DIFFRACTION' OF IJADIO WAVES BY A PARABOLIC CYLINDER 421 



the analogy with the radio case, these two polarizations will be termed 

 "horizontal {)ohuization" (hp) and "vertical polarization" (vp), re- 

 spectively. The incident plane wave for hp is assumed to have an E of 

 unit amplitude. This is indicated by the exp ( — u) in (2.1). For vp the 

 incident wave is assumed to have an H of unit amplitude. The S's 

 are defined by the Fresnel integrals 



e"- + ^1 = ^/7r)'-c 



00 



dt, 



dt. 



J t-> 



h = {2ry'' 



sm 



h = (2/-) 



1/2 



cos 



2 



i = exp (t7r/4), 



where (r, <p) are the polar coordinates sho^^^l in Fig. 2.1 [Si and *S2(0) 

 = — *S3(0) for an arbitraiy angle of incidence are given by Equations 

 (5.3), (5.G), and (5.20) of Section 5]. We use the notation S-M, S,(0) to 

 indicate that these functions are special cases of the functions S-2(h), 

 Ssih) which appear in the analysis for the parabolic cylinder. 



The principal part of the field far to the right of the half-plane, where 

 X is positive, is given by exp ( — ix) + **^i whose absolute value is plotted 

 in Fig. 2.2. The function Si almost cancels the incident wave in the 

 shadow, and then drops down to small \alues outside the shadow. 

 The function aSVO) is always small in the region we shall consider. It 

 becomes large only when (p exceeds x. It then corresponds to the wave 

 reflected from the front (left-hand side) of the half-plane. 



When we are far enough away from the shadow boundary to make 



INCIDENT WAVE 



e-ix 



^\>ru — - 



TRACE OF_, 

 HALF-PLANE 



Fig. 2.1 — Diffraction of a plane wave by a half-plane. 



