GROWING WAVES DUE TO TRANSVERSE VELOCITIES 115 



Let the width of the beam be W. We let 



Thus, for n = 1, there is a half -cycle variation across the beam. From 

 (1.31) and (1.32) 



G = 27.s(^^^\ne db (1.33) 



Now L/uo is the time it takes the electrons to go from one end of the 

 beam to the other, while W/ui is the time it takes the electrons to cross 

 the beam. If the electrons cross the beam A'' times 



iV = ^4 (1-34) 



Thus, 



G = 27.SNnedb (1.35) 



While for a given value of e the gain is higher if we make the phase 

 vary many times across the beam, i.e., if we make n large, we should 

 note that to get any gain at all we must have 





2 . //iTTUlV 

 0)r> > 



(1.36) 



W 



If we increase oop , which is proportional to current density, so that cop 

 passes through this value, the gain will rise sharply just after cOp" passes 

 through this value and will rise less rapidly thereafter. 



.?. A Two-Dimensional Beam of Finite Width. 



Let us assume a beam of finite width in the ^/-direction ; the boundaries 

 lying a,t y = ±^o • It will be assumed also that electrons incident upon 

 these boundaries are elastically reflected, so that electrons of the incident 

 stream (1 or 2) are converted into those of the other stream (2 or 1). The 

 condition of elastic reflection implies that 



yi = -h (2.1) 



Zi = 22 Sit y = ±2/0 (2.2) 



and, in addition, that 



Pi = p2 at y = ±?/o . (2.3) 



since there is no change in the number of electrons at the boundary. 



