 Hi, I'm Zor. Welcome to Inizor Education. Today we continue talking about waves and in this particular case we will talk about pressure waves. Well, pressure waves is primarily, it's about how sound actually propagates through air because the air molecules are changing the pressure in certain areas and that's exactly the most important example of pressure waves. Now, I will explain it slightly differently on a more kind of elementary level but nevertheless it's basically about how sound distributes in the air. Now, this lecture is part of the course called Physics Proteins presented on Unisor.com. I suggest you to watch this lecture from the website because if you are going through the website you have the menu because it's a course which means there are certain topics which are logically related to each other and obviously I would encourage you to go through topics in sequence as it is presented on the website, not because you just found it somewhere on YouTube or somewhere else. And also there is a prerequisite course called Mass Proteins, the same website. Now, the website is completely free, there are no advertisements, no strings attached so that's definitely more preferable way to systematically learn the subject. So, pressure waves. First of all, there is a word which I would like actually to use but it's a very long word and I really don't like it quite frankly. It's called Longitudinal, Longitudinal Oscillations. Now, Longitudinal, well, it implies that there is a length actually, right? Long means something in length. So, that's the oscillation which are propagating through a direction and the waves are also propagating into this direction which is different let's say from the surface of the water when water goes up and down but waves go across. In this particular case waves of pressure go into the same direction as the propagation of waves and molecules are moving in exactly the same direction. Molecules are moving along this line direction and the waves are going through the same direction. So, it's a higher pressure, low pressure, higher pressure, low pressure and the molecules are moving this way. Well, they are not exactly moving that way because molecules of let's say air are chaotically moving in all directions. However, we can always talk about the probability of moving into certain directions and if in the air where there are no oscillations, no sounds all the molecules are moving in all directions with basically the same probabilities but whenever there is a sound which means there are certain pressure waves which are going through the air then the probability to go into this direction of the propagation of the air is slightly greater and then the molecule goes back and goes back to its own chaotical movement until the next move of the pressure comes. And that's what I will be explaining in a simple example. Now, my simple example is consider you have a cylinder, a long one, with a piston on the top. We are in control of this piston, we can move it up and down. Now, there is air inside of this cylinder. Now, the length is sufficiently long so we don't really take length into consideration and the diameter is not really very big because the bigger diameter means there is some distortions from the one direction. I would like to concentrate only on the vertical direction in this particular case, in this example. So, as the piston is moving up and down, I would like to investigate how the air molecules will move inside that cylinder. That's what's important. So, that's why I don't want to have a bigger diameter. It's a relatively small diameter of a cylinder, but long one so I don't really take the length into consideration right now. So, what happens right now under the piston? The piston is in the top position. What happens? All the molecules of air are chaotically moving. Their chaotic brown movements actually sometimes it's called. So, this chaotic movement actually means that you don't really have any kind of notion about where exactly each molecule moves, but you can just consider that the direction every molecule moves has certain probability and the probabilities in all directions are the same. So, they're going chaotically inside the cylinder in all directions. They bump into each other obviously, they reflect. So, that's what chaotic actually means. Now, what happens if I slowly, very slowly move the piston down? Well, that actually reduces the volume. Now, you know the Boyle's law that the volume times the pressure is a constant. Whenever you increase the volume by factor of two, let's say, pressure decreases. If you decrease the volume, the pressure increases, but their product is constant if amount of air and the temperature are the same. Because if there is a temperature, there is a more universal law, but this is the Kelvin temperature. So, this is the constant. But right now, our temperature we're assuming is the same. So, we are talking about basically increasing the pressure if decreasing the volume and decreasing the pressure if increasing the volume. Okay, proportion, inversely proportional to each other. Okay, now, that's what happens if we are slowly moving it down. When we're slowly moving down, the molecules will redistribute