430 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



where the subscripts " — " and "-|-" denote opposite points on the il- 

 himinated side and shadow side, respectively, of the half-plane. These 

 relations follow from (2.20).* The radius vector r from the origin is equal 

 to —y on the half -plane. Equations (2.20) follow quite readily from (2.3). 

 The values of J and J^ for an arbitrary angle of incidence and /i = 

 are given by expressions (6.6) and (6.22), respectively. 



All of the curves in Fig. 2.4, even | ^qJ \ iov h = 1,000, eventually 

 approach the value 2 far down on the illuminated side. It may be 

 shown that | /„ | and | fo-/ | for the half-plane decrease like {irrY^'^ and 

 l/(27r^'V^'^), respectively, deep in the shadow. Hence as we go to the 

 right in Fig. 2.4, the dashed curves will eventually cross over and lie 

 above the solid curves, which decrease exponentially. The larger h, 

 the lower and flatter is the curve for | ^qJ \ . 



Now we turn to the diffraction field at points far to the right of the 

 cylinder. When 1 1/' ] « 1, so that we are not too far from the shadow 

 boundary, and h is large, the field is given by (2.10), or by its analogue 

 for vertical polarization. In order to get acquainted with (2.10) we first 

 examine the field when | ;/' | « 1 but \p''p ^ 1. 



When we are so far behind the cylinder that i/'"p ^ 1 even though 

 I i/' I <3C 1, the asymptotic expressions (2.4) show that 



c{p)/\l/, \p negative (shadow) 



[e-^^ + S,l 



Substitution of (2.21) in (2.10) gives 



+ — — , yp positive 



(2.21) 



E 



cip)h"' U(t) + ^) exp (ir'/S), lA < 



c{p)h"' ( ^(r) + - ) exp (irVS) + e-'\ ^ > 



(2.22) 



The presence of c{p) shows that the total wave may be regarded as the 

 sum of the incident wave and a wave spreading out from the crest. The 

 crest wave is the analogue of the edge wave, c{r)/(p, for the half-plane. 

 In fact, when we are in the region where the l/r in (2.22) is the most 

 important term, the crest wave is approximately 



c{p)/^ (2.23) 



* They also follow from superposition and consideration of the symmetry of 

 the currents produced on the half -plane ?/ < 0, x = when —/I is impressed. Here 

 A denotes the system of currents which flows in the upper half-plane ?/> 0, x = 

 when the incident wave falls on a complete plane at x = 0. 



