Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 52 / 4 / b / Solution

At $V=0$, the particle budget gives $\varphi =1$, so $g^{\prime }=g_0^{\prime }$ is constant. The primitive momentum conservation equation integrates toIntroduce $q=Q/Q_0=1-Wx/Q_0$, $y=h/h_0$ and $\epsilon ={\mathrm{Fr}}_0^2=Q_0^2/(g_0^{\prime }{h}_0^3)\ll 1$. The exact relation on the subcritical flow branch isExpanding uniformly for $0\le q\le 1$ givesThe depth increases slightly because horizontal kinetic energy decreases as fluid is removed. In fact the depth equation has $h^{\prime }\ge 0$ on this branch, andThus the small-Froude number approximation becomes better downstream, including as the flux vanishes. At leading order $u=(Q_0-Wx)/h_0$.