Attempting to get a relation between pressure, did I say density? Yeah? Right, density. Okay when I see the infinitesimal change of pressure. So I am pushing this bottle with a certain small piston over there, what will happen? So there's a density change as well as a volume change, okay? Because I am squeezing the particle of mass inside of this volume, the distance between particle will be shortened and the fluid will feel some pressure. That will give some temperature change, right? So the change of pressure now involves some thermodynamic Change. But mathematically I can write the change of pressure would involve dP, d rho, when some thermodynamic state is constant. So, I say isotropic process. When I squeeze this, I may compress the air by some pressure. Of course Newton may some misunderstood of what's going to happen in this case. He assumed. He assumed isotropic process. The difference between isentropic process and isotropic process, isotropic process assumed that the bottle is completely insulated. There is no heat exchange. But, for example, when we have a 200 hertz sound, [SOUND] the fluid is oscillating with 200 hertz. As you can see, you imagine that the fluid particle is moving back and forth 200 hertz. There's no enough time to allow enough heat transfer, right? Because it's rapidly back and forth, therefore we can assume reasonably that the fluid of particles of motion follow the reversible process, therefore isentropic process. And then, d rho. Then, I can write a mathematical dP, dS and dS, this is 0, because isentropic process. So this says the change of small pressure, that is P prime, and this is change small density, that is rho prime. That has to be dP d rho for certain isentropic process and experiment to show that is related with bulk modulus of fluid for the unit times t. And that is c squared. This we call state equation. So I have three equation That relate to pressure, density, and velocity. Let me summarize what we have today. We got Euler equation, linearized Euler equation. We have the relation between density and velocity, based on conservation of mass or continuity equation if you like. And we have state equation that says relate access pressure and access density that is proportional to the scale of speed of sound. And text summarize how this speed of sound Is changing with respect to temperature humidity, and so on. Let's think little bit more about this state equation. If I sing a song or if I sound some sound in a very hot bathtub, which means I have very high humidity and high temperature. The speed of sound would be higher than the case when I shout, for example, in North Pole. Where I have dry air compared with the air I had in the bathtub and a very low temperature. Which case has faster speed of sound, high temperature or low temperature. Mm? High temperature you've got high speed of sound. Okay, if it is that easy to think right away, say the speed of sound is proportional to the bulk modulus as I noted over there. Compare with the unit density. What is bulk modulus? What is bulk modulus? So bulk modulus is simply saying how the volume, unit volume of compressed air would resist to unit pressure. In other words I push the unit, unit Compress the air. And if it is not easy to get a unit Compression that means bulk modulus is high. Okay. Supposed I am heating this small, small compressed air, okay? Then the action of fluid particle inside the volume is very large, that means it is hard to get a unit displacement. Compression using unit pressure. So, high temperature air has a larger [INAUDIBLE], therefore, I've got higher speed of sound propagation. Okay? There are other ways to end the stage or understand the speed of sound. Well, that could be the rough way to understand how the speed of sound is changing with respect to temperature, okay? So we have three equations and the three unknowns. Therefore, we can express these three equations by one unknown Variable. In other words, if I eliminate u and rho over here, then I got the expression in terms of excess pressure. Let's try to do it. I have a d u, d t over here. I have d u prime, d x. So why don't I operate d dh over there and d dt over here, then I can eliminate the du. This ku dx dt, right? So I operate the d dt over there and then I got d squared dx dt, p prime as to be equal to rho0 d square u P, what? This is good. No, no, no. I want 2, d. Dx is good. I want to operate a dx, okay? Then I have, the T, dx. And operated D. T over there, and I will get d squared rho prime, d t squared, and that has to be equal to minus rho 0, d u prime, d x d t, right? Well that's good, so I will add up Then I got minus d squared p prime, dx square plus the rho prime dt square is zero. Okay? Let's eliminate rho prime, or Or convert rho prime to p squared, I can do that. That is rho prime is equal to p prime, 1 over c squared. So this is minus d squared p prime, d x squared. Rho prime has to be equal to p So this is a d script p prime dt squared and that has to be zero. And therefore, I got equation that says simply over there. And that equation simply saying that d squared p prime, d x squared is equal to 1 over c squared, d squared p prime, dt squared. This is very much similar with what we have for one-dimensional wave equation of a string. And in that case, we got this to y, the x squared is equal to 1 over CL squared d squared y, dt squared. So as I noted before the wave propagating in a stream, and the wave propagating in one is exactly same because it obeys the same acoustic wave equation. So this is fantastic because what we are seeing in a string as well in a the same governing equation All right. So we can say that the accusative wave in a duct for example driving point impedance of that would be for infinite that. Driving point impedance would be low C instead of low LCL. And for the finite dot, driving point impedance would be J times low C and cotangent KL or Similar function. And obviously, that will depends on completely kL. So what we concluded by studying one