Helmholtz equation


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

2f+k2f=0,

or in vector form is

2𝐀+k2𝐀=0,

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

2ψt2=c22ψ

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

ψ(𝐱,t)=e-jωtϕ(𝐱)

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

(2+k2)ϕ=0

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