Dam-break flows at a jump in the width of a rectangular channel

The problem of dam-break flow at a jump in the width of a rectangular channel is studied in the first shallow-water approximation. Two cases where the upstream channel width is greater or smaller than the downstream channel width are considered. It is shown that in the first case, the problem is uniquely solvable under the assumption that the total energy of the flow is conserved at the jump in the channel width, and in the second case, a solution of the problem for some initial data exists only provided that the total energy of the flow is lost at this jump.