  This paper presents a study of the two-stream instability of an electron beam
propagating in a finite-size plasma placed between two electrodes. It is shown
that the growth rate in such a system is much smaller than that of an infinite
plasma or a finite size plasma with periodic boundary conditions. Even if the
width of the plasma matches the resonance condition for a standing wave, a
spatially growing wave is excited instead with the growth rate small compared
to that of the standing wave in a periodic system. The approximate expression
for this growth rate is $\gamma \approx
(1/13)\omega_{pe}(n_{b}/n_{p})(L\omega_{pe}/v_{b})\ln (L\omega_{pe}/v_{b})[
1-0.18\cos ( L\omega_{pe}/v_{b}+{\pi }/{2}) ]$, where $\omega_{pe}$ is the
electron plasma frequency, $n_{b}$ and $n_{p}$ are the beam and the plasma
densities, respectively, $v_{b}$ is the beam velocity, and $L$ is the plasma
width. The frequency, wave number and the spatial and temporal growth rates as
functions of the plasma size exhibit band structure. The amplitude of
saturation of the instability depends on the system length, not on the beam
current. For short systems, the amplitude may exceed values predicted for
infinite plasmas by more than an order of magnitude.

Plasma Physics