# Space-time Fourier transform

When dealing with a problem defined in a restricted region of space and in a time interval, $f=f (r,t)$, it can be useful to calculate the space-time Fourier transforms. The correlated space parameters are:

$k_x = \frac{l\pi}{L}$
$k_y = \frac{m\pi}{W}$
$k_z = \frac{n\pi}{D}$

where L, D and W are the dimensions of the space region and l, m, and n are the integers.

$f\left(k,\omega\right) = \int_\Omega \int_T \sin(k_x x) \sin(k_y y) \sin(k_z z) \exp(-i\omega t) \, dt \, dx \, dy \,dz$

T is the time interval and $\Omega$ is the volume of the concerned region.