PART II — THE UHF NOISE SPECTRUM 867 



Vo = {Vr , Ve , v^) (1) 



where 



Vr = r-f(z), or — ' = -^ (2) 



or r 



ve = rd = r |^ (3) 



V, = u. (4) 



Here, r, 6, z are the polar cylindrical coordinates, Wc = r]B the angular 

 cyclotron frequency corresponding to the longitudinal focusing field B, 

 and j{z) a function describing the amplitude and spatial periodicity of 

 the beam ripple. The experimental data indicates that the potential 

 variations along the beam axis are negligible, permitting the assumption 

 that the longitudinal velocity, xi, is constant. MKS units are used. 



Consistent with the distribution pattern of Fig. 6, the ac field can 

 be represented by an axially-symmetric potential function, similar to 

 that for the smooth Brillouin-flow beam: 



V ~ 7o(7r) expjXwf - ^z), (5) 



E = -grad V. (6) 



The ac equations of fluid motion are obtained by adding a small ac 

 increment to each of the steady-state velocity components. In addition 

 to the space-charge field, the ac electric field contributes forces acting 

 on the charged medium; those contributed by the ac magnetic field 

 are neglected: 



{vo + y) = -77[-grad V - grad Vo + (vo + fJ) X B]. (7) 

 at " - - - 



The ac velocity is distinguished by a tilde, and the dc velocity by a 

 zero subscript. Here ri = e/m is the charge-mass ratio of the electron, 

 a positive quantity. As all ac quantities are functions of spatial positions, 

 their time differentiation (indicated by a dot) is equivalent to multi- 

 plication by y(co — 0u), written jojfc for brevity. 

 The components of the force equation are expanded as follows: 



^ + (^. + v.) ^ (.. + Vr) + (U + V.) ~ (Vr -f r-,.) - ^^^1±^ 



dt dr az r 



dV , dVo f , . 



dr dr 



(8a) 



