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# 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 strings, sound waves, and
electromagnetic waves.

The wave equation in one dimension 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 $n$ dimensions 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 $\nabla^{2}=\frac{\partial^{2}}{\partial x_{1}^{2}}+\frac{\partial^{2}}{% \partial x_{2}^{2}}+\cdots+\frac{\partial^{2}}{\partial x_{n}^{2}}$.

## Mathematics Subject Classification

35L05*no label found*

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