in a smaller volume again evenly because we are doing it slowly. They have the time to redistribute among the whole volume and the pressure will be increasing proportionally to decreasing of the volume, obviously, here. If I move it in such a way that I will reduce the volume by half, my pressure will double, obviously. So, that's what happens if we do it slowly. What happens if we are maybe moving a short abrupt push down? Very short, but very quick. Boom, like this. Well, let's just consider now the molecules. They have mass, which means they have certain inertia. Because of that inertia, the molecules immediately under the piston, these molecules, will be compressed, but they will not have time to redistribute down the volume. So, you can consider that there is a small volume here under this volume, under the piston, which basically contains the same amount of molecules. Molecules are not escaping down. But this volume, when we're short and abruptly move the piston down the cylinder, this volume is reduced. So, the same molecules under the thin layer, under the piston, will be squeezed, really squeezed. And what it means? Well, it means that the pressure of these molecules will increase. So, whenever we start moving, basically, we are moving this immediate layer under the piston, we are squeezing it, which creates a higher pressure. Well, what happens when this higher pressure actually develops? Well, higher pressure means that the molecules are moving more intensely under the piston than the rest of the molecules. Now, they don't have any way to go up. So, the higher pressure obviously pushes them. Now, the molecules in this thin layer, since they are more energetic, they are moving faster or whatever, or they're in a shorter distance between them, so they're kind of bumping into each other more oftenly. It means that they will push the next layer. Well, I am basically separating the whole area of the cylinder into the area. Obviously, in the real life, it's a gradual modification of the pressure. But in any case, it's really helpful to explain that there is a layer and then there is the next layer and the next layer. So, the layer of the higher pressure, since it bumps into each other much more often than down, it will bump into the next layer and actually pushes the next layer. Now, as soon as it pushes the next layer, the layer immediately under the piston will expand because it pushes the other layer down and the pressure basically evens as it was before. Now, what happens with the second layer? Well, the second layer, getting a push from the upper layer will also be squeezed a little bit. As soon as it squeezes, the pressure is increasing and it pushes down the same thing. The same thing as the first to the second, the same thing from the second to the third. So, this particular pushing will go down, down, down, down. The speed of this distribution of energy, we are talking about energy distribution, right? The energy of the piston is distributed down the line. Well, it will distribute with certain speed. Now, what speed depends on? Well, there are many factors. For instance, initial temperature, initial pressure, amount of air, the quality of the air itself, I mean how molecules interact with each other. So, any kind of a medium, and I'm talking about air in this case, but it can be any gas or liquid, it has its own properties. So, the speed of this distribution of the energy is very much dependent on what exactly kind of a medium is here. And there is a very important and I think very understandable analogy to explain this process. Consider you have a train with cars. Now, if this is a locomotive which is supposed to go this way. Now, there are links here in between, obviously. All these cars are linked into some kind of mechanism, link mechanism. Now, there is always a small gap. So, it's like this. And there is some kind of a gap between the link, part of the link which goes to this car and to this car. Why do we need this gap? Well, because if the locomotive is trying to start, for instance, if there are no gaps here, it has to really overcome the friction of all the cars against the rails. If there is a gap here and here, then the locomotive will start and it will move only the first car. When it moves the first car, since there is a gap here, the first car will map, will go after the locomotive. This gap will close and then it will start with a little delay, the second car and then the third car, etc. So, the cars will not be moving at the same moment, all the cars will not be moving at the same moment. The first locomotive moves, then with a slight delay will be the second car, then with a slight delay, the third, etc. So, the efforts, the energy of the locomotive is transferred gradually to the very last car as a wave. This is exactly the same mechanism as the waves of the pressure go through the layers of the air. So, these gaps play exactly the same role as gaps between the molecules and every car is like a molecule. So, whenever you have certain push, now it can be either beginning of the movement or it can be, for instance, when you are slowing down. When you are slowing down, the locomotive slows, this car goes by inertia with the old