So I can write E_reverberant is equal to E_0, initial energy, and exponential minus t and tau, which is, if I write again tau is four divide by speed of sound and V and open area window. Then the next step would be subtracting out and getting out the meaningful time. And later on, the research said the reverberation time or reverberation periods defined as 60 dB, the time required to have energy decreased by 60 dB that we call the reverberation period, reverberation time sometimes. So, how long does it take? That means the energy reverberant compared with the initial energy is a 60 dB down. That means what? That means E_reverberant is E_0 divide by 10 to the six. Why? Because if energy is decreased by 60 dB, then as P_i would be decreased by 60 dB. Therefore, that means, the definition of dB is 10 log 10, p squared average and the square of the reference pressure. That is, 20 micropascal. So, decreased energy 60 dB now means this energy is decreased 60 dB. Therefore, it has to be 10 to the six. So that's good. So we can calculate the T_60. Therefore, initial energy is decreased by 60 dB. So that has to be E_0 exponential T_60 divide by tau. Therefore, I can write, if I take the natural log in both side, then I have log 10 to the six minus has to be equal to minus T_60 divide by tau. Then, simple mathematics gives me the following solution. Hence, I got the reverberation time is equals to log 10 to the six multiply four divide by speed of sound and volume and the open area window. And that is approximately 0.161 V divide by open area window. That's interesting. And what is the assumption? What is the assumption, I mean, behind of this the famous formula? In other words, in which case we can use that formula? First, sound field has to be diffuse or very close to diffuse, least reverberant. And what other assumption? I mean, the open area window is relatively small. In other words, the A_s open area window is sum of alpha_i A_i. The A_i is the area that has a certain absorption quotient. Therefore, if alpha_i is sufficiently small, then open area window is small. That means, in this case, we have very quick reverberation. If open area window is infinity large, we don't have a reverberation. And this Sabine's equation is valid when alpha_i is small. And if alpha_i is relatively large, we have to use a different type of the reverberation period formula. We are preparing some experiments that we can measure this. Let's estimate the reverberation period of this room. How to measure T_60 of a the reverberation period. First, we need sound source that can excite the space with a band that we have interest. So, practically, many people use this kind of sound clap because this kind of impose, if it is ideal, then it will excite all frequency. But obviously, this one would not excite a very low frequency. You will hear the sound is decaying. If you go outside, if you clap in the corridor, you will hear the sound is like woooo. But if you clap in your bathroom, you'll hear different sound decay. Some people use this kind of sound as pu. I cannot do it very well in front of the camera. You can use pu. This sometimes easy to excite a very low frequency. And therefore, the use of balloon and then use the small needle to explode the balloon and then measure the time it decay, 60 dB. But as you might anticipate already, it's not easy to measure the sound it decays 60 dB. For example, what you get. In the beginning, you will see big energy, and then looks like that and with a lot of background noise, it has to be six dB has to be somewhere around here at least, but not always, okay? It's really hard to get a good signal. So, often we measure the time required to have a 20 dB decay, or 30 dB decay, and then extrapolate to get the reverberation period, okay? Let's roughly estimate the reverberation period of this room, and then check whether or not our calculation is correct, which is very brave. Okay, reverberation period and the volume and As. First, we have to estimate the volume of this room. So easy, okay? The height of this lecture hall would be approximately what? I'm two meter height. So this is about twice, right? So, it would be I think about four meter. So, height is four meter, and this room look like this. So, what is the length from here over there, can you estimate? I often play golf sort of so I can estimate the distance. This would be about 10 meter. And from here to there, so again about 15 meters. So, that is the volume approximately. And because I have to subtract out this amount of volume, then I would say that would be two multiply by, over here to there's a 15, and this has to be a eight and because that is the half, so I have this. So the volume would be a 40, 15 and that is 600, and minus this is 120, so that is about 480 cubic meter. Okay. Now, we have this, and the As is difficult. And As has to be alpha i, Ai. Okay. We have this kind of seat and those things. They will absorb a lot of energy, and assuming that alpha i is 0.15, and what about the area of this stuff? Okay, this is I assumed 15, and the length from here to there would be about eight, and also, I have a wall over there, and that has a very small absorption coefficient compared with that. And then, I'll assume 0.1. And then, total surface area would be the height is four, and then, what would it be? This is 10, 15, and 10, so 20, 35. And I have another one on the top, that would be the absorption coefficient, our top case would be a little bit higher than this, so I use 0.11, and total surface area would be the width, would be 15. This one we assumed 10. So, we included this wall, and that wall, and that wall and absorption coefficient of this surface, I will say this is about 0.1, and the height from here to there is a four, and the width, we assume this is 15, so that would be 13. And that approximately covers the whole of the open area window, and what is the value? Okay. He is calculating now, but I can do it faster than you. This is 120, multiply 15, what is it? So that is a 72, and this is 120 multiply 0.1, so that is 12, and this is 15, 150, 0.11 so that would be about 16. And this is 52 multiply by 0.1, so that is about five all together, what is this? 72, 12 is 84, 16 is 100 and 100 five, 105. So that is 105, and so, our calculation has to 480 divided by 105, what is this? This is not five, both four, some time some in between. So I would say that is about 4.8, and then, if I multiply that with 0.161 by 4.8, in that regard, that is a five, so that is about 0.8 seconds. That is what I estimated very roughly, the reverberation period of this room. And let's measure it whether my rough estimation is correct or not. You see, what I demonstrate to you is, you can actually calculate very roughly the reverberation period of a room, which you have interest. Okay, what they measure is 0.7 seconds, but I roughly estimate this 0.9 second, I think. You see what I did is, I didn't spend any money and the total amount of money for having this kind of instrumentation, I can estimate this would be about $5,000. And so, it's way above I think roughly, $6,000 or $7,000. I didn't spend any money. So, there are some small error, I don't know which one is correct, but it is very close. So, I encourage you guys to estimate the reverberation period for your room by roughly calculating the 0.161 multiply by the volume and divide by open area window, and you do not know very precisely what it means by absorption coefficient, and we'll continue to talk about how to measure absorption coefficient next lecture. Okay? So, let me summarize what we've learned today, and we introduced the actually what Sabine did in a long time ago. The father of architecture acoustics, and room acoustics, the founder of room acoustics, and assumption behind the Sabine's lecture is that he regard the sound field that can be assumed to be reverberant or diffused. And you learned about what it means by diffuse sound field, and direct sound field, and a reverberant sound field, and those will be depending on the surface in periods discontinuity. And that discontinuity will absorb sound energy, and reverberation period he argued that ought to determine the quality of a room that is proportionate, that is equal to 0.161 multiplied by the volume of a room divided by open area window. And we demonstrated how to use that formula very, very roughly, extremely roughly, to calculate the reverberation period. Okay. This summarizes today's lecture.
