Solution
= Solution
Horizontal <momentum> and incompressibility give
$$
\widehat p=\rho_0\frac{\omega}{k}\widehat u
=-\rho_0\frac{\omega m}{k^2}\widehat w.
$$
Therefore
$$
\langle\overline I_z\rangle_0
=\frac12\operatorname{Re}(\widehat p\widehat w^*)
=-\frac{\rho_0\omega m}{2k^2}w_0^2.
$$
For upward radiation take $\omega=-Uk$ and $m=\sqrt{N^2/U^2-k^2}>0$, obtaining
$$
\langle\overline I_z\rangle_0
=\frac12\rho_0U^2kh_0^2\sqrt{N^2-U^2k^2}>0.
$$