 Solution

Apply the shearing-wave ansatz to each disturbance. Acting on its phase givesThus a spatially uniform amplitude system exists whenFor a nonzero horizontal wavenumber $k=|\mathbf{k}|$, the razor-thin disk Poisson kernel follows from $({\partial }_z^2-k^2){\tilde{\Phi }}_d^{\prime }=4\pi G{\tilde{\Sigma }}^{\prime }\delta (z)$. The potential decays above and below the sheet as ${\tilde{\Phi }}_{d,m}^{\prime }{e}^{-k|z|}$. Integrating across the midplane gives the jump $-2k{\tilde{\Phi }}_{d,m}^{\prime }=4\pi G{\tilde{\Sigma }}^{\prime }$, henceSubstitute this potential and the phase evolution into the linearized equations. The barotropic shearing-wave gravitational amplitude system isThe self-gravity sign is attractive and opposes the pressure response. The uniform $k=0$ mass density shift requires its own gravitational reference and is not obtained by dividing by $k$.