470 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



e = 7r/2. Also, from (5.3) and (5.G) 



•'-" (8.4) 



= (2ry" sin iW2). 



The subscript r is used to denote correspondence to a half-plane with 

 its edge at r = 0. 



When h is large, physical reasons lead us to expect a similarity be- 

 tween our field and the one behind a half-plane with its edge at the 

 crest of the cylinder where p = (see Fig. 2.3). The main part of this 

 field is the analogue of (8.4): 



(e-« + s,), = {i/nye-'^ r e~''' di, T, = (2p)^'^ sin (^/2), (8.5) 



J— 00 



/here the subscript p indicates that [exp ( — ix) + S^p corresponds to 

 Jiffraction behind the half-plane just mentioned. 



In order to make use of the similarity between the field behind the 

 cylinder and the half-plane with its edge at the crest of the cylinder, 

 we change the polar coordinates from (r, (p) to (p, \}/). From 



pe'^ = re'^ - ih (8.6) 



it may be shown that, when h Ir is small, 



(e-^'^ + Si). - (^""''^ + *Si)p = {il-K^-e-'^ r e-'^' dl 



(8.7) 



where M\ is obtained by putting d = 7r/2 in (7.4). 



When we combine (8.7) and the expression (7.19) for *S2i the 

 2iMi/(^ — 1) terms cancel leaving 



/ —ix 1 t< \ 1 o / — '-f 1 O N I ^1-1*1 1 ih sin w i .1*12 



(e + Si)r + *S2i = (e + Si)p 4- e "^ + i — 



/3 - 1 9 (8.8) 



+ 0(h'//'') -t- 0[Mi exp (-2Th)]. 



The sum of the terms involving Mi and Ah may be expressed in a 

 form which contains the expression c{r) defined by (2.5) and the quan- 

 tity b defined by 



