1348 THE BELL SYSTEM TECHXICAL JOURNAL, NOVEMBER 1954 



velocity and the ac velocity are larger than terms involving the square 

 of the ac velocity. The product terms may be negative or positive. 



We can understand the negative energy of the slow wave qualitatively 

 through a simple argument of a somewhat different sort. In the slow 

 wave, the charge density is greatest in regions of less-than-average 

 velocity and least in regions of more-than-average velocity, so that the 

 electron beam has less total kinetic energy in the presence of the slow 

 wave than it does in the absence of the slow wave. How does this come 

 to be? Suppose that we move with the wave; we then see electrons 

 moving in an electric field which is constant wdth time, and hence, as 

 electrons move through the field their velocities vary as the square root 

 of the potential. Relative to the Avave, the electrons move slowest in the 

 low-potential regions, and correspondingly, they are bunched together 

 in regions of low potential. Now, for the slow wave the total electron 

 velocity is the arithmetic sum of the wave velocity and the electron 

 velocity relative to the wave, so if the electrons are bunched in regions 

 of lowest velocity relative to the wave they are necessarily bunched in 

 the regions of least total electron velocity, and the kinetic energy of the 

 slow wave is thus negative. 



In the case of the fast wave, the electrons travel backward relative to 

 the wave. The total electron velocity is the arithmetic difference between 

 the wave velocity and the electron velocity relative to the wave. Hence, 

 the total electron velocity is greatest at the bunches, where the velocity 

 relative to the wave is least, and the kinetic energy of the fast wave is 

 positive. 



THE KLYSTRON 



We can explain the operation of a number of types of vacuum tubes in 

 terms of space-charge waves. Consider the klystron, illustrated in Fig. 2. 

 The voltage produced across the input resonator by the input signal 

 sets up on the electron beam both the slow and the fast space-charge 

 waves in equal magnitudes and so phased that the velocities v, or the 

 voltages U add, while the currents cancel. Thus, just beyond the input 

 resonator, the beam has an ac velocity; it is velocity modulated, but it 

 has no ac convection current. 



Because the two space-charge waves, one mth negative power flow and 

 one with positive power flow, are set up with equal magnitudes, the ac 

 power flow in the beam between the input and the output resonators is 

 zero. The input resonator neither adds power to nor subtracts power from 

 the beam. 



Because the two wa\'es have different phase velocities, their relative 

 phase changes as they travel along the beam. If we go along the beam a 



