Klippel Near Field Scanner on a Shoestring

[Dave Zan]

Originally Posted by gedlee

...I've done my stint in acoustics, time to move on. I'll advise and answer questions...

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

[gedlee]

Originally Posted by Dave Zan

Efficiency is where I have a conceptual problem at the moment.
To fit a uniformly spaced set of data is not much more difficult that a Fourier transform.
It is not clear to me how to optimize measurement points when the practical interest is much more in the forward direction.
And once we have optimized but uneven spaced data I am not sure exactly how to fit it.
IIRC Earl Williams actually wrote that Fourier Acoustics would be fairly simple if it wasn't for real world issues like this, and truncation effects, numerical stability and so on.

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.

Originally Posted by Dave Zan

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

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.

[aslepekis]

Originally Posted by Dave Zan

I think it is possible to explain the maths in physical terms.
The B&K probe is an all-purpose intensity probe.
There are inevitable physical constraints, if they increase the spacers for low frequency sensitivity then they lose hi frequency accuracy, and conversely.
We have extra information, we know the source is inside the scan surface and the echoes are outside so we can use the fact that the solution should be in the form of a Bessel (or, equivalently Hankel) function.
So we can fit the data to a Bessel function rather than use a linear approximation.
This is more accurate, as the reference shows, we have a better trade-off between hi frequency accuracy and low frequency sensitivity so some of the limits of the B&K probe are not relevant.
So we can exceed the 50 mm limit of the linear approximation but 500 mm is probably excessive.
This is essentially what the Klipppel patent is all about.

That makes sense to me; only focusing on under (say) 1kHz gives more freedom in how much separation we can go with.

Originally Posted by Dave Zan

It is complicated by reflections off the speaker itself, which break our assumption.
It is not clear to me yet how much of a problem this is, Earl says he never noticed this but it may be that it simply wouldn't be evident with his method.

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.

[Dave Zan]

Originally Posted by gedlee

...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...

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

[gedlee]

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.

[Dave Zan]

Originally Posted by gedlee

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...

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

[jcx]

probably not 'shoestring' Microphone Arrays - acoustic-camera.com

[scott wurcer]

Originally Posted by jcx

probably not 'shoestring' Microphone Arrays - acoustic-camera.com

NASA has several papers on using cheap capsule mics in these large arrays.

[Dave Zan]

Originally Posted by jcx

Microphone Arrays - acoustic-camera.com

Originally Posted by scott wurcer

NASA has several papers...

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

[aslepekis]

Originally Posted by jcx

probably not 'shoestring' Microphone Arrays - acoustic-camera.com

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.

Originally Posted by scott wurcer

NASA has several papers on using cheap capsule mics in these large arrays.

I'd like to read more about that, do you have any links?