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

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 $t/t_0>(3/2{)}^{3/2}$.