WAVE PICTURE OF MICROWAVE TUBES 1345 



across the beam, or we can deal with peak or effective vakies much as in 

 the case of voltages and currents in waveguides. 

 These ac quantities are assumed to contain a factor 



That is, they vary sinusoidally with time and with distance, and (as- 

 suming /3 to be positive) propagate in the -\-z direction. The phase con- 

 stants |S of the two waves \\all be called /3i and /So . For beams of mod- 

 erate charge they are very nearly 



CO . Wn 



iSi = - + -^ 



Bo = — - ^ 



Here iio is the electron speed, w is the operating radian frec^uency and w^ 

 is the effective plasma radian frequency. 



The plasma frequency of the electron beam cop is given by 



e 



-po 

 2 m 



Up = 



e 



Here e/m is the charge-to-mass ratio of the electron, po is the charge 

 density and e is the dielectric constant of vacuum. In terms of Wp , Wg 

 may be expressed 



COg = R(j)p 



Here R is a, factor somewhat less than unity which depends on the 

 geometry of the electron beam, on co and cop , and on the velocity distribu- 

 tion of the electrons (see Appendices A and C). 



Let us consider the simple case in which R is unity and the effective 

 plasma frequency is equal to the plasma frequency. The phase velocities 

 Vi and V2 of the two waves, which are co divided by /3, are 



Vi 



V2 = 



Uo 



1 + "^ 



CO 



Uo 



1 



0}p 

 CO 



Thus, the first wave has a phase velocity less than that of the electrons; 

 it is a slow wave, and the second wave is a fast wave. 



