The resonant coupling of linear fast and Alfv¿n modes is considered in a cold plasma permeated by a curl-free background magnetic field. The medium is assumed to possess an invariant coordinate (e.g., slab or axisymmetric geometry). We present the problem in terms of a set of orthogonal wave functions which describe the wave fields. Perturbations are Fourier analyzed along the invariant direction with wave number k&bgr;, which is subsequently employed as a second expansion parameter with which to expand the governing equations. These equations are then solved using time-dependent perturbation theory, familiar in quantum mechanics. Our calculations provide generalizations to the results of previous authors, such as the excitation of resonant Alfv¿n waves and the damping of the fast mode. To test our novel formulation we compare our results with numerical solutions of the ''box'' magnetosphere. For small azimuthal wave numbers (<3--4) our lowest-order estimates of the cavity mode damping rates in excellent agreement with previous calculations. ¿ American Geophysical Union 1992