AVAVE PICTURE OF MICROWAVE TUBES 



1357 



Fig. 11 — Because of coupling of the space-charge waves to the forward circuit 

 wave, gain is produced near /3c = j3, , while the curves sheer off from one another 

 near jic = 13/ . 



space-charge wave and the circuit wave we can have a stop band, while 

 for coupHng between the circuit wave and the fast space-charge wave we 

 cannot. 



The consequences of the couplings between the circuit wave and the 

 space-charge waves near the intersections of /3c with jSs and (3/ are il- 

 lustrated in Fig. 11. 



We see from Fig. 11 that near synchronism between the circuit wave 

 and the fast space-charge wave ((3c = /3/ for no coupling) these waves 

 combine so that for any given value of u/uo there are always two dis- 

 tinct real values of /3. This is typical for coupling between modes with 

 power flows of the same sign. At /3c = /3/ each of the two mixed waves has 

 equal energies in the circuit and in the electron stream. 



Near synchronism between the circuit wave and the slow space-charge 

 wave (/3c = /3s in absence of couplmg) these two waves combine so that 

 over a range of oj/uq near /3c = /3s , /3 has two complex values with the 

 same real part and with equal and opposite imaginary parts. We can 

 write this as 



/3 = /3i ±ja; -jfi = -j/3i + a 



This corresponds to an attenuated and a groAAing wave with the same 

 phase velocity. In Fig. 11, i3i and a are plotted as dashed lines. 



