650 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



direction of electron flow. We might have deduced this from the fact that 

 the phase velocity for the wave is greater than the electron velocity (see 

 (2.9)). 



While the wave which increases in the direction contrary to electron 

 flow has its power flow in the direction of increasing amplitude, it is a back- 

 ward wave and hence not suitable for producing gain. 



The disturbance on the non-propagating ladder is closely related to a 

 passive or cut-off mode of a waveguide excited at a frequency less than the 

 cutoff frequency for the mode in question. In this case, the analogue of the 

 circuit power VI* is the integral of the Poynting vector over the guide 

 cross section. When electrons flow through a waveguide these cut-off modes 

 are perturbed much as indicated by (4.19) and (4.12). Because the per- 

 turbed modes have a "wavelike" character in that the propagation con- 

 stant is no longer purely real, and because the amplitude may increase in 

 the direction of electromagnetic power flow, some workers have proposed 

 to obtain gain from these "growing waves."^ 



V. Further Considerations Concerning Boundary Conditions 



How necessary is it to fit boundary conditions in order to deduce what 

 will happen? The suspect waves we have examined so far might be rejected 

 as increasing in a direction contrary to the direction of electron flow,* or 

 as having electromagnetic power flow in a direction opposite to the direc- 

 tion of growth. Can we find some method for separating waves useful in 

 producing gain from waves which are not, without consideration, explicit 

 or implicit, of boundary conditions? 



Let us consider the problem of fitting boundary conditions for a circuit 

 plus an electron stream. Imagine that the end of the circuit near to the 

 electron source ("near" end) is connected to a load impedance Z and that 

 the end away from the electron source ("far" end) is driven by a voltage V. 

 Let the wave which increases most rapidly in the direction of electron 

 flow vary in amplitude as exp (az). Suppose that the length of the circuit 

 is great, so that aL is a large number and exp (aL) is a very large number. 



At the near end the various wave components must be so related that i 

 and V are zero and that the circuit voltage is ZI. At the near end, all four 

 waves must be used to fit the boundary conditions at a given voltage level. 

 Disallowing very special values of Z, we would expect that at the near end 

 the four waves will have comparable amplitudes (the amplitudes are re- 

 lated by linear simultaneous equations). Thus, at the far end of the circuit, 

 the wave which increases most rapidly with distance should strongly pre- 

 dominate. It seems that the most rapidly increasing wave is naturally con- 

 nected with excitation of the circuit at the far end. 



•Though rejected only through considering boundary conditions. 



