What if I have wall that can be regarded as the wall that has only mass, which I say limp wall. And then what will happen, we want to see. Again we write incident wave like that, and the reflected wave like that, and over here somehow look different. Four but that is j omega t minus omega over c general acts therefore that's the same as the four we've used before. And there is transmitted one. And we assume that the wall is vibrating with the frequency of omega. And let's see what's going to happen. First, At x = 0 the accidental force will move the wall. Okay for unit area, this is the force per unit area acting on this wall. There is the pressure due to instant and the reflective on this surface? And there is a pressure due to the transmitted and of this pressure and that pressure. One pressure is on a pressure at this side will act it this way and the pressure at that side will act that way, therefore this one minus and that one is in that pressure acting in positive x direction. And that has to be equal to, according to Newton Scandal mass times, that's mass pre-unit, unit are times he acceleration that's the The other time derivative of this. So this would give me minus j omega, and minus j omega twice than I have minus omega sky. So this is what I get. This is the of force continuity, or I would say the force balance between unbalanced force and the motion of the plate per unit area. And that has to be, also, another constraint or condition that has to be satisfied. One thing is this, that simply says the velocity, due to this fluid particle, that is one component of pr over z0. That is the velocity of instant y and that is the velocity of reflected wave, and a minus sign, again, to reflect the velocity of a reflected wave should be opposite signal of instant wave. And that has to be same as the velocity of this partition. Mass limp wall, that is, minus j omega y, because I assume that the placement to I is merited on y explanation minus JOM of T. Take a derivative of this displacement with respect to time will give me this. Physically, I say again, this mathematical expression says fluid parts go over here. The velocity of the fluid particle over here is the same as the velocity of partition. And this expression says, velocity of this fluid, this partition has to be same as the velocity of the right inside of the partition. Again, this is the below still partition because Pt over Z0 is below default plane rate case, and this is simply the derivative of this assumed displacement. Now, we have three equations, and the three unknowns, which is unknown is Pr over Pi and Pt over Pi, and also, Y is unknown. So therefore, we can solve it. So the solution look like this. And then power transmission coefficient look like that. Therefore, the magnitude scale of the transmission coefficient look like this. It is very interesting. Why? For example, if omega is very high, then Tau approach to what? Omega is very high, Tau approach to zero. That is no transmission. If m is very high, in other words mass is very high then again no transmission. That makes sense. Okay, I am shouting here and in front of me there is a big thick mass, then no transmission. That is very likely cope with what I can imagine. But also there's an interest and this is a function of omega is very high. That means I have wall, when I generate high frequency, [SOUND]. If I generated low frequency [SOUND], then the transmitted amount of transmission is strongly depends on frequency. Therefore according to this formula. As the frequency goes high then transmission will be reduce that physical it means that if you have a war. When there's somebody is talking then what you hear would be Very much associated with a low frequency, that's why often in your lab or dormitory, what you can hear the sound which supposed to come from the next door or the do not like [SOUND] or [SOUND], very low frequency. Somebody is turning on some music just next door but what you can hear during the night you can hear only low frequency [SOUND]. High frequency component. Right, that's very interesting. Those things comes from this mathematical expression, That we call mass law. Mass law, this is mass law. This is mass, this is well known mass law. For design purpose we would like to have cow minimum right, as small as the top. When we make some practition or a library curtain. To have a minimum transmission, in other words when you are walking in the factory you have lousy noise and then you want to make a some quite office place inside a lousy, I mean noisy factory do you want to have minimum transmission? Minimum transmission. So for design purposes, we would like to have one octave magnitude scale. And because we are measuring sound dv scale, we'll put ten over ten over there, and that, we call transmission lost. So for the design purpose, we would like to have maximum transmission loss, okay? Now the because the transmission loss magnitude is equal to to the zero omega m scale, plus two, z, zero. I think it was magnitude of scale, or two, z, zero scale. I, off side on this, omega, m, square. Plus 2z0 squared divided by 2z0 squared, that has to be actually magnitude, then it look like that. I divide this. This is 1 and so it turns out when omega m over 2Z0 is very large compared with 1, it looks like that. Therefore, this formula says, when omega m increase twice. That means I have 20 log 10, omega m, over two Z zero, but if omega is increased twice, that look like that, then what I will have because of the log behaviour, I have 20 log 10 to the 2 plus 20 log 10 to omega m divided by 2Z0. So that means I have increased the transmission loss by 20 log over 10 by 2. Now, what is this? This is 6 dB. That means, as I increase omega over m twice, then I obtain about 6dB transmission laws so that was interesting. So I have a work over here. If I increase the mass per unit area by twice, then I can obtain 6dB transmission loss gain. That means the noise will be reduce device 6dB. If the wall follow mass law. And what kind of wall really follow the string in mass law, mass law is the word that has only mass. What does it mean? Do you have a word that has only mass m? if you can find me, let me know then I would buy it. [LAUGH] So today's lecture, I attempted to explain the analogous between this string transmission and the reflection. And the transmission and reflection where the plain way when we have a two medium that has a flat surface of this continuity and then we went to, up to mass law. And I will continuously talk about mass law in the next lecture. And the next lecture, we will emphasize how to use mass law in practice.
