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

$${\nabla }^2𝐀+k^2𝐀=0,$$

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 $\omega$,

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

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