Radiation from the Two-Stream Instability

Coupling between longitudinal and transverse waves is examined for an unbounded homogeneous cold plasma in which the several plasma constituents are in relative drift with respect to one another. For mutual relativistic streaming along the direction of a homogeneous dc magnetic field $B$, a dispersion relation is derived. For the special case of two equal counterstreaming electron beams, the extraordinary wave-dispersion law is compared with Lee's nonrelativistic result. For $B=0\mathrm{}$, the dispersion relation is analyzed in some detail, especially as regards the polarization of the unstable eigenmode, and compared with results of Neufeld. We show, both analytically and numerically, that radiation effects are relatively insignificant, except for highly relativistic streaming velocity and for a beam-plasma frequency comparable to that of the background plasma.