So let me start the lecture number three. Review in the last lecture, we talked about one dimensional wave, typical one dimensional wave. We have been handle the waves in a string. The governing equation can be written like that. Here's cs squared is the speed of propagation that is related with string tension divided by pl. And typical one dimensional wave that satisfy discovering equation, wave equation. Can be written as g x minus c t plus h x plus c t as we learn g stands for the wave in right going direction. And the h stands for the wave for the left going direction. And we found that. Also the characteristic of impedance that characterize this medium in this case, that is royal cl. That is the characteristic impedance that define the character or property of the wave of that propagating in the industry. And also we found that the driving point impedance for the wave of infinite strength has the same as the characteristic impedance of the medium. And also they found that driving point impedance for finite case that has lengths of l that is low l, c l, and the J is the measurement of j squared is equal to minus one. And the rest part is cotangent kl and emphasize that kl is a measure of the scale of a finite scale that is l. With the respect to wave lengths lambda. Okay. So essentially, the driving point impedance drive how the wave is propagating for the finite case that has a lengths of l. And that is totally depending on the kl and the cotangent kl and that's what we studied in the last lecture. And today we will see the, chapter, we will go to chapter 2, has to do with acoustic wave equation and it's basic. So we will go to acoustic wave equation. And we will also learn about the basic physical measures. What it essentially means is that we will study acoustic wave equation. Now there's a, we studied one measure of stream wave equation. But now we want to expand our understanding to three dimensional acoustic waves. And we want to study also what would be the basic curve, basic physical measures that represent acoustic wave equations or acoustic wave propagation, okay? So let me have very simple demonstration using this. [NOISE] [SOUND] Which is about 200 hertz and you can hear the sound coming from this small bottle. [SOUND] And what you can imagine from this demonstration there will be some oscillation. With 200, approximately 220 hertz. Therefore, by using very ordinal dispersion relation, we can see the lambda of this wave would be, that is more than. That is larger than what? One meter. [SOUND] So one meter wave is generated over there and propagating into space. So what is happening right now is that there is some compression, the frequency of 220 hertz, okay? So compression indues some change, pressure change. Because there is a compression, there is a density change. Also because there is the pressure change and the density change, there's some change. Of velocity of a fluid particle in the land. So, we can imagine that the typical physical measure would be the density change and the pressure change and the velocity change. And note that the velocity is a vector. And the pressure is a scalar. And density is a scalar. So we are trying to understand how these three physical measures are interrelated each other, in this course. Okay. First, let's see simple case. Again, we'd like to use one dimensional case. Let's look at small segment of this fluid. One dimensional case Then we can imagine that there would be some force acting on this plane. Say this at x and this is x plus delta x. And then we can also easily anticipated the force acting on this surface, x plus dx would be ps plus dps dx dx using taylor expansion. And in this force will act off on this surface like that. Because there is a force difference acting on this mass, using Newton's second law, we can say this mass that will be load times S, that is the that is the, I'm sorry. And the delta x that would be the mass, okay, of this particle. This mass will move with a certain change of velocity, okay? That I could write the velocity change would be du dt, okay? Plus some term that is due to convection. Let's see what is this. Okay. The velocity change. With respect to time has to be written like that. And using chain there I can say this is the du dt plus, they cannot see. So we'll write over there du dx and the dx dt, all right? And this term says how the velocity, how the movement of the particle in x direction. With respect to time. And this is u. So I can write du dt is, has two contributions. One is DU DT plus and the other one is u times DU DX. This is due to the convection of fluid particle and this we call material durability. So D DT again they cannot see it, so I will move this action. So d dt that is d dt plus u d dx, that is the material derivative. So first term, actually seeing the change of velocity with respect of time. And the second term is the seeing the change of velocity as if I am riding on the fluid particle. As we call convection too. But if you, if you cannot understand this stuff don't worry because we will neglect this term later on because this is contribute high order time, but imagine and we say what it means. So to conclude as a result of this diversion, okay? I can write that, okay there is ps, that is a force acting on this side, and minus ps plus the d ps dx dx. Because there's a force acting on this surface, okay that could be represent like that. And that has to be equal to row times dx as the amount of mass that has to be accelerated due to the unbalanced force, okay? And then that has to be multiplied du dt plus u du dx, okay? So what we can see in this Newton's second law applied to the chunk of fluid the particle. Gives us that there is minus dps dx dx that has to be rho dx s ddt plus u du dx. So what I get is taking out the common top that is dx and cross section dx so I have minus dp dx has to be equal to rho times acceleration that is du dt plus u du dx. So this has the very familiar form we learned. When we apply Newton's second law on the fluid particle. So what this means, physically means, the. The distance is dp dx. Okay, that. Essentially says the force of the unbalanced force acting on this fluid particle. So dp dx so what is dp dx? It's a rate of change of pressure with respect to space. That makes sense. Reason why we have minus over here is simply because the pressure acting on this surface. Okay. Increment of pressure change on this surface has to be towards this direction. Therefore, there is minus. Okay? And the other term is load times du dt and u du dx. And this term is due to convection, okay. And notes that. I'll show this results. Denotes that pressure has two components. One is ambient pressure and access pressure P prime, acoustic pressure or so when you experience pressure over here, there is ambient pressure P0 or atmospheric pressure. And then access pressure P prime. This we call excess pressure or acoustic pressure. This is ambient pressure. Okay, and also the rho has a two component, rho 0 plus rho prime. Rho 0 is a static, static constant pressure. And rho prime is the density change due to acoustic pressure. Okay, again, when I exale in this bottle, [SOUND]. Before I exhale a bottle, there is only rho 0 and P0 for remaining sight. Rho pry, p pry. And notes that this equation only describes, of course, the velocity u which we are considering is u prime. Unless there is a mean flow. If there is a mean flow, if somebody blow the air and then each side then there is u0 plus u prime. But if we think that there's no such steady velocity, then there is only u prime. That means prime stands for the everything related with acoustics. Okay, if you look at this equation, this equation only describes the relation p, and velocity u. We need some equation therefore that relate pressure and density and velocity and density. So we need two more equations so that we have a completed description between pressure, velocity and velocity and the density. So, this equation we call Euler equation. [LAUGH] Euler equation. Okay.