wave equation
The wave equation is a partial differential equation
which
describes certain kinds of waves. It arises in various physical
situations, such as vibrating , waves, and
electromagnetic waves.
The wave equation in one is
The general solution of the one-dimensional wave equation can be obtained by a change of coordinates: , where and . This gives , which we can integrate to get d’Alembert’s solution:
where and are twice differentiable functions. and
represent waves traveling in the positive and negative
directions, respectively, with velocity . These functions can be
obtained if appropriate initial conditions and boundary conditions are given. For example, if and are given, the solution is
In general, the wave equation in is
where is a function of the location variables
, and time . Here, is the Laplacian
with respect to the location variables, which in Cartesian coordinates is given by .
Title | wave equation |
---|---|
Canonical name | WaveEquation |
Date of creation | 2013-03-22 13:10:12 |
Last modified on | 2013-03-22 13:10:12 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 10 |
Author | Mathprof (13753) |
Entry type | Definition |
Classification | msc 35L05 |
Related topic | HelmholtzDifferentialEquation |
Related topic | SphericalMean |
Defines | d’Alembert’s solution to the wave equation |