Surface modes of an incompressible disk

For ${\omega }^2>{\kappa }^2$, set $Q=kH\omega /\sqrt{{\omega }^2-{\kappa }^2}$. Even-pressure and odd-pressure modes have $p^{\prime }\propto \cosh (Qz/H)$ and $\sinh (Qz/H)$ and obey ${\omega }^2={\Omega }_z^{2}Q\tanh Q$ and ${\omega }^2={\Omega }_z^{2}Q\coth Q$, respectively. At large $kH$, both become free-surface gravity waves withTheir opposite-parity splitting is exponentially small as the two interfaces decouple.

// Target: astrophysics.bigb