Show Your Scattering Coefficients

The phenomenon of acoustic scattering, when a sound wave bounces off a surface and is redirected in different directions because of the surface texture, has been recognized since ancient Greece. Accounting for acoustic scattering when designing a built space, however, can be tricky. In this episode, we speak with Michael Vorländer (RWTH Aachen University) about his work to develop a general guideline for estimating the effects of acoustic scattering from a given surface.

Associated paper: Michael Vorlaender and Stefan Feistel. "Show your scattering coefficients." Proc. Mtgs. Acoust. 50, 015003 (2022) https://doi.org/10.1121/2.0001816.

Read more from Proceedings of Meetings on Acoustics (POMA).

Learn more about Acoustical Society of America Publications.

 
Music Credit: Min 2019 by minwbu from Pixabay. 

Transcript

ASA Publications (00:26)
Today we're talking to someone who's appeared on the podcast before, Michael Vorlaender. We'll be discussing his recent POMA article, Show Your Scattering is based on a presentation he gave at the 183rd ASA meeting in Nashville, Tennessee. Welcome back and thanks for taking the time to speak with me. How are you today?

Michael (00:43)
Thank you, Kat. I'm fine. And you?

ASA Publications (00:45)
Pretty good.

So first tell us a bit about your research back.

Michael (00:52)
Well, I'm professor at RWTH Aachen University, that's one of the major engineering universities in Germany. I studied physics and my university affiliation for the last 28 years is in electrical engineering. Our institute works in hearing technology and acoustics. My research focus is rather broad, but computer simulation of acoustic environments is the main research area. It started with architectural acoustics indoors,

But during the last years we have focused more on outdoor sound, environmental and urban acoustics and on surface material properties. So it changed  a little bit, but finally everything is about computers and acoustics.

ASA Publications (01:34)
Okay, So to just get started, what is surface scattering and why is it important when considering the acoustics of an architectural space?

Michael (01:43)
Yeah, good question. Actually it's a quite complex physical phenomenon, but it's best to be explained with an optical analogy. Look into a mirror and behind you is a light bulb. What do you see? Yourself and the light bulb. They are mirror images. The light is perfectly reflected.

And according to the law of incident angle equals angle of reflection. This is the same thing as you would try when playing billiards against a smooth railing. Put the ball on the perfect path. The reason for this perfectly so-called specular geometric reflection is the smoothness of the mirror surface, which consists basically of a metal foil behind a glass pane. This is perfectly smooth.

What does smooth mean? We have to think about the size of the surface waviness, the surface corrugations or ripples, whatever it is, compared to the sound wavelength. So the mirror is smooth since its corrugations of the metal foil are smaller than the wavelength of light, even if these are only a few hundred nanometers in size. Now our mirror is now replaced by a white sheet of paper.

So just take out the mirror and put a white sheet of paper. What do you see now? Nothing but a white blank sheet of paper. No more yourself and no more the light bulb. What has happened? The surface corrugations of the paper or the fiber structures in the paper are in the order of magnitude of the light wavelength. And now the light waves are scattered. The incident angle no longer equals reflection angle.

All the waves are scattered, diffused, typically in many directions. Like a single drop on the ground results in a water splash. All information contained in the reflected image, we can say the image was scattered. The unique information was destroyed. This is scattering in light. Now the same happens when sound hits a non-flat surface. The problem is more complicated.

However, because the sound wavelength, at least in air for human hearing, may range some yards long or just an inch, just talking about airborne sound and not about underwater sound. And interestingly for acoustics, the corrugation to wavelength discussion is more complex since we are covering three decades of wavelength, because we are covering three decades of frequencies, from bass frequencies to treble frequencies.

So it is important to consider this in an architectural space, since scattering happens everywhere around us. Think about all the surfaces which are around us. They are not flat. So we can say that a hall of mirrors does not exist in acoustics. And so we need to think about it, how to implement this in our theories.

ASA Publications (04:39)
Okay, okay. So that's why when you have like an anechoic chamber, it's all got all those bumps and like the foam material with all the little crevices in it and that kind of thing.

