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

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.

Before a signal traveling left at speed $c_0$ can reach the wall, the portion $-a<x<-c_{0}t$ remains at rest with $c=c_0$, $h=h_0$. In the rarefaction fan, right-running characteristics bring the unchanged invariant $Q_{+}=2c_0$. The left-running family originates at the removed dam and has speed $u-c=x/t$. Solving these two relations givesAt the dry front $c=0$, so its speed and location are $2c_0$ and $2c_{0}t$, consistent with the stated constant-speed condition. Therefore, for $0<t<a/c_0$,The velocity is zero in the untouched region and $2(c_0+x/t)/3$ in the fan. Ahead of the front there is no fluid, so $h=c=0$ and a fluid velocity need not be assigned. In particular the wall remains at $u=0,c=c_0$ until time $a/c_0$.

Inside the fan, $u+c=(4c_0+x/t)/3$. Hence a right-running characteristic satisfiesMultiplication by the integrating factor $t^{-1/3}$ givesIntegration yields $x_{+}=2c_{0}t+C{t}^{1/3}$. Requiring $x_{+}(\tau )=-c_0\tau$ determines $C=-3c_0{\tau }^{2/3}$, soThe invariant $Q_{+}=2c_0$ stays constant along it until it enters a region affected by new boundary data.

The head of the rarefaction reaches the wall at $(-a,t_0)$ with $t_0=a/c_0$. The right-running characteristic through that event is obtained by setting $\tau =t_0$ in the preceding formula:It bounds the region reached by the first wall-induced change in $Q_{+}$. To its right the characteristic information remains the original dam-break information, soBehind it, the impermeable wall must keep $u(-a,t)=0$. Continuing the original fan there would instead give $2(c_0-a/t)/3>0$ for $t>t_0$. Reflected characteristic data therefore change $Q_{+}$ and invalidate the simple fan formula. The shaded region in the characteristic diagram represents this changed domain of dependence; it includes $0<x<x_b(t)$ once that interval becomes nonempty. Its first positive extent occurs when $t/t_0>(3/2{)}^{3/2}$.