Waves in Collisionless Plasmas. II: Longitudinal Waves

Abstract

The problem of longitudinal waves in an infinite one-dimensional plasma is treated by expressing the charge density in terms of the exact solution of the collisionless Boltzmann equation and then solving Poisson's equation. For an initially homogeneous plasma the electric field is determined exactly. For an initially inhomogeneous plasma an iterative procedure is developed which in principle determines the electric field for an arbitrary initial inhomogeneity. If the initial inhomogeneity is small with respect to the homogeneous part of the distribution function the problem can be treated by perturbation theory. To each order the field is found to obey a linear integral equation which is readily solved in terms of a Green's function, Γ(z, t), which maps the known nth-order source function into the nth-order field. For t ≠ 0, the Fourier transform of Γ(z, t) is a sum of damped oscillations whose complex frequencies are determined for all 0 < t < ∞ by the plasma-dispersion relation. A general procedure for solving this dispersion relation is developed. For a Maxwellian plasma an infinite number of frequencies are determined for large k in addition to the usual Landau results for small k. A physical interpretation of Landau damping is developed which explains the decrease of E with time in terms of the drifting dispersal of an initially localized inhomogeneity.