## Question about surface impedance

Post and discuss acoustic topics, Studio design, construction, and soundproofing here

### Question about surface impedance

In Cox and D'Antonio's new book on absorbers and diffusers, chapter 1 is an introduction to the methods used for calculating the absorption values of porous absorbers.

I have a question about the following statement made in the middle of page 18 as almost a throw away comment.

"The surface impedance is often split into the real term (resistance) and imaginary term (reactance). In general, the real term of the surface impedance is associated with energy losses, and the imaginary term with phase changes. So a simple inspection of the surface acoustic impedance can give more insight into the absorbing properties of a material than the absorption coefficient."

How can more insight be obtained?

I have plotted the real and imaginary parts of the surface impedance against the final absorption coefficients of the absorber, and I cannot see an obvious correlation. So I guess I must have missed something.

Can anyone please explain how to interpret the value of the surface impedance in this manner?

Thanks

Chris W
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

Chris,

I think that's a question for the science group.

If you can get Kari to answer.
He (Kari is a Finnish man's name) is very good in defining those things.
But you know there are others too...
I think I must buy me that book. Your always talking about something I (still) don't have.

Kind regards
Eric
Eric.Desart
Moderator

Posts: 2461
Joined: Tue Feb 10, 2004 4:29 am
Location: Antwerp, Belgium

Hi Eric

If you are referring to alt.sci.physics.acoustics, then I have cross posted this question there as well.

If the Kari that you refer to is Kari Pesonen, then I hope he still reads that NG; he doesn't seem to have posted anything there since February.

BTW, buy the book! I know I'm asking lots of questions about it, but I have also learnt a huge amount about absorber and diffuser design.

Still studying...

Chris W
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

It's the potential and kinetic energy thing.
A moving pendulum has a velocity (kinetic) maximum in the lowest position and the potential energy is then lowest; when it is swinging to it's higest position the potential energy is higest and the speed lowest. The two forces are then 90 degrees out of phase because one force counterforces the other.
The reactive part is here max, the spring is loaded so to say with potential energy.
Electrical, mechanical etc. Impedance is composed out of forces and the way these forces react is frequency dependent.
All materials, speakers, mechanical devices etc can be analyzed as a system of electrical elements like coils, capacitors and resistances. Or mechanical of masses, springs and resistance. This is just a choise you make.
When you know all the properties of an absorber, loudspeaker, coil, horn etc you can calculate it's behaviour because you understand how it is dealing with energy, you know how the reactive parts (springs), masses and the resistance interact.
Using these elements with the proper equations in a speadsheet or a math program like mathcad, mathematica, etc, enables you to make a pretty adequate model of for example a loudspaker.
bert stoltenborg

Posts: 4574
Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

Hi Bert,

I understand that mechanical and acoustic systems can be simulated using various electonic components, but in this example, I am asking about the statement in Cox and D'Antonio's book that says the real part of the impedance (resistance) corresponds to energy loss, and the imaginary part (the reactance) corresponds to phase changes.

Using the graphs attached to this post, how can the values be interpreted to give a better understanding of the material's behaviour than looking simply at the absorption plot?

From an acoustic point of view, the impedance of a material is the ratio of particle pressure within the material to particle velocity through the medium. For air, this value contains only a real part, but for porous absorbers, there is often also an imaginary part.

The attached plots were made in Excel for a 25mm layer of porous absorber mounted directly on a rigid backing (Normal incidence sound). The first plot is of the absorption coefficients, and the second is of the real and imaginary parts of the porous absorer's impedance.

As you can see, the imaginary part of the impedance is very negative at low frequencies, only crossing zero at about 3KHz, 6KHz, 9.5KHz and 12.7KHz. I was under the impression that the zero crossing of the imaginary part indicates that there will be resonance at that frequency.

The zero crossings that coincide with peaks in the real part of the impedance (maximum energy loss?) are those where the imaginary part crosses zero in the positive -> negative direction. However, these peaks do not coincide with absorption peaks.

Do you know how it is possible to interpret simply the real and imaginary parts of the impedance in order to better understand the absorptive properties of the material?

