Wave properties of a cycloidal stream

Integrodifferential equations are formulated for small HF displacements and HF velocities of cycloidal electron streams in M-type tubes with a cathode in the interaction space. Computations of the wave number of disturbances propagating in the stream are reduced to an algebraic equation under the assumption that the height-distribution of HF disturbances of the electron stream is sinusoidal. Attention is given in particular to a class of synchronous waves whose velocity is close to the mean drift velocity of the stream. There are always evolving waves among these synchronous space-charge waves, a fact which testifies to the instability of the electron stream.