Parametric excitation of Alfvén and acoustic waves

The nonlinear interaction between two Alfvén waves and a sound wave is studied, using the normal-mode approach. This leads, in a simple way to a set of coupled equations, and consequently to a dispersion relation for the waves under consideration. It is shown that a large-amplitude Alfvén wave can give rise to two distinct types of parametric instabilities, namely the oscillating and the purely growing waves. In each case, the expressions for the threshold pump intensity, the frequency shift and the growth rate of the excited waves are obtained. In particular, the results for a propagating pump under perfect frequency matching conditions are compared with those of Sagdeev & Galeev.