Past exam of the mathematics and the environment course of the University of Oxford / 2011 / 3 / ii / Solution

Let $B$ be the constant canal width, large compared with water depth. Then $A\simeq Bh$ and $l\simeq B$, so friction is $f{u}^2/h$. Choose a reference uniform water depth $H$, normal-flow speed $U$ satisfying $f{U}^2/H=gS$, and scalesThe continuity equation becomes ${\hat{h}}_{\hat{t}}+(\hat{h}\hat{u}{)}_{\hat{s}}=0$. Dividing momentum by $gS$ and dropping hats givesThe Froude number compares velocity to long gravity-wave speed. At the reference state, $F<1$ is subcritical flow, $F=1$ critical flow, and $F>1$ supercritical flow. In dimensional variables the wave characteristics travel at $u\pm \sqrt{gh}$: one can propagate upstream in subcritical flow, while both move downstream in supercritical flow. Away from the reference state the local Froude number is $F|u|/\sqrt{h}$.