PART II — THE UHP NOISE SPECTRUM 875 



be written as follows: 



( 



dvo\ ve' _ . . 



dt r - -7 - - "^'« ' ^^^^ 



ve = and UcV. (34) 



Thus, the radial "balance" conditions (32) are consistent with either of 

 two values for ve , of the equivalent stream with single-valued velocities. 

 As it develops that either of these values leads to the same result, the 

 first one will be used here for simplicity, ve = 0. 



An ac traveling wave along this beam cannot have any 0-dependency, 

 because the beam has no single value of angular velocity 6, which might 

 remain in synchronism with that of the wave. Thus, the perturbed 

 dynamics equation (7) can be expanded, wdth the assumptions of an 

 axial-symmetric ac field given by (5) and (6), a stream with steady- 

 state velocity (0, 0, u), constant space-charge density po , and no space- 

 charge forces, as follows: 



dVr , d~r dV 



dt az or 



dve , dve 

 dt az 



dv^ , dv^ dV 



dt dz dz 



These are solved for the ac velocity components: 



Ub — ojc or 

 v,= -t^v,, (36) 



V, = -^ V. (37) 



COb 



With grad po = div ^o = 0, the charge-conservation equation (11) can 

 be solved for p: 



A. = - Vo- grad p - po div v, 

 at 



^ = ^^ div .- = ,p. r^^^, - ^1 



03b [_Oib — Wc 0}b J 



At very large ripple amplitudes, it is a fair assumption that the density 

 of non-crossover electrons is negligible relative to that of crossovers in 

 this region. Poisson's equation (19) can then be combined with the 