Regards

Chris W
Attachments
Absorption.gif
Absorption of 25mm porous absorber of 10,000 rayls flow resistivity
Impedance.gif
Real and imaginary part impedance of 25mm porous absorber of 10,000 rayls flow resistivity
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

I'm a loudspeaker guy.
The radiation resistance of speakers is often expressed as a kr or ka ratio, where the dimension of the speaker is compared to the radiated wavelenght. Here you see that the radiation impedance steeply falls when the wave dimension exceeds the speaker dimension. In practice the imaginary part is often neglected.

In your example the ratio between material thickness and wave lenght could make things clearer, although I am surprised about the low freq extension of the real part of the graph.

I came to the acoustical science group because I didn't really understand how they come to these impedances (they screw around with Bessel functions etc), and I still don't. None of the engineers and scientists I know seem to really understand, they use equations they get from old text books and some shapes, like a rectangular membrane, can not be received with analitical methods, you have to get 'm from numerical methods (try and error).
Angelo Campannela, the owner of the acoustical science groep, answered my question by giving a citation from a book by Raleigh, but that didn't clear me up much.
I get the feeling that it is like with elementary particle physics or gravity: "The particles apply a force at each other...." "Yeah dude, but how?" "Ehhhhhh......".
:-)
bert stoltenborg

Posts: 4574
Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

bert stoltenborg wrote:I'm a loudspeaker guy.

And a damned good, clever and nice one.
Still hope to meet you some day ...)

A southern neighbor.
Eric.Desart
Moderator

Posts: 2461
Joined: Tue Feb 10, 2004 4:29 am
Location: Antwerp, Belgium

bert stoltenborg wrote:The radiation resistance of speakers is often expressed as a kr or ka ratio,

Unfortunately, the impedance of a porous absorber is more complex simply that kd where k=wavenumber and d=absorber thickness.

Just calculating the wavenumber of a porous absorber requires some funky calculations using formulae derived empirically by Delaney & Bazley back in the seventies.

bert stoltenborg wrote:In practice the imaginary part is often neglected.

Unfortunately, this is a critical value in acoustic calculations. For a resonant system such as a Helmholtz resonator, resonance occurs when the imaginary part of the impedance = 0.

bert stoltenborg wrote:In your example the ratio between material thickness and wave lenght could make things clearer, although I am surprised about the low freq extension of the real part of the graph.

Me too. How to interpret...

bert stoltenborg wrote:I came to the acoustical science group because I didn't really understand how they come to these impedances (they screw around with Bessel functions etc), and I still don't. None of the engineers and scientists I know seem to really understand, they use equations they get from old text books and some shapes, like a rectangular membrane, can not be received with analitical methods, you have to get 'm from numerical methods (try and error).

The only time I've needed a Bessel function is for calculating the absorption of a micro-perforated panel. I.E. a panel where the holes are <1mm in diameter. At these scales, the viscous air/panel boundary layer (which is about 0.2mm thick) starts to play a significant role. In fact, you don't need any absorbent layer behind a micro-perforated panel, because the holes themselves provide the absorbency. It is this principle that allows you to create a transparent absorber.

This book by Cox and D'Antonio that I keep refering to explains many aspects of the theoretical side of acoustic prediction. However, there are some statements that have left me wondering how best they should be understood.

Hence my original question...

Anyone, anyone, Bueller...

Chris W
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

I ordered the Cox and D'Antonio book a few weeks back, and was informed today that they are out of stock of them in north america, but are expecting new stock in a few weeks.
Bob

Posts: 4360
Joined: Sun Feb 08, 2004 4:37 am
Location: Oakville, Ontario, Canada

(You make me blush, Eric)

Chris,

I am interested in your question.
I've been quickly reading some references.
Pierce (Acoustics) does not seem to come up with equations, although I have to admit that I do not understand much of his book.
Beranek, Ver etc (Noise control) comes with some equations but they seem to neglect imaginary components
Beranek (Acoustics) only gives coefficients
Olson (Acoustical engineering) gives equations and graphs that ca ( heh heh) are similar to the ones you show; but he warnes that measurements differ from calculations. The reason for this has been explained exhausively by Jeff and Eric.
When I have time I'll put the eq's in mathcad and see what the effect of neglecting the imaginaries is.
You have shown here not te be stupid, so when you don´t understand the book you bought, you have in my opinion the right, or even the duty, to mail mister d`Antonio at RPG and ask him for an explanation :-)

Regards,

Bert
bert stoltenborg

Posts: 4574
Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

Hi Chris,
Could it be as simple as once you have real and imaginary values you can calculate magnitude and phase, which may be another way to evaluate the material?

