Normal modes of plasma oscillation

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Abstract
The present paper is concerned with the well-known problem of electrostatic waves in a uniform collisionless plasma, in the linearized approximation. Although much work has already been done on this problem, a number of questions still arise. In particular a slightly unstable wave corresponds to an exact normal mode solution, whereas a slightly damped wave has a much more limited meaning: it is encountered by solving the equations with initial conditions in the manner first introduced by Landau. In this paper a new type of normal mode is introduced which can represent damped waves as well as unstable ones, thus preserving the continuity between both kinds of waves. This is possible by considering the distribution function as a linear functional, or as an analytic function defined for complex values of the velocity. A complete system of such normal modes is formed. The spectrum consists of discrete eigenvalues corresponding to waves, either growing or damped, together with a continuum that may be chosen to have an imaginary part such that it corresponds to rapidly damped transients. An expansion formula is derived and the expansion given by Van Kampen and Case is recovered as a special case. This expansion is used to solve the equation for the distribution function and the relation to the solution by Laplace transformation is discussed.

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