468 THE BELL SYSTEM TECHNICAL JOTTRNAL, MARCH 1954 



(only ii contributes to Un(zo) and only to to Wniz'^) at the saddle point 

 nil of the second integral in (7.38)). 



So far in this section the parabolic cylinder has been assumed to possess 

 infinite condu(;tivity. When the cylinder has a finite (but very large) 

 conductivity, it may be shown that the field far out in the shadow is 

 approximately 



hi sm irn TV„(4n) + a 'Wn\ZQ) 



Equation (7.57) is suggested by (7.14) and (4.25). The analogue of 

 {1 .bl) for vertical polarization may be obtained by replacing E, a in 

 (7.57) by H, t so that 'Vniz'^) + rVnizo) appears in place of Vniz'a) 

 + 0""^ 'V{z'(t), and so on. 



When the parabolic cylinder functions are replaced by Airy integrals 

 according to (13.21) and (13.24), equation (7.57) may be written as 



E ~ f'M, f fe'^P <-"f .71'''m "^ '''^'"'' *» (7-58) 



JL'i At{a + k) 



where g and M2 are given by (7.20), a by (7.23) and 



k = -{ihr"'UU- (7.59) 



I fc I is small compared to unity. L4 is a path of integration in the a plane 

 which encloses the zeros of Ai{a + /c) in a clockwise direction. Changing 

 the variable of integration in (7.58) to u = a -\- k enables us to con- 

 clude that 



E for finite 

 conductivity 



^'^p ' -^/j 



E for infinite 

 conductivity _ 



(7.60) 



Since we have assumed d = ■7r/2, the relation (7.60) holds in the region 

 where the angle ^p defined by Fig. 2.3 is negative. 



The analogue of (7.58) for vertical polarization is obtained by re- 

 placing E by H, omitting the i"'^, and replacing the ratio of the Airy 

 integrals by 



Ai'[{a + (/a)l"]/Ai'{a + (/a) (7.61) 



where 



1= -iihf'UU. (7.62) 



Even though h is large, f/fo is assumed to be so small that ( is small 

 compared to unity. The path of integration L4 must now enclose the 

 zeros of Ai'{a + l/a) which are close to those of Ai'{a) at a = a's^s = 1, 