speed, which means it will gradually close this gap between these two, then this thing will close gap and then this will close gap and that's how the whole cars stop one by one with this wave of stopping really propagating through the train of the cars with certain speed, which depends on many factors. Again, that's exactly the same mechanism. Now, let me just continue. Now, when we are moving the piston down, we are initiating the wave of pressure which goes from layer to layer. Now, what happens when we stop? So, the pistons made a short motion and stopped. Well, at that particular moment, the molecules immediately under it by inertia will go slightly down, which means what? Which means that the volume of these molecules in this layer, in the first layer under the piston, the volume will increase. The same molecules will move further, which means there will be rarity in this particular case. So, there will be low pressure. Okay, as soon as there is a low pressure, the molecules under the first layer, the molecules of the second layer, which have the old, higher pressure, will start moving into the upper part immediately under the piston. The piston stopped, right? So, the low pressure which immediately developed under it will be compensated by higher pressure from underneath. The molecules will go more to the upper, will push basically upwards, which means that they will decrease the number of molecules in the next layer. And the next layer will have a lower pressure. So, my point is that as soon as the piston stops, the low pressure will start propagating in exactly the same way as the higher pressure. Same thing as the train. When you start, you're moving from one to another, from the first wagon to the last one. And when you stop, again, the first one stops the first one, so the stop really propagates down. Same thing here. The higher pressure propagates down, and as soon as it stops, the low pressure propagates down. Okay. So, that's basically an explanation of how high and low pressure are distributing down whenever we are moving this piston in a short move down and stop. Okay. First, when it moves, you will have a higher pressure going down. When it stops, there is a low pressure going down. So, obviously, the next step in our logical explanation of how sound propagates means that we will probably have to make this piston making the reciprocal back and forth up and down, actually, movements with a relatively high speed. Whenever we are moving down, we initiate the higher pressure, and the higher pressure goes down the cylinder. Whenever we move the piston up, the low pressure, well, the stop and moving it up. Same thing, actually. Whenever we're moving up, the whole thing goes to that direction again, and waves will be propagating all the way down. Low pressure, high pressure, low pressure, high pressure. And that's what actually makes the sound. Because whenever we have something like I'm talking right now, it means the air around my mouth starts basically vibrating, and then it goes to all the direction. Now, why I chose this cylinder and the piston? Because it's easier to explain we have only one direction of distribution. Whenever something is sounding in the room, obviously, the sound goes into all directions, so it's kind of a spherical distribution of energy. It's a little bit more difficult, but it doesn't really matter. The whole thing is basically the same. So, I have chosen this to have basically a flat waves. So, the flat waves are layers of high pressure, low pressure, high pressure, low pressure. Okay, now we have to really talk about some terminology. So, if you have areas of high pressure and low pressure, then there is always a distance between the areas of the peak pressure. And this distance is called wavelength. Usually, the Greek letter lambda is used for this. So, lambda is the distance, well, in meters, centimeters in whatever unit you choose, the distance between the crests of the pressure. So, the high pressure has a peak at this point, and at the same time there is a peak here. And obviously it depends on the oscillations of our piston in this particular case. So, this is the wavelength. Now, if you are standing in one particular point, let's say in this point, you can measure a time between crests of the pressure pass by. So, the time is called a period. Now, obviously the distribution of the waves has a certain speed, and if this particular distance is covered by this particular time, then you have a speed B of distribution of the wave. This is the speed of the wave. Now, sometimes people are using frequency. That's number of periods basically in one second. So, that's 1 over T, that's frequency. Sometimes there is an angular frequency or angular speed, which is actually 2 pi over T omega. That's the number of full cycles per minute. So, one full cycle per time that goes to angular frequency or angular speed, whatever. So, these are terminologies. Now, I wanted to express how the whole air oscillations are looking graphically. And I put the graph inside the description for this lecture on unidisor.com. Now, the graph in this particular case, I would like to represent it as a two-dimensional, two arguments. So, it's a