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Assume nonzero upstream speed, so $F>0$. For steady constant-density flow, mass conservation gives the constant volume fluxThe Bernoulli equation states that $u^2/2+p/\rho +gz$ is constant along each streamline of a steady inviscid constant-density flow under gravity. It need not have the same value across unrelated streamlines in a general rotational flow. Here uniform upstream velocity supplies the same head, and the slowly varying, nearly horizontal channel permits hydrostatic pressure $p=p_{\mathrm{atm}}+\rho g(h-z)$. Neglecting losses, the Bernoulli equation therefore givesSubstitute $u=UWH/(wh)$ and set $s=h/H$, $F=U^2/(gH)$, the square of the upstream Froude number. Rearrangement givesFor positive depth, this cubic increases to its maximum at $s_{\ast }=(F+2)/3$ and then decreases until it reaches zero at $(F+2)/2$. Its maximum is $G(s_{\ast })=(F+2{)}^3/(27F)$. Consequently the critical width of a rectangular open-channel contraction isAt equality the two positive-depth branches coalesce and the local Froude number equals one.

When $w>w_c$ everywhere, a smooth solution connected to the upstream state cannot change branch without passing through that coalescence. At $w=W$ the upstream depth is $s=1$, with $G^{\prime }(1)=2(F-1)/F$. If $F<1$, this point is on the descending, subcritical branch. At a wider location $w>W$, the left-hand side falls below $1=G(1)$, so the depth on that branch must increase: $h>H$ for subcritical upstream flow $F<1$. If $F>1$, the upstream state is on the ascending, supercritical branch and decreasing the left-hand side decreases $s$: $h<H$ for supercritical upstream flow $F>1$.

For $F=1$, $w_c=W$ and the upstream state is critical. A literally constant-width upstream region then conflicts with the strict assumption $w>w_c$ everywhere. If the upstream state is interpreted only as an asymptotic critical limit, a widened channel has one solution with $h<H$ and one with $h>H$; a further branch condition is required. Merely solving the cubic at one location, without retaining the upstream-connected branch, would not determine the physically continued depth.