Let me begin with what we learned in the last lecture. We learned that an acoustics the basic so called representative parameters in acoustics as the pressure. Actually pressure, we mean excess pressure or acoustic pressure which is fluctuating a small pressure. And we also talk about the density, which is mean density and the fluctuating density that due to the fluctuating excess pressure. And also fluctuating small velocity, those are what we call representative acoustic variables. And the relation between excess pressure and acoustic pressure was describe by the Euler equation. That simply says, the pressure difference across the small distance in space will induce the fluid to be accelerated that we called linear Euler equation. And the relation between density and the velocity of course, governed by the conservation of mass simply says that the change in density with respect to time and space has to be balance by the mass flux across the volume through the surface. So that is minus d row dt has to be, dx [INAUDIBLE] u. And also the relation between pressure and density is determined by the so called state equation. That is, p prime over row prime has to be proportional to velocity squared. Where the velocity in this case is speed of propagation squared. The last night I attempted to find the meaning of density, that's what we learned in the last lecture. In the last lecture, we also discuss about the measure of pressure for example, how do we measure the pressure? So we introduce the decibel scale. But before we review the decibel scale, I thought that we need some more discussion about the intensity. So last night I start with the one dimensional Euler equation, which I already wrote down over there. This is mathematical expression of Euler equation, a physical meaning of Euler equation is rather straightforward. Meaning that if you have a force balance or pressure balance in space, the fluid will be accelerated. And this describes the motion, this describes the unbalance pressure. So that is interesting. Of course, I'd like to remind you that based on this equation, I mentioned about the principle of measuring intensity. So [COUGH], what if I attempt to calculate the intensity from this equation? What I can do is the intensity in one dimension is p multiplied by u, this simply means that the power flow, right? Power flow, physical meaning of power flow is how much energy is per unit time is coming through this surface, okay? Later on I will also talk about the vector information of the intensity. So rearranging this equation to get a pressure lead us by integrating pressure with respect to space. So that is p and then I move this rho 0 du/dt, I move minus sign over there. And then I will integrate across the space, that is pressure. And then if I multiply the velocity to get a power flow, then I can write pu is minus rho 0 u du/dt dx, okay? Then I can rewrite this, pu that is power flow or one dimensional intensity is minus rho zero. And I can write, this is du square dt, dx, then I have to use one half over there and this one is and this one is same mathematically. But physically, this means this term essentially means that is d dt, sorry, d dt,and one half rho zero u squared. And then everybody knows what it means. This is the acoustic kinetic energy. So by just simply manipulate the gain, the Euler equation, we obtain that the intensity is very much related with rate of change of acoustic kinetic energy through this surface. So that is very interesting. So let us look at some more details about acoustic intensity by looking at the Looking at the acoustic intensity for two cases. One case, case 1, we want to see how the acoustic intensity propagate in one dimension, when there is an infinite duct. In other words, the termination of this duct is non reflective boundary condition. There will be no reflection. In two tubular, we can think that the intensity, there is a very the direction of intensity has to be this direction all the way. What about the intensity fluctuation with respect to time? And what about the intensity fluctuation with respect to space? How does it relate it with the intensity what we have been learning, and case 2, Is the case when you have oscillating source over there, where we have [INAUDIBLE]. With this case and that case, is two extreme cases. One is no reflection, the other one is perfect reflection. General case, for example, when we have an impedance boundary condition over there, then the physical behavior of that general case would follow somewhere in between these two cases. That's why we would like to see these two extreme cases. So let's see how the intensity varies with respect to time. I want to say two things over here. The pressure we assume that the pressure fuel inside will propagate like that. In complex notation, I would like to list would be p0 and then exponential minus j. Omega t minus k x. That is complex notation. Later on, I will introduce the complex notation of intensity, which is somewhat cumbersome and somewhat not easy to understand. And I will address the reason why it is not easy to understand later on. It is quite natural to write pressure in this form. Reason, as I mentioned before, because we're handling linear acoustics, okay? And also we are very much often operating the pressure change with respect to space that dvd tdx and so on. If we operated the change of pressure with space, that means I operate tdx over this expression. Then if I use the cosine and sine form. Then every time when I take a derivative with respect to time or space, the cosine and sine form will change and the sometimes it produces a minus sign, plus sign, and so on and so on. So it is rather bothering, very much us, because of that kind of [INAUDIBLE] trained. So we introduce complex notation, and then by just taking real parts, then we obtain the pressure like that. So what I will attempt to do is, because we are handling linear acoustics, we try to express pressure or velocity in complex form