= Surface gravito-inertial wave
A <surface gravito-inertial wave> combines <free surface> gravitational restoration with the <Coriolis force>. For a uniform incompressible <fluid> occupying $z<0$, take <gravity> downward, <rotation> $\boldsymbol\Omega=\Omega\mathbf e_z$ normal to the surface and $f=2\Omega$. A mode proportional to $e^{i(kx-\omega t)+qz}$ must decay downward, so $q>0$.
The linear rotating <momentum> equations and <incompressibility> give
$$
q^2=\frac{k^2\omega^2}{\omega^2-f^2}.
$$
The <kinematic boundary condition> is $w=-i\omega\eta$ at the surface. The dynamic condition is $p'=\rho g\eta$, so vertical <momentum> gives $\omega^2=gq$. Eliminating $q$ yields
$$
\omega^2(\omega^2-f^2)=g^2k^2,
\qquad
\omega^2=\frac{f^2+\sqrt{f^4+4g^2k^2}}2.
$$
For $f=2\Omega$, this is
$$
\omega^2=2\Omega^2+\sqrt{g^2k^2+4\Omega^4}.
$$
For nonzero horizontal wavenumber, this branch has $\omega^2>f^2$ and an evanescent vertical structure. With no <rotation> it reduces to the <deep-water gravity wave> relation $\omega^2=g|k|$.
The orientation and incompressible deep-fluid assumptions matter. A tilted <rotation> axis, finite depth or compressible <polytropic atmosphere> has a different <eigenvalue> problem; the displayed formula is the stated incompressible model.
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