Michael (04:48)
Yeah, but that kind of thing is absorbing sound. So that's still another issue, but we talk these kind of shapes reflecting sound, but in a crazy way, not in a specular way, like the incident angle, reflection angle law, but in a stochastic way, just splashing, diffusing the sound around us. And that's an interesting thing point is that many, many papers we still find in journals, they use a very simplified acoustic modeling approach based on the specular reflection only, so the mirror effect only. This would mean that we get a sound field as in a hall of mirrors. But I always say that this doesn't exist. So scattering is everywhere around us, everywhere where the sound wavelength, which goes from way long extension to very small,
very tiny, parts of inches. This is all the surface structures which are around us. So we need a solution for this.

ASA Publications (05:51)
Okay, okay. So how have acousticians typically predicted or simulated the effects of scattering on sound propagation?

Michael (05:59)
Yeah, in architectural acoustics for room reverberation prediction, we use usually statistical approaches, ray tracing, radiosity and such things. And we use input data for the material properties, usually based on random incidence. We do this for the absorption, which you just mentioned for the anechoic chamber, but also for scattering random incidence.

So in prediction models it's then calculated how much energy of the sound wave is reflected with the specular effect. So incident angle is reflection angle. And how much of the sound is scattered into other directions like the water splash effect, so other directions. But this is calculated only on average and for random incidents which means the sound can hit the surface from any direction. This is mostly okay because in a room, in a classroom, in a concert hall, the waves in rooms hit the walls many times, from one wall to the other wall to the other wall and so on. So many times, hundreds and more. And the entire sound field is generated by superposition of a lot of reflected waves. And this includes a sequential scattering process from one wall to the next during all the reflections. And this leads to an increased mixing and de-correlation and stochastic mix of the sound in the time domain and also in the space domain. We call this finally diffuse field. That's a very well-known term in architectural acoustics, the diffuse field. Diffuse means just it's not focused, it's not a spot, it's just a blurry diffuse thing. And this is our sound experience in a room actually.

But we also know that the early reflections, which are reflected only once or twice, let's say from an instrument in a concert hall, hitting only the sound, hitting only one surface, this is also very important concerning the perception of music in a concert hall. And knowing about scattering is also helpful there, because we can apply this for designing concert halls, classrooms, recording studios.

You mentioned my talk in Nashville. So in Nashville they have very nice recording studios. It's the city of music, of course, and of recording of music. this is why the session was there and I was talking about scattering there. Recording studios is really one of the key application fields for this. so scattering is important in that. But also to get rid of defects such as a flutter echo. This happens to be between two parallel walls when the sound is bouncing back all the time, falling back. And if I clap my hand or if I speak a loud plosive like a P, it would be PRRRR. This is flutter echo. goes repetitions one after the other. And with a scattering effect on the wall, you can avoid this.

You can also avoid a focus effect like in a burning glass, burning lens. So the sound is no more focused on one point but it's splashed around. So in getting rid of these defects, also scattering surfaces can be used.

ASA Publications (09:13)
Hmm.

Okay, how far do scattering series trace back historically?

Michael (09:30)
Many things go back to the ancient Greek, but I think the first who mentioned scattering also wrote about scattering was Lord Rayleigh in the 19th century in his famous book, Theory of Sound. And then Manfred Schröder at Bell Labs invented the so-called Schröder Diffuser in the 1970s. I think this was the main starting point to think about scattering also in terms of economic effects and products.

Schröder's invention was based on phase gratings, on number theoretical calculations, very fancy things, very interesting. And the calculations happen between the surface shapes, the corrugations. He talked about wells, little holes in the surface and little mountains. And so these effects, these surface corrugations, they interact with the phases of the wave, which comes from a certain direction.

And if the phases are shuffled in the way by the surface structure, a specular wavefront cannot build up again. And instead, there is a desired directional pattern generated, like a fan. So the sound comes in, in a peak, in a ray, and then it's splashed out like in a fan, if you would open a fan, just to get a little bit of fresh air.

ASA Publications (10:44)
Right.

Yeah.

