Kirchhoff formula

For the three-dimensional wave equation $u_{tt}=c^2\Delta u$ on all of ${\mathbb{R}}^3$, with $c>0$, smooth initial displacement $f$ and initial velocity $g$, define the normalized spherical meanFor $t>0$, the Kirchhoff formula isThe factor $t$ and the normalization of the mean are essential. Equivalently, $t{M}_{ct}g=(4\pi c^{2}t{)}^{-1}{\int }_{|y-x|=ct}g(y)d{S}_y$. At $t=0$, the limiting value and time derivative recover $f,g$. For instance constant data give $u=f+tg$, which directly checks the coefficients.^[ref]

The solution uses spherical data rather than the full interior ball, giving the Strong Huygens principle. The differentiated displacement term also samples its normal variation, so this is not merely an average of two lists of pointwise boundary values. The method of descent for the wave equation changes the formula in two spatial dimensions to a disk integral with an interior tail.

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.