I am working with a surface that's both spatially and temporally periodic.
It's defined by
$$\zeta(x,y,t) = \sum_{mnl = -\infty}^\infty P(m,n,l)e^{ia(mx+ny) -iwlt} $$
where $P(m,n,l)$ is the coefficient of the $m,n,l$th Fourier component, $a=2\pi / L$ (where $L$ is the spatial period in both the $x$ and $y$ directions) and $w=2\pi/T$ where $T$ is the temporal period of the surface.
In the paper I'm reading (First-Order Theory and Analysis of MF/HF/VHF Scatter from the Sea - D.E.Barrick, January 1972), we "define the average spatial-temporal spectrum $W(p,q,\omega)$ of the surface height in terms of the Fourier coefficients as $$ W(p,q,\omega) = \frac{1}{\pi^3} \iiint \langle \zeta(x_1,y_1,t_1)\zeta(x_2,y_2,t_2) \rangle e^{ip\tau_x + iq\tau_y - i\omega \tau}\,d\tau_x\,d\tau_y\,d\tau" $$
where $p=am$, $q=an,$ $\omega=wl$, $\tau_x = x_2-x_1$, $y_2-y_1$, $\tau = t_2-t_1$ and $\langle \rangle$ denotes a statistical average.
I gather from the Wiener-Khinchin theorem that the Fourier transform of the autocorrelation function of the surface gives the spectrum. However the factor of $\frac{1}{\pi^3}$ in their definition of $W(p,q,\omega)$ confuses me as I am unaware of any version of the Fourier transform where $\frac{1}{\pi}$ is used.
So my question is where does the $\frac{1}{\pi^3}$ come from?
Any ideas or suggestions are most welcome.
Thanks for reading, Rachael