Let's move, let's use our PowerPoint from now on. Okay, as I said, there is two component, one is propagating in x direction, the other one is propagating in negative x direction, and this is the assumed wave propagating in string number two. And then as I said the velocity in y direction has to be same. Okay, there is a boundary condition, as I explained, is the velocity on the string number 1 has to be same as the velocity on string number 2. Okay, force acting on string number 1 in y direction, this is the force acting on string number 1, and this is the force acting on string number 2. Okay, there is no mass between string number 1 and string number 2, therefore this term is 0, all right? And we can say that at string number 1, there is wave propagating in x direction, that is g1. And note that I changed to observe wave in terms of time, instead of space. This is a little bit tricky, but this is better, this is convenient. Okay, this is the wave propagating in positive x axis, but I am observing the wave to the respective time. And this is the wave propagating in negative x direction, right, observing the wave in time. And this the wave, At string number 2. Again, that is the wave propagating in positive x direction. Okay, then putting this assumed wave to this boundary condition, then I obtain the following, okay, yeah. Thank you. Okay, the following equation. So the velocity, Velocity in y direction, it has to be dy1/dt at x = 0. If I operate that, that will be g1' + gh1' at x = 0. Velocity in y direction on X0 +, that means the velocity of string number 2 can be written like that. And these have to be same. Because of the kinematic boundary condition. And this is force, the boundary condition, and then this will bring us this equation as TL/c1 g1', that is force acting on string number 1 due to the propagating wave in x direction, in positive x direction. This is force due to the propagating wave, but in relative x direction than the TL/c1. Okay? And this is the force acting on string number 2 in y direction and TL/c2 and the g2'. Okay, next. So, this is the expression that says the velocity in y direction at this continuity has to be same. This is the force boundary condition at this continuity. And this reveals that the following set. Okay, the force in y direction and the velocity in y direction, Will be TL/c. Okay? This can be obtained very simply, you plug fy, which is derived before, and uy, which is expressed before, and divide that fy and uy, you will get TL/c. That means force over velocity is TL/c, this is impedance of a string. So how much force will be excited given our unit's velocity. As a new measure, and that we call impedance. And that impedance is related with two different physical measures. One is the tension along the string, and the other one is speed of propagation. Okay. That is Z1 and Z2, Z1 is a TL/c1, and z2 is a TL/c2. And how much transmitted and how much reflected is totally determined by the characteristic impedance of string, which I said TL/c1 for Z1, and the other one is TL/c2, that is Z2, okay? So what I am arguing at the moment is the reflection coefficient is totally determined by the characteristic impedance of a string. And transmission coefficient is totally depends on, again, the characteristic impedance of a string. Okay, this is very exciting. Another one, Is to look at the string dynamics. And this is the force acting on the string, and this is the angle theta, and the angle will be changed if you use, Taylor expansion. And that has to move like that, as I explained before. And derivation show that using this assumption, the angle is small, therefore, I can approximate sine theta to theta and cosine theta is 1. Then we will get this apprximation. As I said before, ds can be approximated by dx because dy, dx is small. Famous linearization and the square of the small quantity is much, much small. Like we have 0.01, and if you square that, that would be 10 to the -4, so, very small. And also, we can write the small change of tension, dTL, like this, again, Taylor expansion. And then, we will have the wave equation. And it is very interesting that this is the wave equation we obtained before, and that this is wave equation we got by considering this small element of a string. As a conclusion, we can say the c square, which is the speed of wave propagation of a string, Is equal to TL over rho L. Well, that is interesting. That means if you have, can you come up? You can come up, can you come up? The speed of propagation, the string, yeah. Speed of propagation of a string. Speed of propagation of a string. You can see the propagation right now. Can you see it? Okay, now you see the speed of propagation. Okay, if I push harder so that I am increasing the tension, then speed of propagation increased, okay? If I make a small TL, then speed up propagation, Just like that, Don't lose the string, then the string will come and hit me, okay? Very careful. So I push, so I'm increasing TL, it's dangerous, as you can see, the speed of propagation is large. Okay, maybe I will increase rho L. Okay, I increase the rho L twice, and according to this formulation, I increase the rho L twice, okay? Then, Okay? Can I have one of this? No, no, we have to keep our TL as before, okay? Okay, you feel TL. And you will decrease rho L. No, no, give me this. No, no, you have to keep TL. Okay, that's okay. I can feel it. Has increased, right? So this demonstrate that c squared L is proportionate. Okay, thank you very much. Thank you very much for not hitting me by the string. So, interesting that speed of propagation thus depends on the mass per unit less of the string and the tension. Well, that is interesting, right? So, we have a medium. The TL is the tension per unit lens. Increasing TL means, I am having harder medium, okay, then speed of propagation increase. Okay? If I have a softer medium, that means TL is reduced. Then the speed of propagation is reduced. If I have a low L, that means I have a medium that has a mass. More mass than I have to oscillate. Then the speed of propagation will be decreased. So, it is very interesting, and that is the speed of propagation, okay? Now, let's move on another, physics, our string. So Z is TL/c. And because cs square is TL/rho, using that, we can find the specific impedance of a string as a simply lower cs. Okay, rho c, the specific impedance of medium is rho c. Wow, that's great. That is great. Let's see how, why this is great later on, okay? Remember, first, how much reflected, how much transmitted depends on the characteristic impedance of string that is rho L c, okay. And speed of propagation of the string inversely proportionate to per unit lens and proportional to TL. Okay, that's the two key things you have to understand. All right, next, let's move on, driving point impedance, which is new concept. Suppose I am oscillating the string at one point. Okay, let's invite another assistant. Okay. He's invited not because he's most handsome boy, but because he had a military service in, Marines [LAUGH]. In Korean, [FOREIGN], so I believe he can assist me very well, okay. So how much of the reflected and transmitted does depend on characteristics of impedance of medium? But, The impedance represents the ratio between force and velocity. When I oscillate here, I am exciting the force this medium, and I do experience the velocity. So I want to investigate, what's the impedance at this point that I call driving point impedance? Okay? Okay, I feel the force and velocity over there. At some frequency, the velocity and force I am experience, those depends on, of course, the frequency. Right, and I want to see the more detail. So, Suppose that I am exciting the system, where the frequency omega and the magnitude of f. Okay.