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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:

Here c is a fixed constantIn mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed. Constant number The most widely mentioned sort of constant is a fixed, but possibly unspecified, n, 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 soundThe speed of sound varies depending on the medium through which the sound waves pass. It is usually quoted in describing properties of substances (e. see the article on sodium). More commonly the term refers to the speed of sound in air. The speed varies. For the vibration of string this can vary widely: on a spiral spring (a slinkyA Slinky is a coil-shaped toy, invented by Naval engineer Richard James and his wife, Betty James. Slinkys come in various sizes, but are usually no larger than a grown adult's fist. The shape is a simple spiral, or coil design, of a ribbon of material, o) 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. is the Laplace operatorMultivariate calculus In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. It is denoted by the symbol Δ. Since it can be calculated 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 equationIn mathematics, a linear differential equation is a differential equation Lf g where the differential operator L is a linear operator. The condition on L rules out operations such as taking the square of the derivative of f but permits, for example, takin 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 transformThe Fourier transform named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). 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

The general solution to this is a Fourier seriesIn mathematics, a Fourier series named in honor of Joseph Fourier ( 1768- 1830), is a representation of a periodic function (often taken to have period 2π in a sense, the simplest case) as a sum of periodic functions of the form : which are harmonics o: an infinite sum of sine waves. These are the harmonics that the sound of a string being plucked is composed of.

In two dimensions:

More complex and realistic forms of the wave equation allow for the constant to change with the frequency of the wave. These equations tend to be non-linear.