the crest of a thin cylinder which is almost, but not quite, a half- 

 plane. Of course, r would have to be interpreted as shown in Fig. 6.1. 



In order to study J when the radius of curvature of the cylinder is 

 large compared to a wavelength we consider the case 6 = 7r/2, i.e. w = 1, 

 in which the incident wave comes in horizontally. In this case most of 

 the variation of the current density occurs near the crest of the cylinder 

 where, as it turns out, ^ is of the order of T?o^'^ vo being large. 



At the beginning of the investigation rough calculations of the inte- 

 grand in (6.2), based on the asymptotic expressions of Section 12, sug- 

 gested that for small ^ and large r?o: 



(a) JNIost of the contribution to the integral (6.2) comes from the 

 neighborhood around point C shown on Fig. 6.2 where m = n -{- 1 = 

 zo^/2 = —irio~/2 = —ih. Point C is a critical point associated with the 

 asymptotic behavior of W„(zo). 



(b) The path of steepest descent for (6.2) roughly corresponds to the 

 line ACD of Fig. 6.2, C being the high point of the path. Along this 

 path Im [fit,) - f(h)] = where /(/) = -T + 2zf - m log /, m = n + 1, 

 is the function entering the definition (10. 1) of the parabolic cylinder 

 functions, and h, h are the saddle points of exp [/(/)]. This path in the 

 n-plane separates the regions in which ir„(^ii) has different asymptotic 

 forms. It is the boundary of region III' in Fig. 12.2 and has been studied 

 in Sections 11 and 12. 



Once (6) is verified the truth of (a) follows almost immediately since 



