Wave Equation

The wave equation is the important partial differential equation

 del ^2psi=1/(v^2)(partial^2psi)/(partialt^2)

that describes propagation of waves with speed v. The form above gives the wave equation in three-dimensional space where del ^2 is the Laplacian, which can also be written

 v^2del ^2psi=psi_(tt).

An even more compact form is given by

  square ^2psi=0,

where  square ^2 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


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.

See also

Wave Equation--1-Dimensional, Wave Equation--Disk, Wave Equation--Rectangle, Wave Equation--Triangle

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Cite this as:

Weisstein, Eric W. "Wave Equation." From MathWorld--A Wolfram Web Resource.

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