Growing plasma oscillations for symmetrical double-humped velocity distributions

The problem considered here is that of growing longitudinal waves which propagate in a hot plasma parallel to any magnetic field which may be present (i.e. the magnetic field is neglected in the Vlasov equation). The necessary and sufficient condition for stability was obtained by Penrose (1960) and growth rates for plasmas obeying a Maxwell zero-order velocity distribution were computed by Fried & Conte (1961).