This important theorem in acoustics is a corollary of the fact that the pressure distribution in one plane is a Fourier transform of the velocity distribution in another one that is far away and parallel to it, as discussed in the previous section. For simplicity, let us consider a two-dimensional system in the xz plane of infinite extent in the y  direction. We then multiply the Fourier transform of an arbitrary velocity distribution f(x) by that of a line source at x ₀ as follows:

where δ(x   −   x ₀) is the Dirac delta function and we have used the property that

which is simply the Fourier transform of the original distribution f(x) shifted to a new origin at x ₀ as illustrated in Fig. 13.41. This is somewhat analogous to the principle of amplitude modulation whereby multiplying a baseband signal by a single tone in the time domain produces a modulated tone with “sideband” spectrums on either side of the tone in the frequency domain. Here the space domain is analogous to the frequency domain and the spatial frequency domain is analogous to the time domain, as the product is taken in the latter. Seeing that the far-field pressure of a planar source is the Fourier transform of the source velocity distribution, we can use the product theorem to derive the far-field pressure for a transducer array by simply taking the product of the directivity function for a single transducer and that of any number of point sources located in the array positions, as illustrated in Fig. 13.42.