# Difference between revisions of "Dictionary:Helmholtz equation"

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where <math>\kappa=\omega/V</math>, <math>\omega</math>=angular frequency, and ''V''=velocity. | where <math>\kappa=\omega/V</math>, <math>\omega</math>=angular frequency, and ''V''=velocity. | ||

+ | == Derivation of the Helmholtz equation == | ||

+ | |||

+ | Given the homogeneous form of the scalar wave equation | ||

+ | |||

+ | <math> \left[ \nabla^2 - \frac{1}{V^2(\boldsymbol{x})}\frac{\partial^2}{\partial t^2} \right] \Psi (\boldsymbol{x},t) = 0 </math>. | ||

+ | |||

+ | Here <math> \boldsymbol{x} \equiv (x_1, x_2, x_3 ) </math>, <math> t </math> is time, <math> V(\boldsymbol{x}) </math> is the wavespeed, and <math> \Psi (\boldsymbol{x},t) </math> is the wave field. | ||

+ | |||

+ | If we replace <math> \Psi (\boldsymbol{x},t) </math> by its [[Fourier transform]] representation | ||

+ | |||

+ | <math> \Psi(\boldsymbol{x},t) = \frac{1}{2 \pi} \int_{-\infty}^\infty \psi(\boldsymbol{x},\omega) e^{-i \omega t } \; d\omega </math>, | ||

+ | |||

+ | noting that the second derivative of <math> \Psi(\boldsymbol{x},t) </math> with respect to time | ||

+ | |||

+ | <math> \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 </math> | ||

+ | |||

+ | the following Fourier integral form | ||

+ | |||

+ | <math> \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 </math> | ||

+ | |||

+ | results. | ||

+ | |||

+ | Because the only way for this Fourier integral representation to vanish is if its integrand vanishes, | ||

+ | the Helmholtz equation appears | ||

+ | |||

+ | <math> \left[ \nabla^2 + \frac{\omega^2}{V^2(\boldsymbol{x})}\right] \psi (\boldsymbol{x},\omega) =0 </math>. | ||

==External links== <!--T:4--> | ==External links== <!--T:4--> |

## Latest revision as of 15:26, 5 January 2021

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

where , =angular frequency, and *V*=velocity.

## Derivation of the Helmholtz equation

Given the homogeneous form of the scalar wave equation

.

Here , is time, is the wavespeed, and is the wave field.

If we replace by its Fourier transform representation

,

noting that the second derivative of with respect to time

the following Fourier integral form

results.

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

.