Spectral representation of plasma oscillations

The linearized Vlasov–Poisson system for small amplitude plasma oscillations is studied by the method of characteristics. Integration over velocities leads to a Volterra equation which is solved by a new method using ± Fourier representations. It is shown that for the full range of k λ D , the perturbed density can be expressed as a sum of stationary modes the spectrum of which is derived in terms of initial conditions. Some points of the theory of Landau and other previous authors are discussed and the ansatz of Van Kampen referring to the velocity distribution of each stationary mode is proved.