When you're operating with a single central tweeter there is no combing. I am interested in what frequencies the arrayed tweeters operate and where we should should expect problems in theory.
I wrote "n points in a plane", not polynomials in a plane. All points lie in one 2D plane, which we can conveniently choose to be the x-y plane, which in turn allows a polynomial in one variable (y = sum{An x^n}) to pass through all of them.
Okay, I guess I wasn't clear, let me try one more time.
You want to generate a function based on properties of Bessel functions. If you have "n" speakers, you are going to calculate a specified phase shift at those "n" points.
Suppose you didn't know the function that was used to generate those "n" specified phase shifts. You could fit a polynomial to them; if you had an infinite number of points to fit, you would end up exactly re-creating the original function used to generate those "n" phase amounts.
(In your example, you would end up with a polynomial that passed exactly through an infinite number of values of Jn(1.5). It would not pass through an infinite number of points picked using some other generating function. So you can tell the original generating function uniquely.)
With an infinite number of speakers, then, tiny subtleties in the way individual phases are chosen are at least mathematically significant: they point back exactly at the function you used to generate them. Therefore there is some reasonable expectation that the sound field generated by this infinite array of speakers would actually be specific to that initial phase generating function; if someone came up with a better way to set those initial phases, there is at least a small chance that it might actually do something at the listener's ear.
However: when you don't have an infinite number of speakers, but only a small number of them, this is no longer true. A limited number of points only allows accuracy up to a polynomial of order (n-1).
If you have only three drivers, and therefore three specified initial phases (one for each speaker), and you're trying to tell if those phases came from a Bessel function or something else, you can't: with three points, there isn't enough resolution to tell the original generating function to any order higher than quadratic. You can't tell if the points were originally generated by a cosine, a Bessel function, or a simple quadratic.
This applies also to the acoustic performance of the speaker. Any subtleties (in the performance of the speaker) that you were hoping for due to a more sophisticated initial phase shift generating function, will not exist; you might just as well have used a quadratic as a Bessel to generate those initial phases.
So: Fine refinements in the exact initial-phase generating function don't matter if you only have a small number of drivers (which is to say, a practical number of drivers.)
Make sense now?
Most of the math used in these speaker calculations goes back a long way, to classical wave optics, maybe 150 years ago. The revolutionary new speaker array designs I hear about usually come down to tiny tweaks of inital phases and path lengths that produce rather small changes in the eventual wave superposition at the listener's ears. And if there are only a few drivers, then those tiny tweaks become entirely inconsequential.
-Gnobuddy
With all due respect, your description of a (finite) Bessel array is not correct. In particular, it is not constructed by polynomial interpolation. Maybe we are not talking about the same thing?
Regarding the age of these ideas, Bessel arrays aren't that old (I believe the derivation via generating function originated at Philips in the 80s). But yes, I agree that much of classical acoustics has been around for a while. Rayleigh's treatise "The Theory of Sound" is now over 120 years old. But it is also dated -- I have both volumes but never look at them. Kinsler, Beranek and Morse (in that order) are the most useful IMO. My impression is that the "modern era" of acoustics began with Olson's books in the 40s.
Depends on the tweeter design, but in terms of most modern tweeters like the Satori (or a goodly proportion thereof), the current fashion is to accept rising 2nd harmonic < 2KHz for significantly reduced higher order across the range. Scan & Vifa (as was) arguably started that with the XT25 and by carrying over relatively large roll surrounds to many domes. Obviously, if you push them out of the nominally linear range distortion of all types increases, but for tweeters of this type 2nd still tends to dominate at the low end, albeit with significantly elevated 3rd also.
Agreed.
Except I think its possible that each 7-tweeter cluster is a co-axial MT (with the central tweeter a tweeter and the 6 ring tweeters acting as a mid). This would make some sense as implemented in the upper section of the Double Impact. In the case of the Ulfberht, however, using two of these coaxial MTs seems to make less sense.
I imagine he already is. 
I very much doubt they've done anything complicated whatsoever with these things, and in fairness to them, if you eject the drivel and gibbering bollocks of the patent etc., you shouldn't really need to to get reasonable results. The phrase there being 'reasonable results' which is emphatically not a synonym for 'state of the art performance'.
It's a 4-way, yes? So if we assume 'simple', then it's a straightforward WMMTMMW configuration. Woofer pair for the LF. Quad of midrange units for the lower mids. 14 tweeters for the upper mids; if we assume the Scans or SBs (model depending) that's an Re of ~3 ohms. So, 2 parallel packs of 7 series-wired drivers for roughly 10.5ohm impedance, which is reasonable enough. And a single central tweeter for the HF.
Can that configuration be improved on? Yes, obviously. Is it completely unnecessary? Yes, obviously. If you select the filter frequencies & slopes well, can it give reasonable results? Should do. It's just a BW limited variation on the Sweet-16.* I suspect the filtering in the Tekton is simple & personally I'd do it differently, but presumably it works well enough not to kill HF units on a regular basis & avoid warranty repairs.
