738 



BELL SYSTEM TECHNICAL JOURNAL 



If is found that this solution obtains for large and small values of b. For 

 very large and very small values of b, either y = — ^ (51) or y = ± / (52) 

 The wave given by (50) is a circuit wave; that given by (51) represents the 

 travel down the tube of electrons oscillating in the magnetic field with cyclo- 

 tron frequency. 



Fig. 1 — Plot of parameters giving velocity and attenuation of the three forward waves 

 vs. a parameter b proportional to electron speed with respect to the undisturbed wave. 

 A positive value of x means an increasing wave; a positive value of y means a wave travel- 

 ing faster than the electrons. This plot is iorp = (no magnetic field). 



In an intermediate range of b, we have from (47a) 

 x= ±V2y(y + b) - iP - f) 



and 



b = -2y± VP - l/2y. 



(53) 



(54) 



For a given value of P we can assume values of y and obtain values of b. 

 Then, x can be obtained from (52) or (53). In Figs. 1-6, x and y are plotted 

 vs. b for/2 _ Q^ 5^ 1, 2, 4 and 10. It should be noted that xi , the parameter 

 expressing the rate of increase of the increasing wave, has a maximum at 

 larger values of b as/ is increased (as the magnetic focusing field is increased). 

 Thus, for higher magnetic focusing fields the electrons must be shot into the 



