# 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: $(x,t)\u27f6(\xi ,\eta )$,
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 |