Andrew
Andrew Steel

Posts: 139
Joined: Sun Feb 08, 2004 9:38 pm
Location: Brisbane Australia

Chris

Trying to remember all the theory I'm suppose to know but Andrew has said it.

A complex number which has both magnitude and angle used in analysing AC sinewave circuits.

I know it best from my ham radio antenna projects which has been replaced by room acoustics now!

A resonant antenna would be 50 ohm j 0 which means that it has a 50 ohm resistive part (power consume) and no phase shift, a purely resistive circuit.

A circuit with reactance would be 50 ohm +j 43 which would cause the current to be out of phase with the voltage by so many degrees. The +/- j operator indicates if it is capacitance or inductance and thus lagging or leading.

You were right about the book Chris, interesting reading. Bob, you can get it as an eBook from the publisher if you don't mind not having it in your hands.

RJ
notallthere

Posts: 23
Joined: Tue Apr 20, 2004 11:10 pm

RJ and Andrew,

what I (and I guess Chris) am curious of:
when you look at the graphs, how can you interpret them in real life terms?

The phase and imaginary stuff (theoretical) I understand, but I don't know how to translate it into pactical terms. I asked other smart people, but they don't come up with something I find satisfactory.

In theoretical acoustics there is much emphasisms put on these theorectics, so I would like to see them translated into something I (stupid ingnorant) understand.
Seems I'm to lazy or dumb to figure out myself.
bert stoltenborg

Posts: 4574
Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

Thanks for all the responses folks. I'll try to respond in order...

To Bert
I ordered the book directly from the publishers http://www.sponpress.com and recieved it in a couple of days. But then I live in the UK!
The graphs I attached to earlier posts were one's I have calculated by re-implementing the MATLAB scripts contained in the book's appendices in Excel. I also have MATLAB plots available which show the same results.
I don't believe the imaginary component of the impedance can be ignored because it is a measure of the porous absorber's reactance. I.E. It is a term that describes the addition of virtual mass to the absorber due to the viscous boundary effects created as air flows through the pores of the absorber. This in turn is related (in some way that I don't fully understand) to a phase change that takes place as the wave travels through the absorber.

However, as you can see from the attached graphs, the imaginary part of the impedance is very negative at low frequencies, and is still negative when the actual absorbency reaches a peak at about 1.5KHz
As for directly contacting the authors, Peter D'Antonio and Trevor Cox, I have already done this in respect to an earlier question I had about one of the MATLAB scripts in the book generating negative absorption values.

I'm afraid the response was not what I was hoping for. Prof D'Antonio has not responded at all, and Prof Cox refuted any mention of a script generating negative absorption values, and said he was too busy to spend any time looking at the details of my question.

Well, I appreciate that both men are probably very busy, but on the other hand, I would have expected a more positive response to someone who takes a genuine interest in their research, and asks questions based on a real desire to learn. I don't know what their reasons are, but I am rather disappointed.
Finally, your comment about needing all the abstract theoretical stuff translated into practical, understandable stuff is largely addressed by Cox and D'Antonio's book. It has been written to balance practical application with theory, which is why I am finding it very helpful (but also challenging!)

To Andrew
I think you are probably right that the impedance can be interpreted in terms of magnitude and phase, but I don't exactly know how to proceed here.

