The structure of magnetosonic eigenmodes was investigated analytically within the framework of simple two- and three-dimensionally inhomogeneous models. Asymptotic (in the limit m≫1, where m is azimuthal wave number) expressions were obtained for the eigenfrequency spectrum of magnetosonic oscillations in the cavity resonator under the plasmapause, as well as for outer magnetospheric cavity modes, the outer boundary for which is represented by the magnetopause. These oscillations were found to be damped ones. Furthermore, the damping of the eigenoscillations in the cavity under the plasmapause is caused by the escape of some of their energy through the potential barrier to the outer magnetosphere. The damping of the outer magnetospheric cavity modes is due to the fact that the magnetopause represents a partially reflecting boundary, through which a part of the oscillation energy escapes to the solar wind. In terms of an axisymmetric model of the magnetosphere, asymptotic (when m≫1) expressions were obtained for damping decrements of the eigenmodes. The spatial structure of magnetosonic eigenmodes was investigated in a simple three-dimensionally inhomogeneous model of the magnetosphere. The asymptotic expression was obtained for the spectrum of their eigenfrequencies. ¿ 2000 American Geophysical Union