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wave equation
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(Definition)
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The wave equation is a partial differential equation which describes certain kinds of waves. It arises in various physical situations, such as vibrating strings, sound waves, and electromagnetic waves.
The wave equation in one dimension 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 starting or boundary conditions are given. For example, if
 and
 are given, the solution is
In general, the wave equation in  dimensions 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
 .
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(view preamble)
Cross-references: Cartesian coordinates, Laplacian, variables, boundary conditions, functions, negative, positive, represent, differentiable functions, solution, change of coordinates, general solution, partial differential equation
There are 10 references to this entry.
This is version 6 of wave equation, born on 2002-11-21, modified 2007-06-26.
Object id is 3614, canonical name is WaveEquation.
Accessed 15939 times total.
Classification:
| AMS MSC: | 35L05 (Partial differential equations :: Partial differential equations of hyperbolic type :: Wave equation) |
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Pending Errata and Addenda
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![$\displaystyle u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\mathrm d s. $](http://images.planetmath.org:8080/cache/objects/3614/l2h/img15.png "$\displaystyle u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\mathrm d s. $")
