1368 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



mean square deviation {u ) about the mean value Wo . If cor/o is the effective 

 plasma frequency for {u") = 0, then taking the velocity spread into 

 account, the effective plasma frequency co, is given approximately by 



OiQ — 03,0 I i -r O TT 



W50/ / 



If the velocity distribution is the same as for electrons accelerated 

 individually by a voltage Vo , that is, if electron interactions do not 

 affect the velocity distribution appreciably (as they probably do not) 



{u') _ 1 /A-r.V _ 1 



ul 4:\eVo/ 4\ll,600Fo 



Here k is Boltzman's constant and 7\ is cathode temperature. Thus, 

 from this assumption 



Following our wave picture, we can take into account the thermal 

 velocity spread by using this corrected value for the effective plasma 

 frequency in all our formulae. For all practical purposes, the change in 

 effective plasma frequency due to thermal velocities is negligible. 



In a paper which wiW appear in the Journal of Applied Physics, D. A. 

 Watkins has used a somewhat different approach in treating the effect of 

 thermal velocities on the operation of traveling- wave tubes. 



Appendix D 



phase and attenuation curves for coupled modes 



When two unattenuated modes of propagation are coupled together 

 periodically in a lossless manner, they combine to form two new modes. 

 For each of these new modes the amplitude is changed in one period of 

 the coupling structure by a factor 



where M is a root of 



M' - 2VTWT^ cos (^^ - ^+0y- ^3) ^1 = (D2) 



Here A- is a coupling coefficient which is zero for zero coupling. The 

 upper sign applies if the power flow in the two modes have the same 



