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Bessel arrays

Loudspeaker systems that use relatively large numbers of identical drive units tend to create a tight beam of sound and frequency 'holes' can occur. A Bessel array can result in even distribution. A proprietary system patented by Philips, lending itself to use with arrays of speakers and microphones, gives a near-spherical radiation pattern with as few as five transducers. Planes can be created using the same technique.

Simple Bessel series with five identical drive units satisfying the power distribution function ratio of A:B:C:D:E = 1:2:2:-2:1. Note polarities, indicated by bullets, and distance d being only slightly larger than loudspeaker diameter.

A Bessel series with seven weighting factors (function ratio of A:B:C:D:E:F:G = 1:2:2:0:-2:2:1) needs only six drive units because D = 0.

Nine drive units will have a function ratio of A:B:C:D:E:F:G:H:I = 1:2:2:0:-2:0:2:-2:1, thus two may be omitted.

A stereo layout does not require two banks.

Arrangement to obtain continuously variable base width setting. k = 0 gives superstereo, k = 0-0.5 expanded stereo, k = 0.5 normal stereo, k = 1 mono.

To reduce the cancellation effects, in expanded/superstereo, of low omni-directional frequencies contained in both channels, an input for bass reinforcement can be derived by combining LF left and right signals before the mixer.

Identical arrays placed side by side give a particularly horizontal radiation pattern (cinemas, theatres). Note how some senses are shown inverted compared to quoted ratio (ringed) showing variance in published data.

A spherical radiation pattern requires identical rows and columns.

A seven-by-seven Bessel array, suited to architectural (eg; ceiling) uses (some of the polarities given by the Philips paper can be questioned, as in the array below). Suggestions for wiring this array for stereo are invited.

Twenty-five units can be combined in a fairly simple manner, the weightings being rearranged to reduce the number of drive units to eleven.
Variances arise in 'Bessel panels - high-power speaker systems with radial sound distribution', Philips Technical Publication 091, 15/3/83 and 'Bessel arrays', elektor, p14-15, Jul-Aug 89. The US patent (4399328, 16/8/83, Google search for same) contains two certificates of correction.

A discussion of a compact practical implementation can be found here. However, 'normal' multi-way hi-fi speakers, which ideally should be identical, can be used instead to construct arrays. Microphones deployed according to these distributions will yield a super-sensitive microphone with a spherical radiation pattern.

Apart from the interest in my John Linsley Hood web-pages, this is the one area that I probably receive the most e-mails about. Most enquire about the factors, those shown above being checked thoroughly against the original publications. Later patents and designs employ differing weighting factors and layouts. Chris Ziolkowski kindly points out a good example, 'Bessel array with full amplitude signal to half amplitude position transducers' (US pat 2006/0159289 A1, 20/7/06, Google search for same) by Stiles, Turnmire and Calderwood which contains lots of constructional detail for a number of variants and is recommended since this approach reduces the variance between individual power factors. As Chris has indicated, Drg 77 (wall-mounted, surround sound) is worth a look, although he would wire as a bipole rather than dipole (below). Any problems with pressure differentials could be overcome by a central partition between the 6 horizontal drivers. Offsetting the speaker axes would help prevent standing waves and the addition of a tweeter array might be useful.

Initial thoughts consider a 'club' environment and a unit employing 'rock' speakers (12-15") 2-3m behind a solo vocalist / saxophonist whose output drives the array.

A Google search for Bessel arrays.
Speakers | Width

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