# Dictionary:Helmholtz equation

Other languages:
English • ‎español • ‎中文

The space-dependent form of the wave equation for a wave that is harmonic in time:

${\displaystyle \left(\nabla ^{2}+\kappa ^{2}\right)\psi =0}$,

where ${\displaystyle \kappa =\omega /V}$, ${\displaystyle \omega }$=angular frequency, and V=velocity.

## Derivation of the Helmholtz equation

Given the homogeneous form of the scalar wave equation

${\displaystyle \left[\nabla ^{2}-{\frac {1}{V^{2}({\boldsymbol {x}})}}{\frac {\partial ^{2}}{\partial t^{2}}}\right]\Psi ({\boldsymbol {x}},t)=0}$.

Here ${\displaystyle {\boldsymbol {x}}\equiv (x_{1},x_{2},x_{3})}$, ${\displaystyle t}$ is time, ${\displaystyle V({\boldsymbol {x}})}$ is the wavespeed, and ${\displaystyle \Psi ({\boldsymbol {x}},t)}$ is the wave field.

If we replace ${\displaystyle \Psi ({\boldsymbol {x}},t)}$ by its Fourier transform representation

${\displaystyle \Psi ({\boldsymbol {x}},t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\psi ({\boldsymbol {x}},\omega )e^{-i\omega t}\;d\omega }$,

noting that the second derivative of ${\displaystyle \Psi ({\boldsymbol {x}},t)}$ with respect to time

${\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\Psi ({\boldsymbol {x}},t)\equiv {\frac {1}{2\pi }}\int _{0}^{\infty }(-i\omega )^{2}\psi ({\boldsymbol {x}},\omega )e^{-i\omega t}\;d\omega }$

the following Fourier integral form

${\displaystyle {\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left[\nabla ^{2}+{\frac {\omega ^{2}}{V^{2}({\boldsymbol {x}})}}\right]\psi ({\boldsymbol {x}},\omega )e^{-i\omega t}\;d\omega =0}$

results.

Because the only way for this Fourier integral representation to vanish is if its integrand vanishes, the Helmholtz equation appears

${\displaystyle \left[\nabla ^{2}+{\frac {\omega ^{2}}{V^{2}({\boldsymbol {x}})}}\right]\psi ({\boldsymbol {x}},\omega )=0}$.