DIFFRACTION" OF RADIO WAVKS m A r\i;\H()I,IC ( ^I.IM)1;K rJ.) 



of order 3-^. As in (2.3), the fractional i)o\voi-s of i arc made precise by 

 takiiii;- / = e.\p {iir/'2). 



Tln-ee kinds of appro.ximations Ikiac l)een made in the derixation of 

 (2.10), namel}^ those associated with the assumptions (1) that the angle 

 ^ is small, (2) that h is large, and (3) that r/h' is large. The terms in \/^ 

 and 1/r do not cause trouble at i/' = because their infinities cancel 

 each other. 



The counterpart of (2.10) for vertical polarization is obtained from 

 (2.10) In' replacing E by H and ^(r) by ^r(T), where the subscript v 

 stands for "vertical"; and ^,.(r) is given by (2.11) when Ai{v) and Bi{u) 

 are replaced by their derivatives with respect to ;/. 



Series for '^(r) + 1/r and ^,.(r) + 1/r which converge for negative 

 (shadow) values of r are given bj^ Equations (7.31) and (7.53), respec- 

 tively, with (j in place of r. Table 2.1 gives values of "^(r) and ^i.(t) 

 which were obtained from the series for r negati\'e, and from numerical 

 integration of (2.11) and its analogue for r ^ 0. 



When T is large and positive Equations (7.35) and (7.55) show that 



(/7rr)'''exp (-2tV12), 



(2.12) 



^(r) + 1/r 



^,(r) + 1/r ~ (?Vr)''"- exp {iir - trVl2). 



When r is large and negative the leading terms in (7.31) and (7.53) 

 give 



^(r) + 1/r ~ - i'^ 2.03 exp [(2.025 + i 1.169)r], 

 ^,(r) ^ 1/r i^^ 3.42 exp [(0.882 + i 0.509)r]. 



(2.13) 



Xow that we have expressions for the field what do they tell us? For 

 one thing, they may be used to show that the field near the shadow 



Table 2.1 — Values of ^(r) and ^^(r) 



