Free boundaries in one-dimensional flow

Several problems are discussed regarding flow in a horizontal channel. The channel bed may be impervious to infiltration, or may allow a constant rate of infiltration, or may be impervious to the left of some point in the channel bed and allow a constant rate of infiltration to the right. The free boundary is the time history of the motion of a piston at which the water height $\mu \left ( x \right )$ has been specified. The case $\mu \left ( x \right ) \equiv 0$ is the dam breaking problem. In the dam breaking problem in which the channel bed allows a constant rate of infiltration a more general form of a centered rarefaction wave is required, i.e., the characteristics are not straight lines. Two problems are formulated for channel beds that are partly impervious and partly porous. Shock formation may arise here. This possibility is exhibited in a single nonconservation equation.