WAVES IN ELECTRON STREAMS AND CIRCUITS 651 



On the other hand, assume that the far end of the circuit is terminated 

 in some impedance Z. Consider the case in which the electron stream is 

 velocity modulated at the source end and no exciting voltage is applied 

 at the far end. We would expect that the required boundary conditions at 

 the source end could be satisfied by using the waves excepting the one which 

 increases most rapidly with distance. At the far end, to make F = /Z it is 

 necessary to add a component of the wave which increases most rapidly with 

 distance, a component of magnitude comparable to the sum of other com- 

 ponents present at the jar end. However, this added component is so small 

 at the near end that there it can be disregarded. Thus, the manifestation 

 of large forward gain comes not from the mere presence of a wave which 

 increases in the forward direction, but from special properties of the waves 

 and/or the terminating impedances which can be determined with cer- 

 tainty only by fitting boundary conditions. 



Are not these arguments at variance with the usual analyses of opera- 

 tion of the traveling-wave tube? Suppose, for instance, that the helix is 

 terminated in an arbitrary impedance at the input (near) end and that a 

 voltage V is applied at the output (far) end. What wave will predominate? 

 For a lossless helix, the true answer is that the increasing (forward) wave, 

 not the unattenuated backward wave, will predominate. This can be 

 avoided only by (1) choosing a particular (matched) value of source im- 

 pedance or (2) making the helix lossy enough so that the backward wave 

 "increases" more rapidly in the -|-z direction than any forward wave does. 

 In tubes with a uniform loss along the helix, expedient (2) is adopted; 

 when a center lossy section is used, both (1) and (2) are invoked, (1) in 

 the output section and (2) in the center lossy section. 



It is dangerous to consider the solutions of the linear differential equa- 

 tions of a physical system singly rather than in the combination which 

 satisfies the boundary conditions. This sort of reasoning might lead one to 

 believe that the problem of obtaining high voltages can be solved by find- 

 ing a solution of Laplace's equation (say V = 1/r) for which the potential 

 goes to infinity at some point. 



Cautions against neglecting the problem of boundary conditions apply 

 equally well to problems of instability (increase of disturbances with time) 

 as to problems of amplification. Thus, electron flow may be unstable when 

 none of the waves grows with time for real values of ^. On the other hand, 

 in criticizing the work of Bohm and Gross,^ R. Q. Twiss has shown^ that 

 electron flow is not necessarily unstable merely because some of the waves 

 grow exponentially with time for real values of /3. 



5 R. Q. Twiss, "On the Theory of Plasma Oscillations" Services Electronics Research 

 Laboratory, Extracts from Quarterly Report No. 20, Oct. 1950, pp. 14-28. 



