= Solution
The volume alone does not specify the early release geometry. For a concrete sketch, use a rectangular lock initially occupying $0<x<\ell_0$ with depth $h_0=V/\ell_0$, at rest, and remove its gate at $t=0$. Let $c_0=\sqrt{g'h_0}$ and impose a <gravity-current front condition> $u_f=F\sqrt{g'h_f}$, with fixed $F$. The <Benjamin deep-ambient front condition> gives $F=\sqrt2$ for the usual deep-ambient idealization.
A <rarefaction wave> propagates back from the release. Before information returns from the closed wall, its invariant is $u+2c=2c_0$. With $s=(x-\ell_0)/t$, the fan has $u=2(c_0+s)/3$, $c=(2c_0-s)/3$. The front-adjacent shelf has
$$
c_f=\frac{2c_0}{F+2},\qquad u_f=F c_f,\qquad h_f=c_f^2/g'.
$$
The fan extends from $s=-c_0$ to $s=u_f-c_f$; a uniform shelf connects its tail to the finite-depth head at $x=\ell_0+u_ft$. A small nonhydrostatic head displaces ambient fluid; the hydrostatic interior model does not resolve its overturning structure.
The fan head first reaches the rear wall at $t_r=\ell_0/c_0$. The wall condition is $u=0$, and the returned positive <characteristic curve> begins to alter the interior. In the still-undisturbed fan it solves $dx/dt=(4c_0+(x-\ell_0)/t)/3$, hence
$$
x=\ell_0+2c_0t-3c_0t_r^{2/3}t^{1/3}.
$$
It reaches the fan's uniform shelf at $t_*=t_r[(F+2)/2]^{3/2}$, then propagates with speed $u_f+c_f$ to the head at $t_{\rm hit}=2t_*$. These times refer to this particular initial lock and front closure. After that first arrival, the finite supply influences the front: the constant-speed slumping stage gives way to deceleration and eventually the <finite-volume inertial gravity current> with $L\propto t^{2/3}$, depth proportional to $t^{-2/3}$ and speed proportional to $t^{-1/3}$. The initial fan and reflected signal are shown below; later characteristic trajectories change as the current adjusts.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-43-lock-release.png]
{title=Finite-lock gravity current, initial rarefaction, reflected information and late-time spreading}
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