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BELL SYSTEM TECHNICAL JOURNAL 



It will also be convenient to define a sort of mean velocity Wo 



uo = (7) 



th + ^2 



We may also let Vo be the potential drop specifying a velocity mq , so that 



tio = V2vVo (8) 



It is further convenient to define a phase constant based on uq 



CO 



/3o = - 

 We see from (6), (7) and (9) that 



/3i = ^o(l - b/2) 

 /32 = /3o(l + b/2) 

 We shall treat only a special case, that in which 



Ji Ji 



Jo 



3 

 Ui 



3 — 3 



W2 Wo 



(9) 



(10) 

 (11) 



(12) 



Here Jo is a sort of mean current which, together with Uo , specifies the 

 ratios Ji/ui and /2/W2 , which appear in (4) and (5). 



In terms of these new quantities, the expression for the total a-c. charge 

 density p is, from (4) and (5) and (8) 



p = Pi + p2 



/or 



2uo Vo 



.W'-l)-^I^W'+0-^L 



(13) 



Equation (13) is a ballistical equation telling what charge density p is 

 produced when the flow is bunched by a voltage V. To solve our problem, 

 that is, to solve for the phase constant /3, we must associate (13) with a 

 circuit equation which tells us what voltage V the charge density produces. 

 We assume that the electron flow takes place in a tube too narrow to propa- 

 gate a wave of the frequency considered. Further, we assume that the 

 wave velocity is much smaller than the velocity of light. Under these 

 circumstances the circuit problem is essentially an electrostatic problem. 

 The a-c. voltage will be of the same sign as, and in phase with, the a-c. 

 charge density p. In other words, the "circuit efi'ect" is purely capacitive. 



Let us assume at first that the electron stream is very narrow compared 

 with the tube through which it flows, so that V may be assumed to be con- 

 stant over its cross section. We can easily obtain the relation between 



