= Gravitational potential of a sinusoidal sheet
{title2=$\Phi=-\frac{2\pi G\Sigma_k}k\cos(kx)e^{-k|y|}$}
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.
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