Michael (10:52)
These diffusers are very important and they have been successfully designed and produced as products for recording studios, broadcasting studios, Nashville, just to mention Nashville again, but also for concert halls and so on. And they support a very fast sound mixing in the room and they can help to avoid these regular reflections such as the brrrr, the flutter echo. And the broad application in history required a measurement method and a standardized scattering metric. And this started about 25 years ago with my colleague Eckard Mommertz and me. We introduced an initial proposal for a measurement method for random incidence scattering coefficients. And this method is similar to the well-known reverberation rule method for absorption. So you measure in a very reverberant room, the reverberation time, how long takes the time to be inaudible, let's say, after a loud sound has been exciting the room, then you switch the sound off and then you listen to the decaying sound. And you do this twice. You do this with an empty room do this with a room with some specimen, with an absorbing material, mineral wool or wooden panel with holes, such things. With this you can measure the absorption by using the two decay times. The methods for measuring scattering is quite similar. The specimen is rotated. It's a little bit complicated to explain the whole procedure. We measure two decay times again, but with this method it's possible to separate the specular, the mirror effect in the sound reflection and the splashing, the scattering effect. We can separate those two parts. When we can separate them, we know how much sound was scattered and how much is still in the mirror direction, right? So that's the method about. And then Peter D’Antonio and Trevor Cox developed a method in an anechoic chamber. That's an alternative method for measuring the directional scattering pattern.

Also, they characterize the smoothness of this pattern and they call it a diffusion coefficient. It's important to distinguish between the two, the diffusion coefficient and the scattering coefficient. The diffusion coefficient is another design parameter where you can optimize your surfaces, your surface shapes. Both quantities are today standardized in the International Standardization Organization, ISO and in the Audio Engineering Society, AES. So we have these standards and people can go for, they can develop a product and ask a test laboratory, please measure these data for us. And that's good to have because designers, consultants, they need this information.

ASA Publications (14:04)
Right, right. It probably is extremely helpful for when you're planning a room or planning a product. So you mentioned this. What are scattering coefficients and how do they play into these calculations?

Michael (14:07)
Yeah. Yes.

Talking more about the measurement method, it determines the amount of energy which goes not into the specular, the mirror direction. So any other direction, but not just the one in the mirror direction. And this energy is obviously scattered away from the specular path, from the specular direction. Again, we can use the optical energy. Imagine a somewhat milky glass mirror. It's a milky glass.

What happens when the scattering coefficient is between 0 and 1? 0 means it's a mirror, it's no scattering, and 1 means a white paper. It's all diffuse. So we could say that the scattering coefficient describes the milkiness of the surface. In acoustics we can apply this in calculations, in room acoustic computer methods.

ASA Publications (15:03)
Okay, okay, yeah.

Michael (15:12)
That's my field, right, computer simulations. There, the reflected energy is then separated and partially sent into the specular direction and the other energy is sent to other directions. For example, in a ray tracing process. It's like, ray tracing is like a vector method where you have a certain direction of sound with an arrow and then this arrow hits a surface and then the question is where does the arrow go next?

ASA Publications (15:13)
Mm-hmm.


Michael (15:41)
So like in billiards, we take the path along an arrow, just how the ball is rolling, and then the ball is rolling to another direction with another arrow as the rolling direction. Also in the scattering railing, would be that you hit the ball to go into a certain direction, but it doesn't follow what you like to do because it comes back.

ASA Publications (15:41)
Mmm.

Michael (16:08)
Or it goes just parallel to the ball because the railing is not smooth anymore. So that would be fun. Would be funny pool billiard, right? people don't know where the ball goes. Yeah, right. yeah, in the computer simulation, we know where the sound is going and then we can implement this in software tools. But yeah, I could... I could go into more detail, but I think we keep it on the fundamentals for the moment. Because the problem is that only the relative energy is considered between the specular sound and the scattered sound. So the milkiness is just an energy quantity. We don't know the specific directional distribution of the reflected sound. Because the measurement method doesn't give us this information.

It's just about a relative energy ratio, but not a spatial information about how the fan looks into where the sound goes. This is a simplification. But in the model, in the application, at least we assume the so-called Lambert's law. Lambert's law is the perfectly diffusely reflecting surface. This actually is equivalent to the sheet of paper. This is following the Lambert's law.

It says that the surface looks the same brightness for all illumination directions and for all observer directions. If next time you have a white sheet of paper, just turn it a little bit and observe that it looks the same white, independent of the angle under which you observe the paper. So this is Lambert's law. And this is implemented in the acoustic modeling for the moment.

And yeah, that was 25 years ago a big step because it wasn't known at that time that scattering was that important. And it was the only way to get started with the scattering theory for practitioners.

And it was good to be able to represent the broad range of sound waves from low frequencies up to very high frequencies and their wavelengths in a little more correct way.

