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 over a horizontal boundary with depth $h(x, t)$ and velocity $u(x, t)$, where $g$ is the acceleration due to gravity.

Show that the Riemann invariants $u \pm 2\left(c-c_{0}\right)$ are constant along characteristics $C_{\pm}$satisfying $d x / d t=u \pm c$, where $c(h)$ is the linear wave speed and $c_{0}$ denotes a reference state.

An initially stationary pool of fluid of depth $h_{0}$ is held between a stationary wall at $x=a>0$ and a removable barrier at $x=0$. At $t=0$ the barrier is instantaneously removed allowing the fluid to flow into the region $x<0$.

For $0 \leqslant t \leqslant a / c_{0}$, find $u(x, t)$ and $c(x, t)$ in each of the regions

$$\begin{gathered} \text { (i) } \quad c_{0} t \leqslant x \leqslant a \\ \text { (ii) }-2 c_{0} t \leqslant x \leqslant c_{0} t \end{gathered}$$

explaining your argument carefully with a sketch of the characteristics in the $(x, t)$ plane.

For $t \geqslant a / c_{0}$, show that the solution in region (ii) above continues to hold in the region $-2 c_{0} t \leqslant x \leqslant 3 a\left(c_{0} t / a\right)^{1 / 3}-2 c_{0} t$. Explain why this solution does not hold in $3 a\left(c_{0} t / a\right)^{1 / 3}-2 c_{0} t<x<a .$