In the terrestrial magnetosphere, the inhomogeneous magnetic field and plasma density give rise to a continuous spectrum of field line resonant frequencies. Compressional disturbances with characteristic frequencies lying within the range of the spectrum may couple to transverse oscillations of resonant field lines. The coupling is of particular interest for global compressional modes trapped in the magnetic cavity. These modes decay in time through the coupling, even in the absence of dissipation. The importance of the process is that, through the damping of the global modes, large-scale motion can drive localized field line resonances. In this study, we investigate the mode coupling and examine the parameter dependence of the damping rate of the global mode. The problem is discussed as an initial value problem in the box model which retains most of the significant physics yet remains mathematically tractable. To treat the coupling, we use the analogy of Landau damping in a homogeneous plasma. From the Laplace transform approach, we obtain the complex frequencies of the compressional wave by finding the singularities of the associated Green's function. Once the complex frequency has been found numerically, we obtain the corresponding waveforms in the box. Many observed wave properties can then be obtained. The calculations agree well with other simulation work and correspond to results obtained for the reflection of radio waves from the ionosphere and for plasma heating by absorption of radiation.