WAVES IN ELECTRON STREAMS AND ClRCmTS 641 



III. Fitting Boundary Conditions; Gain 



So far the discussion has been concerned with a differential equation and 

 wave-type solutions of it. Let us now consider an overall problem. Suppose 

 that we inject an unmodulated electron stream into a circuit of some finite 

 length and apply a signal to the end of the circuit nearest the source of 

 ele.nrons. Suppose that we adjust the output termination so that there is 

 no backward wave.* How will the field strength vary along the circuit? 

 To answer this question, we must find out what combination in phase and 

 amplitude of the three forward waves corresponds to these conditions. 

 In terms of solving differential equations, we must fit the boundary con- 

 ditions. 



From Section I we have 



(1.2) 



or 



with which we couple 



In terms of 



these relations become 



(3.1) 



(2.5) 



(3.2) 



(3.3) 



These relations hold for each of the waves separately. Now, let us denote 

 by El , Ezj Ez the fields of the three waves, and by E the actual field on the 

 circuit. Then at the beginning of the circuit, where £ is £o , the applied 

 field, the ampUtudes £io , £20 , £30 of £1 , £2 and £3 must satisfy 



£10 + £20 + £30 = £0 (3.4) 



* This is a very special case, requiring a unique impedance terminating the 4-z end of 

 the output circuit. See Section V. 



