Extra credit: include a Coriolis force

If we include rotation (Coriolis parameter $f$), the horizontal momentum equations become

$$\frac{\partial u}{\partial t} - f v = -\frac{\partial\phi}{\partial x},$$ (37)

$$\frac{\partial v}{\partial t} + f u = -\frac{\partial\phi}{\partial y}.$$ (38)

For plane waves with propagation purely in $x$ (or for two-dimensional $x$–$z$ problems where $\partial/\partial y=0$ so the $v$-equation decouples except for Coriolis terms) the algebra becomes lengthier but follows the same elimination procedure. One obtains the inertia-gravity-wave dispersion relation modified by $\alpha$; for the classical nonhydrostatic Boussinesq ($\alpha=1$) the dispersion relation for two-dimensional inertia-gravity waves (with $k,m$ horizontal and vertical wavenumbers) is

$$\omega^2 = \frac{f^2 k^2 + N^2 k^2}{k^2 + m^2} = f^2\frac{k^2}{k^2+m^2} + N^2\frac{k^2}{k^2+m^2}.$$

With the $\alpha$ modification the denominator becomes $\alpha^2 k^2 + m^2$, and the numerator picks up the $f^2$ contribution arising from coupling between $u$ and $v$. The resulting dispersion relation (for waves with no $y$-variation) can be written as

$$\boxed{\;\omega^2 = \frac{f^2 m^2 + N^2 k^2}{\alpha^2 k^2 + m^2}\;},$$

which recovers the non-rotating result when $f=0$ and the classical inertia-gravity relation when $\alpha=1$. (Derivation omitted for brevity but follows straightforward elimination of $u,v,\phi,b$ as above.)