WAVK I'HOI'AGATION ALOXG AN ELECTRON BEAM 405 



The wave (Mniations (1) and (4) for /s, and II, now reduce to the 

 following. 



These equations have solutions for E^. and Hz , which are finite at r = 0, 

 of the form A-Io(yr), where .4 is an arbitrary constant and /o the modi- 

 fied Bessel function of zero-th order. 



It is not without interest to remark that the same pair of solutions, 

 given by (32) and (39) — (40), has been found by L. R. Walker for a 

 beam of arbitrary cross-section, with the same longitudinal velocity and 

 space-charge density at every point, in the absence of any impressed do 

 magnetic field. 



Due to the radial component of electron motion, the beam surface is 

 rippled. For a steady-state radius h, this rippling can be expressed, in a 

 linear approximation, by the perturbed radius: 



r(b) = 6 + rib) exp j(oot - yz) (41) 



The rippled beam is equivalent to a uniform cylindrical beam with an ac 

 surface charge density por, or a surface current density whose components 

 are: 



Gz = poruo (42) 



Gg = pordob (43) 



The total ac convection current may be written in a form which applies 

 equalh' well to the cylindrical beam with purely rectilinear flow: 



Ic = f ./.27rr dr + 2Trbp,n^r{b) 



•'0 



= -jioe-R-2Trb-A-h{yb)/y 

 = -j.eRl E.2.rdr ^^^^ 



where jR is a beam propagation function which will prove convenient: 



(co - yuoY {yb - m' ^ 



and 



^e = Oi/Uo , (3p = OOp/Uo (46) 



