NONLINEAR PLASMA OSCILLATIONS WITH TRAPPED ELECTRONS

Propagation of stationary longitudinal waves in a hot (Maxwellian) plasma is investigated, when the wave amplitude, phase velocity, and electron temperature are such that the relativistic effect is negligible but trapped electrons must be taken into account. The treatment is based on an exact solution of the nonlinear collisionless Boltzmann equation compatible with the equilibrium electron distribution. The existence, propagation, and amplitude limit of a dimensionless periodic potential wave ψ are discussed in terms of the behavior of the first integral Y(ψ) of the Poisson equation. It was found that, for α = mW2/2κT (W is the wave velocity) less than a critical value αe = 0.854, no stationary wave can exist, irrespective of its amplitude. For α slightly greater than αe, wave propagation is limited to small amplitude and low frequency. As α is further increased, waves of progressively larger amplitude can also propagate; but their amplitude is limited when Y(ψ) becomes tangent to the ψ axis at ψmin because of the excessive number of electrons trapped in the wave troughs. An expression for the maximum wave amplitude as a function of α is derived and plotted. Wavelengths for different values of α and amplitude levels are computed numerically and plotted as dispersion curves. A typical example for wave form is also displayed to show the progressive distortion from pure sinusoids due to nonlinearity and the effect of trapped electrons.