DIFFKACTIOX OF RADIO WAVKS \i\ A l-\l(\H<)Mc (">M\DKR 433 



(loser to the crest is perhaps blocked out by the cuiAnliirc of Ihe cylinder. 

 For comparison ^vith the horizontal polaiization case we note that E 

 at (p, 4/) lor an infinitel}^ long current filament at the origin p = is 

 given by 



l^/fo/ I ~ .5(2tp)-"' (2.28) 



where / is the current carried by the filament and the frequency is 

 is such that X = 'Iir. There is some difficulty with the pictui-e for vertical 

 polarization because the current element at T points directly towards P 

 and hence should produce very little field there. This is perhaps as- 

 sociated with the fact that the (vp) ratio in (2.27) is smaller than the 

 (hp) ratio by appro.ximately the factor /^^''^ 



We now leave the shadow region and consider the field at points well 

 inside the illuminated region. Fig. 2.5 shows that for large positive 

 A'alues of T the amplitude of the crest Avave tends to increase with t. 

 The asymptotic expressions (2.12) show that when r is large and positive 



I ^(r) + 1/r I ^ I ^„(t) + 1/r I ~ (xr)''" = (TnPfV", (2.29) 



and hence the amplitude of the crest wa\e deep in the illuminated 

 region is, from (2.22) and its analogue, 



(hp) I E - e-'^ I ^ (27rpr"' (TrrPh)"\ 



(vp) I H - e-'" I -- (2xp)-''- (x#)'''. 



(2.30) 



Since (2.30) is derived from the general expression (2.10) it is subject 

 to the restrictions mentioned just below eciuation (2.11), In particular 

 the angle xj/ should be small (but we must still have \f/~p » 1 as assumed 

 in (2.22)). AVhen \l/ is positive, an application of the laws of geometrical 

 optics to determine the reflection from the curved surface of the para- 

 bolic cj'linder leads to the expressions^' 



(hp) 1^ 

 (vp) 



'h tan (iA/2)" 



P 

 'h tan (\l//2)' 



sec (ip/2), ^ > 



(2.31) 

 sec (^/2), xl^ > 



p 



for the reflected wave. When xp is small these expressions reduce to 

 (2.30) as they should. 

 Expressions (2.31) may also be obtained from our analysis by start- 



'2 In our two-dimensional case the calculation of the required radius of curva- 

 ture, etc., is not difficult. General theorems dealing with jjroblems of this sort 

 and references to earlier work are given in the paper, ,\ General Divergence 

 Formula, II. J. Riblet and C. B. Barker, J. Appl. Phys. 19, pp. 63-70, 1948. 



