I do have one question from your stint in acoustics.
The core idea for the scanner is that the sound field can be separated into components.
First we separate out the time variability with phasor notation.
Then we are left with a radial component and a directivity component.
The idea is manifested in the maths as "separation of variable" in the differential equations.
Separation of Variables also came up in your discussion with Putland about Oblate Spheriodal co-ordinates.
You initially claimed that S. of V. meant that a 1 parameter solution of the wave equation was possible in OS co-ords.
(for convenience I will use r, theta, phi for the co-ordinates in OS despite the fact they are not the same r, theta, phi we use in cylindrical co-ords)
The idea was a solution of form F(r)G(theta)H(phi) could be equal to a solution of the form F(r)*constant -with appropriate series expansion in the theta and phi functions.
You later revisited the subject and dropped that claim.
I remember when I first read the claim that it was not clear to me but it seemed plausible.
Do you have a physical or intuitive explanation why it doesn't work?
This may help me clarify S. of V. as we use it in this context.
Best wishes
David
Yes, one can simply interpolate polar data on a regular grid and if this grid is fine enough the results will be accurate. But I can reduce the number of points required from about 90 to about 13 with the same resolution. This is not at all intuitive, but doable, and I also get several other features such as independence of measurement distance and the ability to reconstruct the sources velocity profile. So its not the "real world" problems that are the limitation, but the core fundamental math/physics understanding of the radiation problem that is the limitation. I should note that these same techniques are what made quantum mechanics possible. The electron shell structure is defined by exactly the same set of functions as the sound radiation one. It was Weinreich, a quantum mechanics physicist at Michigan who realized this connection and hence his paper.
When I first started to study sound propagation in horns I quickly realized that for a one parameter wave (1P) to exist the contour had to lie along a separable coordinate system. This indeed remains a necessary condition for analysis, but it turned out NOT to be sufficient. In all but a few of the separable systems the coordinate functions are coupled through the eigenvalues. This means that in three dimensions, only the spherical coordinates (a conical horn) allow true 1P waves. I noted this in my original paper and Putland proved it in his. In all other coordinate systems, specifically the OS, there is a coupling of the radial wave into the angular ones, hence no 1P is possible.
So I never changed my claim, I only added a further restriction to it that applies in some (most) cases.
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That makes sense to me; only focusing on under (say) 1kHz gives more freedom in how much separation we can go with.
I have been thinking about how we might get an idea of what effect the secondary reflections off the speaker; perhaps a ground plane measurement with a second speaker set up as the only reflection source. I have friends that have an open field in the country with power near enough that I've used it for speaker measurements before, and it's getting warmer out so it's an experiment I could do if it'll get us useful data.
How big of an effect the secondary reflections are likely depends on the size of the speaker being measured, with a small satellite probably not providing much in the way of a reflective surface until above 1kHz or so.
Not very intuitive to me, I know there is theory for non uniform Fourier transform but it has slowed me down while I study it.
Can you explain how you have done it?
I understand you take closer samples near the axis, I haven't yet worked out how to calculate the polars from this.
If the number of samples can be reduced sufficiently then it is more realistic to avoid a complicated CNC scanner.
Best wishes
David
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If you want to understand the details then you need to dig into the detailed math. It is not feasible to show that here, so you will have to take my word for it, but here is the gist of the thing.
In the radiation mode domain we have a sound field that is a sum of radiation modes. Each mode has a cut-in frequency below which it does not contribute. In essence we thus have a problem of N samples to fit M modes. At the lowest frequencies I only need 1 point to fit the monopole mode, a little higher I need two for the dipole mode coming into play. Klippel shows this well in his slides. It takes about 13 modes to get good resolution up to 10 kHz from a normal sized speaker (size isn't critical though as doubling it only adds the need for one more mode) so I need 13 data points to fit these 13 modes.
The non-uniform Fourier transform plays no role in any of this. It is more closely associated with the Hankel transform, but that is still not the way its done.
The paper linked earlier shows this linear algebra approach very well.
In the radiation mode domain we have a sound field that is a sum of radiation modes. Each mode has a cut-in frequency below which it does not contribute. In essence we thus have a problem of N samples to fit M modes. At the lowest frequencies I only need 1 point to fit the monopole mode, a little higher I need two for the dipole mode coming into play. Klippel shows this well in his slides. It takes about 13 modes to get good resolution up to 10 kHz from a normal sized speaker (size isn't critical though as doubling it only adds the need for one more mode) so I need 13 data points to fit these 13 modes.
The non-uniform Fourier transform plays no role in any of this. It is more closely associated with the Hankel transform, but that is still not the way its done.
The paper linked earlier shows this linear algebra approach very well.
I meant "Fourier transform" in the broad sense, with some set of orthonormal functions as a basis - sine/cosine, Bessel/Hankel or whatever.
Spherical Harmonics in this case.
Best wishes
David
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Thank you both, this stuff is pretty neat and interests me.
It's kind of the flip side of the Klippel, it shows a 'picture' of the source whereas Klippel is optimized for the far field behaviour, polars and the like.
Closely related maths but a different application.
It would be fun to actually 'see' the source intensity, cabinet panel resonances, port turbulence noise and all.
Maybe a separate project, I haven't even fully worked out the maths for this one yet.
Best wishes
David
Not shoestring, but interesting!
