Remarks on some nonconservation problems

A horizontal channel of uniform cross section has an impervious channel bed to the left of $x = 0$ and allows infiltration at a constant rate to the right of $x = 0$. Initially there is water at constant depth and zero velocity. There are left and right moving interfaces and, between them, water with positive velocity. At a certain time there will be a water edge to the right of which there is no water in the channel. The time history of this water edge is a free boundary. The solution of this problem, which is nonconservation because mass and momentum are carried away by infiltration, is discussed below. A single equation, which is also nonconservation, has an explicit solution. The characteristics of this single equation have a geometry similar to that the ${C_2}$ characteristics of the channel problem.