Wave equation

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

$u_{tt} = c^2\Delta u$ : disturbances travel at speed $c$ . Sound, vibrating strings, seismic waves and light obey it; games use discretized versions for water ripples and room acoustics.

Formulas

$\frac{\partial^2 u}{\partial t^2} = c^2\,\Delta u, \qquad u(x,t) = f(x - ct) + g(x + ct)$
d'Alembert: in 1D, two shapes travelling in opposite directions

The mathematics behind it

Laplacian★★★★★fundamental
Waves are $u_{tt} = c^2\Delta u$ .
Partial differential equations★★★★★fundamental
The prototype hyperbolic PDE.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.