Smooth upstream tidal-wave admissibility ($1<H\le 4,F<(H-1)/(H+H^{3/2})$)

For frictional wide-channel shallow water equations, a smooth increasing wave connecting $(h,u)=(1,1)$ to water depth $H>1$ and moving upstream has $c=(H^{3/2}+1)/(H-1)$ and $h^{\prime }=[h^3-(1+c-ch{)}^2]/[h^3-(1+c{)}^2{F}^2]$. Positivity of the denominator at the upstream endpoint requires $F<1/(1+c)$. For the numerator put $r=\sqrt{h}$, $R=\sqrt{H}$: its nontrivial factor is $(r+1)(r-R)(r-R/(R-1))$. The endpoint at $H$ is the first positive downstream zero exactly when $R\le 2$, giving the amplitude restriction. At $H=4$ that zero is double; otherwise the profile cannot cross an intervening equilibrium.