 Surface modes of a self-gravitating incompressible slab

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.