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Beam-driven instabilities are considered in a pulsar plasma assuming thatboth the background plasma and the beam are relativistic J\"uttnerdistributions. In the rest frame of the background, the only waves that cansatisfy the resonance condition are in a tiny range of slightly subluminalphase speeds. The growth rate for the kinetic (or maser) version of theweak-beam instability is much smaller than has been estimated for arelativistically streaming Gaussian distribution, and the reasons for this arediscussed. The growth rate for the reactive version of the weak-beaminstability is treated in a conventional way. We compare the results with exactcalculations, and find that the approximate solutions are not consistent withthe exact results. We conclude that, for plausible parameters, there is noreactive version of the instability. The growth rate in the pulsar frame issmaller than that in the rest frame of the background plasma by a factor$2\gamma_{\rm s}$, where $\gamma_{\rm s} = 10^2 {\rm -} 10^3$ is the Lorentzfactor of the bulk motion of the background plasma, placing a furtherconstraint on effective wave growth. Based on these results, we argue thatbeam-driven wave growth probably plays no role in pulsar radio emission.

Wave dispersion in pulsar plasma: 3. Beam-driven instabilities