Let's review what we learned in the last lecture. [SOUND] Basically we learned that in acoustics, access pressure, and density, and velocity, are the major parameters that describes the property of compressive or acoustic wave. And we found that the relation between the pressure and acoustic velocity can be returned as so-called Euler equation, that physically says that the pressure difference along infinitesimal element has to be balanced by the fluid density multiplied by the velocity. And the relation between access pressure and density, P over rho, is characterized by so-called State Equation P prime over rho prime has to be equal to c squared. And he relation between density and the velocity of fluid follows the law, conservation law. Okay, this follows the conservation law, conservation. Of mass. Okay, and that can be written as d rho dt has to be balanced by the flux of mass through the surface of the element that we have interest. Okay, and all this relation can be summarized by the form of acoustic wave equation. Which can be written like that, where p is access Pressure. Or acoustic pressure. All right? Then the next question is, what other acoustic measures has a significant meaning? The first candidate would be energy. We will like to know acoustic energy per unit volume, for example, okay? Then I can write acoustic energy would be composed by two components. One would be potential energy And the other one is kinetic energy. Okay, let's start with the very simple one first. Kinetic energy Per unit volume. Per unit volume can be expressed as one-half rho zero, rho zero is the mean acoustic density, times velocity, U squared. Of course one can argue that there is, I'm strictly speaking acoustic energy, has to be one-half rho zero plus rho prime u squared. But the contribution of energy, due to the fluctuating component of density, is relatively smaller than this contribution due to the mean density. Okay, what about the potential energy? Acoustic potential energy. So let's see how much energy the compressive fluid can store by changing units or infinitesimal Volume change, okay? To start with, let's consider the one-dimensional case. Of getting dunked. Suppose I have a pressure over here. This will push the surface in this direction, and I have some pressure, P0 plus P prime, that is access pressure. This access pressure will shrink This amount of volume or land. Let's call this as delta l. And then let's use coordinate from here to that is x. This is a positive direction, okay? And this is negative compression. The compression is done by the displacement in this direction. So I put minus over there, not to confuse our necessary things. Okay, then the acoustic energy stored in this volume. Can be studied by looking at this graph, okay? So pressure and compression per unit length, okay? The relation between the pressure which is a force and delta l over l is the compression per unit lens. That is very akin with the spring constant, if you like. And the store of the energy is this. So I can write the potential energy per unit biome would be one-half P times minus delta L over L. Okay, rest the things is to convert this variable to one of this variable, p, u or rho, because that I call prime variable. Prime variable, or in the text, I said that is representative acoustic variable. So let's convert this to those prime or representative value. How to do it? The compression has to follow what law? Conservation of mass. So that means, regionally, the mass of this volume has to be, this is the length l arrow from here to there, that is the length l. Then I can say, in the beginning, when there is no access acoustic pressure, the mass in this volume will be rho zero S and l. Okay, that mass is changed by this compression, okay? That is rho zero is increased, rho prime if you'd like, or rho. And then l, delta l, and S. Okay, I put plus over there. What I'm saying over here is regionally, this is the total mass. When I have delta l in this case, I am following the coordinate of this. That is. Elongated length, delta l, multiplied by S and rho zero plus rho. Has to be equal to the rho zero s l. Okay this relation gives us rho zero s l equal to rho zero l s plus rho zero delta l s plus rho l s plus rho delta l s. Therefore this one go out, and I got, I got, I got, what I got. Okay, Rho zero l plus rho l plus rho delta l has to be equal to zero. This one has to be cancelled, right? [LAUGH]. Rho zero Delta L, okay this is what I have, and rho L. Okay, and the rho Delta L, this is small quantity of any small quantity. Therefore, this is negligible compared with other stuff, so what I got is l delta l minus has to be equal to rho over rho zero, here rho is rho-prime. So, now I have this has to be equal to one-half p rho zero rho prime rho. And I am using state equation that says p over rho has to be c squared. So I got, I plugged this rho, should be p over c squared, so potential energy has to be equal to well-known forms. So then therefore, potential energy, one-half p squared over rho zero c squared. That is acoustic potential energy, so in summary, what I have, so total acoustic energy has two component and the one is one-half P rho zero c squared, and the other one is one half rho zero u squared. p squared. All right, that's interesting. This is potential energy and this is kinetic energy. What other energy could be possible in the compressor fluid? Of course, another possible energy would be the energy due to viscous force, or anything related with friction. Question? Okay, so we are relating the, anything related with the viscous damping. Plus damping term Due to friction. So the fluid we are handling does not have a damping term, in other words, when I blow, when I blow whistle you see here, [SOUND] then, then inside of this pipe, because there is no damping, if I close this, then you can hear same sound like 100 years later. Because there is no damping, okay? If you want to consider damping, you are more than welcome, but that makes our acoustics very complicated. So this is the acoustic energy. What other physical measures has a significant meaning on acoustics?