The problem of noise‐ (and, as a special case, signal‐) wave propagation along a drifting, one‐dimensional electron stream with a half‐Maxwellian velocity distribution is analyzed. The integral equation resulting from the linearized Boltzmann equation is intractable in its exact form and so, heretofore, has succumbed only to machine computation. In this paper closed‐form solutions are obtained through the use of a simplified kernel. This approximation, although it limits the analysis to beams with small relative velocity spreads and small ratios of plasma frequency to operating frequency, is yet innocuous enough not to mask the main effects.
The noise current and voltage are found to consist of three wave terms: a ballistic wave describing the injected noisy ``test particle,'' a slow space‐charge wave and an aggregate of fast space‐charge ``waves'' which decays, en masse, with distance because of phase mixing. This aggregate is approximately representable by a single fast wave with exponential damping. Far downstream the noise power spectra (ψ,Φ,Π,S) lose their familiar sinusoidal pattern and settle to constant values. Along the entire beam the positive rf noise power, S+Π, is invariant.