Past exam of the theoretical mechanics course of the University of Oxford / 2010 / B / 6 / a

For positive depth put $c=\sqrt{gh}$. The equations becomeAdding or subtracting twice the second equation to the first givesThus the two characteristic speeds are $\boxed{dx/dt=u\pm c}$ and the corresponding conserved Riemann invariants are $\boxed{Q_{\pm }=u\pm 2c}$. At a dry front $c=0$ the two speeds coincide; the positive-depth formulas are applied up to that limit.