# Helmholtz equation

The *Helmholtz equation* is a partial differential equation^{} which, in scalar form is

$${\nabla}^{2}f+{k}^{2}f=0,$$ |

or in vector form is

$${\nabla}^{2}\mathbf{A}+{k}^{2}\mathbf{A}=0,$$ |

where ${\nabla}^{2}$ is the Laplacian.
The solutions of this equation represent the solution of the wave equation^{}, which is of great interest in physics.

Consider a wave equation

$$\frac{{\partial}^{2}\psi}{\partial {t}^{2}}={c}^{2}{\nabla}^{2}\psi $$ |

with wave speed $c$. If we look for time harmonic standing waves of frequency $\omega $,

$$\psi (\mathbf{x},t)={e}^{-j\omega t}\varphi (\mathbf{x})$$ |

we find that $\varphi (x)$ satisfies the Helmholtz equation:

$$({\nabla}^{2}+{k}^{2})\varphi =0$$ |

where $k=\omega /c$ is the wave number.

Usually the Helmholtz equation is solved by the separation of variables^{} method, in Cartesian, spherical or cylindrical coordinates^{}.

Title | Helmholtz equation |
---|---|

Canonical name | HelmholtzEquation |

Date of creation | 2013-03-22 13:09:09 |

Last modified on | 2013-03-22 13:09:09 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 11 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 26B12 |

Classification | msc 35-00 |

Synonym | Helmholtz differential equation |

Synonym | reduced wave equation |

Related topic | WaveEquation |

Related topic | PoissonsEquation |