THE NATURE OF THE WAVES 



401 



This equation is a quadratic in b, and, by assigning various values of y, 

 we can solve for b. We can then obtain x from (8.12). 



In this fashion we can construct curves of x and y vs b. Such curves are 

 shown in Fig. 8.1. 



VVe see that for 



b < i3/2){2y" 



there are two waves for which ;v ^ and one unattenuated wave. The in- 

 creasing and decreasing waves (.r 5^ 0) have equal and opposite values of 

 X, and since for them y < 1, they travel more slowly than the electrons, 

 even when the electrons travel more slowly than the imdisturbed wave. It can be 



Fig. 8.1 — The three waves vary with distance as exp (— J/3e + j0eCy + ^tCx)z. Here 

 the x's and y's for the three waves are shown vs the velocity parameter b for no attenua- 

 tion {d = 0) and no space charge {QC = 0). 



shown that the electrons must travel faster than the increasing wave in 

 order to give energy to it. 



For b > (3/2) (2) , there are 3 unattenuated waves: two travel faster 

 than the electrons and one more slowly. 



For large positive or negative values of b, two waves have nearly the 

 electron speed (| y \ small) and one wave travels with the speed of the un- 

 disturbed wave. We measure velocity with respect to electron velocity. 

 Thus, if we assigned a parameter y to describe the velocity of the undis- 

 turbed wave relative to the electron velocity, it would vary as the 45° 

 hne in Fig. 8.1. 



The data expressed in Fig. 8.1 give the variation of gain per wavelength 

 of the undisturbed wave with electron velocity, and are also useful in fitting 