ASA Publications (18:29)
Okay, okay. So it sounds like it was, like you were saying, it's like, if you're looking at a white paper from any angle, it'll still look like a white paper, but like, in real life, if you have like a holographic little foil paper type deal that like changes the image a little bit when you look at it a different way, it's sort of like that. Like in real life, the sound will sound a little different depending on where you are in the room, or...

Michael (18:51)
Kat, you are a genius. This is a very nice example. Actually, yes. Yes, so we just turn it to a different angle and then we see something completely different. That's a perfect example.

ASA Publications (18:55)
Okay, cool, awesome, fantastic. So what challenges arise when estimates of scattering coefficients are used in simulations?

Michael (19:13)
There are two challenges. One is the lack of datasets of scattering coefficients for a large variety of shape, so all architectural shapes we could imagine. So we have some examples in textbooks and in publications, but not so many. And the other problem is lack of information about the actual directional pattern. So the holographic pattern, in which direction is going what? Since we assume just the generalized pattern, the Lambert's law. And our work from the Nashville paper was focusing on the first challenge, collecting more data and possibly finding a guide for architectural acoustics practice.

ASA Publications (19:52)
Okay. Okay. So it sounds like your goal is fairly clear, but what was the overall goal for the work you presented in this paper?

Michael (20:00)
The test results for surface shapes are sparse. The task for a practitioner could be to estimate the amount of scattered energy for a particular surface structure. Let's say a row of parallel battens or a setting of a randomly arrangement of hemispheres on an otherwise flat surface or something like that. Any kind of designed artistic thing is possible.

We wondered whether it would be possible to develop a general guideline for practitioners. So if we don't have actual test results for a particular surface, can we at least develop a rule of thumb?

ASA Publications (20:39)
Okay, so you went into some theoretical background for your work. Can you tell us a bit about this?

Michael (20:46)
Already knowing from Lord Rayleigh from his 19th century book, we know that the transition frequency between a smooth surface, a small scattering effect surface, and a large scattering effect surface is where the surface corrugation, the surface shape ripple, waviness, corresponds to half of the sound wavelength. And we call the sound wavelength usually lambda, the Greek lambda. And the surface corrugation that let's call L, just L. So for small ripples, for small L, small ripples on a plastered wall, this transition frequency would be above 20 kilohertz. For a very large floor equipped with office desks and people and chairs, this would be below 100 hertz. This is why we use a relative scale. And the main idea is we determine the average size of the surface corrugation L, meaning the ripple or hump distance, whatever the surface structure looks like. And then we set our frequency scale according to L divided by lambda. So a relative scale, geometric size relative to the wavelength lambda. And below divided by lambda of 0.5, there is no scattering. We would say the wall is smooth, acoustically smooth, maybe not optically.

And not if you touch it, it's not smooth at all, but acoustically it's still smooth. And this means all the energy goes into the regular reflection direction. So incident angle equals reflection angle. And above L over lambda is 0.5, we have the complex interference pattern with all these diffraction effects. And we can associate this condition at 0.5 with the transition frequency.

As it's a relative scale, it's not one frequency, but it's a scalable frequency just depending on our ratio L divided by lambda. And we can keep in mind that large structures refer to very low transition frequency and small structures to very, to quite high transition frequencies. And large structures, in our actual work, we talk about building facades. They can be structures
made of balconies. So and balconies may have several feet distance between each other or windows, settings or windows. So and then we have scattering and we have no scattering for very low frequencies as well. that's and on the other end we can talk about a plastered wall where the little roughness features are just a part of an inch. So this explains a little bit how complex the problem might be in acoustics because of all these different wavelengths.

ASA Publications (23:44)
Right, right. So let's go into that a little bit more. How do the size, shape, and periodicity of corrugated surface affect scattering?

Michael (23:52)
Yeah, this is the main question and it takes a long answer as well. The theory goes, I said this too, it goes back to Lord Rayleigh. He showed that a plane wave incident at an oblique angle can be considered equivalent to a surface wave, something like equivalent to a surface wave. If the surface is flat, the equivalent surface wave produces a plane wave front in only one direction.

But when the surface has corrugations, waves' phases get out of sync, so to speak, and the superposition results in a sum of outgoing waves in various directions. So that's the main principle. And the exact solution for a specific surface is very difficult. And it can be solved by numerical methods in big computers, discretized surface meshes and so on.

