Dictionary:Helmholtz equation

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

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

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

Derivation of the Helmholtz equation

Given the homogeneous form of the scalar wave equation

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

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

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

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

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

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

the following Fourier integral form

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

results.

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

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

External links

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