400 BELL SYSTEM TECHNICAL JOURNAL 



The meaning of this will be clear when we remember that, in the pres- 

 ence of electrons, quantities vary with z as (from (7.10)) 



-;/3e(l+;C5)z 



If V is the phase velocity in the presence of electrons, we have 



coA = (a,/«o)(l - Cy) (8.7) 



If Cy « 1, very nearly 



V = Mo(l + Cy) (8.8) 



In other words, if y > 0, the wave travels faster than the electrons; if 

 y < the wave travels more slowly than the electrons. 



From (8.6) we see that, if x > 0, the wave increases as it travels and if 

 .T < the wave decreases as it travels. In Chapter II we expressed the 

 gain of the increasing wave as 



BCN dh 

 where N is the number of wavelengths. We see that 



B = 2()(2x)(logioe)x 

 B = 54.5x- 

 In terms of x and y, (8.1) becomes 



(^2 _ y2^(y ^ b) -{- 2x^y +1 = (8.10) 



xix^ - Sy'^ - 2yb) = (8.11) 



We see that (8.11) yields two kinds of roots: those corresponding to 

 unattenuated waves, for which x = and those for which 



x'' = 3y2 + 2yb (8.12) 



li X = 0, from (8.10) 



f(y + 6) = 1 



(8.13) 

 6 = -y + l/y^ 



If we assume values of y ranging from perhaps -|-4 to —4 we can find the 

 corresponding values of b from (8.13), and plot out y vs b for these unattenu- 

 ated waves. 



For the other waves, we substitute (8.12) into (8.10) and obtain 



2yb^ + Sy^b + 8/ + 1 = (8.14) 



(8.9) 



