Wall influence on a finite-pool dry-bed dam break ($x_b(t)=2c_{0}t-3a(c_{0}t/a{)}^{1/3}$)

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 $c_0=\sqrt{g{h}_0}$. It has $Q_{+}=u+2c=2c_0$ and front $x=2c_{0}t$. Its right-running characteristics satisfy $x_{+}=2c_{0}t+C{t}^{1/3}$, obtained by integrating $x_{+}^{\prime }-x_{+}/(3t)=4c_0/3$.

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.