Theory of the Parametric Oscillation in an Electron Beam-Plasma System

The density oscillations in a system composed of a one-dimensional cold electron plasma and a cold electron beam are considered where the beam is assumed to be modulated sinusoidally and also to be bounded by a repeller in the plasma. A nonlinear differential equation which describes the density oscillations in the beam-plasma system is derived from the Vlasov and Poisson equations. The equation does not take the so-called van der Pol-type, but is, as one of the linear approximations, reduced to a Mathieu-type equation with an inhomogeneous term: d²ρ⁄dt²+ω_e ²[1−σ cos (ωt)]ρ=σ(ω²−ω_e ²) cos (ωt). This inhomogeneous Mathieu equation is analyzed by the method of Bogoliubov and Mitropolsky. The spectrum of the characteristic frequency obtained from the analysis is compared with the experimental results.