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Wave equation

The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. It arises in many different fields, such as acoustics, electromagnetics, and fluid dynamics. Variations of the wave equation are also found in quantum mechanics and general relativity.

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.

The general form of the wave equation is:

{ \partial^2 u \over \partial t^2 } = c^2 \nabla^2u

Here c is a fixed constant, the speed of the wave's propagation. For a sound wave in air this is about 300 m/s, and we refer to the speed of sound. For the vibration of string this can vary widely: on a spiral spring (a slinky) it can be as slow as a meter per second.

u, = u(x,t), is the amplitude, a measure of the intensity of the wave at a particular location x and time t. For a sound wave in air u is the local air pressure, for a vibrating string it is the physical displacement of the string from its rest position. \nabla^2 is the Laplace operator with respect to the location variable(s) x. Note that u may be a scalar or vector quantity.

The basic wave equation is a linear differential equation which means that the amplitude of two waves interacting is simply the sum of the waves. This means also that a behavior of a wave can be analyzed by breaking up the wave into components. The Fourier transform breaks up a wave into sinusodal components and is useful for analyzing the wave equation.

The one-dimensional form can be derived from considering a flexible string, stretched between two points on a x-axis. It is

{ \partial^2 u \over \partial t^2 } = c^2 { \partial^2 u \over \partial x^2 }

The general solution to this is a Fourier series: an infinite sum of sine waves. These are the harmonics that the sound of a string being plucked is composed of.

In two dimensions:

{ \partial^2 u \over \partial t^2 } = c^2 \left ({ \partial^2 u \over \partial x^2 } + { \partial^2 u \over \partial y^2 } \right )

The wave equation is the prototypical example of a hyperbolic partial differential equation .

More realistic differential equations for waves allow for the speed of wave propagation to vary with the frequency of the wave. These equations tend to be non-linear.

See also

Last updated: 02-08-2005 20:29:14
Last updated: 02-22-2005 16:15:51