Linear plasma oscillation described by superposition of normal modes

The existence of steady-state solutions to the linearized ion and electron Vlasov equation is demonstrated for longitudinal waves in an initially stable plasma. The evolution of an arbitrary initial perturbation can be described by superposition of these solutions. Some common approximations to the full set of equations can be solved in the same way. In some special cases, relevant, for instance, for single-ended Q machine experiments, a problem with given boundary conditions can be solved by superposition of normal modes.