Surface modes of a self-gravitating incompressible slab
= Surface modes of a self-gravitating incompressible slab
{title2=$\omega_{e,o}^2(k)$}
For $q=kH$, the even and odd scalar-potential branches have $\omega_e^2=(2\pi G\Sigma/H)[q\tanh q-1/(1+\coth q)]$ and $\omega_o^2=(2\pi G\Sigma/H)[q\coth q-1/(1+\tanh q)]$. Even potential symmetry moves the surfaces oppositely; odd potential symmetry bends them together. Hydrostatic free-surface <pressure> gives the positive term and perturbed self-gravity the negative term.