Let us see now. We have Governing equation Okay, and then we know the relation between p, rho, and u. And now we know the relation between energy and intensity. Those threes are the major mathematical or theoretical description that involves an acoustic wave. When there is no convection. Also the pressure is small enough to be linearized. And the velocity is small enough to be linearized. And the access density is small to be linearized. All right and then what about the solution, typical solution? Of this wave equation. One typical solution is, as I mentioned before, is a plane wave. Okay, plane wave. Plane wave means that the wave front is a plane. So one dimensional plane wave can be written as p(x, t), x is the coordinate in this direction. And I can write there is a certain magnitude. Maybe I will say cosine (kx- omega t). That is one possible plane wave. Okay? This can also be written as, The real part of Complex amplitude of P0 hat exponential j (kx- omega t). Okay I introduce complex Real part. Real part of this P0. Is complex. Okay? Now, because we are handling linear differential equation, because we are handling linear differential equation. If P0 hat exponential j (kx- omega t) satisfy Governing equation, wave equation, then real part has to be satisfied, or the imaginary part has to be satisfied. Therefore, instead of using cosine and sine, now we can use complex notation. Everywhere in linear acoustics, because we are handling acoustics that is governed by this linear acoustic wave equation. In other word I can put real over there, Still it works. Or imaginary over there. It works. So, we need to express everything in acoustics in complex, Okay? And the second example will be P three-dimensional wave, okay, P(rit). That has certain magnitude P0 hat. But decaying with respect to r and exponential j (kr dot- omega t). Okay now I introduced wave number vector k vector. What is k vector? K vector has three component. One is wave number in x direction, and the other one is wave number in y direction, and the other one wave number in z direction. In other words, this wave is propagating In space Like In a space like that. But the magnitude is decreasing 1 over r and this is called monopole. In other words, pole simply means something that has singularity at a certain position. Here is the case this one has a singularity at r equals zero. At r equals zero, the pressure it goes to infinity. And the mono stands for, it does have only one pole. If I plot this one to wave equation We can immediately see that this one satisfy the wave equation. Okay? Let's check it whether this service file equation. So what is the key of this? Not easy. Right. We have to operate one of our, as well as j k r, and so on, and so on. Okay? Okay. What are the other possible solutions? [NO_AUDIO] Maybe. A di pole. In other words there is two poles which is separated in certain, certain, certain distance. These kind of things can all, all satisfy the wave equation. Therefore what you can see in the text or theory is that okay, the any pressure field can be described by the sum of these two the monopoles, dipoles. For the reports, so on, so on, so on. As recalled the much poor expansion method. Later on we will have a chance to go detail about it. Okay. This concludes the topic I'd like to handle in chapter one in the beginning. What I said is, linearized the acoustic wave equation covering all the ways. Second, I introduce energy relation between energy and intensity. And I also introduce three prime variables, which is pressure, velocity, and density. Okay that is a physical value But we have to see, or we have to get some standard, or unit that can make those variables come to us. In other words, there is a physical value of P rho u, energy and intensity. Put, to get a feeling we have to somehow put their value in terms of units, right? So let's see the units of those variables first pressure. Unit of pressure. Pressure is the force divided by area. And unit of a force, what is it? Unit of a force. An NKS unit is Newton. An area is when m unit meets a square. So the pressure is Newton divided by meter square, and this is called Pascal. Okay, 1 Pascal is simply means that 1 Newton per meter square. Suppose how much Pascal will experience the surface just beneath of my foot? I calculate, my weight is about 50 kilograms, believe it or not. Let's assume my weight is 50 kilogram. And that the force exerting to the floor would be 50 kilogram multiplied by 9.8 me to scare gravitation and acceleration. And I have to divide two [INAUDIBLE] by foot, okay? The lengths of my foot is approximately. I don't know. What? >> [INAUDIBLE] >> 30 centimeter, that large? >> [INAUDIBLE] >> Including shoes, okay. For easy calculation, let's say that is .3 meter, and this is approximately. 0.1 meta, and I have 2, so this has to be 0.3 x 0.1 x 2 = 0.06. So, I have to multiply 10 to the square over there, and this divide by 6, and I have 6, 9.8 divded by 6. So it would be 1.6. 1.6 multiplied 50 would be 80. So 80 x 10 to the one hundreds. So 8,000 Pascal. Wow, 8,000 Pascal. Anybody had a feeling about Pascal? No. Therefore, the pressure scale is measured By using a relative scale. Because as I demonstrate, this pascal does not really provide us real feeling. So we use scale of decibel and decibel is a measure of acoustic pressure. This stand for 10 over here and then we are using log10, log10 not log of e log10, and then P square avg, that is Mean of square, mean square, mean square. So that is I integrate the pressure square for certain period of time. And divide by T. That is mean scale value. Okay, mean scale value, all right? Mean scale value. For example. I have [SOUND]. If I square this, what I get is square and then square and then square and then square and then square and then square and then square. Therefore, these square value will be somewhere over here. That is p scale average. Mean scale value, so mean scale value of [SOUND] 1 kilohertz is not 0. Okay you got it? And it does have a physical meaning. Now and we look at this miss value with respect to scale value of reference. And a p reference is P reference is 20 micropascal. What is this strange number? What is a 20 micropascal? Who invented this? Okay. Because of acoustic pressure we can sense. The range of acoustic pressure is just so wide. Anybody hear