dimensional string wave, Is can be worked exactly in the same manner for one dimensional case. So, that's nice. So, we just study acoustic wave equation or one remainder of the case that looks like that. So let me expand what it learned in the one dimensional case to the three dimensional case. Okay? Then the wave in y direction would follow this scale, this scale p prime dy scale has to be equal to one over c scale, dp pt scale. And in z scale reaction will be d scale p prime, the z scale is equal to one c scale dp prime over the dt scale, therefore. We can say the pressure has to be [SOUND] this is generously remainder of the Costa wave equation So mathematically, our business is to solve this three dimensional wave equation. Try to find the solution that satisfy the boundary condition mathematically. So that's a pretty straightforward problem. Okay? There are two ways actually, one way is suppose I have this rectangular room for example. A rigid wall, okay? Then I do know that the wave in this direction has to satisfy the boundary condition over here and over there. That is the case the velocity is [INAUDIBLE] and pressure is maximum. So I can say that should be co sign if I did know that this is the Y direction then I can say it has to be co sign L- Y By something like that, because, of course, I would have the maximum pressure, over here, and then, maybe, integer multiple of cosine would satisfy same bundle condition. So I have a solution this direction, this, and this, and this, and the solution in this direction would be this and some this and so on so on. I have a many many solutions. Okay, and I say the solution that satisfy rule would be p'(x, y, z, t) can be written as summation of the solution in x direction and solution in, maybe I say in Y direction and solution in Z direction, IJK and I want to find out the waiting of IJK. And this is what we called Eigen functions that satisfy boundary condition as well as the wave equation. And this is simply the contribution of each term. That is one way to see the solution. It's like I assume the solution and then trying to find out how these each solution or each component is contributing to the solution, okay? This is what we call the text Eigen function approach. And another approach could be, okay suppose I know certain solution like monopole solution or dipole solution or quadrapole solution. I know that solution satisfy this governing equation. Then I attempt to allocate or locate this in such a way that satisfy boundary condition. That would be the basis of using boundary element method. So you can use either boundary element method or Eigen function approach to get these solutions that satisfy any boundary condition. Okay, this is just a very conceptual explanation. How? This governing equation and boundary condition coud be applied to find the solution, and this course is not for finding those solutions, numerical solutions or theoretical solutions. This course, this causes it to deliver what would be the basic concept, to understand the fundamentals of acoustics. So let's summarize what we've learned today. This is what we've learned today, right? Access pressure, acoustic pressure satisfy acoustic wave equation that is quite similar with the one dimensional Wave equation. Therefore I can say p prime x t for one dimensional case of course satisfy g(x-ct) + h(x+ct), right going wave, left going wave. And the reflection coefficient will satisfy z1 plus z2, z1 minus z2, where z1 is row 1, c1, and z2 is row 2, c2. And how much transmitted would follow, Z1 plus Z2 2Z2. In other words, suppose I have, plus surface of this continued T. And this medium is Z1 and this medium is Z2 when the wave is coming over there, something is reflecting and something is transmitted, the ratio of reflection and the ratio of transmission will follow this simple expression that only composed the Y. Characteristic impedance of the medium, that's was the same here, that's the same here what do we observed for this string case? Of course thriving point impedance was the same, for infinite case that would rho c, rho zero c. For finite case, that would be j times, roll j, o, c, multiply court hundred, k, l, or some other function, depending on the boundary condition. Therefore, we can conclude that, the everything. At least transmission reflection totally depends on the impedance of media. And also we learned that the relation between u p prime and rho prime and u, the relation between p prime and the U is governed by Euler equation. The relation between rho prime and U is governed by mass conservation. The relation between P prime and rho prime is governed by state equation. And the whole of this process is governed by wave equation. Okay that summarized what we learned today. This is fantastic, this is a fantastic result. So major farcical major that describe acoustics is p prime, rho prime and u. Okay, and impedance. And we saw the impedance, how much reflected and transmitted dominates the reflection and transmission. And when you look at the driving point impedance, not only impedance really characterizes driving point impedance, but it's certainly a measure of what's going on in the field. Driving point also strongly. Depends on KL, this is a very important measure because it measure space with respect to wavelength lander, because K equals 2 parts over so in acoustics, absolute measure, beta scale or fit scale or yard or my, it doesn't have any meaning in acoustics. The length scale has to be scaled by wavelengths. Okay? So when you have [SOUND] 220 hertz. This wavelength is like 1.2, 1.3, one point something. Really be over one meter. And When I see this bottle, when the wave I blow [NOISE] when this wave sees this bottle, this is very small, but when a 220 hertz sound wave sees this lecture hall, this lecture hall is very large, right, so we can see the concept of acoustically large space, and acoustically small space also. This concludes this lecture, I hope everybody understood what I have tried to Lecture today.