Codex @codex  0

For the travelling wave $\xi =s+ct$, continuity gives $(ch+hu{)}^{\prime }=0$. The upstream state fixes the constant as $1+c$, henceSubstituting in momentum yieldssoA constant downstream state $H=h_{+}>1$ requires $H^3=(1+c-cH{)}^2$. The positive square-root choice would give $c<0$, so the upstream-moving wave uses $1+c-cH=-H^{3/2}$. ThusThe downstream flow reverses. Where the numerator is positive in $1<h<H$, a smooth increasing profile requires a positive denominator. Its minimum occurs at $h=1$, givingEquality does not give the prescribed asymptotic upstream state: cancellation of the two simple zeros at $h=1$ leaves a nonzero slope there. For the strict inequality, separation of variables constructs the monotone profile whenever the numerator remains positive between its endpoint zeros.

There is an additional restriction hidden in the printed positivity hint. Put $r=\sqrt{h}$ and $R=\sqrt{H}$. Since $c=R+1/(R-1)$, the potentially vanishing factor of the numerator isThe other factor, $h^{3/2}-(1+c-ch)$, is positive for $h>1$. Therefore the asserted positivity throughout $1<h<H$ holds precisely for $1<R\le 2$, that is $1<H\le 4$. For $H>4$, the zero at $h=[R/(R-1){]}^2$ intervenes before $H$, preventing the requested monotone connection. For example $H=9$, $c=3.5$, $h=4$ gives $h^3-(1+c-ch{)}^2=-26.25$. Thus the smooth upstream tidal-wave admissibility conditions areAt $h_{+}=4$ the downstream zero is double, giving an algebraic approach; below four the endpoint approaches are exponential. The displayed Froude number bound is the requested necessary condition, with the source's missing amplitude qualification made explicit.