Analytical Analysis - Displacement vs Sound Pressure

[René - Acculution.com]

- Introduction -

In a recent interview with Erin of ErinsAudioCorner (https://www.audiosciencereview.com/...ns-with-rene-christensen-of-acculution.28103/), I showed that for steady-state conditions, a loudspeaker moving outwards creates a lower overall pressure, not higher as is often claimed. When doing measurements, you see how pressure relates to voltage only, but when doing simulations you have a much better view into the phase relations along the ‘transductance chain’, and you can come across effects that may go against your initial intuition.

I the following, I will show in very few steps, how you can analytically arrive at the same conclusion, looking at a flat circular piston in an infinite baffle.


- Details -

The complex sound pressure phasor (bold p) relates to the complex velocity phasor (bold U0) as [see e.g. Fundamentals of Acoustics, Kinsler and Frey] for the situation with a piston of radius a in an infinite baffle is:

I have adapted the expression so that z is in the axial direction and r is in the radial direction, and a is the radius, k is the wavenumber, and we have density and sound speed related to the medium.

When putting the observation point directly in the middle of the piston, the expression simplifies to:

We only look at low frequencies where we have a flat frequency response and so we consider that ka is low. We can then approximate the exponential term as:

and inserting back into the original expression we get:

or

Since acceleration relates to velocity as

we can finally arrive at

and we see that the pressure is in-phase with acceleration, which in turn is in anti-phase with displacement. Hence, at the piston’s outermost position into the medium, the acceleration is negative and so is the sound pressure.


- Conclusion -

A piston in a baffle creates the lowest total pressure (negative sound pressure) when it has moved all the way out, not in. This is also seen in more typical electrodynamic driver setups.

You should always be careful when carrying around knowledge that came out of intuition or “it is well-known in this industry that…”-thinking, where you trust what is being said within the company that you are in. Once in a while, it can affect the engineering decisions in ways that can cost a lot of time and money to course correct once the problems show up. Try and work from first principles whenever you can.


- About me -
René (no need for Dr title, please), BSEE, MSc (Physics), PhD (Microacoustics), FEM and BEM simulations specialist. Own consultancy company Acculution ApS, and blog at acculution.com/blog

 

[fieldcar]

Thanks for the post. It's a bit over my head, but It's spiked a bit of curiosity.

a loudspeaker moving outwards creates a lower overall pressure,

Overall pressure of the room, or near field measuring point? I've always seen those room simulations showing positive or negative pressures depending on the wavelength in a room. I think this is why I've always assumed the traditional mantra.



Is this dependent on enclosure/baffle type? I imagine that an infinite baffle setup would react differently than an open baffle.

Thanks for the post. I'll have to watch the youtube interview. It looks really interesting.

 

[ppataki]

Thank you for the post; also way above my head but would like to ask one question if I may:
What is the practical usage of this? I mean how can we/shall we take this into account when building a loudspeaker into a given room?
Many thanks

 

[René - Acculution.com]

Thanks for the post. It's a bit over my head, but It's spiked a bit of curiosity.



Overall pressure of the room, or near field measuring point? I've always seen those room simulations showing positive or negative pressures depending on the wavelength in a room. I think this is why I've always assumed the traditional mantra.

View attachment 170836

Is this dependent on enclosure/baffle type? I imagine that an infinite baffle setup would react differently than an open baffle.

Thanks for the post. I'll have to watch the youtube interview. It looks really interesting.

Think near-field, or far-field with the phase change from the distance taken out of the picture. No room is considered here. While the molecules near the piston do follow the motion of the piston, it is not true that when the piston moves out, the pressure increases, it is the other way around, when playing into open space.

Looking at a color plot like the one you show here, you can see the phase of the pressure on its own (red is in anti-phase with blue), but it is not always related back to some reference phase, especially if it is a mode that you are visualising.

In front of the piston, the conclusion holds no matter what the enclosure is, but the overall piston/driver displacement will of course will be affected by the rear side if there is coupling included.