I ran across those early in my investigations into the NFS, I think something like that would have some interesting applications for enthusiasts like ourselves.
I'd like to read more about that, do you have any links?
Still don't have a copy of 'Fourier Acoustics' but I have worked a little more on the maths, specifically the positions of the sample points.
A uniform sample grid is the easiest mathematically but wastes time with measurements behind the speaker that don't usually matter much.
A non-uniform grid should be faster but has mathematical complications.
For a scan in one plane a-la-Geddes the extra complexity is not too bad but a full spherical non uniform scan grid is close to my limit.
There doesn't even appear to be a 'that's the way everybody does it' standard technique AFAIK.
I have at least reassured myself that it should be possible to do this without too much trouble from numerical sensitivity in the measurement and calculation.
(some of the near field calculations do suffer from this problem so it's not a trivial question.)
One option is to use a math routine library to handle the details, as Earl initially did.
There is one available for Matlab.
Matlab is expensive for a full licence but a student license is cheap ~$50.
Can you be a student? Or do you already have another preferred software solution?
It should be possible to port the Spherical Harmonics library to freeware GNU Octave open source, if there is not already one available.
Best wishes
David
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I still use many routines from Numerical Recipes. I'd never write a routine that was already available.
Given the possibility that someone may want to use our scanner to measure speakers that aren't "forward firing", maybe a uniform sample grid is ok. And having less complications to face in development is a bonus. That being said, not knowing how much faster a non-uniform grid would make the scan makes it hard to say if the trade off is worth it.
That solution sounds best to me. It keeps this project as accessible as possible.
I do think we need to allow for rear ports and the like but I don't think we need as detailed resolution in the rear.
But spherical harmonics are inherently based on symmetry, I don't yet have an intuitive feel for how to reconcile this.
I have finally started to work out approximately the required number of points and the trade-offs.
The step from just horizontal polars to full spherical harmonics is harder than I expected.
I kind of expected I would just have to find the answers, not work them out
I have no experience with either Matlab or Octave but they look like the most suitable software for the job.
So it's likely to be Octave.
Best wishes
David
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Agreed, but with the exception of dipole, omni and other speakers with more exotic radiation patterns... but then again, since I don't know the math, maybe that's not a problem. Or maybe it's just not worth designing for something this thing may never be measuring.
Me either. I just downloaded Octave and now I'll have to make some time to take a look at it.
Back in post #70, Kessito mentioned being skilled at Matlab.
Yes, I noticed that and favor Octave because it is compatible with Matlab, but open source.
So now I'll have to make some time to look at it too.
Hopefully I can use it to help develop more sense for how the maths works.
It turns out to be quite similar to the quantum mechanics of electron orbitals, which sounds impressive but, unfortunately, I was rather poor at.
Best wishes
David
Personally, I don't think that Matlab or Octave is the way to go. I started out with MathCad, which has all of the spherical harmonics functions, but eventually gave up on that as it wasn't flexible enough to do what I wanted. These "canned" packages work fine for development work - proving out the techniques - but when you want to make a specific application it gets problematic. Also, when coded, the run times were almost ten times faster than with MathCad.
The thing that needs to be resolved is if you will "scan" or do individual points. If you scan then the number of points is irrelevant. If you do discrete points, especially if you rotate by hand, then the number of points becomes a BIG factor. There will also be a very big factor in complexity and cost between scanning and manual rotation. No DIY is going to be able to implement a scanning system, so I don't know why this is even considered here. A system like that would take a "Klippel" level of resources.
The thing that needs to be resolved is if you will "scan" or do individual points. If you scan then the number of points is irrelevant. If you do discrete points, especially if you rotate by hand, then the number of points becomes a BIG factor. There will also be a very big factor in complexity and cost between scanning and manual rotation. No DIY is going to be able to implement a scanning system, so I don't know why this is even considered here. A system like that would take a "Klippel" level of resources.
This isn't really true, even harmonics are symmetric and odd are anti-symmetric and the combination of the two can handle any level of complexity. It is true that symmetry allows for massive reductions in complexity. Completely arbitrary configuration will require a very very complex system.
Also, the number of points depends on frequency and resolution. For example a LF monopole only requires a single point, a HF beam might require 40-50 points to get the same resolution.
Have you tried Mathlab or Octave? I'd like to have the opinion of someone with some experience of that software before I commit to learn it.
I understand that Mathlab is focused on numerical solutions of matrix problems, pretty much what we want here.
MathCad seems to have a somewhat different orientation, which may explain why it was rather slow in your application.
In any case, "fine for development work" will satisfy me.
Not sure I understand what sort of scan you mean, do you mean a continuous position sweep or just automated movement to fixed points?
I assume we will measure at a number of fixed points with a time windowed acoustic measurement technique.
Initial idea is a smallish number of points with manual movement.
If this works then the next step is to automate the movement to increase the practical number of points.
It should not be too hard to automate the mechanical structure I proposed because it only needs rotary movement, fairly easy with a simple stepper motor.
This is only a bit less flexible than the Klippel implementation but far simpler, less massive and less expensive.
At the moment I feel acutely the lack of Klippel level intellectual resources.
A nice fresh PhD level mathematician plus a multi year experienced acoustics veteran to work full time would be handy.
Best wishes
David
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