Movement and modification of an advancing front on an initially dry infiltrating surface is modeled mathematically by a kinematic description of surface flow and a kinematic shock. Criteria for the goodness of approximation are presented, and the model is shown to describe two common hydraulic problems; border irrigation advance and dry channel flood wave movement. Two numerical methods for solution of the equations involved are developed and compared. Results from the kinematic model are compared with published data and other methods of predicting border irrigation advance. Sensitivity of advance rate to the infiltration relations is demonstrated, and a constant mean depth is shown to be a reasonable assumption as a first approximation. The model may also be used to route floods through a dry infiltrating channel, and correspondence of properties of measured floods with kinematic shock waves is demonstrated.