and then when we got the result we just take the real part, okay? That is good, but not always that is nice. Because if you look at intensity, intensity is pressure times velocity. So now you involved the multiplication. Or if you involve the impetus, then you have to handle with a p divided by velocity, that is not straightforward. So you have to be very careful to use complex notation, when you handle intensity or the impetus. Okay now, because we are handling two acoustic variables, pressure and velocity, of course it has a physical meaning, that means power flow per unit area, and that certainly provide those very useful physical unit. But you have to be very careful. So, let's see, let's see how it is useful and what's the things we have to care about when we use this notation. Okay, now pressure is P 0 cosine omega t minus k x. And I can write this using complex notation, real part, our P Exponential -j(omega t- kx). Okay, then the velocity can be written like that. This velocity's obtained from pressure using oil equation. Okay? And it is interesting now. If you see the result, the pressure is P0 cosine omega t- kx, and ux is also u0 cosine omega t- kx. In other word, the pressure and velocity, most of physical quantity has cosine form. In other words, the pressure and velocity t has the same phase. In other words, when I push fluid with some pressure p0, cosine omega t, then fluid particle will move. U0 cosine omega t- k x. In other words, physical meaning that when I push the fluid, the fluid particle will move with the same face as I excite the fluid. This is the case when we have infinite medium. Infinite medium, meaning that there is no reflection, meaning that In other words when I push the fluid because there is no reflection from the boundary, the fluid or energy per unit area has to be effectively the media. No reflection. So that indicates that Intensity. If you look at the intensity, that it is possible to characterize the medium you are exciting. This is one reason why we often look at the intensity in space. Okay, let's see some more details. The first column, first row, I'm sorry, shows that the pressure at x equals 0 with respect to time. In other words, I want to see the pressure right at this point with respect to time. And mathematical expression is available over here so I popped x equals zero over there then I got p0 cosine omega t. That's simply. This p zero co sign on [INAUDIBLE]. This is time. So, yeah there is a pressure change, when I served the pressure right over here. There is a pressure change following co sign omega t. What about the velocity at x equal zero? Over there we will see the velocity would be huge arrow cosine [INAUDIBLE]. So it has the same fluctuation as the pressure. Another words as I emphasize the pressure and the velocity has the same face. What about the intensity? P multiplied by velocity, okay? This is simple. I multiply p and velocity, that will be p0u0. That is the magnitude of intensity, and I have cosine squared. Omega t minus kx, And the cosine scale alpha has some dc value as you all know and also the argument of a cosine scale. Omega T has to be doubled to two omega T. So there is some DC value. And there's a song fluctuation, in this case omega T changed to two omega T, so the important one is If there is no reflection one [INAUDIBLE] case, but there is a mean intensity indicating that there is a net power flow from this side to that side. Also fluctuating intensity is too [INAUDIBLE]. So frequency is changing, of course that is very obvious because we have a pressure that is proportioned to co-sign of P, and velocity is proportioned to co-sign P too. So much prolation, will induce two Intensity is not the linear function of the pressure of velocity, you have to be very careful. Okay? It is very natural you have a double infrequency, because P multiplied by U, is non-linear of pressure. That's why when we handle intensity using complex variable notation, you have to be very careful because it is nonlinear operation. Now let's look at pressure and velocity and intensity with respect to space. So, all this case of [INAUDIBLE] intensity at, time t equals zero. The instance of time, when it start to drive the fluid. Along x, so I am seeing Pressure velocity and intensity along this axis. Okay? Then this means that I am doing spatial observation. Then what we observe, is this. Because the mathematical expression available over here. What I do is, I put t equals 0, then I have a p0 cosine minus kx. And a cosine minus r plays the same as a cosine so I got this. Pressure fluctuation. And here, this is x, okay? And kn omega, there's a relation between kn omega. K has to be equal to omega over c, and omega is 2 pi f. And k is two pi over lambda. So, using that relation, we can see that this is one half wavelength, and this is one wavelength. And interestingly, the acoustic intensity alone, space is also double, right? And that has to be 2kx instead of 1kx. Okay, also interesting, in space the acoustic power or intensity that has mean value as well as a fluctuating value. So, this is the result when we have a tube, one dimensional case, tube that does not have a reflection. Okay, what if we have the perfect reflection over here at x equal l. There is a [INAUDIBLE] reflection. Okay, if you have a [INAUDIBLE] reflection, then the spacial reflection facial solution that satisfied this boundary condition is simply that there will be no velocity over here. X equal l, the pressure has to be maximum at x equal l. Right? That's pretty much intuitive observation. So, if we get a solution that gives us maximum pressure at x equal l, then that would have the form of cosine and then l minus x because form gives me when x equal l the cosine argument will be zero and that gives me f. Why did I use l minus x instead of using x minus f? Vertically, (L- x) means I want to see my position with respect to error. Say L is 1 meter and I am, for example, in the position x=7 centimeter. So what that