 Method of descent for the wave equation

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.