Landau Damping of Space-Charge Waves

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Abstract
The current modulation excited by a sinusoidal voltage across a narrow‐gap resonator on an electron beam with a Maxwellian distribution of velocities is treated on the basis of the linearized Boltzmann equation without collisions. The numerical part of the analysis reveals that the current modulation as function of distance has the form of a damped sinusoid, with finite minima, and approaches a constant level at large distances. These results are in qualitative agreement with experimental observations reported elsewhere. The analytical results show that the fast space‐charge wave of monovelocity theory is replaced by a continuum of waves (for ωp<ω). This fast‐wave packet decays with distance because of phase mixing, so that only the slow space‐charge wave survives at large distances. The spatial decay of the fast‐wave packet is the equivalent in multivelocity beams of temporal Landau damping in stationary plasmas.