= Solution
Long-wave geometry makes vertical acceleration small, so <pressure> is approximately hydrostatic, $p=p_{\rm atm}+\rho g(h-z)$. The leading horizontal <velocity> is nearly depth-independent. Integrating <incompressibility> through the depth and using the impermeable bed and kinematic free-surface condition gives $\boxed{h_t+(hu)_x=0}$. The horizontal <Euler equation> then gives $\boxed{u_t+uu_x+gh_x=0}$.
Linearizing about a uniform resting layer gives wave speed $\boxed{c=\sqrt{gh}}$. Since $h=c^2/g$, the nonlinear equations can be written $c_t+uc_x+(c/2)u_x=0$ and $u_t+uu_x+2cc_x=0$. Combining them verifies
$$
\boxed{[\partial_t+(u\pm c)\partial_x](u\pm2c)=0.}
$$
Let $c_0=\sqrt{gh_0}$ and $V\geq0$ be the receding wall's speed, so its position is $x=-Vt$ and its fluid <velocity> condition is $u=-V$. The outward-moving rarefaction has the other invariant fixed at its resting value, $u-2c=-2c_0$. In its varying part, <self-similarity> gives $x/t=u+c$. Therefore
$$
\boxed{c=\frac{x/t+2c_0}{3},\qquad u=\frac23(x/t-c_0).}
$$
The wall-adjacent state has $u=-V$ and $c=c_w=c_0-V/2$. Its right-going <characteristic curve> speed is $c_0-3V/2$, whereas the undisturbed head travels at $c_0$. The complete solution in the fluid $x>-Vt$ is
$$
\boxed{(u,c)=\begin{cases}
(-V,c_0-V/2),&-V<x/t<c_0-3V/2,\\
(\tfrac23(x/t-c_0),\tfrac13(x/t+2c_0)),&c_0-3V/2\leq x/t\leq c_0,\\
(0,c_0),&x/t>c_0.
\end{cases}}
$$
Both edges match continuously and the wall <velocity> condition holds. The assumption $V^2<4gh_0$ means $V<2c_0$, giving positive wall depth and placing the fan tail to the right of the wall by speed $c_w>0$. The depth everywhere is $h=c^2/g$. This is a <shallow-water rarefaction from a receding wall>.
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