Swing wave-wave interaction: Coupling between fast magnetosonic and Alfv{\'e}n waves

We suggest a mechanism of energy transformation from fast magnetosonic waves propagating across a magnetic field to Alfv{\'e}n waves propagating along the field. The mechanism is based on {\it swing wave-wave interaction} [T.V. Zaqarashvili, Astrophys. J. Lett. {\bf 552}, 107 (2001)]. The standing fast magnetosonic waves cause a periodical variation in the Alfv{\'e}n speed, with the amplitude of an Alfv{\'e}n wave being governed by Mathieu's equation. Consequently, sub-harmonics of Alfv{\'e}n waves with a frequency half that of magnetosonic waves grow exponentially in time. It is suggested that the energy of nonelectromagnetic forces, which are able to support the magnetosonic oscillations, may be transmitted into the energy of purely magnetic oscillations. Possible astrophysical applications of the mechanism are briefly discussed.