Nonlinear Theory of Space-Charge Wave in Moving, Interacting Electron Beams with Application to Solar Radio Noise

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Abstract
A consideration of the complete equations of interaction (without the linear approximation) of two moving electron beams of given densities (with sufficient numbers of ions to make the charges macroscopically neutral) shows that propagation of steady-state space-charge waves is possible in such a medium. The period of the space-charge wave is a function of its amplitude and phase velocity. For small amplitudes the oscillation is simple harmonic, and the characteristic dispersion equation of the first order theory is obtained. For a given phase velocity of the wave, the oscillation becomes increasingly anharmonic with increase of amplitude. Beyond a particular value of the amplitude (which is a function of the phase velocity of the wave), the wave form of the oscillation becomes discontinuous.