On reexamining the hydrodynamic instability, Yu [1] showed that when the fully dynamical interactions are duly accounted for, and proper mathematical analysis is carried out, the positive feedback between the wave and evolving current can initiate and sustain rip current circulations with scales comparable to field observations on alongshore uniform beaches. In this study, we extend that analysis to consider nonplanar beaches, and to include a new branch of unstable modes that correspond to alongshore propagating horizontal circulations with the magnitudes of the flow growing in time. The latter has not previously been studied. These propagating unstable modes have typical time periods of tens of minutes and alongshore propagation speeds of a few centimeters per second. The physical implications of their spatial and slow time oscillations are discussed, as of relevance to occurrence and recurrence of transient rips, alongshore migration of rip currents and very low frequency pulsations in surf zone eddy circulations.