Transverse ridges on the surfaces of catastrophic mass flow deposits such as landslides, debris flows, and lahars may be the remnants of surface waves that originated during transit. Such waves can form as a result of a flow instability or from irregularities or pulses in the release of material at the source. We first consider a steady state release of material into the flow. By considering classic fluid dynamic stability criteria, we show that flow instabilities known to lead to roll waves seem likely for the mass flow deposits studied on Earth and Mars. Motivated by the plausibility of dynamic wave formation, we consider the more realistic case of a time-dependent release of material into the flow by performing a detailed mathematical analysis of the momentum and volume conservation equations. The results show that for some regimes of flow conditions, any disturbances in the source conditions grow downstream without bound, much like a classical fluid dynamic instability. It is conjectured that this regime may lead to analogs of roll waves. In other regimes, however, any irregularities in the boundary conditions at the source of the flow decay, as in a stable fluid flow. One critical parameter delineating the regimes of stability and instability is the time constant for the decay of the flow depth at the source. When surface waves are preserved in the deposits of long runout, catastrophic mass flows, our results provide a new diagnostic for constraining the emplacement conditions and the rheology of the flow during transit.