424 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



(obtained by setting the angle of incidence 6 equal to 7r/2 in equation 

 (5.4)) involving parabolic cylinder functions. When /i = the parabolic 

 cylinder reduces to a half-plane and S'i{h) reduces to the value *S?(0) 

 appearing in (2.3). The symbol S^Qi) represents an integral much like 

 S^Qi) except that it contains derivatives of the parabolic cylinder func- 

 tions. As we might expect, S2Q1) and S^Qi) behave in much the same 

 way as does »S2(0). In particular, they are small compared to exp { — ix) 

 + »Si at the shadow boundary, and their asymptotic expressions anal- 

 ogous to (2.6) hold as (p and \l/ pass through zero. 



S2{h) and SzQi) have been put in a form suitable for computation in 

 two cases, (1) when /i = 0, which is the half-plane case already discussed, 

 and (2) when h and r/h are very large. In the second case it is conven- 

 ient to introduce new polar coordinates (p, \}/) with their origin at the 

 crest of the parabola as shown in Fig. 2.3. In these coordinates a circular 

 cylindrical wave spreading out from the crest is asymptotically pro- 

 portional to 



c{p) = i"'' (2Tpr"' e-'". (2.9) 



In Section 8 it is shown that 



exp(zrV3), (2.10) 



c{p) , r N7 1/3 



E ^ [f-'^ + S,l - -^ + c{pW' 



Hr) + - 



T 



is an approximation which gives the field (for horizontal polarization) in 

 the region of the shadow boundary far behind a large cylmder. Our r is 

 an approximation to the g used there. Here the subscript p on [exp ( — ix) 

 + >Si1p means that the quantity within the brackets is to be computed 

 as though it corresponded to a half-plane with its edge at the crest of 

 the parabolic cylinder, so that p, \l/ are to be used in (2.3) instead of ?•, (p. 

 Also, 



Jo Ai(u) — iBiiu) 



(2.11) 

 r" Ai{u) exp {(ut) flu 



Jo Ai{u) + iBi{u) 



where Ai{ii) and Bi{u) are Airy integrals defined by equations (13.12) 

 and (13.16), and tabulated in reference." In this paper we find it con- 

 venient to use the Airy integrals instead of the related Bessel functions 



"The Airy Integral, Brit. Asso. Math. Tables, Part — Vol. B (Cambridge, 

 1946). 



