 Self-gravitating incompressible slab

A uniform mass density slab with free surfaces at $z=\pm H$ has ${\rho }_0=\Sigma /(2H)$, internal gravity $4\pi G{\rho }_{0}z$ and hydrostatic pressure $2\pi G{\rho }_0^2(H^2-z^2)$ above ambient pressure. Its surface modes of a self-gravitating incompressible slab separate into thickness and bending parities.