406 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1954 



Thus we note that wave propagation along a cylindrical beam with 

 Brilloiiin flow is accompanied by swelling and contracting of its boundary, 

 with constant space-charge density, rather than by space-charge bunch- 

 ing. The second interesting result is that the dynamics and field ecjua- 

 tions for the focused beam are identical with those for a beam with zero 

 dc magnetic field, except for the angular component of surface current 

 density Ge ■ 



SPACE-CHARGE WAVES 



We now consider the given beam, of radius h, in a concentric conduct- 

 ing tube of radius a > h. The boundary problem consists of matching 

 the TM wave admittances inside and outside of the beam, at its boun- 

 dary. (The TE fields are of no interest in the drift-tube problem, as 

 they are not excited at the ends of the tube, and are not coupled to the 

 TM fields.) Let I refer to the beam region < r < 6, and II to the space 

 between beam and conductor b < r < a. Then, at r = b, 



He + Gz _ He 



tjz til. 



(47) 



The beam admittance on the left is evaluated with the aid of (3), (7), 

 (36), and (42): 



Y 



7 /o(7o) 



(48) 



where 7io and Ki are modified Bessel functions of the second kind. The 

 wave admittance at r = 6 in II is therefore: 



i c 



'h{lb) - {C/B)-K,{yb) 



(49) 



7 L/o(7&) + (C/j5)-Ko(t6)J 

 At r = a, E^' = or: 



C/B = -7o(7a)/i^o(7«) (50) 



Equating beam and circuit admittances (48) and (49), we obtain: 



_ hiya) 



R = ^ ""^'^1 . ■ --, (51) 



