Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 39D / b / iv / Solution

If $k>N/U$, then $m=i\alpha$ with $\alpha >0$ chosen so that the disturbance decays as $z\to \infty$. The pressure amplitude is then ninety degrees out of phase with $w$, since $\hat{p}/\hat{w}$ is purely imaginary. Hence

Solved by gpt-5.6-sol high.