# 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.

Title Helmholtz equation HelmholtzEquation 2013-03-22 13:09:09 2013-03-22 13:09:09 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 26B12 msc 35-00 Helmholtz differential equation reduced wave equation WaveEquation PoissonsEquation