Diffuser field

[Tibor Kovari]

Hello Everyone on this site!

I have just found an interesting approximative connection between the reflection factor of a line diffuser and its polar far-field pressure characteristics.
It is published in (9.7) 'Acoustic Absorbers and Diffusers' by Cox and D'Antonio.

The R(x) is the reflection factor function of the surface, theta is the angle of reflection, psi is the angle of incidence, and k is the wavenumber.

Could anyone share deduction for this? Is that done the same way as in electromagnetics?

[bert stoltenborg]

Yeah.
R(x) is the reflection factor.
But how is that defined?

[Scott R. Foster]

The compliment of the absorption coefficient?  0.1 becomes 0.9?

[Tibor Kovari]

This is the pressure reflection coefficient, the ratio of the reflected and incident pressure. Seems to me much like the voltage reflection factor, Γ in electomagnetics. A complex function of the only dimension (x) of the surface of the diffuser.

[Tibor Kovari]

Yes, α can be expressed with the magnitude of R, α(x) is a scalar function.

[bert stoltenborg]

What I mean is:
can in practice a complex diffusor respons be modelled in a formula?
With speakers theoretically this is possible, but in practice it's more difficult and you have to do measurements.

[Tibor Kovari]

If the incidence is a plane wave from straight angle this complexity makes no problem, in cases of other angles there is a linear time delay along x (line diffuser). The surface impedance is static, so the continuous reflection in time will also rise no problems. The way of elimination of the problem of the uneven surface is as follows:

The shape itself is represented by a line, and the surface impedance is calculated by impedance transformation from well depths and planned absorption, just like in the case of electromagnetic supply lines. The wave is considered as a plane wave along the elements of the surface. Of course these elements can be as small as you like, but usually we consider the raster size of the diffuser if it has one.

The reflection factor is given by the calculated impedance, this way we know the reflected component of surface pressure in one line, just if it were an array of phase and amplitude modulated primary half-space point sources*.

The problem is the rest, the far field radiation characteristics of half-space point sources with given phase and magnitude.

Have you seen such a deduction somewhere?

Update: If we consider a 3D problem instead of a 2D one, the sources will be line sources, so we must calculate with an 1/2rπ-style pressure characteristics.

[bert stoltenborg]

In Beranek or Olson there are formulae for several types of sources.
If you want to model a real world multiple driver sound source with arbitrary position, phase, amplitude etc you could use Akabak.

[Tibor Kovari]

I have looked at the campus library catalog here, but no Olson books. Which one of Beranek do you think about?
Akabak: no numeric solvers, thanks, this is a theoretical question. I would like to understand the limitations of this approximation.

[bert stoltenborg]

Beranek's Acoustics, the famous one.

[Tibor Kovari]

Thank you, Bert.

The book does deal with point arrays, but it does not talk about line sources. Sorry, I was sloppy with the 'point sources'. Those are points only in this particular 2D interpretation.
However, I managed to deduce the first (and more simple ;) part of the problem, please correct me, if you encounter mistakes. The incident wave is a plane wave from a far point source or a plane source. The magnitude of the incident component of the surface pressure is constant, the phase depents only on the difference of the distance (from a chosen wavefront considered to have a phase of 0˚).
I uploaded a fast sketch on this.

If then we modulate this with R(x), the given reflection coefficient, we can get the reflected component of pressure on the surface of the reflector element.

As I remenber, the field of a line source can be counted as

p(r)=p0/(2rπ)exp(-jkr)
and multiplied by 2 for the half space impedance

p(r)=p0/(rπ).

I could not find a reliable source for this so far.
As r cannot be 0, this must be expressed some other way e.g. let this be only applicable to a non-zero long element, and let's express r by a sum of two components, a radius, that is measured form the surface (r'), and an other one, that is the rest to the concentrated virtual source (r'').

o|---------r'-------I--r''--¤

r=r'+r''

o| stands for the microphone, I is the element, thougth to be a 'window', and ¤ is the concentrated, virtual source.
(This is just the display of the summation, and the length of the element is less than λ/6, so do not look for geometrical sense in it, diffraction and so on.) Hopefully later any r can be eliminated of course.

