Gravitational potential of a displaced cylindrical interface (${\Phi }^{\prime }(R_0)=-2\pi G{\rho }_0{R}_0\eta /m,m\ge 1$)

For a uniform-mass density cylinder with surface displacement $\eta e^{im\varphi }$, the interior and exterior potential perturbations solve Laplace's equation and are proportional to $R^m$ and $R^{-m}$. Continuity of potential and the Poisson derivative jump ${\Phi }_{R,\mathrm{out}}^{\prime }-{\Phi }_{R,\mathrm{in}}^{\prime }=4\pi G{\rho }_0\eta$ give the displayed surface amplitude. The surface mass density term must be retained even though bulk incompressibility makes the interior mass density perturbation zero.