the method of spherical means

The wave equation on a line is easy (d'Alembert). In two or three space dimensions it looks much harder — until you use a clever trick: instead of tracking the wave at each point, track its AVERAGE over spheres of every radius. This averaging is the method of spherical means, and it magically reduces the hard multidimensional problem to a one-dimensional one you already know how to solve.

Here is the idea in plain steps. Fix a centre point x. For each radius r and time t, let M(r,t) be the average value of the solution u over the sphere of radius r around x. Two facts make this useful. First, as r shrinks to zero, M(r,t) returns to u(x,t) itself — so if you know all the averages, you recover the actual solution. Second, and remarkably, when u solves the wave equation, its spherical average M satisfies a wave equation in the single variable r (the Euler-Poisson-Darboux equation; in three dimensions, after multiplying by r, it becomes the plain one-dimensional wave equation for r times M). So you solve a 1D wave equation for the averages, then let r go to zero. Out drops Kirchhoff's formula in 3D and, by a descent trick (treating a 2D problem as a 3D one that does not depend on the third coordinate), Poisson's formula in 2D.

This is the engine room behind the explicit higher-dimensional wave solutions and behind Huygens' principle. It is also where the deep odd-versus-even-dimension behaviour comes from: the reduction to a clean 1D wave equation works beautifully in odd dimensions (giving sharp wavefronts), while even dimensions require the descent and pick up the filled-in integral that produces a wake. The same spherical-averaging idea reappears for Laplace's equation, where the mean-value property says a harmonic function equals its own spherical average exactly.