WAVE I'UOPAGATION ALONG AN KLKCTHON BIOAM 



415 



z4 



o 



Q. 2 



5 



12 3 4 5 6 7 



7a 



Fig. 4 — Comparative values of impedance reduction factor for several kinds of 

 Ijcams, of the same relative radii hja. 



(2) ("Ui.sing electrons to move preferentially into regions of retarding 

 longitudinal tields, a process analogous to bundling. 



CONCLUSIONS 



Field solutions have been presented for the magnetically-focused 

 cylindrical beam, when modulated by a small ac signal. Two types of 

 beam enclosure have been treated: the concentric drift tube and the 

 ideally thin (sheath) helix. 



There are two pairs of unattenuated space-charge waves in the drift- 

 tube: one with arbitrary amplitude distribution, and another pair which 

 is coupled to the external field (Fig. 1). The space-charge wavelength 

 of the latter pair is greater than that of space-charge wa\'es in a similar 

 beam with rectilinear flow (Fig. 2). 



The solution of the helix prol)lem consists of the afoicmcnlioiicd two 

 space-charge waves with arbitrary amplitude, as well as the usual f<uu" 

 wa\es of traveling-wave tub(» theory, or six waves in all. In order to com- 

 pute the gain constant of the growing wave, the field solution has been 

 re-written as the admittance equation of a thin beam in an artificial 



