= Solution
Let $B$ be the constant canal width, large compared with <water depth>. Then $A\simeq Bh$ and $l\simeq B$, so <friction> is $fu^2/h$. Choose a reference uniform <water depth> $H$, normal-flow speed $U$ satisfying $fU^2/H=gS$, and scales
$$
\ell=H/S,\qquad t_0=\ell/U,\qquad h=H\widehat h,\quad u=U\widehat u,\quad s=\ell\widehat s,\quad t=t_0\widehat t.
$$
The <continuity equation> becomes $\widehat h_{\widehat t}+(\widehat h\widehat u)_{\widehat s}=0$. Dividing <momentum> by $gS$ and dropping hats gives
$$
\boxed{h_t+(hu)_s=0,\qquad F^2(u_t+uu_s)=1-\frac{u^2}{h}-h_s,\qquad F=\frac{U}{\sqrt{gH}}=\sqrt{\frac Sf}.}
$$
The <Froude number> compares <velocity> to long gravity-<wave speed>. At the reference state, $F<1$ is <subcritical flow>, $F=1$ critical flow, and $F>1$ <supercritical flow>. In dimensional variables the <wave> characteristics travel at $u\pm\sqrt{gh}$: one can propagate upstream in subcritical flow, while both move downstream in supercritical flow. Away from the reference state the local <Froude number> is $F|u|/\sqrt h$.
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