Wave Equation

The wave equation is the important partial differential equation

(1)

that describes propagation of waves with speed . The form above gives the wave equation in three-dimensional space where  is the Laplacian, which can also be written

(2)

An even more compact form is given by

(3)

where  is the d'Alembertian, which subsumes the second time derivative and second space derivatives into a single operator.

The one-dimensional wave equation is given by

(4)

As with all partial differential equations, suitable initial and/or boundary conditions must be given to obtain solutions to the equation for particular geometries and starting conditions.