But usually people don't know and people don't apply these methods, so they are looking for measurement methods. This is why we have the standard anyway. And a lot of assumptions can be made in the theoretical solution, such as a small perturbation condition and other conditions. And for these few cases, we can have even analytic solutions.

At the time of Lord Rayleigh, of course, was about analytic solutions and not computers in 1877. But the analytic solution is very fundamental, as it should be. And the key finding is that the directional scattering pattern depends on the shape function of the surface itself. This is what we could expect if the shape function is more like rectangular battens or hemispherical things or sinusoidal shape or... whatever the shape is. important to remember is that we only talk about the surface scattering on an extended surface, nominally infinitely extended. There's also, not to create confusion now, but there's also for single objects. So scattering at a sphere or a cube or something like a single object.

ASA Publications (25:59)
Mmm.

Michael (26:02)
Our work, our discussion of the random incident scattering coefficient deals with nominally infinitely extended surfaces which have these surface shapes. That's also very important. It started with Lord Rayleigh and he gave us lot of insight into the shape effects and even analytic solutions. In his book he presented the solution for a sinusoidal wave.

It's like a roof structure, like a sine wave, which you see on some roofs.

ASA Publications (26:35)
Yes, got it. So you also collected data from publications in which random incident scattering coefficients were published. What did this entail?

Michael (26:47)
We have tried to investigate the data we have. So in a literature search, we found a number of articles and data sets for random incidence scattering coefficients according to the ISO standard. we just, we all plotted the results into one diagram. And then we saw that, we can confirm Rayleigh's theory. And Rayleigh's theory predicts an S-shaped curve.

So with a low value on the left side and then an S-shape increase like a ski jump. And then there is a maximum value on the right side and in the middle there is this turnover point And this L over lambda is 0.5. So here we are. So we can then use this as a transition frequency below that scattering is low. Above that scattering has some value. And then we categorized further the surface shapes into groups of 1D structures, which is more like stripes or lines or just battens, and 2D structures, which would be a two-dimensional pattern on the surface with different aspects in both directions, right? Not only one. And then we furthermore divided them into round shapes, more like smooth, round shapes like mountains with round tops and with edges like really steep like skyscrapers or skyline shapes, angular shapes. And so we have put them into these boxes and then we studied the average value, the mean values, and the standard deviations in these categories

ASA Publications (28:18)
Okay, that sounds like a lot of So how did you use this information to develop a simplified estimation method for scattering coefficients?

Michael (28:42)
Putting all curves together, the mean values and the standard deviations are already interesting because they confirm Rayleigh's theory and the S-shaped curves. We found that round shapes show more smooth scattering curve over frequencies, so the ski jump looks like a smooth jump. And angular shapes show more fluctuations and larger standard deviations. But, by the way, this is also predicted by the theory.

But overall the trend is all the same. It's a ski jump curve in S-shape. We hoped that this would be already useful because we had evidence from other work that the human perception excuses large uncertainties of scattering coefficients. Which means if we play computer generated sound with scattering included and we have simulated a reflection with the medium scattering and then with the very high scattering. People cannot hear the difference. means that the human hearing excuses large differences and it's a scattered sound. At least it's not a mirror effect, it's not the specular effect, which sounds, by the way, very much crispy.

So if we would listen to a room which was simulated with no scattering, it sounds like crispy and very artificial. But as soon as we add a little bit of scattering, it doesn't matter if it was 0.3 or 0.5 scattering. It sounds the same.

ASA Publications (29:57)
Okay. Well, that kind of makes sense because from the human perspective, like, you'd have to be able to perceive speech with different levels of scattering around right? So, yeah. Yeah. Okay. 

Michael (30:27)
Yes. Yes.

ASA Publications (30:33)
So, how accurate do scattering coefficients need to be for simulations to accurately represent what we as listeners perceive?

Michael (30:40)
We expected that the human hearing excuses large uncertainties, but we need to know better actually how much. So it should be quantitative way to describe this effect. I suspect this effect comes from the mixing which takes place in room acoustics and although we don't find a perfectly diffuse sound field the mixing of the sound waves and with random phases and energies. This cannot be resolved by human hearing. And there was research by two independent people, one in our lab, one in Italy, and they found that the so-called just noticeable difference, JND, just noticeable difference, in changes of the scattering coefficients resulted in very large values.

