 So, tell me is a mass m kept over here fine, now this length is very small x tends to 0 ok, from here to here this length is l, mass per unit length is mu, find out how much time it will take for a disturbance to travel from here to here, you have created a disturbance here, how much time this disturbance will take to go from here to here. Sir, mu is given. Mu is given, it is straight forward, so velocity of the disturbance is root over t by mu, so the time is the l by root over t by mu, so this one, total mass of this rod is m and length is l, you need to find out how much time a disturbance from here will take to reach here, is tension same everywhere, tension is not uniform, draw the free body diagram to find tension at a distance x away from here, what is the tension over here. Sir, what is root of 2 by g? Dimensally incorrect, but I doubt whether it is correct. What is the tension at a distance x away from the base, anyone got that, t is equal to what? Draw the free body diagram of this portion, tension will pull it up, this is t, what is the mass of this portion? m by l x, m by l x, so this times g will be downward force, the tension is what? m g by l times x, this is the tension at a particular x, right, so the velocity is under root of t by mu, t is m g x by l divided by mu is m by l, so the velocity of the wave comes out to be root over g x, yes or no, velocity of wave depends on the where you are finding it, because tension is not uniform, this velocity and velocity is also d x by d t, velocity is not uniform, that is why you have to integrate to get the answer, you integrate this, 0 to t, 0 to l, t will come out to be how much, so this is the time, methods are there, whether you got the wrong answer also, it is fine, but method should be correct, this came in j advance couple of years back, okay, okay, lika bhai, lika, lika, lika, speed of longitudinal wave, longitudinal wave typically it is inside a medium, it is not on the surface of the medium, on the surface of medium there will be transverse wave mostly, okay, so once you are inside the medium, there is nothing like mass per unit length, okay, so what kind of inertia property we can say about the medium, mass per unit volume, okay, so my inertia property is density, okay, my elastic property is the bulk modulus b, okay, what is the formula for bulk modulus, anybody knows it, minus p by delta v by delta v, delta p by delta v is the bulk modulus, okay, so please use the dimension analysis, dimension analysis to find out what are the values of a and b, if velocity is depending on that, so if we hear the elastic is bulk modulus, then why don't we use shear modulus for transverse waves, that is not shear modulus, it is tension, so it is tension, but why don't we use shear modulus, this is bulk, see it will not fit in like this, you can use shear modulus also, no problem, you can use it, but the equation will not look like a simple equation, okay, anything that represents elasticity, what the value of a and b, a is minus half, b is half, even dimension analysis you will cut it, so velocity is root over b by root, remember this is the formula for the longitudinal waves, and if it is, see bulk modulus is usually calculated for the liquid or for the gas, okay, so if we are talking about solid, then bulk modulus is replaced by the Young's modulus, please write down, this is in case of solid, this is in case of fluid, fluid include gas, now what is the most common longitudinal wave modulus, sound, sound is the most common longitudinal wave, so let's try to find out what is the velocity of sound, so basically we are trying to determine the bulk modulus of air, okay, we are trying to apply this formula for the air, write down v is equal to b by rho for sound, what is sound, sound is a wave that travels in a, in air, okay, you have to stop doing that, don't play with the pen, it distracts me also, fine, so let's try to find out what is the bulk modulus, so bulk modulus is minus of v delta p by delta v, I can write it as minus of v dp by dv, okay, now when it comes to the atmospheric gas, can I treat it like an ideal gas, I can, I mean roughly it is an ideal gas because gas molecules are very far away from each other, it is not a compressed form of the gas, so more or less it behaves like an ideal gas, okay, so it follows ideal gas equation p v is equal to nRT, okay, now at the same time when the wave is travelling, what will happen, there will be compression and then expansion, it is a longitudinal wave, so travel like that, travel like that, okay, so what kind of thermodynamic process it is, I need to understand that also, so is it an isothermal process, adiabatic process, isocorect process, what kind of process it is, isothermal, isothermal comes in our mind, the most common because atmospheric temperature has to be constant, okay, so if it is an isothermal process, what is the process equation, p