I see is I see the finish from 30 centimeter from 1 meter at the boundary. So I prefer to use this notation. And then because this solution will satisfy also the other solution like this solution would satisfy. Therefore, I employ the n pi over L, over here, to express every possible solution that n can be 1, 2, any value. This L is just the scaling, the distribution of some pressure along this length L. That has a physical medium. And then, of course, there should be some time fluctuation component that has the exponential e to the minus j omega t, and P0. I will take real part of this solution. So I am arguing that this from satisfy acoustic wave equation, as well as boundary equation. Of course, I wrote down the solution based on our intuition, that because the solution, where expresses the boundary condition, which means that the pressure at x equal L has to be maximum. So now we have this solution, that is pressure. And using all equation again we can obtain the velocity solution like that. The velocity solution rather complicated, it compared with the infinite pipe case. But if you look at some details about this solution, there is a sign over that t, and there is a cosine over that t. Wow! Compared with the previous case, time fluctuation of pressure, and the time fluctuation of velocity, time fluctuation of pressure is cosine of Lrt. Time fluctuation of velocity is sine omega t, that means there is a phase difference of 90 degree. What it means physically? When I drive the fluid like that, then fluid does not respond in phase to me, but responds to me with a 90 degree phase difference. Why? Because we do know there is a different boundary condition, rigid end. The fluid certainly rigidest or I would like to use the word react. React to the force to represent by boundary condition. There is a certain reaction due to the presence of boundary condition. That means if I had rigid n, when I push the fluid there is a reaction from the fluid to me. That is interesting. Let's look at another variation with respect to space. One is cosine n pi over L (L- x). The other one is sine m Pi over L (L- x). Again, spatial distribution of pressure and velocity, one is cosine, the other one is sine, okay? There is 90 degree phase difference, wow. So if we have have a rigid end and when I drive the fluid pipe, the whole fluid with respect to time and space, react to me by having 90 degree phase difference. So that means when I measure pressure, and when I measure velocity, therefore I am able to measure intensity if the intensity has this kind of form. In other words, if the pressure and velocity has 90 degree phase difference, then I can say that this fluid is very reactive. In other words, this fluid has certain rigid end boundary condition. On the other hand, if I have the pressure and velocity that has same phase, then I can anticipate that this fluid behave as if infinite dot. In between if there is a boundary condition which is compliant had a certain compliance or impedance, then what I observe would be somewhere in between. So let's look at some details about the intensity distribution of finite dot. First one sees the intensity to respect to time, but observing at 7L over 8. In other words, I'm seeing the first modes. Then it look like that and the velocity look like that. Obviously, this velocity and this pressure has as you can see right away is 90 degree phase difference. And this is a highlight. If you look at the pressure multiplied by velocity, that is intense fluctuation. Observing at this point, what you see is the fluctuating intensity that has two omega fluctuation? But interesting as you can see over there, there's should not been intensity 0 intensity distribution with respect to time. What it means? It's not possible to pour My energy into this system. Because there is no mean intensity. Intensity is just fluctuating within two omega in time. Right? So, if we have this kind of fluid system, it's not possible to pour my energy into this system because it is pretty perfectly reactive system. All right? So similar behavior you can obtain the intensity with respect to space. Let me generalize what we observed from here, for example when I whistle over here, [SOUND] one kilohertz sound. Okay? And if you measure intensity over here If you measure intensity over here. Or if you measure intensity far away from me, over there. Far away from me, as you can see in the text in the previous lecture, because you are in a far field. The pressure and velocity has the same phase. But if you measure and test it over here, you will see there is a phase difference between pressure and velocity. And we call this its near field. So in near field, pressure and velocity does not have the same phase. It has some phase difference and therefore it is reactive. Basically that is imaginable. Suppose we have many, many fluid, fluid parts. But suppose you have small bubbles over here and I'm pushing the bubble by pressure, okay? Because it has a certain curvature over there, when I push this fluid particle and that fluid particle at the same time. This fluid particle has to move the other fluid particle that has more fluid particle, right? Because it has certain curvature. But in the far field, the curvature, I mean the real front, is almost straight. So the fluid particle over here will move the same fluid particle, same amount of fluid particle. Therefore, if I fill the case, it's the same as the pipe that does not have reflection over here. Because I have to move more fluid particle then some fluid particle would move as if, I mean move with no phase difference. But some fluid particle will have a certain phase difference, in other words is reactive. So in the near field we call the medium is reactive and the far field medium is resistive. I didn't explain about the resistive, but resistive means that the intensity has a real part and there is an average intensity. Or in other words the pressure and the velocity is in space, then recall resisitve.
