Two-wave interaction in ideal magnetohydrodynamics

The weakly nonlinear interaction of sound and linearly polarised Alfv{\'e}n waves propagating in the same direction along an applied magnetic field is studied. It is found that a sound wave is coupled to the Alfv{\'e}n wave with double period and wavelength when the sound and Alfv{\'e}n speeds are equal. The Alfv{\'e}n wave drives the sound wave through the ponderomotive force, while the sound wave returns energy back to the Alfv{\'e}n wave through the parametric (swing) influence. As a result, the two waves alternately exchange their energy during propagation. The process of energy exchange is faster for waves with stronger amplitudes. The phenomenon can be of importance in astrophysical plasmas, including the solar atmosphere and solar wind.