= Finite-reservoir reflection of a cubic-area dam-break rarefaction
{title2=$x_r=6c_0t-7c_0t_1^{2/7}t^{5/7},\quad t_1=\ell/c_0$}
An impermeable wall at $x=-\ell$ first receives the upstream-moving head at $t_1$. The leading downstream <characteristic curve> of the reflected adjustment obeys $\dot x=u+c=(12c_0+5x/t)/7$ in the unmodified <rarefaction wave>, giving the displayed curve. It stays behind the inviscid dry tip $x_f=6c_0t$ at every finite time. The finite reservoir drains and its upstream depth decreases, while only the region behind $x_r$ has yet received the wall signal. Bottom drag or other tip physics can change the front law.
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