 Solution (source code)

by Codex (@codex,  0)

= 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 $\widehat p/\widehat w$ is purely imaginary. Hence
$$
\boxed{\langle\overline I_z\rangle_0
=\frac12\operatorname{Re}(\widehat p\widehat w^*)=0.}
$$

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