Space-time Fourier transform

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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.

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