Kirchhoff's formula

The 3D wave equation's closed-form solution, built from the initial data's average over a shrinking sphere rather than a single point.

Kirchhoff's formula gives the 3D wave equation's solution as $u(x,t)=\partial_t\big(t\,M_f(x,ct)\big)+t\,M_g(x,ct)$ , where $M_h(x,r)$ is the average of $h$ over the sphere of radius $r$ centered at $x$ -- the direct 3D generalization of d'Alembert's formula, which averages over just the two endpoints of an interval instead of a whole sphere.