 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 requiresIf $\theta$ is the angle between the wavevector and the rotation axis, $\cos \theta =k_z/|\mathbf{k}|$, andThis 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.