WAVES IN ELECTRON STREAMS AND CIRCUITS 633 



Each of these four components is a solution of the diferential equation. The 

 solution of an actual physical problem will be the sum of the four compo- 

 nents, or, if we like, the real part of that sum, and the amplitude factors 

 Ai — Ai, which are in general complex, will depend on the particular 

 physical problem which is solved. 



What has been the purpose of this argument? First of all, it is intended 

 to indicate how the waves get into the picture. The differential equations 

 for a long beam of constant average velocity uq and charge density po were 

 Unearized by neglecting terms in which the products of a-c. quantities ap- 

 peared. By this means a linear partial differential equation with constant 

 coefficients which relates i and E was found. This was combined with the 

 linear partial differential equation for a uniform transmission-line circuit, 

 and an overall partial differential equation for V was obtained, linear and 

 with constant coefficients. Such an equation could be solved by any means, 

 but it is known to have wave-type solutions, and the solution of the original 

 physical problem must be a sum of all such solutions. 



In general, we will not expect so simple a relation between i and V or E 

 as (1.12), that for a simple transmission line. Further, for broad electron 

 streams the electronic behavior cannot be expressed so simply as it has been 

 in (1.8). Nonetheless, we will find wave solutions in which all quantities vary 

 with time and distance as 



as long as 



(1) the d-c. beam properties (the undisturbed electron flow) and the 

 circuit properties do not vary with z. 



(2) the signal amplitude is low enough so that terms involving products 

 of a-c. quantities can be neglected. 



When this is so, the solution of a physical problem can be expressed as 

 the sum, or the real part of the sum, of such wave solutions, taken with 

 the proper amplitudes.! 



n. The Component Waves 



Once we are convinced that the solution of our problem can be expressed 

 as the sum of a number of waves which are solutions of a linear partial dif- 

 ferential equation, it is simplest to use this fact directly in finding certain 

 properties of the waves of which the solution is to be made up. 



Let us, for instance, let E in (1.8) contain the factor 



jut —j?z 



e e 

 t An additional overall condition is that the electron flow has no velocity distribution. 



