 Gravitational potential of a sinusoidal sheet

A sinusoidal surface-density component ${\Sigma }_k\cos (kx)$ on $y=0$, with $k>0$ and translational invariance in the remaining direction, produces the displayed gravitational potential. Laplace's equation away from the sheet separates into a cosine in $x$ and a decaying exponential in $|y|$. Continuity across the sheet excludes a potential jump, while integrating Poisson equation for Newtonian gravity across it gives $[{\Phi }_y{]}_{0^{-}}^{0^{+}}=4\pi G{\Sigma }_k\cos (kx)$. For a potential $C\cos (kx)e^{-k|y|}$ that derivative jump is $-2kC\cos (kx)$, giving $C=-2\pi G{\Sigma }_k/k$.

More generally a specified jump $J\cos (kx)$ gives potential $-J\cos (kx)e^{-k|y|}/(2k)$. Each spatial Fourier mode decays on the length scale $1/k$, so fine density structure has a shorter-range field. The sinusoid changes sign and is naturally interpreted as a density perturbation component. General sheet distributions are built by superposing their Fourier modes; decay alone does not restrict a harmonic potential to a single mode.