= Solution
At the crest the two depth branches meet at \b[$h=h_c=h_1F^{1/3}$]. For a smooth nondegenerate maximum of the bed, $D^*-D$ is proportional to $(x-x_c)^2$, while expansion of the depth relation gives $D^*-D\sim3(h-h_c)^2/(2h_c)$. A differentiable flow through the crest therefore switches branches rather than following the cusp obtained by staying on the same branch. Subcritical upstream flow becomes supercritical downstream; supercritical upstream flow becomes subcritical downstream.
When $F<1$ and the bed returns to zero, put $y=h_{\rm down}/h_1$. The head equation becomes $2y^3-(2+F)y^2+F=0$, or
$$
(y-1)(2y^2-Fy-F)=0.
$$
The root $y=1$ is the original subcritical branch. The other positive root is smaller and lies on the supercritical branch, giving
$$
\boxed{h_{\rm down}=\frac{h_1}{4}\left(F+\sqrt{F^2+8F}\right).}
$$
This is the smooth transcritical, shock-free solution. A downstream <hydraulic jump> would require further downstream conditions and energy loss. A degenerate flat crest can also require a branch-selection condition; the standard critical-control interpretation uses a nondegenerate smooth crest. For $F=1$, $D^*=0$, so a positive hump is incompatible with the prescribed upstream critical head. If $F=0$, the head equation is the still-water relation $h+D=h_1$; a crest of height $h_1$ dries out, so the wet transcritical-flow interpretation requires $F>0$.
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