TRAVELING-WAVE TUBES 13 



The quantities v, p, and i are assumed to vary with time and distance as 

 exp(;c<j/ — Tz). 



One equation we have concerning the motion of the electrons is that the 

 time rate of change of velocity is equal to the charge- to-mass ratio times 

 the electric gradient. 



d(uo + v) dV 



dt ^ dz 



(2.11) 



In (2.11) the derivative represents the change of velocity observed in fol- 

 lowing an individual electron. There is, of course, no change in the average 

 velocity uo . The change in the a-c component of velocity may be expressed 



dv . . . . 



in terms of partial derivatives, — , which is the rate of change with time of 



dt 



dv . . . . 



the velocity of electrons passing a given point, and — , which is variation of 



dz 



electron velocity with distance at a fixed time. 



dv dv , dv dz dV ,^ . -X 



— - = — + — =17 — l^-l-^j 



dt dt dz dt dz 



Equation (2.12) may be rewritten 



^, + — (mo + t') = fl -^ (2.13) 



dt dz dz 



Now it will be assumed that the a-c velocity v is very small compared with 

 the average velocity Mq, and v will be neglected in the parentheses. The reason 

 for doing this is to obtain differential equations which are linear, that is, 

 in which products of a-c terms do not appear. Such linear equations neces- 

 sarily give a wave type of variation with time and distance, such as we 

 have assumed. The justification for neglecting products of a-c terms is that 

 we are interested in the behavior of traveling-wave tubes at small signal 

 levels, and that it is very difficult to handle the non-linear equations. When 

 we have linearized (2.13) we may replace the difi'erentiation with a respect 

 to time by multiplication byj'co and difi'erentiation with respect to distance 

 by multiplication by — F and obtain 



(yw - Mor)i' - -r]VV (2.14) 



We can solve (2.14) for the a-c velocity and obtain 



. = -^^^ , (2.15) 



<j^e - r) 



[where 



0, = oi/uo (2.16) 



