634 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



In Other words, let E in (1.8) be one of the wave components of a solution. 

 Then (1.8) becomes 



(— ca^ + 2«oco/3 — ul^^)i = — Jw — poJS 



m 





(2.1) 



Here c, the dielectric constant of vacuum, has been introduced for reasons 

 which will become apparent later. It is further of interest to introduce other 

 simple parameters. 



- - - = 0)^ (2.2) 



m € 





= /3p (2.3) 



= iSo (2.4) 



6) 



The quantity cop is called the plasma frequency (a radian frequency), cop 

 is positive because po is negative. jSo would be the phase constant of a wave 

 travehng with the electron velocity. While /3p would be the phase constant 

 of a wave traveling with a phase velocity equal to the electron velocity, 

 and having a frequency cop , we may merely regard jSp as a convenient 

 parameter which increases as the beam current is increased. In terms of 

 /3p and j8o 



-■ - ~^' ^, O^) (2.5) 



(^0 - ^) 



This may seem a strange form in which to write the equation. It will 

 perhaps seem less strange, however, if we recall that the current density / 

 in a dielectric medium is given by 



/ = icoeE 



Thus, we see that for real values of jS the electron convection current den- 

 sity i is that which would correspond to a negative dielectric constant or a 

 negative capacitance. Its magnitude depends on /3p , which is proportional 

 to the d-c. beam current density; and the magnitude becomes very large 

 when the phase velocity of the wave approaches the velocity of the elec- 

 trons, that is when j8 approaches jSo . 



