Kirchhoff formula

In three spatial dimensions, the solution with initial position $u_0$ and velocity $u_1$ iswhere $M_t$ is the spherical mean.

Extend lower-dimensional data independently of an extra coordinate and apply the higher-dimensional wave equation solution. Projecting a three-dimensional sphere to the planar disk yields the kernel $[2\pi \sqrt{t^2-r^2}{]}^{-1}$ for the two-dimensional solution with zero initial displacement. This method of descent for the wave equation explains why three-dimensional wave propagation can depend on a sphere while two-dimensional propagation depends on its interior.

The strong Huygens principle says that the three-dimensional wave solution at $(t,x)$ depends only on initial data on the sphere $|y-x|=|t|$. A compactly supported disturbance therefore leaves no tail in the interior of its light cone.