The depth-averaged hydraulic equations augmented with a suitable bed-load sediment transport function form a closed system which governs the one-dimensional flow in an alluvial river or channel. In this paper, it is shown that this system is hyperbolic and yields three families of shock-wave solutions. These are determined to be temporally stable in restricted regions of the (H,F0)-plane, via the Lax shock inequalities. Further, it is demonstrated that this criterion is equivalent to the energy dissipation criterion developed by Needham and Hey (1991).