472 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1951 



we obtain 



E = {c-" + *S'i)p + c(p) 



H = (.-'^ + S{), 4- c(p) 



1-1 + 

 2 'A 



+ ^(g)}frexpiigyd) 



+ 



+ - +M^.(^) r^expOWB) 



(8.15) 



(8.16) 



+ 



The terms neglected in (8.15) and (8.16) are the "order of" terms in 

 (8.10), plus those neglected by virtue of \l/ being small, plus the errors in 

 (8.14) The errors in (8.14) are of two kinds namely those of ()(Ji/r^'') 

 and those due to approximating the parabolic cylinder functions by 

 Airy integrals. 



It is interesting to observe the forms assumed by (8.15) and (8.16) 

 when /i = even though they are not supposed to hold for small values 

 of h. In this case p, \p go into r, (p and the right hand sides of (8.15) and 

 (8.16) become the same, namely 



ie-'' + S,)r + c(/0/2. (8.17) 



The half plane results given in Section 2 become, for small values of (f, 



|| = (^-''^ + S.)r ± c(r)/2 (8.18) 



where the upper sign corresponds to E and the lower one to H. Com- 

 parison of (8.17) and (8.18) shows that (8.15) for E reduces to the proper 

 value but (8.16) for H fails to do so because the signs of c(r)/2 do not 

 agree. 



The discrepancy is apparently related to the approximations we have 

 made in obtaining the expression (7.19) for S-2i from (7.18) and to the 

 errors introduced by approximating the parabolic cylinder functions 

 by the Airy integrals. As we let /i — ^ in the more complete expression 

 (7.18) for *S2i, the value obtained for *S2i — ^ <». This is explained by the 

 fact that the upper limit of integration — 1 — ih (at point C) approaches 

 the pole of the integrand of (7.17) at ?i = — 1 . This large value of S21 tends 

 to be cancelled by the large value of *S23 (for horizontal polarization). 

 On the other hand our approximation (7.19) yields via (7.21) the value 

 iMi for S-n and ^Ssi when /i = and |8 = 1. The factor li'^ in Mo makes 

 our approximations for ^22, S-n, Sz2, Sss in terms of Airy integrals vanish 

 when /i = 0. 