*Edit: I've just noticed somebody mentioned in the YouTube video linked to above somebody else has mentioned the Sweet16.
I very much doubt they've done anything complicated whatsoever with these things, and in fairness to them, if you eject the drivel and gibbering bollocks of the patent etc., you shouldn't really need to to get reasonable results. The phrase there being 'reasonable results' which is emphatically not a synonym for 'state of the art performance'.
It's a 4-way, yes? So if we assume 'simple', then it's a straightforward WMMTMMW configuration. Woofer pair for the LF. Quad of midrange units for the lower mids. 14 tweeters for the upper mids; if we assume the Scans or SBs (model depending) that's an Re of ~3 ohms. So, 2 parallel packs of 7 series-wired drivers for roughly 10.5ohm impedance, which is reasonable enough. And a single central tweeter for the HF.
Can that configuration be improved on? Yes, obviously. Is it completely unnecessary? Yes, obviously. If you select the filter frequencies & slopes well, can it give reasonable results? Should do. It's just a BW limited variation on the Sweet-16.* I suspect the filtering in the Tekton is simple & personally I'd do it differently, but presumably it works well enough not to kill HF units on a regular basis & avoid warranty repairs.
*Edit: I've just noticed somebody mentioned in the YouTube video linked to above somebody else has mentioned the Sweet16.
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I don't think we are, but having tried twice, I'm not sure if I can come up with a third different way to explain. But I'll give it one more try.
Consider Jn(1.5), for n from 0 to infinity.
Plot Jn(x) for a reasonable number of n (say n = 0 to n = 20.)
Set x = 1.5 (draw a vertical line at x = 1.5).
Write down the value of J0(1.5); let this have a value y0. I now have two coordinates, x=1.5, y = y0, which define one point in the (x,y) plane.
Write down the value of J1(1.5); let this be y1; I now have a second point in the (x,y) plane.
Repeat for n = 2, 3, ........20.
That array of yn values is going to be used to generate the initial phase shifts for "m" different drivers in an array, yes?
If m=3 (drivers), I have only 3 points: (x0,y0), (x1,y1), and (x2,y2).
We would like to believe that the use of Bessel functions Jn to choose this set of initial phases has endowed our speaker with some improved characteristic (better dispersion, less combing, compared to some other form of generating the initial phase.
This will only be true if the three points (x0,y0), (x1,y1), (x2,y2), are special and unique compared to 3 points chosen using some other generating function; they point back at the unique function used to generate them in the first place, i.e., starting with these three points, we should be able to go backwards mathematically, and say "Yes, these three points were generated using Jn(1.5) for n = 1,2,3, and not any other function.
So now we fit a function to the 3 points we have, and look at it to see if we have re-created Jn(1.5), or not.
But we cannot do this, because with only 3 points, the original generating function could have simply been A0+ A1 x + A2 x^2 (and not a Bessel function at all.) All you have to do is pick the right A0, A1, A2.
Therefore our initial phases do not have special properties unique to the use of Bessel functions to generate them.
Therefore our 3-driver system does not have special properties resulting from the use of Bessel functions to pick those initial 3 phases.
Things get better as we have more and more drivers, which constrain the original generating function more and more tightly.
That any better, or am I still clear as mud?
It was the latter I meant - first Young's double-slit experiment, then extending the math for a diffraction grating with N lines. That part is old.
Adding special initial phases (Bessel or whatever) is the only new part.
You reminded me that in the 1990's I read a paper about light beams that did not diffract, in the sense that the beam's transverse profile remained constant as the beam propagated through space. For this to happen, the beam had to be monochromatic (laser), and created with a specific radial intensity profile at the source - which was generated with the same Bessel functions Jn(kr), where k = 2*pi/lambda, lambda being the wavelength of the light.
The beam would then propagate forward, without widening sideways at all. But it would get dimmer and dimmer (presumably through absorption, since there was no beam widening), until it was effectively gone.
This seems to be more recent work on the same phenomenon: Jim Durnin - NonDiffracting Beams
And this on a number of ways to cause atypical behaviour of light beams (including beams that bend):
https://onlinelibrary.wiley.com/doi/pdf/10.1002/lpor.200910019
I mention this in this thread, because whatever you can do with light waves, you can do with sound waves...imagine a sheet full of pinholes placed in front of a loudspeaker, with the pinholes spaced to produce the right Bessel intensity distribution. Would you then get a beam of sound that projected straight in front of the speaker with little diffraction to the sides? If you did, would it have any applications (reduced room reflections, say?)
-Gnobuddy
This is a beautiful expression, but I am afraid it needs to be uttered with a British accent
No, sorry, its not any better.That any better, or am I still clear as mud?
So, what are your three points (x,y), exactly, for a 3-element Bessel array? How would you write the total pressure of this array? You can assume monopole radiators.
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