= Solution
The force on the mean flow and the wave-activity tendency obey
$$
-\partial_z\overline{u'w'}
=\partial_z\mathcal F_{\mathcal A}
=-\partial_t\mathcal A.
$$
Hence growth of positive wave activity produces a negative mean-flow force, while decay deposits negative momentum into the mean flow. Since part iii found a negative force for a growing, upward-radiating wave with $U>0$, its wave activity is positive.
If $U<0$, stationarity still gives $\widehat\omega=-kU$, but upward propagation now selects $km<0$. For $k>0$ one takes $m<0$, and therefore
$$
\overline{u'w'}=-\frac12mkU^2h_0^2>0.
$$
The integrated force reverses and accelerates the mean flow in the positive $x$ direction. The activity of a growing wave correspondingly has the opposite sign and is negative.
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