= Shallow-water rarefaction from a receding wall
{title2=$u=\frac23(x/t-c_0),\quad c=\frac13(x/t+2c_0)$}
A resting <shallow water> layer has depth $h_0$ and <wave speed> $c_0=\sqrt{gh_0}$. A vertical wall receding into negative $x$ at speed $0<V<2c_0$ creates a <simple wave> with invariant $u-2c=-2c_0$. Inside the expansion fan $x/t=u+c$, proving the displayed profiles. The wall-adjacent state is $u=-V$, $c=c_0-V/2$; the fan occupies $c_0-3V/2<x/t<c_0$, with the resting state ahead. The wall lies at $x/t=-V$, strictly to the left of the fan tail because $c_0-V/2>0$. Thus the assumption on wall speed is exactly the condition for positive depth and continued contact between fluid and wall. The depth follows from $h=c^2/g$.
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