DIFFRACTION OF RADIO AVAVES RY A PARABOLIC CYLINDER 423 



2.2. In the shadow only the edge wave is present and there is no inter- 

 ference. Si is not the edge wave in the shadow. 



The edge does not radiate uniformly in all directions. The <p in the 

 denominator of c{r)/<p indicates that the edge sends out its strongest 

 wa^'e in the direction of the shadow boundary where ^ = 0. This ac- 

 counts for the decreasing size of the oscillations in Fig. 2.2 as <p becomes 

 more and more positive. It likewise accounts for the steady decrease as 

 <p becomes more and more negative. 



That »S2(0) is small in comparison with exp ( — ix) + Si follows from 



»S2(0)~c(r)/2cos| 



(2.6) 



and the fact that this is small compared to the c(r)/2 sin {<p/2) in 

 (2.4) when tp is small. 



So far, we have been discussing a special case of Sommerf eld's results. 

 Xow we turn to the case of the perfectly conducting parabolic cylinder 

 shown in Fig. 2.3. Here, as in Fig. 2.1, the incident plane wave exp ( — ix) 

 comes in from the left. The fields for the two kinds of polarization are 

 gi\-en by 



(hp) 

 (vp) 



E = {e-'^ -f Si) + Sm, 

 H = (e-" + Si) + S,ih), 



(2.7) 

 (2.8) 



where, just as in the half-plane case, the fundamental vectors E and H 

 are perpendicular to the plane of Fig. 2.3 and the incident waves are of 

 unit amplitude. 



The [exp ( — ix) + Si] in (2.7) and (2.8) is exactly the same Fresnel 

 integral (2.3) as for the half-plane. S^ih) is a rather complicated integral 



INCIDENT WAVE 



Fig. 2.3 — Coordinates used in the discussion of the parabolic cylinder. The 

 coordinates such as (0, h) refer to (x, y). The origin of coordinates is at the 

 focus of the parabola and h is the height of the vertex or crest, above the origin. 



