110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Actual electron flow as it occurs in practical tubes can exhibit trans- 

 verse velocities. For instance, in Brillouin flow, ' • if we consider electron 

 motion in a coordinate system rotating with the Larmor frequency we 

 see that electrons with transverse velocities are free to cross the beam 

 repeatedly, being reflected at the boundaries of the beam. The trans- 

 verse \-elocities may be completely disorganized thermal velocities, or 

 they may be larger and better-organized velocities due to aberrations at 

 the edges of the cathode or at lenses or apertures. Two-dimensional 

 Brillouin flow allows similar transverse motions. 



It would be difficult to treat the case of Brillouin or Brillouin-like flow 

 with transverse velocities. Here, simpler cases with transverse velocities 

 will be considered. The first case treated is that of infinite ion-neutra- 

 lized two-dimensional flow with transverse velocities. The second case 

 treated is that of two-dimensional flow in a beam of finite width in which 

 the electrons are elastically reflected at the boundaries of the beam. 

 Growing waves are found in both cases, and the rate of growth may be 

 large. 



In the case of the finite beam both an antisymmetric mode and a 

 symmetric mode are possible. Here, it appears, the current density 

 required for a growing wave in the symmetric mode is about ^^ times 

 as great as the current density required for a growing wa^•e in the anti- 

 symmetric mode. Hence, as the current is increased, the first growing 

 waves to arise might be antisymmetric modes, which could couple to a 

 symmetrical resonator or helix only through a lack of symmetry or 

 through high-level effects. 



1 . Infinite two-dimensional flow 



Consider a two-dimensional problem in which the potential varies 

 sinusoidally in the y direction, as exp{—j^z) in the z direction and as exp 

 (jut) with time. Let there be two electron streams, each of a negative 

 charge po and each moving with the velocity ?/o in the z direction, but 

 with velocities Wi and —ih respectively in the y direction. Let us denote 

 ac quantities pertaining to the first stream by subscripts 1 and ac quan- 

 tities pertaining to the second stream by subscripts 2. The ac charge 

 density will be denoted by p, the ac velocity in the y direction by y, 

 and the ac velocity in the z direction by i. We will use linearized or 

 small-signal equations of motion.^ We will denote differentiation with 

 respect to ?/ by the operator D. 



The equation of continuity gives 



jupi = -D(piUi + po?yi) + j|8(piWo + pnii) (1.1)1 



jcopo = -D{-p-iHi -\- pi)lj':d + il3(P2''o + Poi2) (1.2) 



t; 





