Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 4 / vi / Solution

The force on the mean flow and the wave-activity tendency obeyHence 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 $\hat{\omega }=-kU$, but upward propagation now selects $km<0$. For $k>0$ one takes $m<0$, and thereforeThe 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.