1362 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



excite the unattenuated wave, for that wa^-e carries power. Let us assume 

 that it excites the decreasing wa^■e alone, which, when present alone, 

 carries no power. So far there is no contradiction, and we can believe that 

 it is possible to arrange matters so that the current alone does not excite 

 the increasing wave. 



Suppose we have so arranged matters that i does excite the decreasing 

 wave only. Consider what happens when / = 0. Can v alone excite the 

 decreasing wave only if i alone excites the decreasing wave onl}^? If it 

 can, then v and i together must excite the decreasing wa^'e onl3^ But 

 suppose V and i are of the same frequency and in phase. Then the beam 

 carries power. But, the decreasing wave alone cannot carry power, and 

 hence what we have assumed is impossible. If the i excites the decreasing 

 wave only, then v must excite at least a component of the growing wave. 

 Hence, we cannot cancel out the noise from the beam completely. 



FINAL COMMENTS 



We have seen that the properties of space-charge waves and the 

 behavior which must follow when space-charge waves are coupled to 

 other space-charge waves or to circuit waves can be used to explain the 

 operation of seemingly diverse types of microwave tubes. The wave 

 picture gives a clear and quantitative picture of energy relations and 

 power flow. It enables us to understand simply the effect of thermal 

 velocities on the operation of tubes through their effect on the phase 

 constants of the space-charge waves. It is useful in detailed considera- 

 tions of noise, and in one case it has enabled us to draw a general con- 

 clusion without resorting to formal mathematical manipulation. It may 

 well be that the wave picture can be of further use both in calculating 

 detailed behavior of tubes and in understanding their general properties. 



Appendix A 



SPACE-CHARGE WAVES 



Consider a narrow electron stream in which we may assume that elec- 

 tron velocity and charge density do not vary across the stream, and in 

 which the electrons are free to move in the ^-direction only. An axially 

 symmetrical electron focusing system immersed, cathode and all, in a 

 strong magnetic field approximates this. 



Let all ac quantities contain the factor 



—jSz jut 



e e 

 and let the total charge densitj^, current and electron velocity be made 



