structural:reformatting, linguistic:paraphrasing, information:classification, structural:expansion
General aperture

The wave that emerges from a point source has amplitude ψ {\displaystyle \psi } at location r {\displaystyle \mathbf {r} } that is given by the solution of the frequency domain wave equation for a point source (the Helmholtz equation), ∇ 2 ψ + k 2 ψ = δ (r), {\displaystyle \nabla ^{2}\psi +k^{2}\psi =\delta (\mathbf {r}),} where δ (r) {\displaystyle \delta (\mathbf {r})} is the 3-dimensional delta function. The delta function has only radial dependence, so the Laplace operator (a.k.a. scalar Laplacian) in the spherical coordinate system simplifies to ∇ 2 ψ = 1 r ∂ 2 ∂ r 2 (r ψ). {\displaystyle \nabla ^{2}\psi ={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(r\psi).}