into v is constant, so at any moment wherever the wave is travelling, pressure into volume should be a constant, so assuming that it is an isothermal process, p v is constant, can you find out the value of bulk modulus, this one, try finding it out, just differentiate it, this expression differentiate with respect to v, what you will get, this is what you will get, so what will be the bulk modulus minus of v dp by dv is p only, p is what atmospheric pressure, okay, so the velocity of the sound should be equal to root over p by rho, okay, p is the atmospheric pressure, the value of atmospheric pressure you know is 1.01, 10 is power 5 Pascal, okay, density of the gas is roughly 1.29 kg per meter cube, okay, so can you roughly calculate the velocity of sound, anyone, just a rough estimation, what are the answers, 334, no, what calculate this, how much it is, so this is roughly 4 or not, under root of this 4 to the 8, no this is approximately 2.9, approximately 2.9, let us say 2.9, 2.8, so this is around 280 meter per second, roughly, okay, so the actual answer is yeah 280, so this is the velocity of sound, but experimentally velocity of sound is found to be 333, 333, 333 like that, so the assumption of it being an isothermal process is wrong, Newton has assumed that it is an isothermal process and he derived exactly like this, during those days nobody knew the speed of sound exact value, so everybody expected that this is the speed of sound, alright, later on when people understood that the speed of sound is not this, it is roughly 320 or 330, so they were wondering what Newton has done wrong here, so the only assumption Newton has made in entire process is this, that it is an isothermal process, okay, so Laplace is another scientist who came up and told everyone that it is not an isothermal process, rather it is an adiabatic process, how it can be an adiabatic process, there is no insulation anywhere, so how can it be adiabatic, it is open to the atmosphere, everything is open right there, if your system is this, where sound is there, it can exchange heat from surrounding as well. Is it because energy of the wave is conserved, like potential kinetic here or the potential kinetic here? Because the process happens very fast, it happens very quickly, so the heat doesn't get time to get transferred from one point to the other point, so it is a good assumption that it is an adiabatic process, because it happens very quickly, okay, so if you assume it to be adiabatic process, what will be the process equation like? P v raise to power gamma is constant, now tell me what is the bulk modulus, find out. Why is the isothermal assumption wrong? Because it is when you suddenly compress something, temperature will go up. That is in a very short time. So what? There is a change in temperature. It is because energy is conserved between an issue or another. Which energy? Energy of the wave you are talking about. You are assuming that wave will not die off, in reality wave will die off, if the same loudness level near your mouth is not same after some time, you are assuming that the temperature will go up and down. Temperature can suddenly go up and down, but heat can't suddenly get exchanged, heat needs time to get exchanged. P gamma v to the power gamma minus. Bulk modulus is how much? Bulk modulus is gamma p. How will you say gamma p differentiate that? P into gamma v gamma minus 1 plus v gamma dp by dv is equal to 0. So v dp by dv minus of that is gamma p. What is the value of gamma for the air? 1.4. Why? Diatomic. Mostly diatomic. So gamma is 7 by 5. Gamma is 7 by 5. So whatever answer you have got earlier, which was 280, this is multiplied by root over 7 by 5. So this should be the velocity. Quickly tell me how much it is. Now the answer come out to be 331.3 till Laplace correction. So why are we saying gamma is? Laplace. Oxygen iron. Okay. Any doubts till now? Anything? Anything we have done since start? So what was the equation? Time k. No sir, from the beginning. From the beginning. No, the wave equation is written in such a manner so that 2 by k becomes lambda and 2 by 5 omega becomes time period. So k has no right. You can say omega is angular frequency of SHM of the particles and k is referred as wave number because it is basically inverse of lambda but 2 pi is multiplied on it. So somebody doesn't have any like this? No, no. You can't actually find out in a wave it's given so this is k. Is there an SHM like one particle goes up and then the next? Is there will there be friction between the two? Like is there any friction? Elasticity is there. Elasticity. So because of the elasticity when I move some particle up this will automatically move up. If the medium is very brittle in nature then wave will not travel because elasticity is zero. This particle if you put a disturbance here this will not affect the neighboring particles. It will not move anywhere.