= 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 $p'\propto\cosh(Qz/H)$ and $\sinh(Qz/H)$ and obey $\omega^2=\Omega_z^2Q\tanh Q$ and $\omega^2=\Omega_z^2Q\coth Q$, respectively. At large $kH$, both become free-surface gravity waves with
$$
\omega^2\sim\tfrac12\left[\kappa^2+\sqrt{\kappa^4+4\Omega_z^4(kH)^2}\right]=\Omega_z^2kH+\tfrac12\kappa^2+O((kH)^{-1}).
$$
Their opposite-parity splitting is exponentially small as the two interfaces decouple.
// Target: astrophysics.bigb
Back to article page
