Wave equation

The wave equation is the prototypical hyperbolic partial differential equation, governing the propagation of disturbances through a medium with finite speed. In one spatial dimension, it reads ∂²u/∂t² = c² ∂²u/∂x², where c is the wave speed. Unlike the diffusive heat equation, the wave equation preserves information: disturbances propagate along characteristic curves without dissipation.

The wave equation is distinguished by Huygens' principle, which states that waves in odd spatial dimensions propagate on sharp wavefronts, while waves in even dimensions leave trailing wakes. This dimensional dependence is not a curiosity but a deep fact about the geometry of Minkowski spacetime. The wave equation connects to Fourier analysis through the decomposition of solutions into superpositions of plane waves, and to quantum mechanics through the Schrödinger equation, which is formally obtained by analytic continuation of the wave equation from real to imaginary time.

The wave equation is democracy in physical form: every point in the medium participates equally in propagating the disturbance. Yet the superposition principle allows complex patterns to emerge from simple waves — interference, standing waves, resonance. The wave equation is linear, but the world it describes is not.