And they gave a quantitative result. From this we can conclude that the random incident scattering coefficients in room acoustics don't need to be determined precisely, but a range of plus minus 0.3 is acceptable. So you can say that if the scattering, the true result is 0.4, you can use 0.3 more, so 0.7. 0.3 less, 0.1, and it all sounds the same, which is really, really surprising, this large range. But for us, it's good news. So this means that our rough categorization approach with the mean values and a very large standard deviation might be already sufficient, at least as a first approximation. And the conclusion is that a very simple estimation is mostly sufficient, and this is good news for us.

ASA Publications (32:06)
Yeah, that sounds very exciting. What are the next steps in this line of research?

Michael (32:32)
The next logical step is to provide the scattered sound with more exact directional information. So when we say the sound is scattered away from the specular direction, we know how much of the sound in terms of energy is scattered, but we don't know where it goes. And this depends very much on the surface shape itself. So the sound can even be retro-reflected. It can go back to where it came from. There are surfaces which do this.

And for some applications this might be interesting to send the sound back where it came from. So we need more directional information. There's nothing against Lambert's law. It's very nice for the white sheet of paper, but it has its limits in acoustics. So we are now working on another method, on a free field measurement method, which can deliver this angle dependent results. This is actually nothing new because earlier work also 25 years ago even earlier by Eckard Mommertz, my colleague at that time, Peter D’Antonio and Trevor Cox and others, they wrote a very nice book about diffusers by the way, they have described also the main components of the method already, but we extended this. The method so far in literature is 2D, so it's just a method which happens to be applied in a plane arrangement, but not in 3D. So we extended this to a hemispherical arrangement and we can measure the full directional scattering pattern a shape of a flower in 3D for a large number of incident angles and the whole procedure takes about an hour, which is not too bad. It's long, but I think the robot will do this.

We have some stepping motors where things are moved and then we can measure the whole pattern in one hour. And the particularly novel part is the definition of a directional scattering coefficient based not the random incidence, single number, but all the directions in a five-dimensional data set. Because we have the azimuth and elevation incident angle, we have the azimuth and elevation scattering angle, and we have frequency.

So that's a quite complex thing. But we can, I mean with computer method, can define data formats where this five-dimensional data set can be stored and later used for application. And we can use this, of course, again in room acoustics, in indoor spaces, concert halls, classrooms, recording studios. But our project focuses on the shapes of building facades.

And this is about sound in urban environments. So sounds in urban environments; they can be from vehicles, trucks, trains. So this application now is outdoors. But it can also be indoors. Yeah, anyway, our work is on environmental noise. And the relevance for the actual directional pattern is higher than in rooms because this big mixing effect doesn't happen outdoors. Because there's only a few reflections and then we have the open sky. So there is not that big reverberation and diffuse mix as in a room. And our question was also, for example, can a facade shape direct the street sound upwards into the sky? So can this be used to reduce noise in cities by sending the sound up to the sky? And how would this shape look like?

ASA Publications (36:07)
Yeah.

Michael (36:10)
So these are interesting questions and they are new. Directional scattering coefficients, which we are working on now, implemented in models of urban sound propagation could be interesting in future. So let's see.

ASA Publications (36:24)
Yeah, that sounds really fun and very useful. Do you have any closing thoughts?

Michael (36:30)
Yes, I have for the young people out there. Young people come to study acoustics. Acoustics is not just science, but it's really fun. It's a lot of physics involved, so you must know a little bit about math and physics. But it's not just dry theory, since the results are not abstract equations and numbers and all these things. They are tangible, they are audible. And hearing is also very important.

ASA Publications (36:34)
Yes.

Michael (36:59)
Noise control is important, not only to protect people, but also to protect the animals in the sea, for example. And not to forget acoustics for human and animal communication. So that's really, really helpful. When you study acoustics, it does help you and it helps the society. We could do another podcast on the importance of acoustics to global challenges. Maybe next time. 

Michael (37:27)
But anyway, science and freedom of science are more important than ever. And everyone who listens to this podcast, please defend and support science.

ASA Publications (37:37)
I agree with that. Well, thank you so much for taking the speak with me today. It was really interesting learning about scattering coefficients and just how much goes into considering a built environment and scattering absorption and all of that. Hopefully the insight you provided in this work will help other acousticians when they are designing spaces. And thank you again for speaking with me. Have a great day.

Michael (37:39)
Thank you, Kat. Excellent. And have a nice day.