 

[Inner Space]

René (no need for Dr title, please), BSEE, MSc (Physics), PhD (Microacoustics), FEM and BEM simulations specialist. Own consultancy company Acculution ApS, and blog at acculution.com/blog

Thanks, Rene. Off this topic, but since you're here, may I ask another question? I remember you did simulations about waveguide screws. Along the same lines, do you have a view about how seamlessly a soffit-mounted speaker needs to be mounted into the wall? How much discontinuity will be noticeable?

 

[René - Acculution.com]

Thank you for the post; also way above my head but would like to ask one question if I may:
What is the practical usage of this? I mean how can we/shall we take this into account when building a loudspeaker into a given room?
Many thanks

It is just one misconception that is so prevalent that most companies I have been in, the engineers have gotten this wrong. When you work with e.g. hearing aids, where feedback is big problem, it can lead to disatrous results, if you miss a sign inversion, because you think something is opposite of what it actually is.

 

[René - Acculution.com]

Thanks, Rene. Off this topic, but since you're here, may I ask another question? I remember you did simulations about waveguide screws. Along the same lines, do you have a view about how seamlessly a soffit-mounted speaker needs to be mounted into the wall? How much discontinuity will be noticeable?

Ooooh, it is the same thing; it will be case dependent, and you will be able to see more and more of an effect, as you stray away from the surface-mounted situation. It can of course be simulated, but it does not make for a good post (IMO) to convey some principle or misconception, which I am mostly into.

 

[KSTR]

Isn't that all just covered with the notion that SPL is portional to acceleration, and acceleration is the derivative of velocity which in turn is the derivative of position? So, for sinusoids, we have sin(x) --> cos(x) --> -sin(x).

 

[René - Acculution.com]

Isn't that all just covered with the notion that SPL is portional to acceleration, and acceleration is the derivative of velocity which in turn is the derivative of position? So, for sinusoids, we have sin(x) --> cos(x) --> -sin(x).

Yes, when considering that free space can be considered mass-like. But still, it is wrong in most videos on how a loudspeaker works, and engineers mess it up as well.

 

[Kvalsvoll]

Yes, this is the case for a small radiator. But when ka>=1, the acoustic load impedance becomes resistive, and thus the resulting acoustic sound field will have p and v in-phase.

Such a situation is present at higher frequencies for many direct radiator drivers, and for horns, and for a radiator at the throat of a horn.

 

[pseudoid]

A piston in a baffle creates the lowest total pressure (negative sound pressure) when it has moved all the way out, not in.

Isn't the total radiated energy (pressure) during the forward excursion still much greater than the reverse?
Can I actually (try to) infer that during this excursion (forward or reverse) the behavior of the cone-caused distortion products vary along the path of the excursion? Appears that the worst (instantaneous distortion) would occur at the extremities (limits) of the excursions in each direction.

 

[René - Acculution.com]

Yes, this is the case for a small radiator. But when ka>=1, the acoustic load impedance becomes resistive, and thus the resulting acoustic sound field will have p and v in-phase.

Such a situation is present at higher frequencies for many direct radiator drivers, and for horns, and for a radiator at the throat of a horn.

Sure, but I would still like engineers to jump to 'acceleration' instead of 'displacement', if they don't want to learn about Laplace Transforms, and Bessel and Hankel functions.

 

[René - Acculution.com]

Isn't the total radiated energy (pressure) during the forward excursion still much greater than the reverse?
Can I actually (try to) infer that during this excursion (forward or reverse) the behavior of the cone-caused distortion products vary along the path of the excursion? Appears that the worst (instantaneous distortion) would occur at the extremities (limits) of the excursions in each direction.

This is non-linear behavior, which is not included in this analysis.

 

[charleski]

Isn't that all just covered with the notion that SPL is portional to acceleration, and acceleration is the derivative of velocity which in turn is the derivative of position? So, for sinusoids, we have sin(x) --> cos(x) --> -sin(x).

Or, in graphical form:

For those unfamiliar with calculus, just think of a particle riding on the displacement wave, then consider how fast that particle is moving at the peak, trough and zero-crossings (its velocity), and then consider how much that velocity is changing at each of those points.

 

[René - Acculution.com]

Or, in graphical form:

For those unfamiliar with calculus, just think of a particle riding on the displacement wave, then consider how fast that particle is moving at the peak, trough and zero-crossings (its velocity), and then consider how much that velocity is changing at each of those points.

