Dynamics of compressional MHD wave propagation is theoretically studied in the dayside and nightside magnetosphere. In order to analytically examine the MHD wave equation, we adopt the quantum mechanical approach which is often used in the problem of the Schr¿dinger's equation. The wave solutions are obtained for dayside and nightside Alfven speed profiles, respectively, without any arbitrary boundary condition at the magnetopause or the bow shock. The result shows that the previous cavity/waveguide models are found to be inappropriate for studying compressional wave properties in the magnetosphere. We discuss the role of virtual resonances which generally represent any possible compressional modes in the magnetosphere. ¿ 1999 American Geophysical Union