= Solution
Put $R=h_2/h_1>1$. The squared <Froude numbers> follow directly from the discharge relation:
$$
\boxed{\mathrm{Fr}_1^2=\frac{u_1^2}{gh_1}=\frac{R(1+R)}2>1,\qquad \mathrm{Fr}_2^2=\frac{u_2^2}{gh_2}=\frac{1+R}{2R^2}<1}.
$$
Thus the jump connects <supercritical flow> to <subcritical flow>. Long <surface gravity waves> have speed $\sqrt{gh}$ relative to the fluid, so the <characteristic speeds> in the stationary frame are $u\pm\sqrt{gh}$. Upstream both point with the flow; downstream one points against it. The jump changes whether long-wave information can propagate upstream through the local flow.
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