Waveless gravity flow over an inclined step

Waveless gravity flows over an inclined step of heightH are studied. The cases $H>0$ and $H<0$ are considered. It is always assumed that the flow is supercritical at infinity on the right, which ensures the existence of a solution with no downstream wave. For the case of subcritical flow, the relation between the Froude number and the step height is determined that ensures a waveless regime. An approximate analytical relation between the step height and the Froude number is obtained from an analysis of numerical data. This dependence is shown to be nearly identical for steps of any slopes. For the case of supercritical flow, it is established that the problem has a two-parameter set of solutions. For the case $H<0$, approximate analytical formulas for the free-surface shape are obtained.