Finite-Froude lock-release rarefaction ($h_f/H=4/(F+2{)}^2$)

A semi-infinite resting layer of depth $H$ and constant reduced gravity $b$ released into a deep ambient develops a rarefaction wave satisfying $u+2\sqrt{bh}=2\sqrt{bH}$. The head closure $u_f=F\sqrt{b{h}_f}$ gives $h_f=4H/(F+2{)}^2$ and $u_f=2F\sqrt{bH}/(F+2)$. A constant-depth shelf lies between the rarefaction tail speed $u_f-\sqrt{b{h}_f}$ and the front speed $u_f$; its width is positive for every finite positive $F$.

A rectangular lock of length ${\ell }_0$ and depth $h_0$ has rarefaction head speed $-c_0$, where $c_0=\sqrt{g^{\prime }{h}_0}$. It reaches a closed rear wall at $t_r={\ell }_0/c_0$. The first reflected positive characteristic curve follows $x={\ell }_0+2c_{0}t-3c_0{t}_r^{2/3}{t}^{1/3}$ inside the undisturbed rarefaction, then crosses the uniform shelf and reaches the front at the displayed time. This marks the end of the initial constant-speed front stage in the ideal fixed-front-Froude model.