It is well known that the far-field radiation pattern of an acoustic
source is proportional to the spatial Fourier transform of the
source's radiating amplitude distribution. In optics, this is called
Fraunhofer diffraction theory (Goodman, 2005). By linearity of the
Fourier transform, it follows that if the speaker array emits valid
sampling kernels near the array (*i.e.*, the radiation patterns overlap
and add to a constant overall gain for a plane wave), then the
far-field patterns will similarly overlap and add to a constant gain.
All points in between must be valid as well, being progressive
diffractions of the source distribution, but a rigorous proof with
quantified approximation error would be nice to see (there are always
terms to neglect, and it is good to be mindful of them).

Download huygens.pdf

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`http://arxiv.org/abs/1911.07575`

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University