= Point scatterer
A <point scatterer> idealizes an object as a localized coupling at one position when its physical size is small compared with the relevant <wavelength> and observation distances. For a three-dimensional scalar <time-harmonic wave>, an outgoing contribution has the form
$$
\psi_s(\mathbf r)=q\,G(\mathbf r,\mathbf r_0),\qquad G(\mathbf r,\mathbf r_0)=\frac{e^{ik|\mathbf r-\mathbf r_0|}}{4\pi|\mathbf r-\mathbf r_0|},
$$
where the source strength $q$ depends on the scatterer's response and the exciting field at its position. This convention gives $(\nabla^2+k^2)G=-\delta$.
The idealization keeps a finite scattering response while discarding the resolved surface geometry. It is useful for dilute assemblies of small objects and models of <multiple scattering>. The <Green function> is singular at the source, so using the total field at that same point requires a prescribed treatment of the self-field, such as <short-distance regularization of a resonant point scatterer>. Replacing a finite object by a point without specifying its response does not define a complete physical model.
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