Acoustic impedance is the ratio of particle pressure to particle velocity. This gives the characteristic impedance (zc) of a medium (which for air is entirely real). Using a transfer matrix, the suface impedance (z) of a porous absorber is given by:
Code: Select all
`z = -j * zc * cot(k * d)`

Where k is the wavenumber within the porous absorber and d is the thickness. Calculating this is straightforward, but how should the real and imaginary parts be interpreted - particularly the imaginary part, which is very negative (several thousand) at low frequencies. How does this relate to phase?

To RJ
As I said to Andrew, the imaginary part is very negative at low frequencies - with values that do not appear to relate to anything measured in radians.

Summary
There's a statement in the first paragraph of section 3.6.2 (page 81) that sheds some more light on what the imaginary term of the impedance represents, but it still puzzles me as to how the statement in my original question should be understood.

For those people who have bought this book (and can remember all the applied maths!), I'd like to have further discussions about the contents. Particularly surrounding the design of hybrid diffusers.

For those of you who don't have the this book (and want to give your brain a workout) - get it! It is very interesting reading.

Regards

Chris W
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

hi Chris,

last remark:
Olson says that there is no satisfactory method to determinate the effective density of the air in the pores of the material. It has to be assumed.
Of course this is a book from 1957.

Maybe you can tickle the very learned authors of your book by saying that it is OK to admit that they don't understand their own book :-)

Regards,

Bert
bert stoltenborg

Posts: 4574
Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

Hi Bert
bert stoltenborg wrote:Olson says that there is no satisfactory method to determinate the effective density of the air in the pores of the material. It has to be assumed.
Of course this is a book from 1957.

That statement was made even before Delaney and Bazley derived their empirical formulae for determining the characteristic impedance and wavenumber in a porous absorber.

D&B's formula (derived in the early 70's) does not require the density of air to be adjusted in order to make an acceptably accurate predition of either the impedance or the wavenumber.

So I think Olson's statement has been superceded by later developments.

bert stoltenborg wrote:Maybe you can tickle the very learned authors of your book by saying that it is OK to admit that they don't understand their own book :-)

Hmmm, I think its more a case of the author making a statement about something he knows he understands, but hasn't yet condensed that understanding down into words so that it can be communicated to others.

I hope that in future, they will have more time to look at my questions - unless of course someone else knows the answer...

Regards

Chris W
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

Chris

I'm no math wiz here. But after reading about the test setup for determining the impedance, around pg 168 in "the book", I would think your graph maybe right.

Looking at the test setup, I would think the pressure at mic 2 would become greater than mic 1 as you lower the frequency. The reverse would happen when you raise the frequency.

Just thinking when I should be working. Shhh, don't tell the boss!

RJ
notallthere

Posts: 23
Joined: Tue Apr 20, 2004 11:10 pm

### Re: Question about surface impedance

Hi Chris

I realise this thread is a bit old....!!

I was very interested to read through this topic as it has also been bothering me for some time. Did you ever get a definitive answer?

Rgds
Gary
Icecube

Posts: 3
Joined: Sat Oct 13, 2012 10:04 am

### Re: Question about surface impedance

Amazing to read this old stuff.
And indeed it seems that Cox and dÁntonio go here over the matter with seven miles boots.

Our friend Jonnesy works with Cox now, AFAIK, and Peter is on linkedin, so maybe you can ask the authors themselfs?
If you view life with the knowledge that there are no problems, only opportunities, you are a marketing manager.......this is my personal philosophy
bert stoltenborg

Posts: 4574
Joined: Sun Apr 18, 2004 11:03 am
Location: Achterhood, Netherlands

### Re: Question about surface impedance

Hi Gary, Bert

No, I never did get an answer.

I mailed Trever Cox about a couple of things such as errors in the MATLAB scripts in the appendix of his book and another small but significant typo in one the equations, but he didn't seem too interested in getting into any sort of discussion. I typically got one or two line replies.

To be honest, I haven't pursued this any further; although now that the question has come up again, may be I'll look into it again.

Chris W
--
The voice of ignorance speaks loud and long,
but the words of the wise are quiet and few.
--
ChrisW

Posts: 153
Joined: Tue Mar 30, 2004 12:42 pm
Location: Brentwood, Essex, UK

Next