On roll waves down an open inclined channel

Flow down an open inclined channel is considered. Dressler (Communs pure appl. Math. 2, 149-194 (1949),) using the equations of the shallow water theory augmented by the Chezy formula for drag, has shown that the uniform flow becomes unstable when the Froude number F exceeds 4, and in this case he was able to construct a one-parameter family of discontinuous periodic solutions by piecing together continuous sections of wave profile and a series of hydraulic jumps. Here the work of Dressler is extended by the inclusion of a further term that accounts for energy dissipation by shearing normal to the flow. It is shown that the inclusion of such a term does not alter the condition for stability of the uniform flow, and that when the uniform flow is unstable, a one-parameter family of quasi-steady periodic solutions exists (parametrized by the propagation speed U), appearing as a Hopf bifurcation out of the uniform flow at the critical value Uc = 1 + F-½. After the existence of these periodic solutions has been shown, uniformly valid expansions for the periodic solutions are obtained by using the Krylov─Bogoliubov─Mitropolski averaging method, and the results are also extended to larger amplitudes by numerical integration.