= Wall influence on a finite-pool dry-bed dam break
{title2=$x_b(t)=2c_0t-3a(c_0t/a)^{1/3}$}
A flat pool of depth $h_0$ between a wall at $-a$ and a removed dam at zero initially develops the <shallow water equations> rarefaction $u=2(c_0+x/t)/3$, $c=(2c_0-x/t)/3$, where $c_0=\sqrt{gh_0}$. It has $Q_+=u+2c=2c_0$ and front $x=2c_0t$. Its right-running characteristics satisfy $x_+=2c_0t+Ct^{1/3}$, obtained by integrating $x_+'-x_+/(3t)=4c_0/3$.
The first rarefaction signal reaches the wall at $(-a,a/c_0)$. The right-running characteristic through that event is the displayed curve. To its right the original fan remains valid; to its left reflected wall data alter the invariant. Continuing the original fan to the wall after that time would give nonzero wall velocity, contradicting impermeability. This is a concrete <domain of dependence> calculation for a nonlinear hyperbolic system.
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