Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 76 / 4 / c

Treat the newly placed sand barrier as impermeable during the initial release. The downstream isolated reservoir occupies $-\ell <x<0$ with depth $H$ and has finite volume $\alpha H^3\ell /3$. The remaining upstream water stays behind the barrier; it does not continuously feed the released region. Assume $\ell \gg H$ so that the shallow-water approximation is appropriate.

Until the rarefaction head reaches the barrier atthe previous dam-break solution is unchanged on the released side. The wall imposes $u(-\ell ,t)=0$. After $t_1$, a reflected downstream family adjusts the flow near the wall. It is a lowering/depletion adjustment, not a new upstream supply. The upstream depth decreases, the finite water volume spreads downstream, and the unmodified simple-wave region shrinks relatively.

The leading characteristic of this adjustment can be determined exactly without solving the whole reflected region. In the still-unmodified fan,Integrating gives the finite-reservoir reflection of a cubic-area dam-break rarefaction:The solution to the right of this curve is still the original fan. Crucially $x_f-x_r=7c_0{t}_1^{2/7}{t}^{5/7}>0$ at every finite time. Therefore the ideal inviscid dry tip remains at $x_f=6c_{0}t$: finite reservoir volume does not by itself justify an immediate change to that front law. Only a vanishingly small amount of fluid occupies its very thin forward tail at late times. Bottom drag, leakage or other front physics changes this conclusion.

The profiles behind the reflected edge are schematic; there the two characteristic families interact and the original one-invariant fan no longer supplies the complete fields. The x–t diagram distinguishes the wall, the incident head, the exact reflected leading edge and the unaffected dry-tip trajectory.