= Solution
For a nonzero <wavevector> $\mathbf k$, substitute the <plane wave> into part (c). Each time derivative contributes $-i\omega$, each Laplacian contributes $-|\mathbf k|^2$, and each second $z$ derivative contributes $-k_z^2$. A nonzero amplitude therefore requires
$$
\omega^2|\mathbf k|^2-4\Omega^2k_z^2=0.
$$
If $\theta$ is the angle between the <wavevector> and the rotation axis, $\cos\theta=k_z/|\mathbf k|$, and
$$
\boxed{\omega=\pm2\Omega\cos\theta.}
$$
This is the <inertial wave> dispersion relation. Incompressibility also requires $\mathbf k\cdot\mathbf U_0=0$; the original first-order equations select the corresponding transverse polarizations. A horizontal wavevector gives the zero-frequency geostrophic case.
Back to article page