Yes. But the link to why the force is related to mass is not there unless going into to details about radiation impedance, whereas the shown math for the sound field from a piston in a baffle is rigorous on its own.

 

[bigjacko]

@René - Acculution.com Thanks for sharing this, may I ask why engineers got this the wrong way in the first place, how did they get ths misinformation from?

I have looked up some acoustic text books about those transducer equations and acoustic equations, but the author always assumes the reader already got enough background knowledge about the math or physics going on. The more basic simpler stuff gets less explained, which makes me very mad. There are just too much high level stuff flowing around that in the end no one can find the basic stuff anymore. Another thing is some textbook just rant on and on, don't make concise and clear statment. Can you recommend some website or text that explains the very basic math, physic and acoustic stuff with very concise words and easy to understand(with picture would be awesome)? I don't need those text to give me all information, but just save the dummys that never heard of the word "science". Thank you very much for your work on ASR, wish you can come here more often and hand out with us!

 

[René - Acculution.com]

@René - Acculution.com Thanks for sharing this, may I ask why engineers got this the wrong way in the first place, how did they get ths misinformation from?

I have looked up some acoustic text books about those transducer equations and acoustic equations, but the author always assumes the reader already got enough background knowledge about the math or physics going on. The more basic simpler stuff gets less explained, which makes me very mad. There are just too much high level stuff flowing around that in the end no one can find the basic stuff anymore. Another thing is some textbook just rant on and on, don't make concise and clear statment. Can you recommend some website or text that explains the very basic math, physic and acoustic stuff with very concise words and easy to understand(with picture would be awesome)? I don't need those text to give me all information, but just save the dummys that never heard of the word "science". Thank you very much for your work on ASR, wish you can come here more often and hand out with us!

Hi bigjacko Well, for one, engineers are taught these things, but they don't always stick in people's brains, especially when it goes against intuition. In many companies, the management does not have a focus on retaining (fundamental) knowledge in the departments, and so it fizzles out. My wife is a medical doctor and they have internal conferences all the time discussing cases, and I tried doing something similar in my previous company but with fundamentals, not on-going problems. And it just was impossible, since management is of course afraid that the time spent does not repay itself somehow (it does, you would work much faster and with fewer 'fire-fighting' panic cases), and the engineers have gotten comfortable in this way of working via analogy and intuition, and live in an echo-chamber, because they never go out and present at conferences or something similar.

So specifically for drivers or anything moving, most people's intution will be displacement->pressure. This can cause a lot of problems in hearing aids, where microphone output comes both from the pressure generated both from wanted sound but also from the hearing aid itself vibrating, and then directly from the microphones being vibrated. If you think 'displacement' instead of 'acceleration' on the pressure side, you have messed up a sign, and now you are in a bad spot for ideas regarding lowering feedback issues. It takes grit to stay on top of these things, and study pretty much every day, but that is what engineers should strive towards...

I have not come upon a good site or book that explains things concisely without getting too much into the mathematical details, and it is tough to do. I have started making videos on Engineering Mathematics on YouTube, but it is difficult to find the time, and there will be a lot of math there too. Most books or videos that tackle complex topics in a simple manner revert to intuition and analogy, and so get some important details wrong. One topic which should not be an issue at all for example, is Phase vs Polarity; most videos and articles on this topic are wrong, so I might work with students that carry these misconceptions also, and it is taxing at times. I don't know what the solution is, but there should almost be peer-review on videos of technical nature.

I would like to have more time to post things, but it is difficult with a newly started company. One thing that I would like is something where many people could gather and discuss things on-line, but with one or more moderators. So that topics like the one discussed in this post could be closed once and for all, and the discussions could serve as a repository for these questions that are discussed on forum over and over again (room modes, room gain, polarity, ...).

Would it be of interest to create a post on ASR just collecting technical topics that people would like to see discussed?

