DIFFRACTION OF RADIO "WAVES BY A I'MJABOIJC rVIJXDEH H)3 



.1/. = 2M,h''Y'-'' - ^'^'''^''' ^^p ^-'' + ^"^'^''^ 



where we have introduced two (luantities which will he used later 



Vtt/ 2^ sin (0/2) ' (7.20) 



g = -h"'\og0. 

 When )8 = 1 



Sn ~ 2iMi(ih + }4). (7.21) 



When h is large most of the contributions to the integrals (7.17) for 

 ^22 and aSjs come from around n = no = —1 — ih, and we may use the 

 approximations 



UM)/Wr.(zo) ~ i-*"Ai(ai-'")/Ai(a), (7.22) 



Vr.(z'o)/Wn(zo) - i"'Ai(ai'yAi{a), (7.23) 



a 



= (ihr"\n + 1 + ih), n - no = a{ih)"\ 



which come from (13.21). Setting these in the integrals (7.17) and using 

 the fact that i'^/sin -wn is nearly 2? around no leads to 



S.,, ilfo / exp( -l'^'ag)Ai{ai-"') da/Ai{a), (7.24) 



Joo exn (i2j-/3) 



/« exp (i2j-/3) 



,00 exp (— i2?r/3) 



S,, f"Mo / exp i-t'''ag)Ai{ai^") da/Ai(a), (7.25) 



Jo 



S22 + *S23 ~ ?:i/2^(^), (7.26) 



where 



^(^) = i [ exp (-i'' a^)Az(ar''') da/Aiia) 



(7-27) 



+ /" exp (-i"'ag)Ai{m'") da/Ai(a). 



Jo 



The expression (2.11) for -^(g) is obtained from (7.27) by changing the 

 variables of integration and using the transformations (13.17) for Ai(a). 

 Thus, when h is large and (3 near unity, (7.19) and (7.26) give 



S.(h) ~ 2i]\h(0 - ly' + Ur-^g-' + ^(g)]. (7.28) 



In the .shadow, where (5 > 1 and g is negative, (7.13) and (7.28) give 



E = exp[-w- sin 6 + //y cos 6] + S^ + S,(h) (7.29) 



~a/2[^"^ + ^i^((/)]. 



This and the series (7.14) for E suggest that 'i'(g) + 1/ry may be ex- 



