Past exam of the theoretical mechanics course of the University of Oxford / 2007 / B / 7

Long-wave geometry makes vertical acceleration small, so pressure is approximately hydrostatic, $p=p_{\mathrm{atm}}+\rho g(h-z)$. 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 $\boxed{u_t+u{u}_x+g{h}_x=0}$.

Linearizing about a uniform resting layer gives wave speed $\boxed{c=\sqrt{gh}}$. Since $h=c^2/g$, the nonlinear equations can be written $c_t+u{c}_x+(c/2)u_x=0$ and $u_t+u{u}_x+2c{c}_x=0$. Combining them verifiesLet $c_0=\sqrt{g{h}_0}$ and $V\ge 0$ 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 $x/t=u+c$. ThereforeThe 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 $x>-Vt$ isBoth edges match continuously and the wall velocity condition holds. The assumption $V^2<4g{h}_0$ 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.