WAVES IN ELECTRON STREAMS AND CIRCUITS 627 



to be approximations to those which would be obtained for more realistic 

 but mathematically more refractory situations. 



Other misinterpretations have arisen through combining non-relativistic 

 equations of motion with Maxwell's equations and then attaching signifi- 

 cance to terms of the order {v/cYJ 



Finally, granting that all else is well, it is unsafe to draw conclusions 

 from the examination of particular solutions of differential equations. In a 

 very simple example, it is impossible to determine the gain of an amplifier 

 tube which uses an electron stream simply by examining various "waves" 

 which can travel on the stream. In solving a physical problem, one must 

 not only solve the differential equation involved but he must satisfy the 

 appropriate boundary conditions as well. 



In all, such confusion as there has been concerning waves in clouds of 

 electrons and ions seems to have arisen not through lack of mathematical 

 ambitiousness but rather through simple errors in physical interpretation. 



The following material concerns itself with some particular types of 

 "waves" and with the importance and consequences of fitting boundary 

 conditions. The work treats a very easy case, simplified and abstracted 

 from a physically realizable system. The case was made so simple in order 

 to avoid painful mathematics which might obscure the actual points to be 

 made. The purpose is to explore this simple case thoroughly, avoiding basic 

 misunderstandings. If it is objected that matters so simple should not be 

 treated at such length, because no one could misunderstand them anyway, 

 I can only reply that I did misunderstand some of the matters recounted 

 herein. 



I. Why Are Waves Introduced? 



We will consider the case of a narrow or thin beam of electrons across 

 which we can assume that the electric field is constant. f In our calculations 

 we assume that all electrons in a given very small region have the same 

 velocity, thus neglecting the thermal velocity distribution. J We assume 

 that the flow is a smoothed-out jelly of charge, J with the charge per unit 

 mass characteristic of electrons; thus, we neglect individual interactions 

 between electrons, and consider only a sort of average effect. 



We will write the quantities involved in the following forms 



velocity = v -\- Uq 



'L. R. Walker, "Note on Wave Amplification by Interaction with a Stream of Elec- 

 trons," Phys. Rev., Vol. 76, pp. 1721-1722 (1949). 



t This is in itself a drastic abstraction. No attempt wiU be made to justify it here, 

 beyond saying that it is useful in considering the problems that follow. 



X Other drastic approximations for which no justification will be given. 



