= Smooth upstream tidal-wave admissibility
{title2=$1<H\leq4,\quad F<(H-1)/(H+H^{3/2})$}
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 $h'=[h^3-(1+c-ch)^2]/[h^3-(1+c)^2F^2]$. 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 $R\leq2$, giving the amplitude restriction. At $H=4$ that zero is double; otherwise the profile cannot cross an intervening equilibrium.
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