Growing Waves Due to Transverse Velocities

109 110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 1 8 Actual electron flow as it occurs in practical tubes can exhibit transverse 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 transverse velocities 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-neutralized 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 Yz times as great as the current density required for a growing wave in the antisymmetric 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.