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.

See also

• Fourier transform
• Sine and cosine transforms