622 BELL SYSTEM TECHNICAL JOURNAL 



instance, in the two-dimensional electrostatic field in which the potential 

 V is given by 



V = Ae'^'^e-'^" (13.28) 



dV/dy = -(3eV (13.29) 



and everywhere 



a2 = ($7^,4>)2 = 1. (13.30) 



Relation (13.30) is approximately true far from the axis in an axially sym- 

 metrical field. 



Consider a potential giving a purely transverse field at y = 



V = Ae~'^'' sinh I3ey (13.31) 



^ = l3Ae~'^'' cosh /3ey. (13.32) 



dy 



In this case, at y = 



a2 = ($7/3,$)2 = 00 (13.33) 



In the case of a purely transverse field we let 



^ - I^ (13.34) 



D' = (£j//3'P)(/o/8Ko) (13.35) 



In (13.35), Ey is the magnitude of the y component of field for a power 

 flow P, and /3 is the phase constant. 



We then redefine 8 and b in terms of D rather than C 



-r = -p, + ^,D8 (13.36) 



-ri= -j%-pM (13.37) 



and our equation for a purely transverse field becomes 



1 = (j^-bW-^p) (13.38) 



In (13.38), 5 and b are of course not the same as in (13.26) but are defined 

 by (13.36) and (13.37). 



13.4 Purely Transverse Fields 



The case of purely transverse fields is of interest chiefly because, as was 

 mentioned in ('hapter X, it has been suggested that such tubes should have 

 low noise. 



