Using a quasi-one-dimensional (self-similar) form of the magnetohydrodynamic equations, a model of the interaction of shocked magnetosheath plasma flow with an indented high-altitude cusp is presented. The model applies only to a closed magnetopause. It is shown that the resulting plasma flow into the cusp indentation is highly dependent on the ratio of the plasma velocity to the magnetosonic wave speeds. If the magnetosheath velocity is in excess of the slow mode wave speed, the flow at the indentation is associated with a plasma compression accompanied by a minimal rotation of the flow into the cusp. The behavior of the density differs from that usually associated with self-similar flows and is due to the relative phase between density, field, and flow associated with the slow mode wave. For flows in excess of the fast mode speed the flow rotates significantly into the cusp by means of a fast mode expansion wave. We propose that the slow mode wave model is applicable for the summertime cusp and that the fast mode one is applicable for the wintertime cusp.