458 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



For positive values of 7 (6.23) and d'Ai(a)/d~a = aAi{a) lead to 



^ exp (t'^ya's) 

 J, ^ -exp i-i^rjo - n /3] Z /, Y '\ (6.24) 



where a'l — —1.019, 02 = —3.248, • • •, etc. are the zeros of Az'(a).* 

 When we use Ai{ax) = 0.5357 the leading term in (6.24) gives (2.19). 

 For large values of s* 



a.: [3x(4s - 3)/8]''', 



4 V '\ / \s/ '\-l/4 -1/2 



Ai{a,) ~ -(-) (-a,) TT . 



The expression (2.17) for J„ when 7 is large and positive may be ob- 

 tained by applying the method of steepest descent to (6.23). The asymp- 

 totic expression for Ai'{a) is obtained by differentiating (13.19). 



When the cylinder is a good conductor, but not perfect, the expression 

 for H analogous to (4.25) leads to an integral for J,,, obtained from (6.23) 

 by substituting Ai'{a) 4- (Ai{a) for Ai'{a), which is ecjuivalent to one 

 given by Fock.j Here ( = — {ihY'^^/^o is assumed to be small compared 

 to unity and f/fo is the ratio of the intrinsic impedance of the cylin- 

 der material to that of free space. Horizontal incidence, 6 = tt/'I, is as- 

 sumed. 



The analogue of (4.25) for H has the same form as (4.21) except that 

 now 'Un(zo) is replaced by 'L^,(^o) -|- rUnizo), etc. The development 

 leading from (4.21) through (5.15), (5.17), (6.19) to (6.23) may be car- 

 ried out just as before. The work is also related to that given at the end of 

 Section 7 where the effect of finite conductivity on the diffracted wave 

 is briefly discussed. 



A series corresponding to (6.24) may be derived from the integral. 

 The exponential terms in this series are approximately exp [i^'\(a's 

 — f/a's)], and are similar to those in (7.63). Since fo = (Mo/eo) " is real 

 and f ?^ {iojix/gY'' when g » we (the notation is explained in connection 

 with (4.24); the g denoting conductivity should not be confused with 

 the g defined by (7.20)) the (juantity —I'^f/a's has a positive part. Thus, 

 the attenuation of ./„ in the shadow is decreased slightly when the con- 

 ductivity g of the cylinder is reduced from infinity to a large finite value. 



7. FIELD AT A GREAT DISTANCE BEHIND THE PARABOLIC CYLINDER 



The field at any point, for the case of horizontal polarization, is 

 given by expression (5.5) with S->(h) given by (5.4). Since S-2(h) is the 



* Reference 11, page 424. 

 t Reference 5, page 418. 



