Helmholtz equation

The Helmholtz equation is a partial differential equationMathworldPlanetmath which, in scalar form is


or in vector form is


where 2 is the Laplacian. The solutions of this equation represent the solution of the wave equationMathworldPlanetmath, which is of great interest in physics.

Consider a wave equation


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


we find that ϕ(x) satisfies the Helmholtz equation:


where k=ω/c is the wave number.

Usually the Helmholtz equation is solved by the separation of variablesMathworldPlanetmath method, in Cartesian, spherical or cylindrical coordinatesMathworldPlanetmath.

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