Electrostatic oscillations in cold inhomogeneous plasma Part 2. Integral equation approach

Abstract

Small amplitude electrostatic oscillations in a cold plasma with continuously varying density have been investigated. The problem is the same as that treated by Barston (1964), and in part 1 of this paper, but, instead of starting from a differential equation, we base our treatment on an integro-differential equation for the transverse component of the electric field. This equation is very similar to the Vlasov equation, and we show this to be the cause of the striking similarities between the two theories pointed out in part 1. The normal-mode analysis utilizes the similarity of the integral equation for the eigenfunctions to the equation of anisotropic neutron transport. The continuum eigenfunctions are found in the form of a generalized van Kampen eigenmode, and their calculation is reduced to solving a non-singular integral equation. Equivalence of the eigenfimctions found in this way and of those found by solving the equivalent differential equation is proved, and the question of orthogonality and normalization is discussed.