The coupling of different MHD waves is important for an understanding of laboratory, space, and solar plasmas. In this paper we investigate fast and Alfv¿n mode coupling in the cold plasma limit. The medium is assumed to have an invariant direction (e.g., slab or axisymmetric models) and carries an arbitrary solenoidal background magnetic field. Wave fields are Fourier analysed with wave number k&bgr; in the invariant direction. The coupled linearized wave fields are then expanded in a power series (of k&bgr;) to yield a decoupled hierarchy of inhomogeneous wave equations. The higher order terms in the series represent wave coupling phenomena such as the resonant excitation of Alfv¿n waves, and the damping of a magnetospheric cavity mode. ¿ American Geophysical Union 1991