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

$$\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}.$$

The general solution of the one-dimensional wave equation can be obtained by a change of coordinates: $(x,t)⟶(\xi ,\eta )$, where $\xi =x-ct$ and $\eta =x+ct$. This gives $\frac{{\partial }^{2}u}{\partial \xi \partial \eta }=0$, which we can integrate to get d’Alembert’s solution:

$$u(x,t)=F(x-ct)+G(x+ct)$$

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 conditions and boundary conditions are given. For example, if $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t}(x,0)=g(x)$ are given, the solution is

$$u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds.$$

In general, the wave equation in $n$ is

$$\frac{{\partial }^{2}u}{\partial t^2}=c^2{\nabla }^{2}u.$$

where $u$ is a function of the location variables $x_1,x_2,\mathrm{\dots },x_n$, and time $t$. Here, ${\nabla }^2$ is the Laplacian with respect to the location variables, which in Cartesian coordinates is given by ${\nabla }^2=\frac{{\partial }^2}{\partial x_1^2}+\frac{{\partial }^2}{\partial x_2^2}+\mathrm{\cdots }+\frac{{\partial }^2}{\partial x_n^2}$.

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