wave equation

The wave equationMathworldPlanetmath is a partial differential equationMathworldPlanetmath 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: (x,t)(ξ,η), where ξ=x-ct and η=x+ct. This gives 2uξη=0, which we can integrate to get d’Alembert’s solution:


where F and G are twice differentiable functions. F and G represent waves traveling in the positive and negative x directions, respectively, with velocity c. These functions can be obtained if appropriate initial conditionsMathworldPlanetmath and boundary conditions are given. For example, if u(x,0)=f(x) and ut(x,0)=g(x) are given, the solution is


In general, the wave equation in n is


where u is a function of the location variables x1,x2,,xn, and time t. Here, 2 is the Laplacian with respect to the location variables, which in Cartesian coordinatesMathworldPlanetmath is given by 2=2x12+2x22++2xn2.

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