DIFFRACTION OF HADIO WAVES BY A PAKAKOI.K CYMXDKI? 435 



ins with ('(Illations (7.30) and (7.5()). Furthernioi'c, it may be verified 

 that tlic phase angles of tlie reflected wa\'es as eoinpiitcMl from (7.30) 

 and (7.50) agree with those computed from geometrical optics when 

 reflection coefficients of —1 and +1 are assumed for ]\\) and vp, re- 

 spectively. 



The amplitudes of the crest waves for h = 1,000 and h = are shown 

 for hp ill Fig. 2.7 and for \'p in Fig. 2.8. Of course, when /; = the crest 

 wave reduces to the edge wave from a half-plane. The curx-es for h = 

 1,000 were computed from equations (2.24) and the curves of Fig. 2.5 

 (or their e(iiii\'alent when r is small). The curves for h = were com- 

 puted from 



1^1- (27rp)-''- 

 (hp) 

 E - C-" I ~ i2Tp)-"' 



' + ' 



2 sin (t/'/2) 2 cos (i/'/2) 



' + ' 



lA < 



2 sin (i/'/2) 2 cos {^p/2) 



1 _ 1 



2 sin {yp/2) 2 cos (i/'/2) 



yp > Q 



(2.32) 



(vp) 



H 



(2tp)- 



1 



1 



2 sin (i/'/2) 2 cos (yp/2) 



xP <0 



4^ > 



which follow from (2.1), (2.2), (2.4) and (2.6). 



From equation (2.21) onward we have been discussing the field for 

 values of i/' and p such that pi/'" » 1. For these values the concept of the 

 crest wave is helpful in visualizing the behavior of the field. Now we 

 consider the field at points close to the boundary of the geometric shadow 

 far behind the cylinder. This is the region in which Artmann was es- 

 pecially interested. His results for the shift of the field may be obtained 

 from (2.10) and its analogue bj" taking | ^ | to be very small. 



At the shadow boundary xj/ = and [exp ( — ix) + Si]p = }/2- Hence 

 the region of interest at present is in the neighborhood of the point 

 point ti = 0, \ exp ( — ix) + aSi | = ^^ of Fig. 2.2. A magnified view of 

 this region showing the shift of the field is given in Fig. 2.9. The figure 

 shows that, for a given volue of p^, | E | for hp is less than | H \ for vp. 

 As Artmann has pointed out, this is to be expected smce the reflection 

 coefficient for E (hp) is roughly — 1 and the reflected wave therefore 



