WAVE PROPAGATION ALONG AN KLKCTKOX HKAM 407 



This cHiuation must ho solved simultaneously with cadi ol' the follow- 

 ing: 



7i6 = 0J) + l3,h/\/Ri 



^ (52) 



Thus, for a j^ixen l)eam and froquoncv, the solution consists of two un- 

 attenuated waves, one faster and the other slower than the beam \elo('ity. 

 The wavelength of the'interference pattern is given by: 



■4:7r 



X. = -^^ (53) 



7i — 72 



For a cylindrical beam, 



(3 J) = 174VP (54) 



where P = I/V^'^ amps/(volts)^'^, the perveance. In practice, P and 

 hence /3p6 are usually so small that we can gain a fair estimate of X^ by 

 assuming 7?i = Ro: 



\s ^ ~— (OO) 



Pp 



Fig. 1 shows the variation of R^'^ with yb for several values of h/a. 

 (The "intrinsic" solution (32) is included as a line at R^'~ = 1.) The 

 ordinates of these curves are approximately proportional to the space- 

 charge wavelength, and the abcissae to the frequency, as y c^ ^e = wA/o 

 for small perveance. 



Space-charge Avaves propagating along a cylindrical beam with rec- 

 tilinear flow have been treated by Hahn and Ramo^. In Fig. 2, their 

 computations have been reformulated in the same way as in Fig. 1, and 

 compared with the results for Brillouin flow, for two values of b/a. The 

 space-charge wavelength is always greater in Brillouin flow, for the 

 principal pair of waves and the same b/a and 76. 



HKLIX PROBLEM 



In place of the drift tube at radius a, we now have a helically conduct- 

 ing sheet of zero thickness and pitch angle \l/. In addition to I (0 < r 

 < b) and II (6 < r < a), we shall use III to identify fields in the region 

 (a < r < x). The boundary conditions at r = b are: 



H/ + a - H/' = 



E/ - E/' = 



(56) 

 HJ - Ge - HJ' =0 



E/ - EJ' = 



