The time-dependent phase structure of Alfv¿n waves on open and closed field lines is studied. In accord with previous observations we find that Alfv¿n waves on near-Earth closed field lines exhibit a poleward phase motion unless they are close to the plasmapause, in which case the motion may be equatorward. Alfv¿n waves generated on tail-like closed or open field lines threading the plasma sheet boundary layer have received much less attention but may be shown to have an equatorward phase motion [Liu et al., 1995>. Phase mixing in the magnetotail proves to have a much richer behavior than that of near-Earth (dipole-like) closed field lines as not only the Alfv¿n frequency varies across the background field lines but the field-aligned wavenumber varies too. The two contributions tend to cancel each other partially for typical tail equilibria. Observations are given of a double oval configuration showing long-period pulsations on the poleward portion of this oval. Equatorward phase motion is observed and supports the theory presented here. These observations illustrate that Pc5 pulsation activity can be much richer than previously thought and can occur at locations not in the dipole-like region, as is usually supposed. The concepts presented in this paper provide a powerful framework with which to interpret observations related to auroral arcs, substorms, and magnetospheric equilibria. ¿ 1999 American Geophysical Union