the sound I make, please raise your hand. I didn't blow, I didn't make any sound. Okay, okay. And then I make a big sound. I didn't make a sound. >> [LAUGH] >> Wuh! And then you hear, right? So the sound we can hear start with a very small one to very large one. In fact, we can normally hear sound little bit more than 120 dB from general decibel. General decibel means 20 micropascal. And 20 micropascal often referred as the sound that very healthy young student can hear. And you guys over there can there can hear very small sound right. When I was a student, my teacher said it's the sound like you drop a small needle from 1 meter above the floor. That's the 20 micropascal that you can hear. I don't know, I didn't test it. Maybe we test like pickup the small metal very thin and 1 meter and then measure it whether it is really 20 micropascal or not. Maybe you guys have a fun out of it? Because that is the smallest of pressure we can hear we always always always measure this scale pressure with respect to smallest sound pressure. Okay, so that is decibel. And the decibel is very interesting. And also 3 dB is minimum pressure that human beings can recognize the difference. But unfortunately 3 dB is in linear scale is twice of the pressure we will see that later on. So when you work on in noise problem, when you reduce the noise level twice. Then your supervisor will not appreciate because 3 dB is just perceivable. Noise difference by human being, okay. So tax you can see the rough reference about to feel about the decibel scale. But I will just say In your lecture hall at the moment. When I do not say anything your lecture hall would be without any sound. About 40 dB. Maybe 42, 45 dB. When you go to cafeteria, when you go to cafeteria, I don't know whether you have a very fancy, nice fascinating cafeteria. But for a usual, normal university cafeteria, it's very loud. It's very noisy. That case, that would be maybe more than 65 dB. If you go to the rock and roll, the concert hall or big music that would be more than 80db, sometimes you feel 100db. Okay, if you hear the sound coming from rocket, for example, that would be more than 120 dB. So variety of this scale you can hear. If you hear the sound coming from a big truck passing by, that would be 90 dB or 85 dB, okay? Now what if I have two sources? P1 and P2. Okay, I have a source that has P scale average of one, That is addvd. An I have another source, P squared average 2, and that is again 80 dB. What is the dB of 1 and 2? Source 1 and 2, what would be That would be 160 dB. No, because that will scale as log of 10 P squared average over physical reference. So let's see how to get the new DB that is 10 log 10 P scale reference and I have a two source. P square average 1, and I have another sort of p square average 2. Okay, and then what is it? This is 10 over 10 p squared over that, plus this okay. So I have 10 log 10, p squared reference, p squared average 1 + p squared reference. [INAUDIBLE] Over p square average 2. What is this? This is p square average. Okay this is 10 log 10, P squared reference over P square average 1. Therefore, P square average over p scale reference has to be i80 divided by 10 is 8. Therefore would be 10 to the 8, is it correct? So I plot this one over there I got Your decibel = 10 log 10 10 to the 8th + 10 to the 8th. That is 10 log 10, 10 to the 8th multiplied by 2, that is 10. That is 10 to the, 10 log 10 to the eighth That is what? >> 80 dB. >> 80 dB. Multiplication will be resolved in logarithmic operation plus, so I have 10 log 10 to the 2. What is log 10 to the 2? Log 10 to the 2 is 0.301, okay? Therefore this is log 10 to the 2, so I have to add 3 so 83 db. So when I have a two source 80 db then resulting sound pressure level is not 160 db, but 80 db. Okay, another interesting one. Because [INAUDIBLE] scale [SOUND] if I have p square average, As p square average big 1 + p square average small one For example I have a big decibel 100 db and another component that is like 50 db, maybe I will make I will make fifty dv and you will make seventy dv. Okay I will make 60 dBand you will make 40 dB. [SOUND] And you can not hear 40 dB right? You cannot hear 40 dB. So in dash scale, if the scale of two different sources is relatively large and a bit different, then you don't have to worry about. The small one because if this is a 100dB, and this is a 50dB then total decibel scale would be 100.1dB. So from those control issues you don't have to worry about the small noise. So what you have to immediately conclude that, if you have noise abatement problem, first the job you have to do is you have to find out the major contribution of noise, and then you develop the certain way to reduce the major contribution, right? Because of the decibel scale. I introduce the significance of the decibel scale, that is defined like that. And a p square average is p squared pressure. And the operation of the decibel scale is not linear. So will have to very much worry about how to operate the decibel scale. I had a letter from one of the judges. Local judges. And he said, okay, the noise level of rifle, he measured. Not he, somebody measured, is 80 dB. And there is the place where the soldier has to practice the rifle. And the number of stands over there is 20. Therefore, ADDB, and we have a 20 rifle exercise stand. Therefore, the would be ADDB multiplied by 20, and that is 1600. And that is so big. He asked to me whether this kind of estimation is correct or not. Absolutely not, right? Because of the dash ville scale that is defined like that. Okay, this concludes our fourth lecture. We studied about linear acoustic waves. And we learned about pressure, density and velocity is a sort of major acoustic variables. And we also studied about the relation between energy and intensity. Intensity is nothing but the power per unit area. I miss, intensity is also vector. Also vector and energy is not vector, it's scalar. The reason why intensity is vector is because the power has a direction of propagation, okay? And also we introduce the unit of pressure that is a decibel reason why we use a decimal, one reason is because our hearing system says the acoustic pressure from very small from very large ones. That's why we are using decimals scale. Next lecture, I will explain about octave scale. And then other important measures, okay?