WAVES IN ELECTRON STREAMS AND CIRCUITS 639 



K we like, we can combine (2.15) with (2.14). Doing this directly, we 

 obtain 



(^ - PoY = o^^-rKM' (^f4^ (2.16) 



Unless the electron velocity is near the wave velocity 03o near to ^i) 

 we will expect two sorts of solutions: one sort, for which /3 is near to /5o 

 corresponding to "space-charge" waves; and the other, for which ^ is near 

 to zb/?! , corresponding to "circuit" waves. If /3o is not near to j8i , we can 

 easily obtain approximate values of jS for these two t3^es of wave. 



To obtain /3 for the space-charge waves we put j8 = /3o on the right- 

 hand side of (2.16) and obtain 



^ = ft ± ^, //f^ (2-17) 



If (2.17) gives a value of jS differing by a small fraction from jSo , then (2.17) 

 is to be trusted. 



To obtain j8 for the forward circuit wave we put jS = /3i on the left of 

 (2.16) and in the numerator on the right. This gives for the forward wave 



To obtain the backward wave, we put jS = jSi on the left of (2.16) and 

 in the numerator on the right, and obtain 



Again, (2.18) and (2.19) are to be trusted as long as ^ as given by (2.18) 

 differs by a small fraction only from j8i . 



We see that according to (2.19) the space-charge waves are unattenuated 

 (real 0) for /3o < jSi , that is, for electrons traveling faster than the circuit 

 phase velocity, while there are increasing and decreasing waves for jSo > i^i , 

 that is, for electrons traveling more slowly than the circuit phase velocity. 

 We see from (2.18) and (2.19) that the circuit waves are unattenuated 

 (for lossless circuits), and travel a little more slowly than in the absence of 

 electrons. 



Further, we see that (2.17) and (2.18) are not to be trusted when 0o 

 is close to jSi , that is, when the electron velocity is near to the circuit phase 

 velocity. As a simple example, let 



/3o = iSi (2.20) 



