The shallow-water equations

$$\frac{\partial h}{\partial t}+u \frac{\partial h}{\partial x}+h \frac{\partial u}{\partial x}=0, \quad \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial h}{\partial x}=0$$

describe one-dimensional flow in a channel with depth $h(x, t)$ and velocity $u(x, t)$, where $g$ is the acceleration due to gravity.

(i) Find the speed $c(h)$ of linearized waves on fluid at rest and of uniform depth.

(ii) Show that the Riemann invariants $u \pm 2 c$ are constant on characteristic curves $C_{\pm}$of slope $u \pm c$ in the $(x, t)$-plane.

(iii) Use the shallow-water equations to derive the equation of momentum conservation

$$\frac{\partial(h u)}{\partial t}+\frac{\partial I}{\partial x}=0$$

and identify the horizontal momentum flux $I$.

(iv) A hydraulic jump propagates at constant speed along a straight constant-width channel. Ahead of the jump the fluid is at rest with uniform depth $h_{0}$. Behind the jump the fluid has uniform depth $h_{1}=h_{0}(1+\beta)$, with $\beta>0$. Determine both the speed $V$ of the jump and the fluid velocity $u_{1}$ behind the jump.

Express $V / c\left(h_{0}\right)$ and $\left(V-u_{1}\right) / c\left(h_{1}\right)$ as functions of $\beta$. Hence sketch the pattern of characteristics in the frame of reference of the jump.