872 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



The gain constants in (29) are independent of frequency. The net gain 

 per ripple wavelength, however, will vary with frequency, depending on 

 how closely both the current maxima and minima coincide with the 

 regions of maximum ± (vr/r), respectively. This is a statement of the 

 "resonance" condition between ripple wavelength and half the space- 

 charge wavelength, which emerges from one-dimensional analyses^" of 

 this gain mechanism based on transmission-line analogies. 



Such analj^ses generally assume small-amplitude sinusoidal variations 

 of the reduced plasma wave number, jS, , along a one-dimensional beam 

 in a longitudinal ac field with no losses. Periodic variations in either beam 

 or wall diameters, or beam velocity, cause the beam "impedance" to 

 vary periodicallj^ imparting to it narrow-band filter-like properties 

 equivalent to narrow-band signal gain. From another point of view, 

 these periodic impedance changes couple the fast and slow space-charge 

 waves to each other intermittently, thereby effecting an energj^ trans- 

 fer from the fast to the slow wave. As this coupling is lossless, Is increases 

 and // decreases with drift distance, in such a way as to keep their product 

 constant. Then the product /max/min increases, and the ratio /max/min 

 correspondingly decreases. In the case of noise-power amplification, two 

 uncorrelated space-charge standing waves are present. Because the two 

 slow waves cannot simultaneously be amplified at the expense of the two 

 fast waves, the product /max/min must remain constant. 



The observed noise-current patterns in rippled-beam amplification, 

 however, are characterized b}- a nearly constant ratio Imax/Imin , and an 

 increase in the product /max/min along the beam, despite the fact that 

 the beam voltage is fixed. This apparent contradiction can be resolved 

 by a closer look at the energy-exchange processes. 



Chu'^ has shown that the kinetic power flow in space-charge waves 

 (the major part of the total power) is equal to the difference in powers 

 carried by the fast and slow waves. This is equally true of beams with 

 transverse motions and fields. In rippled-beam amplification, whether 

 analyzed as a modulated linear beam or at each beam cross-section sepa- 

 rately, as here, the propagation constants are found to be complex con- 

 jugate quantities, whose real parts describe the ordinary fast and slow 

 waves of a uniform beam. From either point of view, therefore, a de- 

 crease in // and an increase in /« signifies an increase in the negative 

 kinetic power flow carried by the waves, or a decrease in the total kinetic 

 energy of the beam. 



As shown in (29), the gain constants are proportional to tv , indicating 

 that the dc energy transferred to the waves when the beam contracts 

 could only have come from the radial kinetic energy, not the longitudinal. 



