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

The exact no-penetration condition on the stationary boundary is that the velocity be tangent to it. Linearized at $z=0$, it isThe terrain is stationary, so $\omega =0$ and $\hat{\omega }=-kU$. The dispersion relation givesA vertically propagating wave therefore exists if and only ifThe upward radiation condition $\hat{\omega }m<0$ selects $km>0$ for $U>0$. Taking $k>0$ gives $m>0$. Lines of constant phase satisfyso they tilt upstream as height increases. The discarded sign of $m$ gives the mirror-image phase tilt and downward energy propagation.