870 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



above expression to obtain the characteristic equation : 



2 



1 "'P 



2 ' ~ 2 2 



_ (jJb — COe 



2 ' ' 



2 

 C06'' 



(39) 



In a frame of reference moving with the stream, u^ is 0, ^'u' is 0, and 

 03b = w. Then, 



^/\2 / 2 2 2\ 2 



and iS' becomes an infinite imaginary quantity when co is Wc . The phase 

 velocity in the moving frame is infinite, as the real part of (3' is zero; 

 therefore the phase velocity Vp in the rest frame is also infinite. Thus, 

 there is no Doppler shift in the "resonant" frequency observed in the 

 rest frame: 



_ ^g _ 



'^'observed — — ^c • / A1\ 



Vp 



As the actual beam has a 2- velocity spread, the field is never perfectly 

 uniform, and as the calculation is valid for small ac quantities only, the 

 discrepancy between this result and the observed "proper" frequencies, 

 which were 1.03 to 1.06 times the cyclotron value, is not unexpected. 



The singularity in (40) is seen to disappear when Wp = 0. This indi- 

 cates that an exact calculation would show the gain constant (— j/S) 

 to increase with po , the density of the crossover electrons. Their tra- 

 jectories, described by (31), and the absence of space-charge forces are 

 such that K = rmax^min ; that is, the greater r^ax , the smaller rmin , 

 the distance of closest approach to the axis. Thus, a larger ripple ampli- 

 tude (permitted by a lower magnetic field) produces a greater electron 

 density in the waist region, and accordingly a greater oscillation ampli- 

 tude at the resonant frequency, as observed. 



The foregoing mathematics describes a form of resonance, the infinite 

 phase velocity corresponding to longitudinal "cutoff" in a waveguide. 

 Unlike a waveguide, however, the disturbance increases rather than 

 attenuates along the axis, due to the transfer of dc kinetic energy (repre- 

 sented by ve^/r) to the ac fields (excited by noise fluctuations at the 

 cathode), at the cyclotron frequency Wc . 



Except for the direction of energy transfer, the situation is analogous 

 to that of a low-pressure gas in a uniform magnetic field, when stressed 

 by an impressed ac field of varying frequency. It has been found that 

 the breakdown field at the cyclotron frequency is very much less than 



