I want to go some of the details about the resistive and reactive term of intensity. Okay, this is not easy, but it is very interesting. What I said in the previous talk, I used two extreme case. At first I said p multiplied by u. It's not linear operator. We are operating p as well as u so you have to be careful. And the intensity physical it means that the net power flow. So intensity certainly provide us how much energy we can pour into the medium. And then by looking at two extreme case, one is infinite medium and the other one is very perfectly refract or the reactive case, we see the meaning of phase difference between pressure and velocity. And I think this is very useful observation you can use later on. Let me attempt to Introduce intensity. Okay, say the pressure with respect to space and time And time fluctuation component is j omega t. And, there is a phase difference p. And the levels are low. The phase difference depends on space, because that is more general case, okay? So this is function of x. Now I attempt to drive, now I attempt to drive The complex intensity, Okay, based on this assumed pressure field. Okay, that is done in the text. Okay, using Euler's equation, that is again, pressure across a small distance is equal to rho zero, add a du/dt. So, let me get this from this assumed pressure field, okay? Let's see. Using the PowerPoint, okay? So, dp/dx, I'm attempting to have dp/dx of this. Then, I have a px, therefore I have to operate the dp/dx over here. And then I have this, right? And I have to operate pdx with respect to this phase, right? That is dp phi pdx and multiply p and then I have a minus j because I have j over there, all right? And the velocity I can obtain from Euler equation. Euler equation simply says that -dp/dx = rho zero du/dt. So what I do is I have to integrate this expression with respect to time and then put the minus. I'm integrating this part with respect to time. Give me this same form, but introduce 1/ -j omega because of this. And that has to be operate over there, and I have this, that is velocity. And the pressure is this. So let's see the difference between this pressure and that velocity. Okay, even if it has the same exponential term, the velocity term has this part as well as this part. Obviously, there is some velocity term that has the same phase with this pressure field, but some pressure field that has phase difference with a pressure field, phase difference of j. And then, I multiply this velocity with this pressure, and only obtaining so called active pod. In other words, the intensity in which the pressure and velocity is in phase, that means I have pressure over here, And this is in phase part, so this is what I can obtain. If I rewrite this one it looks like that. Okay, that is interesting, because this pod is the intensity that has the pressure and velocity has it in phase. As we discuss in the previous example, when we have those not have reflection reactive term. This active intensity has a cosine square omega t minus phi. That makes sense, as we saw before, because have a cosine square, there is a mean intensity, and that means intensity has interesting property that is associated with pressure squared. And remember pressure squared is associated with acoustic potential energy, because the acoustic pressure is p squared over rho zero p squared. So activity intensity is associated with acoustic potential energy and d phi pds, what is this? That is the phase if ip is the spatial phase distribution and a d phi p dx is the spacial phase change with respect to space, okay? So, active intensity certainly represent how the energy of the acoustic potential energy propagate with respect to space? If you look at the reactive intensity. If you look at the reactive intensity, which is written over here. Based on observation, reactive intensity has two sine, two omega t tau, which is fluctuating, time fluctuates with respect to time with 2 omega. And reactive time is proportion of dP squared over dX, Instead of d phi p dX. In the previous case, active intensity is a proportion of d phi p dx. So reactive intensity is certainly expressing how the acoustic potential in a dp squared Is changing with respect to space. So if you have more reactive term you have more potential energy change with respect to space. That makes sense, it's like, for example, you have a balloon over here. Then the balloon has a certain potential energy, right? And if the space is reactive, the balloon is reacting to as opposed to your pressure. So that is this term, this dP scale over dx. And note also that, time average of this reactive term is zero. Before, time average of active intensity is not general. In fact, the average intensity of active intensity has this form. That means active intensity express how the acoustic potential energy is propagating with respect to space. That is d phi p dx. So that is average intensity. So, if you look at average intensity look like that, and the reactive intensity look like that. So look at this term very carefully. If reactive intensity is zero, that means what? There is no potential energy change with respect to space. If active intensity's zero what it means? There is no power flow into the direction of x, all right? So, let's see what would be the complex intensity. So we obtain the intensity as two terms. One is this and one is that. This is average intensity, associated average intensity, that this is associated with the reactive intensity. And if you use a complex function, it look like that And this is complex intensity. I'm sure everybody over there already lost to follow what I try. Because you didn't drive by yourself, I drove by myself. I certainly know what it means. So I suggest to you to drive this complex intensity by yourself. Meaning that there will be some homework [LAUGH]. I ask you to derive this complex intensity. You often remember that the active intensity is. Active intensity is imaginary part of complex intensity or real part of complex intensity in the book, why? You will see why if you drive this complex intensity. This summarize, this concludes the by derivation about the complex intensity.