A canal has uniform width and a bottom that is horizontal apart from a localised slowly-varying hump of height $D(x)$ whose maximum value is $D_{\text {max }}$. Far upstream the water has depth $h_{1}$ and velocity $u_{1}$. Show that the depth $h(x)$ of the water satisfies the following equation:

$$\frac{D(x)}{h_{1}}=1-\frac{h}{h_{1}}-\frac{F}{2}\left(\frac{h_{1}^{2}}{h^{2}}-1\right)$$

where $F=u_{1}^{2} / g h_{1}$.

Describe qualitatively how $h(x)$ varies as the flow passes over the hump in the three cases

$$\begin{array}{ll} \text { (i) } F<1 & \text { and } D_{\max }<D^{*} \\ \text { (ii) } F>1 & \text { and } D_{\max }<D^{*} \\ \text { (iii) } \quad D_{\max } & =D^{*}, \end{array}$$

where $D^{*}=h_{1}\left(1-\frac{3}{2} F^{1 / 3}+\frac{1}{2} F\right)$.

Calculate the water depth far downstream in case (iii) when $F<1$.