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# 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 $\omega$,

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

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

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

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.

Related:

WaveEquation, PoissonsEquation

Synonym:

Helmholtz differential equation, reduced wave equation

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

26B12*no label found*35-00

*no label found*

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