Plasma-Beam Instability in the Hartree-Appleton Approximation

The interaction of a beam of protons of small intensity moving with velocity cβ (where c is the velocity of light) through a stationary plasma is investigated under the assumption that the beam is aligned along the direction of an impressed static magnetic field. Expressions are derived for frequencies and rates of growth characterizing the transverse waves excited by the beam and aligned along the direction parallel to that of the impressed field. It is assumed that the frequencies of the excited waves are large when compared to the proton gyrofrequency and are sufficiently high so that the motion of the plasma ions perturbed by these waves are neglected. It is shown that two waves are simultaneously excited by a beam having velocity cβ < cβmax and a relationship is obtained between βmax and the parameters charaterizing the stationary plasma.