426 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



boundary is almost the same as for a half-plane. Away from the shadow 

 boundary, the field in the shadow can be interpreted as a "crest wave" 

 which reduces to the "edge wave" for a half -plane. The crest wave de- 

 creases as an exponential function of \l/ in the shadow instead of as l/<p 

 in the case of a half-plane. In other words it is much darker behind a 

 parabolic cylinder than behind a half-plane — and the larger the cyl- 

 inder the darker it is. (A glance at Fig. 2.8 shows that this statement 

 must be qualified for vertical polarization by requiring the observer to 

 be deep in the shadow.) As Figs. 2.7 and 2.8 show, deep in the illuminated 

 region the crest wave behaves like the wave reflected (as computed by 

 geometrical optics) from the illuminated portion of the cylinder. 



Now we consider expressions for the surface currents. Let J and J„ be 

 the densities of the conduction currents which flow on the surface of the 

 perfectly conducting parabolic cylinder for the cases of horizontal and 

 vertical polarizations, respectively. J is parallel to E and is perpendicular 

 to the plane of Fig. 2.3 while J„ flows in the plane of the figure. Jv is 

 positive when the current flows in the direction of increasing x. In Sec- 

 tion 6 it is shown that when h is large, J and J„ are given approximately 

 by 



fo-/ 



exp (— ix — iy^/3) 



Jo 



■ —2/3 / -1/3 \ 



t exp (— I uj) 



Ai{u) 



+ 



- iBi{u) 



exp (— iwy) 

 Ai{u) + iBi{u)_ 



(2.14) 



du, 



J D 



i exp (— ix — ij /3) 



r 



exp ( 



\iy) 



Ai'iu) - iBi'iu) 

 exp (— iuy) 



+ 



Ai'iu) + i Bi\u)} 



(2.15) 



du. 



These expressions are obtained when the relations (13.17) for Airy 

 integrals are used in equations (6.16) and (6.23). Here fo is the intrinsic 

 impedance of free space. In the rational MKS system which we use 

 fo = 1207r ohms. The factor fo appears in (2.14) but not in (2.15) because 

 we assume the incident wave for vertical polarization {H = 1, E = ^qH 

 — r207r) to be 120x times stronger than the one for horizontal polariza- 

 tion {E = 1). The primes on Ai{u) and Bi{u) denote their derivatives 

 with respect to u. The parameter y depends upon the coordinate x of 

 the point at which the current is Ijeing observed: 



7 = x/2h 



2/3 



(2.16) 



