Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 1 / 17D / i / Solution

The upstream flow is subcritical flow, with $h_1>h_c$. On this branch $dD/dh<0$, so the depth decreases as the bed rises; the velocity $Q/h$ increases. Since the crest height is below $D^{\ast }$, the depth remains above $h_c$ and never becomes critical. Over the falling side of the hump the depth increases again to $h_1$ downstream. The free surface falls over the hump: $d(h+D)/dD=-{\mathrm{Fr}}^2/(1-{\mathrm{Fr}}^2)<0$ on the subcritical branch.