When the azimuthal wave number is large, the equations describing standing hydromagnetic waves in the magnetosphere can be written as a set of coupled equations describing the coupled magnetosonic and Alfv¿n waves. These equations are decoupled when the field lines are straight. The eigenfrequencies of the decoupled oscillations are computed. For typical conditions in the outer magnetosphere these give periods in the Pc 4--5 band or above. The longitudinal magnetosonic wave consists of oscillations in the plasma pressure, the longitudinal plasma drift velocity and the compressional magnetic field. Higher harmonics of the standing waves have nodes quite near the equator. These higher harmonics have larger fractional pressure perturbations at high latitudes. The compressional magnetic field for all modes, however, is substantially attenuated at higher latitudes, and the theory predicts that compressional oscillations of B are only likely to be seen near the equator. Conditions can be favorable for resonance to occur between the magnetosonic mode and the transverse Alfv¿n mode. The computed results show periods of the right order of magnitude to explain observations of compressional pulsations. The theory has the potential to explain the polarization when coupling is fully taken into account. ¿American Geophysical Union 1987