= Solution
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 at
$$
\boxed{t_1=\ell/c_0,}
$$
the 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,
$$
\frac{dx_r}{dt}=u+c=\frac{12c_0+5x_r/t}{7},\qquad x_r(t_1)=-\ell.
$$
Integrating gives the <finite-reservoir reflection of a cubic-area dam-break rarefaction>:
$$
\boxed{x_r(t)=6c_0t-7c_0t_1^{2/7}t^{5/7},\qquad t\geq t_1.}
$$
The solution to the right of this curve is still the original fan. Crucially $x_f-x_r=7c_0t_1^{2/7}t^{5/7}>0$ at every finite time. Therefore the ideal inviscid dry tip remains at $x_f=6c_0t$: 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-76-finite-reservoir.png]
{title=Finite-reservoir depletion sketches and the exact leading reflected characteristic in the x–t plane}
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.
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