Nonlinear, periodic waves in a cold plasma: a quantitative analysis

The exact theory of a wave of fixed profile travelling with speed c/n (0 < n < 1) through a uniform, cold, collisionless, unmagnetized plasma is investigated for the case in which the electric field has a non-zero transverse component, which lies in a fixed direction. The governing equations, referred to the frame of reference in which there is no space dependence, are studied analytically in each limit n→0, n→1. and by computation for other values of n. It is shown that, for each value of n, there is, in general, just one periodic solution of an arbitrarily given amplitude; and a quantitative description of these periodic solutions is provided in graphical form. A simple ‘dispersion relation’ is obtained for large- amplitude waves.