978 



THE BELL SYSTEM TECHXICAL JOURNAL, JULY 1954 



The characteristic equation is 



/i(u/2) 61 lUun) 



Io(u/2) "^ Co Ko(u/2) 



hiu/2) 

 h{u/2) 



I /S I fo = -u/2 



- AZ^{u) 



(37) 



and can now be discussed in terms of o- at a fixed p. Suppose that p < 1. 

 Then A is negative for o- < 1 — p- In the same range, x < M, so that 

 Z behaves essentially as Ko/Ki . Therefore both numerator and denomi- 

 nator are positive for all u, and the ratio tends to the planar result 



1 + 



€0 



1 -A 



asw-^ oo.Asw— >0, ^ tends to zero along the vertical, as can be shown 

 by an examination of the various functions near u = 0. For a < 1 — p, 

 the course of the 0^ versus ^/-curves is as shown schematically in Fig. 

 16(a), and it is easily seen that the ^ versus fo curves run in essentially 

 the same way, Fig. 12(a) to (c). However, as a approaches 1 — p, the 

 ^ versus u curves steadily fall, until at a- = 1 — p, j8^ = for all finite 

 u, since A = — oo . 



As 0- passes 1 — p, A changes sign and at the same time x passes 3^ 

 so that Zx acquires a zero. As a varies from 1 — p to 1 — p/2, A decreases 

 from + oo to unity. Therefore, while u < Ui , the zero of Z^ , 1 — AZ^ 

 is positive; however as u increases beyond Ui , {Iq{ii/2)/Ii{u/2)] — 

 A2iy{u) eventually passes zero, since ZJ^u) and 7o(w/2)//i(w/2) both 

 approach unity. On the other hand the numerator of equation (37) is 



ZERO OF D = 



t 



X<4r 



(a) (b) (c) 



Fig. 16 — Schematic variation of /3_ with m. a) x < K; b) ]4. < x < H',c) ^ 

 < X < %. 



