= Finite-volume inertial gravity current
{title2=$L^3=\frac{9F^2g'V}{4(1-F^2/6)}(t-t_v)^2$}
For a one-sided fixed-volume <gravity current> in a unit-width channel with a closed rear wall, use front closure $\dot L=F\sqrt{g'h_f}$ with $0<F<2$. An exact late-time <similarity solution> has $u=\dot L x/L$, $h=h_f[1-F^2/4+(F^2/4)(x/L)^2]$ and $h_fL=V/(1-F^2/6)$. Substituting into the <shallow water equations> proves the radius law. The <Benjamin deep-ambient front condition> has $F=\sqrt2$. The virtual time origin $t_v$ is fixed by matching to the finite release, not by volume alone.
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