Of course, the wave will propagate in positive x direction. Suppose the string is infinite for convenience, [INAUDIBLE] as a starting point, and we will see the driving point impedance of this infinite medium. And then that's the real, would you like to see driving point impedance of finite string and it what, is the difference. Okay then, what I can see is the motion why is at x equals 0 can be written like that, okay and then, the motion of the string in the whole direction can be represented as y exponential minus j. Omega t minus kx. All right, and then velocity is simply dy/dt. Because y is the displacement, the velocity is rate of change of displacement. At [INAUDIBLE] that can be written like that. Just to take a derivative of this one in respect to time that gives me minus [INAUDIBLE] I want to know the velocity at [INAUDIBLE] zero. Tha gives me minus j omega y, exponential minus j omega t. And the force at x equal zero, then we assume that's exponential j minus omega t. Okay, that will be minus t. If you look at the force and velocity the racial of force at x equal zero that is it turns out to be See. That means the driving point impedance is exactly the same as the character impedance of string. No reflection. Everything is. From the point at x equals zero. So this is very simple, but let me try to put the sum [INAUDIBLE] on this driving point in. First I could say because driving point in p is the same as the characteristic of impedance of the string. That means when I oxalate over there, what I feel is the characteristic impedance of the medium, which is row lc. So what I actually feel is that the medium is infinite. That is very important. Driving point impedance is the major of impedance at one point. But it describes what is going to happen in whole space.the So that is interesting. So if you want. If you measure driving point at one point but you can see that. Whether or not there is a reflection, or not. By I just measuring one point, you can describe what's going to happen in the whole space. And actually, I can feel that when I oscillate, with a certain frequency, I can feel whether there is a reflection or not. Okay let's consider the other case, suppose I have a string. That has finite termination and X=L and X=L this has to be zero. In other words. Somebody's holding my string. Therefore the boundary condition at x equals l say that displacement has to be zero. Reasons why I use L minus X, instead of X minus L. Expression of L minus X is I am observing the way, observing the vibration or wave. With respect to, L minus X. That means I'm seeing the wave with respect to the lens scale L. X minus L is rather different physically. Okay, then again I can calculate the velocity of the Y direction. By just simply operating, you are the T over there, then minus J omega would come out. Y sign K L and x. I said I want to know the velocity at x=0. Therefore I got this expression. That is the velocity. What about the force? Force Is minus t(DyDx) As we saw in the previous case and this looks like that. And the driving point impedance is this one divided by this one. Wow this is interesting One interesting thing is because I'm dividing this one by that I got kl divided by sign kl. That is one So driving point impedance does depend on kl in the manner of tangent kl. And this term is the same, there's no explanation j or mega t dependency. And this one will exhibit some other interesting thing. So driving impedance of final stream has imaginary part and row arrow Z, that is the characteristic impedance. Okay, we got that, and we got J, that is exactly the same as the driving point impedance. But it has imaginary part J, what does it mean by J. Okay, that is interesting point, I have a force f, and I have a response. If it has an imaginary part, that means that the force and response has first we're going to solve 90 degree, all right? All right what does it mean? If there is no phase difference, like low air Z, that means when I push in the middle with a force f, the medium will immediately respond without having any face difference. Okay this is important. When you have a face difference of j. That means that when I push the medium or in this case when I excite a medium, the medium will respond with the face of 90 degree. There's some face left. Why it has a face left? For infinity medium case. [INAUDIBLE] the medium respond immediately. But because the medium has a finite lenses, there is a face difference. Or in other words, if I measure the medium by driving point impedance. If I find there is a phase difference then I can immediately say this medium is a finite, okay. If I major driving point impedance and if I find a one is real part and the other one is immeasurable part and the real part is. I can say the medium has infinity characteristics. In other words there is something that is going forever in a positive direction. If there is some imaginary pov and I measured a driving point impedance. Then I can say there is something that is related with the final string, okay? I will show you by using the demonstration. This is very important, okay, now. When I excite a system. What you can see is something that is populating and the assumption that is back and forth for example in 3D case, What I, okay suppose you have many different strings. And you're excited when you hear one is direct field, and the one other is something revolver. For the extreme case, let excite what you see in the beginning is direct wave is propagating in that direction that is royal c. And if the other one is assumption back and force that is j lower c Cotangent kL, so essentially this essentially exhibit a how the finiteness of string exhibit in space. It does depends care. What is KM? KM, what is a KM? K is a number, therefore that's two pi over lambda. Therefore, the KL should be represent how the l you have with respect to. So kl is 2 pi over l over lambda. So answering this part says how the finites dimension of over string L has to be scaled by wavelength lambda. So if wavelength is very small compared with the L that means this cotangent kL has many many many zeros and infinities. If lambda is very very large compared with L then this cotangent kL which is 1 over tangent kL and the wave length is very large compared with wave length L then that tangent kL can be approximated as kL. Then this one is rogea, rogea LKR. Let's see some of the details. So, this shows the driving point impedance of finding three. We scale this, J to CS. This is very interesting, at this point driving point impedance is zero. What does it mean? Because driving point impedance is measured between the force acting at one point over velocity. Therefore zero means the velocity is infinite. So this point means that we have resonance. What does it mean? So when I oscillate the string over here, even if I use very small amount of force this point, Exists a lot of oscillation that is a resonance. What about over here? Driving point to impedance infinite that is even if I exert a lot of force, the velocity is almost zero. So, that means that point is antiresonance and that point is that it is very hard to move the string. When? Let's look at some detail. Pi over two, when kL is pi over two, and then k is two pi over lambda, therefore, 2 pi over lambda, L over L. Therefore, what is this? Is a quarter wave lengths. So when I at the the stream was, I mean, is quarter wave lengths a string actually. This case, When a string has half wavelength, we have very large driving point impedance So if you measure driving pointed impedance properly. Then you can measure the whole media whether it has empty resonance behavior or personal behavior. Essentially, driving point impedance can measure the physics of a medium. I summarize many driving point impedance for string, rod, bar, membrane, and so on, so on. Infinite carries and finite carries. Okay. Summarize on this, this is what I intended to say today. In this lecture I delivered two important messages. One is, How much reflected to how much transmitted is totally determined by the characteristic impedance of medium. Which is royal C. And second what I said is drive important impedance certainly exhibit the characteristic of medium. So drive importance impedance of infinite stream. Single lc which is the same as the characteristic impedance of string that means if the media as driving point impedance is same the characteristic impedance of medium the loop there is no reflection at all, that make sense. But. If the medium is finite, of course, there should be some reflection back. And a driving point impedance of that case is J, that is the, that is there is a face difference of 90 degrees. And the royal C and the characteristics of medium is all compressed in the expression of Kohl Tangent KL. And the KL is a measure of space. In other words KL means two part over lambda, I mean, L over two, no, two pi l over so we are seeing the l with the respect of wave lengths. All right? So as we saw in the previous graph. If kl is very small over here, the driving point impedance goes up. Okay? So, we have to see all the lens scale in acoustics, With respect to wave lens Landau. That means kl. And, later on we will see the In any other case, we always see the with respect to wavelength lambda. So please enjoy driving point impedance. As well as characteristic impedance.