FIELD SOLUTIONS 643 



These backward-traveling waves cannot give gain in the +2 direction, and 

 could give gain in the —z direction only under conditions similar to those 

 discussed in Chapter XI. 



14.4 A Special Type of Solution 



Consider (14.25) in a case in which 



^0 « Be (14.52) 



Be « 1 (14.53) 



In this case in the range 



e <de- Va and e > de+ VA (14.54) 



we can replace the hyperbolic tangent by its argument, giving 



^=-(-A) = (^,-l. (14.55) 



This can be solved for 6, giving 



^ = 0, T \/A/{P + 1) (14.56) 



If 



P < -1 



Then will be complex and there will be a pair of waves, one increasing and 

 one decreasing. We note that, under these circumstances, there is no cir- 

 cuit wave, either with or without electrons. 



What we have is in essence an electron stream passing through a series 

 of inductively detuned resonators, as in a multi-resonator klystron. Thus, 

 the structure is in essence a distributed multi-resonator klystron, with loss- 

 less resonators. If the resonators have loss, we can let 



P = {-jG + B)/doV^ (14.57) 



where G is the resonant conductance of the slots. In this case, (14.56) be- 

 comes 



\-jG + {B + doVe/fjd/ 



Near resonance we can assume G is a constant and that B varies linearly 

 with frequency. Accordingly, we can show the form of the gain of the in- 

 creasing wave by plotting vs. frequency the quantity g 



g = Im(-j 4- co/coo)-i/2 (14,59) 



In Fig. 14.7, g is plotted vs. co/coq . 



