The structure of magnetosonic eigenoscillations of an axisymmetric magnetosphere in the direction across magnetic shells was determined within the WKB approximation. The spectrum of their eigenfrequencies was investigated numerically. The model of the medium assumes a dipole magnetic field and a two-dimensional Alfven velocity distribution in the meridional plane which takes into account abrupt changes of the Alfven velocity at the plasmapause and magnetopause. It is shown that eigenfrequencies are complex valued. The imaginary part of the frequency is a damping decrement of the eigenoscillations associated with the escape of some of their energy to the solar wind. It is shown that in a dipole-like geomagnetic field in which field lines concentrated toward the origin of coordinates, the amplitude of magnetosonic oscillations increases from the magnetopause to the ionosphere. While the oscillations penetrating deep into the magnetosphere from the solar wind or generated at the magnetospheric boundary have near the magnetopause a relatively small amplitude, as the Earth is approached, their amplitude can reach a significant value for geometrical reasons. ¿ 2001 American Geophysical Union