Surface density perturbation of a displaced uniform interface (${\rho }^{\prime }={\rho }_0{\eta }_{+}\delta (z-H)-{\rho }_0{\eta }_{-}\delta (z+H)$)

Moving a uniform-density boundary produces a delta-function sheet in the Eulerian mass density perturbation. For a slab displaced vertically by ${\eta }_{\pm }$, ${\rho }^{\prime }={\rho }_0{\eta }_{+}\delta (z-H)-{\rho }_0{\eta }_{-}\delta (z+H)$. Poisson equation then gives a continuous gravitational potential with derivative jumps fixed by those sheets.