Kirchhoff's and Poisson's formulas

D'Alembert's formula solves the wave equation on a line. What about a sound wave in three-dimensional air, or a ripple on a two-dimensional pond? Kirchhoff's formula (three dimensions) and Poisson's formula (two dimensions) are the explicit answers — the higher-dimensional cousins of d'Alembert, writing the solution directly from the initial data by averaging it over spheres or disks.

Kirchhoff's formula (3D) says the value u(x,t) is built from the AVERAGE of the initial data over the surface of the sphere of radius ct centred at x — exactly the points whose signal, travelling at speed c, is arriving right now. Because it samples only the thin spherical shell at radius ct, after that shell passes the contribution drops back to zero: this is precisely why three dimensions enjoy the strong Huygens principle (a clean trailing edge). Poisson's formula (2D) is similar but integrates over the whole filled DISK of radius ct, not just its boundary circle — every point closer than ct still contributes. That filled-in integral is exactly why two dimensions have a lingering wake and no clean trailing edge: old data, from the interior of the disk, keeps feeding in.

The contrast between the two formulas is the whole sharp-versus-diffuse story made algebraic. A surface average (3D) gives clean wavefronts; a solid average (2D) gives a wake. Both are derived by the method of spherical means — averaging the unknown over spheres turns the multidimensional wave equation back into a one-dimensional one (the Euler-Poisson-Darboux equation) that d'Alembert-style reasoning can crack. These formulas are how acoustics, optics, and antenna theory actually compute radiated fields, and they are the rigorous source of finite propagation speed in higher dimensions: u(x,t) depends only on data within distance ct.