 

[Hayabusa]

- Introduction -

In a recent interview with Erin of ErinsAudioCorner (https://www.audiosciencereview.com/...ns-with-rene-christensen-of-acculution.28103/), I showed that for steady-state conditions, a loudspeaker moving outwards creates a lower overall pressure, not higher as is often claimed. When doing measurements, you see how pressure relates to voltage only, but when doing simulations you have a much better view into the phase relations along the ‘transductance chain’, and you can come across effects that may go against your initial intuition.

I the following, I will show in very few steps, how you can analytically arrive at the same conclusion, looking at a flat circular piston in an infinite baffle.


- Details -

The complex sound pressure phasor (bold p) relates to the complex velocity phasor (bold U0) as [see e.g. Fundamentals of Acoustics, Kinsler and Frey] for the situation with a piston of radius a in an infinite baffle is:
View attachment 170827

I have adapted the expression so that z is in the axial direction and r is in the radial direction, and a is the radius, k is the wavenumber, and we have density and sound speed related to the medium.

When putting the observation point directly in the middle of the piston, the expression simplifies to:
View attachment 170828

We only look at low frequencies where we have a flat frequency response and so we consider that ka is low. We can then approximate the exponential term as:
View attachment 170829

and inserting back into the original expression we get:
View attachment 170830

or
View attachment 170831

Since acceleration relates to velocity as
View attachment 170832

we can finally arrive at
View attachment 170833
and we see that the pressure is in-phase with acceleration, which in turn is in anti-phase with displacement. Hence, at the piston’s outermost position into the medium, the acceleration is negative and so is the sound pressure.


- Conclusion -

A piston in a baffle creates the lowest total pressure (negative sound pressure) when it has moved all the way out, not in. This is also seen in more typical electrodynamic driver setups.

You should always be careful when carrying around knowledge that came out of intuition or “it is well-known in this industry that…”-thinking, where you trust what is being said within the company that you are in. Once in a while, it can affect the engineering decisions in ways that can cost a lot of time and money to course correct once the problems show up. Try and work from first principles whenever you can.


- About me -
René (no need for Dr title, please), BSEE, MSc (Physics), PhD (Microacoustics), FEM and BEM simulations specialist. Own consultancy company Acculution ApS, and blog at acculution.com/blog

So what about if you measure this pressure with a microphone?
Do we then get this same effect again so that the voltage from the microphone is back in phase with the displacement?

 

[René - Acculution.com]

So what about if you measure this pressure with a microphone?
Do we then get this same effect again so that the voltage from the microphone is back in phase with the displacement?

When you measure the pressure phasor, its phase relates back to the phasor phase of the input voltage. So we typically have zero phase voltage as reference and whatever pressure phasor phase we happen to have; the one shown in data sheets (if it is even shown). Some microphone/preamp combos flip polarity, and I imagine that this is compensated for. The driver being a minimum-phase component, you know that it should be zero phase somewhere in the pass-band region. Comparing the pressure phase to the phase of the driver requires you to take out the part of the overall phase coming from the distance travelled by the sound to the mic, but this is typically also done in the software. And there you will see that pressure is in-phase with acceleration, not displacement (under the conditions stated). So if the microphone retains phase (positive pressure->positive electrical output) then you will have no polarity flip there, and for the microphone you have the membrane moving into a small chamber, so there pressure will relate to displacement(!) typically, and so that is probably what you are asking for. But the internal mic setup and the preamp will determine if there is an overall sign flip, or not.

 

[KSTR]

Some microphone/preamp combos flip polarity, and I imagine that this is compensated for.

Yikes. Some Gefell measurement mics had flipped polarity for years and that has created a lot of confusion. It became standard procedure for me to test mic chain polarity by popping a ballon (my breakfast paper bag ;-)

As for misconceptions, a very annoying one is the "inverse square law" for sound pressure vs. distance, when it's actually only 1/r. Even renowned text books have this wrong, confusing pressure with intensity.

And for resources, the website of Eberhard Sengpiel (RIP) is a classic -- really a jewel -- with very accurate and condensed info without bloating math. It's main focus is on recording techniques and microphones but of course much of this also directly transfers to speakers.

Best entry point for non-german readers is via the google search presets: http://www.sengpielaudio.com/Searchengine.htm
Also there are many online calculators: http://www.sengpielaudio.com/Calculations03.htm