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For an incompressible perturbation, $\Delta =0$, so $\Psi =g{\xi }_z$. The vertical equation givesIncompressibility together with the horizontal equation also givesEquating the two expressions yieldsThe positive-frequency branch is the surface gravito-inertial waveIts vertical displacement decays as $e^{{\omega }^{2}z/g}$ into the atmosphere $z<0$.

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 $\mathbf{\Omega }=\Omega {\mathbf{e}}_z$ 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 giveThe kinematic boundary condition is $w=-i\omega \eta$ at the surface. The dynamic condition is $p^{\prime }=\rho g\eta$, so vertical momentum gives ${\omega }^2=gq$. Eliminating $q$ yieldsFor $f=2\Omega$, this isFor 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.