114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 



v2 



m 



Fig. 2 

 This relation defines e. Equation (1.27) becomes 



2 2 

 0}J 



1 - e^ 



(1 + e^)^ ^'-''^ 



In Fig. 2, e is plotted versus the parameter y^Ui/oip^. We see that as the 

 parameter falls below unity, e increases, at first rapidly, and then more 

 slowly, reaching a value of ±1 as the parameter goes to zero (as cop' 

 goes to infinity, for instance). 



It will be shown in Section 2 of this paper that these results for infinite 

 flow are in some degree an approximation to the results for flow in narrow 

 beams. It is therefore of interest to see what results they yield if applied 

 to a beam of finite width. 



If the beam has a length L, the voltage gain is 



The gain G in db is 



G = 8.7 '^ € db 



Wo 



(1.30) 



(1.31) 



