Non-Linear Theory of Plasma Oscillations and Waves†

Abstract

ABSTRACT A non-linear equation for plasma oscillations is derived. Only Coulomb interactions between electrons are considered. From an exact first integration the following properties are found : There is an asymmetry in the oscillations expressed by ρ max = a0+a0 2/3n0 and ρ min = −a0+a0 2/3n0 where ρ is the density of electrons. There is also a dependence of the frequency on the amplitude of the oscillations. The uniform equilibrium distribution of electrons is found to be highly stable. A second approximate integration gives the explicit form of the oscillations and the dependence of the frequency on the amplitude ω = (1 − a0 2/n0 2)1/2 ωp. If collisions between elections are ignored, instead of progressive waves there are stationary plasma waves. The form of these is found for the case of spherical waves in an isotropic plasma. A non-linear approximate equation is found and an exact solution of this equation is obtained. A suitable approximation to this solution is ρ = A cos (B/r). The approximations ...