PART II — THE UHF XOISE SPECTRUM 



869 



The laminar-flow rippled beam can be described by the particle 

 trajectories, as follows: 



ri = ro,(l + 5 C0S|S,2), 



where ro, is the maximum radius for the paiticle considered, and 

 < 5 < 1. For this model of the beam, 



(15) 



r 

 r dz 



h _ —boic sin (Sc2 

 h ~ 1 + 5cos/3,2' 



— ajci8c5(5 + cos ^cz) 



(Ki) 



(17) 



(1 + 5cosM- ■ 



The region of interest, judging from the observed peak locations, is 

 not at the mid-plane of the beam waist, where Vr = 0, but on either side 

 of that plane, where | Vr/r \ is greatest. It is readily found that, at these 

 positions (I//-) (dVr/dz) is zero, and (14) can be written 



P = 





7 



J 





0' 



fiu r / 



W6/ > 



(18) 



This can be combined with Poisson's Equation, 



AV = (y' - 0')V = -p/e, 

 to furnish a relation between y and |8: 



(19) 



7 



R 



1 



(jibV/ _ 



= ^2 



ini -J 



. 2 v., 



1 - 



0u r 



1 



mr 



(20) 



where R = Wp/wb and oij = — Tjpo/e, the square of the angular plasma 

 fi'equency. 



At the beam boundary, r = h, the continuity of the tangential field 

 components and the change in radial electric displacement can be ex- 

 pressed in the form of an admittance equation: 



V \ dr 



a 



1 ^J 

 V di 



nil 



(21) 



Here I refers to the beam, ^ ?• ^ b, and II to the space between beam 

 and the concentric conducting tube, b ^ r ^ a. The surface charge 

 layer, a, takes account of the surface ripple, of amplitude r: 



