5 Waves in Higher Dimensions

In two and three dimensions the wave equation \[ u_{tt} = c^2\Delta u \] still propagates disturbances at speed \(c\) and conserves energy, but a new phenomenon appears: the dimension decides whether a signal passes cleanly. In three dimensions a flash of light or a clap is seen or heard for exactly as long as it lasted, and then there is silence; in two dimensions a disturbance leaves a wake that dies away only slowly, as the ripples on a pond show. This is Huygens’ principle in its sharp form. It follows from the explicit solution formulas of Kirchhoff and Poisson, which this section derives.

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