= Solution
For the <travelling wave> $\xi=s+ct$, continuity gives $(ch+hu)'=0$. The upstream state fixes the constant as $1+c$, hence
$$
\boxed{u=-c+\frac{1+c}{h},\qquad u'=-\frac{1+c}{h^2}h'.}
$$
Substituting in <momentum> yields
$$
-\frac{F^2(1+c)^2}{h^3}h'=1-\frac{(1+c-ch)^2}{h^3}-h',
$$
so
$$
\boxed{h'=\frac{h^3-(1+c-ch)^2}{h^3-(1+c)^2F^2}.}
$$
A 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 $1+c-cH=-H^{3/2}$. Thus
$$
\boxed{c=\frac{H^{3/2}+1}{H-1},\qquad u(H)=-\sqrt H.}
$$
The downstream flow reverses. Where the numerator is positive in $1<h<H$, a smooth increasing profile requires a positive denominator. Its minimum occurs at $h=1$, giving
$$
1-(1+c)^2F^2>0,\qquad\boxed{F<\frac1{1+c}=\frac{H-1}{H+H^{3/2}}.}
$$
Equality 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.
\b[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 is
$$
h^{3/2}+1+c-ch=(r+1)(r-R)\left(r-\frac{R}{R-1}\right).
$$
The other factor, $h^{3/2}-(1+c-ch)$, is positive for $h>1$. Therefore the asserted positivity throughout $1<h<H$ holds precisely for $1<R\leq2$, that is $1<H\leq4$. 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 are
$$
\boxed{1<h_+\leq4,\qquad F<\frac{h_+-1}{h_++h_+^{3/2}}.}
$$
At $h_+=4$ 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.
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