# wave equation

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: $(x,t)\longrightarrow(\xi,\eta)$, 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

 $u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\mathrm{% d}s.$

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 $x_{1},x_{2},\ldots,x_{n}$, and time $t$. Here, $\nabla^{2}$ is the Laplacian with respect to the location variables, which in Cartesian coordinates  is given by $\nabla^{2}=\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{% \partial x_{2}^{2}}+\cdots+\frac{\partial^{2}}{\partial x_{n}^{2}}$.

Title wave equation WaveEquation 2013-03-22 13:10:12 2013-03-22 13:10:12 Mathprof (13753) Mathprof (13753) 10 Mathprof (13753) Definition msc 35L05 HelmholtzDifferentialEquation SphericalMean d’Alembert’s solution to the wave equation