Familiarize an oral equation. I think I can use this one. Linearization means that this is a fairly large compared with the p. This is also fairly large compared with p. So plot that plot into there then I have D DX P prime, that has to be equal to [INAUDIBLE] row prime, and I have material derivative Du dt. So I have a du prime dt+ u prime du prime dx. Okay, u prime is a small quantity and the du prime dx is. See the small quantity with respect to change of dx, therefore this is a small compared with that. So I am neglecting this. So, I have here. du dx and this one is a small one, and this one is also a small one. Small one is multiplied, small one is very small one compared with the pu over prime u prime dt. So this, right inside I only have pu du prime dt that has to be equal to a half the rule prime ds no the prime. Sorry, the p prime there. So and usually the amber pressure is constant very slow fluctuation with respect to dx, compared with that. Therefore in this case, dp prime dx is a larger value than dp0 with respect to dx. Therefore, I have Bp prime tx minus this, what we call linearized oil equation. Okay, physical meaning of this oil equation, what it means? So one dimensional Euler equation. Of course linearized. What do you mean by linearized physically? Small pressure. Okay. Later on we will talk about, I mean the limitation of the linear wave equation. Okay. [SOUND] This is one kHz and about 60 dB. You can measure it. And we didn't learn about dB scale but surprisingly later on where we found that this linearized all our equation is valid up to 120 decibel. And 120 decibel and 120 decibel is fairly large acoustic pressure. So because we linearized this acoustic Euler equation valid only for the acoustic pressure of two 120 decibel. Okay, I am rewriting the linearized acoustic Eurer equation. And -dp prime dx means I'm looking for the pressure changing is small. I mean small changing in x direction. So this is unbalanced pressure in other words. That will induce the fluid part to rho0. dU dt this is acceleration. So again this is the Newton's second law for the compressible fluid. And also another interesting part is this. So that means if I measure the fluid of particle pressure using two microphones with a certain distance delta x, okay. Is a p1 and p2. This means that p2-p1 over delta x, okay I measure it And that is [INAUDIBLE] du dt, therefore I. Divide this one by P. Then I have du dt. So what I can do is. If I integrate this part with the respective tie, then I can obtain gradient over DPDX. Okay, therefore I can measure. Pressure based on pressure measurement I can estimate the velocity. So I can measure velocity based on pressure, based on Euler linearized the Euler equation, okay. So every untested probe. Use this all equation to measure in test because in test we'll learn later on, in test is define p times velocity, okay. So linearized all our equation is providing there is tested measurement. All right, so we got the relation between p' and u. Let's try to have the relation Of U and 0 prime, or U and velocity, and density change Okay we are attempting to locate the relation between velocity, and a density change. Okay again let's go back to this famous bottle. I'm trying to, suppose I have a piston under there, and I'm trying to push the air contained in this bottle using piston then you will see that the density will be changed. Right. That's the level change. So let's consider again small segment of pie for one dimensional pie. If I If I supply some density with a certain velocity rho u. Then there will be some density change Mathematically I simply use the of expansion. Or physically I said if there is a sum that is applied into this volume then there will be a sum density change on this surface that would be the rho u plus d rho u dx dx. Okay. If it is not easy to understand, then suppose this is bank. And this is money. Okay? You supply, $1 million at a certain time over here, and if this bank gives some money to other people to get some interest, say $1 million plus $1000. Then there should be a change in the bank accounts, then I can express that is, if this is delta x then I can say delta x times s that's the volume. Okay. Rho. So I said the change of the bank account rho s, dx dt. Okay. If it is a positive, then I have more income than the long. So I would say the the role u, the role u dx, dx multiplied by s. Okay, that’s the amount of money I give you somebody else. And then minus Rho U S. This call conservation of mass. Okay something wrong here? >> [INAUDIBLE] >> Sign is changed, this one has to be over there? Why? Because this is the net increase of a bank account. To get a net increase, I have to get more income. Sorry. Well, so that's a plus, that's a minus and this is plus, okay. Thank you very much. And I will summarize what I expressed based on physical observation. And this will go away and I have this and I have that one. So I have the row S DT delta X that has the equal to minus D row U DX DX, and this is the same so I eliminate Where is s? It should be s2. So eliminate s, then I add d rho / d t, it has to be in equal balance to d rho u / d x. Okay that's mass balance equation. If I rewrite that equation somewhere over here, and I have equation Saying that d rho dt has to be balanced of i-d rho d x. The physical meaning is that rate of change of density in this small volume has to be balanced by net mass flux through two surfaces. Okay? So, mass flux is The money I got from customer, the money I give to the customer to get interest. Okay. And then linearize. Linearize means that the density has two components. One is rho 0 and rho prime and the velocity itself so u prime then what I got? Has to be 0 and I have the row prime has to be balanced by -D dx, and I said before small one, but with small one is a very small one. I have -rho0 du prime dx. Okay so this is conservation of a math. Simply saying that the rate of change of the rate of increase of mass in a unit volume has to balanced by mass flux. Okay, again this means that if I measure velocity. At certain pollution x1 and another pollution that is separated by delta x. Then I can estimate the change in density change. That is interesting, so I will write over there, this is the relation between Density and velocity. So if I got the relation between density and pressure. Then I have complete description between pressure density and velocity.