Landau Solution of the Plasma Oscillation Problem

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Abstract
The controversial aspects of Landau's treatment of the initial value problem are discussed. It is demonstrated that any kind of decaying solution other than that of Landau can be obtained for some initial velocity distribution function. Nevertheless, it is shown that several phenomenologically motivated distributions—Maxwellian functions, Lorentzian functions, and smoothly cutoff functions—yield solutions which are dominated by the plasma wave for long periods of time, although the plasma wave may not be present in the asymptotic limit. The distribution functions that are chosen lead to specific examples of each conceivable type of singularity structure for the integrand in the representation of the solution. It is suggested, therefore, that Landau's conclusion that all long‐wavelength disturbances in a plasma produce a damped plasma wave which dominates the ensuing behavior is physically correct.

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