GROWING WAVES DUE TO TRANSVERSE VELOCITIES 



113 



(u;-/3Uo 



Fig. 1 



Suppose we plot the left-hand and the right-hand sides of (1.26) versus 

 (co — ^Uo)- The general appearance of the left-hand and right-hand sides 

 of (1.26) is indicated in Fig. 1 for the case of two velocities Un . There 

 will always be two unattenuated waves at values of (w — /3wo) > y Ug 

 where Ue is the extreme value of lu; these correspond to intersections 3 

 and 3' in Fig. 2. The other waves, two per value of Un , may be unat- 

 tenuated or a pair of increasing and decreasing waves, depending on the 

 values of the parameters. If 



CO 



pn 



-yhir? 



> 1 



there will be at least one pair of increasing and decreasing waves. 



It is not clear what will happen for a Maxwellian distribution of veloci- 

 ties. However, we must remember that various aberrations might give a 

 very different, strongly peaked velocity distribution. 



Let us consider the amount of gain in the case of one pair of transverse 

 velocities, ±i/i . The equation is now 



2 2 

 7 Ui 



C0„2 



[ 



1 + 



CO — |3wo 



)•] 



[ ■ - (^OI 



(1.27) 



Let 



/5 = ^+i^ 



Wo Wo 



(1.28) 



