A magnetohydrodynamic approach is used to investigate how compressional linearly unstable Kelvin-Helmholtz surface waves on the magnetopause extract energy from the velocity shear and radiate that energy away from the boundary in the separate rest frames of the plasmas on the two sides. On the magnetospheric side, the energy transport velocity normal to the magnetopause may become comparable with the Alfv¿n speed. The energy flux, for typical conditions on the dayside magnetopause, is found to be ≂10-3 ergs/cm2 s, and the total energy flux over the unstable region on the dayside magnetopause is estimated to be ≂1017 ergs/s in magnetic quiet times, or >1018 ergs/s in disturbed times. The discussion of energy input into the magnetosphere illuminates how surface waves on the magnetopause provide the power to drive resonant regions within the magnetosphere at large distances from the boundary.