Abstract
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
U
c
= 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.
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