Waves in a Plasma in a Magnetic Field

The small oscillations of a fully ionized plasma, in which collisions are negligible, in a constant external magnetic field, is treated by the Laplace transform method. The full set of Maxwell equations is employed and the ion dynamics are included. Various limiting cases are considered. It is shown that self-excitation of waves around thermal equilibrium is impossible. It is also demonstrated that for longitudinal electron oscillations propagating perpendicular to the constant magnetic field, there are gaps in the spectrum of allowed frequencies at multiples of the electron gyration frequency, but zero Landau damping. These particular waves are also associated with a nonuniformity of convergence in the limit of vanishing magnetic field which phenomenon, however, is of no physical consequence. When the ion dynamics are included, two classes of low frequency oscillations are found, the existence of both of which has been predicted by the simple hydrodynamic theory, namely longitudinal ion waves, and transverse hydromagnetic waves. The well known results for the propagation of electromagnetic waves in an ionized atmosphere are also recovered, as well as the effects on such waves in various limiting cases of the magnetic field and particle motion. These calculations indicate that in many cases the transport equations are capable of yielding correct results, apart from such things as Landau damping, for a wide class of waves in a collision-free plasma.