62S 



BELL SYSTEM TECHNICAL JOURNAL 



Consideration of both terms presents considerable difficulty as (13.26) 

 leads to fi\^ waves (5 values of 5) instead of three. The writer has attacked 

 the problem only for the special case of 6 = 0. In this case we obtain from 

 (13.26) 



"1 



5 = -j 



52 '^ 52 



^1 



(13.47) 



MacColl has shown^ that the two "new" waves (waves introduced when 

 a = 0) are unattenuated and thus unimportant and uninteresting (unless, 

 as an off-chance, they have some drastic effect in fitting the boundary 

 conditions). 



Proceeding from this information, we will find the change in b as P is 

 increased from zero. From (13.47) we obtain 



db =j 



'2d8 2a 8d8 adf 



Now, if/ = 



If we use this in connection with (13.48) we obtain 



db= -i,df 

 60 



For an increasing wave 



5i = (1 + ($7/3.$)M\/3/2 - j/2) 

 Hence, for the increasing wave 



3(1 + a-) 



(13.48) 

 (13.49) 



(13.50) 

 (13.51) 



(13.52) 



This shows that applying a small magnetic field tends to decrease the gain. 

 This does not mean, however, that the gain with a longitudinal and trans- 

 verse field and a magnetic field is less than the gain with the longitudinal 

 field alone. To see this we assume that not/^ but {^'/(3e^y is small. Differen- 

 tiating, we obtain 



2 



db = -j 



2db 



2a bdb . da' 



(52 -\- py "^ 52 + /2J 



If a = 



53 =. _j 



(13.53) 



(13.54) 



* J. R. Pierce, "Transverse Fields in Traveling- Wave Tubes," Bdl System Technical 

 Journal, Vol. 27, pp. 732-746. 



