Thompson order formula

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.

$f(k,\omega )=\underset{\mathrm{\Omega }}{\int }\underset{T}{\int }\sin (k_{x}x)\sin (k_{y}y)\sin (k_{z}z)\exp (-i\omega t)dtdxdydz$

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

See also

• Fourier transform
• Sine and cosine transforms