We are saying that there are three representative acoustic measures. One is pressure and the other one is density. The other one is velocity. And also we have studied that acoustic energy is important variables. And what others acoustic variables would be important? What do you think could be the next to candidate that has a significant meaning other than energy, pressure, velocity, and density in acoustics? >> [INAUDIBLE] This intensity. >> Intensity! [LAUGH] Very happy. What is intensity? In mechanical engineering, what other significant measure that is similar with intensity, what would it be? And this is power per unit area right? Because the pressure is force per unit area. So intensity is nothing but power per unit area. That's great! So let's see what's the relation between intensity and the other acoustic barrier. Okay, let's consider one dimensional case. Okay, I have volume, That has delta x multiplied by s, okay? If there is some power into this volume, through this surface, that would be p multiplied by u multiplied by s. That is power. As you said, force times velocity. p multiply as is a force. Then what about the power coming out from this volume through this surface x + delta x? That should be, using tail expansion I can write, and that'll be pus + dpus dx. Okay, so if I have more power out through this surface, then I would have a decrease of energy inside of volume. So I can write the energy balance through this surface and this volume. I can write energy balance like that. Increase, Of energy, Inside of this volume has to be balance divided by power in through surface of this and the power out through the surface of this, x + delta x. Okay, If I put minus then that means the power in through the surface of this- power out through the surface that. If it is a positive, then increase of energy has to be positive. Right? Okay, and let's think that this is a bank account. [LAUGH] The money coming in per day is 10 million, and the money coming out per day is 9 million, then increase of the money in the bank would be 1 million. That has to be positive, as I said before. So rearrange what I said over there conceptually by using some mathematical form that I can write. dE, T, that's the energy inside of this volume. Therefore, I write a delta x and s. Thus, the total energy inside of this volume. That has to be balanced by this that is rho us and then- rho us- DP. Sorry, not rho, P. Sorry. u, pus. And pus- dpus, dx. That is? Delta x. Okay, dx. So what we have now is, Rate of increase of energy per unit pile that is de, vt. This will go away, and the dx, dx, s, s, so I have. DPu dx. Note that This is change with respect to space. This is change with respect to time. Basically means that the net intensity flux has to be balanced by rate of increase of energy in the unit volume. So expanding this to 3D, I can write this as de dt that has to be balanced by intensive flux. Why? For y component I have dp v dy, With a z component I might have dpw dk at dz, okay? We can easily expand what we obtained over here to y and x case, and this expresses what we just argued. So make a box over here. And I erase. So another one is energy has potential and kinetic and potential energies one-half p square over rho 0 c square and kinetic energy is one-half rho 0 u squared. Okay, now, Suppose I have wind velocity. Okay, you have a wind is blowing in this direction. Can we use this formulation? Do you think we can use this form for the case when we have wind? Okay, wind is blowing and I am whistling over there. Can you use this formulation? What I'm saying, if I measure the velocity of a wind that is usual. Then I + u0 + u over there. The kinetic energy has to be 1/2 rho 0 u 0 + u squared. Do you think that is correct? And remember, the assumption we made when we got this famous p rho u relation. We assume the fluid is quiescent. In other words, there's no conduction. So we cannot use this term. u0 + u. So everything we got over here is valid under several assumptions, that's what I want to emphasize. First assumption, quiescent fluid. In other words, there is no mean flow, no convection. Second, key that's excess pressure and velocity and rho prime is very small, enough to be linearized, okay. Under this assumption, what we obtain is valid. Okay, so now let's move on. To enjoy or have a further insight about what we have. What if I have a plane wave? What if I have a plane wave? Okay, for plane wave pressure over velocity has to be equal to z0. That is characteristic impedance of medium, right? We saw it, therefore in this case I can write p = rho 0 cu. Then the potential energy which is one-half p squared over rho 0 c square = one-half. I plot this one over there then I have rho 0 c squared u squared, rho 0 c squared. Therefore I got that is one-half rho 0, U squared. That is the same as kinetic energy. So for plane wave, kinetic energy and potential energy, equally likely participate in acoustic wave. That is interesting. That is interesting. But what about for the wave that is not playing? Obviously, we can say that potential energy and kinetic energy were different.