So at this point I'd like to talk a little bit about 3D room acoustics and reverberant sound. So, when we talked about, the duct it's, it's pretty easy to visualize the, the standing waves. As you go to three dimensions, you basically, it's the superposition, if you will, of the standing waves that correspond to each the x, y, and z dimensions. and you could see how, with three dimensions, you're going to have many, many standing waves in that space. in room acoustics, we frequently talk about two components of, of sound. it turns out there are so many standing waves, so many modes in a room that they, they overlap and it becomes so dense, we call it high modal density. and what happens is, is that we tend to, to describe room acoustics. From a statistical energy perspective, as opposed to talking about it specifically from from the modal characteristic. So, you can think about the modes at very low frequency. But as soon as you get up to higher frequency, you know, past the first you know, few modes of the room. As soon as you the, there will be so many modes that you really can't, you can't really think about the response from that perspective. so acousticians tend to talk about the direct sound and the reverberant sound in a given room. And I've sketched here a couple of loudspeakers that are kind of placed out from the corner and away from the walls of the room. And I've sketched it a person here with ears on each side. And then the green line, I show the sound due to the direct path okay. And all of the blue lines here represent sound wave propagation due to reflections. off the boundaries and surfaces in the room. And the point I'm getting at is is that, you know, when the sound wave is first launched, it's, it's launched as you know, we, we, we'll talk about radiation from loudspeakers. But basically, it's going to be launched like this and so you know, it's going to hit the walls off the enclosure. And it's going to bounce around in the room, basically, until it dies out. and it gets absorbed with time. But the listener is actually going to be able to hear first any given sound that comes off this loudspeaker. and this loudspeaker will be heard first by the listener due to the direct path. But then as the sound continues to propagate from the direct path you're going to have these contributions from the reverbrant path coming you know to the ear. And so there's something called the radius of reverberation. And as a listener, you know, I could move back and forth. Between from the speakers to the wall. If I started out here, really close in the plane of the speakers, maybe in this area, and I start moving back. At the point where the sound pressure level due to the direct contribution, the green line that we saw here. Is equal to the sound pressure level due to the, reverberant path. All of the, sound that's bouncing, and reflecting off the walls. When those two are equal, we are at the, point that's called the radius of reverberation. So, you can imagine, you know? As you get closer to the speakers, you hear less of the room acoustics. And as you move further away from the speakers, you hear more of the room acoustics. So, in fact, the room itself influences the experience that you have. And you've probably witnessed that whether you've listened to music in your home, sitting on a sofa. Or you've listened out in a a live venue outside. Or perhaps you've been in a in a bar or a concert hall, someplace that you know, with, with a rigid-wall enclosures. It depends on how much absorptive material, but, you know, acousticians tend to describe rooms as either live being that they have lots of reverberant energy. Or a dead room would be one that absorbs sound pretty quickly. And it turns out that there's the this Sabine Frankler, Franklin-Jaeger Theory of Room Acoustics. and basically it's, it's It's based upon the statistical energy-based approach of, of measuring the decay of sound in a room. And so for any given listener in a room or if you were to place a microphone in the room. Then the sound pressure level is going to decay and t60 is defined to be the time required for the sound to decay by 60 dB. And that's something we're interested in because at that point the direct sound is going to dominate what you hear. as opposed to that portion of the decayed sound so if you would clap your hands in a room you will be able to hear this. This expression is six times the natural log of 10 times the, 4 times the volume of the of the room. And then C is the speed of sound, and A is the equivalent area of an open window so that sound could escape. typically, basically, it's a way of measuring the amount of damping that's in a room. So, you know, if you were to put sofas, and drapes, and carpet, and things in a room. It would basically there'd be some co, there's be some constant that would represent an equivalent area of an open window that corresponded to that for sound absorption. Now this is a pretty good point to go back to one more demonstration from our colleagues web page at Penn State. And in particular, he has a demonstration of reverberation in a small room. And I thought it would be interesting because what he did is he's taken a room that's empty and then he's actually placed carpet in the same room. So let's listen to the reverbent characteristics of the room. So here's the empty room [SOUND]. And here's [SOUND] the carpeted room. You can hear how much more quickly the sound pressure level dissipates. If you you can hear him speaking. >> I am standing in a very reverberant room. >> Okay. And then with carpet. >> I am now standing in the same room, now, the carpet has been. >> All right, so, you know, again, feel free to explore the website. it's a, it's you know, a very effective demonstration. He actually gives the Sabine equation for estimating the the t60, the re-, the reverberation time required for the response to decay by 60 db here. and talks about how you would calculate absorption coefficients and such for rooms. and here you can kind of see a difference in the measured reverberation time. the red curve being for the empty room, and the black curve for a room with carpet on the floor. and so the effect of the carpet's just reduced the reverberant characteristics of the room. So, this expression can be used to to calculate this rever, the, the amount of decay. And it's it's a useful expression and, and this is one of the keys in, in, in room acoustics. And, and it's typ-, traditionally applied by acousticians.