Does anyone have a handbook near that contains the field of a line source as a function of distance?

This is the closest publication to the subject I found, but it is still not applicable directly:
http://www.diracdelta.co.uk/science/source/a/c/acoustic%20sources/source.html

[bert stoltenborg]

Not sure what you mean, Tibor.
A line source is by definition just a source radiating a cylindrical wave, nothing else. It decreases with 3 dB for every doubling of distance, as a point source does with 6 dB.
Such a source is mathematically described by formula 4.11, page 96 in beranek, third printing 1990.

[Tibor Kovari]

You are right, 3dB, indeed. Then I was wrong with the pressure decay too, it will be 1/sqrt(r), not 1/r.

This is formula is just perfect by its approach, as it provides polar characteristics directly, no need for polar conversion.

But it's argument is not pressure, but d, the size of the array, which gives a square function. The Fourier-transformation then gives a nice sin(x)/x shape, as expected. At least we see, that it really starts to behave similar to the Fourier-function in the main formula.

I think the topic starting formula will work best, when d converges to infinite.

But, this is still promising, as an example solution of the function we are looking for, with p(x)=constant case, showing p(θ)=constant.

[bert stoltenborg]

A line array will show lobing, and this is freq dependent.
In practice lobes with amplitudes under 15 dB ref main amplitude are neglected.
Maybe this should be done for diffusers also.

[Tibor Kovari]

This phenomenon can be seen in 3 dimension on this numeric model:
The subject is a plain almost square shaped reflector. The dark blue circle is a point source. There is a symmerty plane defined at z=0, so we can see the half of the subject reflector.
Similar pattern is on the back side, but of course on the blue end of the pressure scale.

[bert stoltenborg]

A speaker line array is of course a device meant to beam sound as much as possible.
A diffuser is the opposite.
A numerical diffuser such as a QRD only works at incident sound.
Under angles you get phase shifts not fulfilling the QRD assumptions.
This is why they place dividers between the wells.
As these will also behave as 1/4 wavelength absorbers/resonators, and you get all kinds of diffraction effects, the real radiation from such a surface is extremely complex, I guess.
What your initial formula says is only: when I measure the polar of a diffuser, I get a lot of freq dependent  amplitude differences.
That's this Rx or whatever function.
To describe this function properly, you need a lot of understanding about wave behaviour and thus some really complex math.
Normally you would use some FEM or BEM modelling software, I think,.

[Tibor Kovari]

This is the point, lobing, as our main equotion says, if we take a sequence known to have an even Fourier transform, as MLS has e.g., there will be no scattered lobes, the reflection characteristics will be a half circle, the diffusion will be perfect.
But this is just true to the extent, as the main equation is true. This is why I would like to deduce and analyse it a bit.

[Tibor Kovari]

The sequences and the way they achieve the proper impedances are belong to the application, not really relevant here. As i know, the separator plates are for making the depth-based diffuser behaving more like the mathematic model, where there are plane wavefronts in the wells, with step changes at the borders. This way the computation of the impedance transformation will require no equipment.

[Tibor Kovari]

Absolutely right, awful, but this is a far-field approximation, I hope, will not be so difficult. Even after we are through with the half of it ;)
So the next step is the proper description far field of the line source.

I just got a hint from my Teacher, that it is in

Morse, Philip McCord, Theoretical Acoustics, 1968.

All the pieces are occupied or can be ordered from a remore stock. Now I try to order it, but I will not get it until tuesday. Do someone have a specimen of it with the formula?

[Tibor Kovari]

Hm. We should try combine your directivity pattern with d as the element size, modulated with p(x), added the missing 1/sqrt(r) -style distance dependant decay, integrate it by θ, and approximate the whole thing to zero element size. This sounds like a process that could give us the eqation.