three-dimensional graph with two arguments. One argument is time, and another argument is position. How far from the piston my point is. So, let's say this is x. Now, at any particular x as time goes by, we have high pressure passing or lower pressure. So, how this function P of x and T looks like. Well, let's just think about it. Let's talk about layers again. So, let's consider a thin layer where this particular point x is. Now, if my piston is making harmonic oscillations, which means its position actually is changing by something like A times sine of omega T. Omega is some kind of angular frequency of oscillation. So, we are assuming this is harmonic oscillation of piston. So, delta y is going a little bit down, a little bit up, a little bit down from the initial position. That's the move from the initial position. Whatever the initial position is, doesn't matter. Now, when it moves this way, it changes the layer. We are assuming that the same layer is here, and then gradually it goes down. So, we are assuming that this thin layer is changing in volume, obviously. Because it's a cylinder, which means the diameter is the same. So, if this is my thin layer and I'm still changing it down and up, my volume is changing proportionally to the height. So, that value, delta value, would be equal to also some kind of a coefficient, which is actually the cross-section of the cylinder times exactly the same sine of omega T. So, the volume of this thin layer is changing proportionally to the movement of the piston, obviously. Now, if the volume is changing this way, you remember this. It means that the pressure inside this particular thin layer would also be changing. So, if this thin layer initially has volume v0 and delta v is a change up and down, then the pressure inside would be equal to, again, some kind of a constant, c divided by volume, which is c divided by v0 plus delta v. So, what is my final formula? V0xg is equal to some kind of a constant divided by v0 plus delta v, which is b sine omega T. So, we started with harmonic oscillations of the piston and we ended with a pressure which looks like this. It doesn't look harmonic. However, number one, it's periodic because sine is periodic, so this function is periodic. So, it moves up and down, up and down, right? There is a period, the same period as this one. There is also maximum and minimum, obviously. The maximum is when the sine is equal to plus one. So, the maximum, I mean, that's the maximum of denominator. So, it's a minimum of function. So, function is between its minimum, which is c over v0 plus b and c v0 minus b. c and b are some constant which depends on cylinder, air pressure, etc. So, it's a periodic function. There is a minimum and maximum. And then I basically draw this function using some tool available online. And I present this graph, which I definitely cannot represent it here because it's three-dimensional waves. And this graph represents actually how the pressure is changing at any point x at any time t. Now, why it's different for different axes? I didn't put x here, right? Yet. So, let's just think about how it depends. Now, as we know, the wave propagates with certain speed, which depends basically on the qualities of the air, initial pressure, and whatever it is. So, what does it mean? It means that into the point x, this wave will come. This is a good actually, this is a good formula for x is equal to zero, immediately under the piston. Now, what happens in any other point at distance x? Well, the wave goes with certain speed, which we don't know, but basically, I mean, in this particular case we don't know. But it exists. It's some kind of a constant, which basically depends on the cylinder and the temperature and the initial pressure, number of molecules, etc. So, it introduces certain delay because the same wave, which is here, will go down and to the point x, it will reach later on. Now, how late? Well, the time shift would be equal to x divided by v. x is a distance, v is the speed of propagation. So, this would be a time delay. How can I introduce the time delay into the formula? Very simple. Omega times t plus tau. That's it. And instead of tau, I can put x divided by v. So, this is basically a formula, which gives you two arguments, x and t. v is a constant, which depends on the qualities of air inside. Omega is the frequency of how we manipulate with the piston, hopefully with harmonic oscillations, as I was saying. B and C are, again, constants, which depend on different factors, temperature, etc., obviously as initial pressure in the air. So, with certain numerical values of these variables, there is a graph in the notes for this lecture, which actually looks like waves on the surface of the water, actually, but it does not represent the water. It actually represents, there are two arguments, c. This is t and this is x and this is p. So, this is a time, this is a distance, and this is the pressure. So, there is a wave surface here. It's drawn, not by me, I can draw it like this, by some software, which represents this formula. Well, basically, that's the end of it. This is the graph of the surface, which represents pressure at any point, at any time. Okay, so that's all I wanted to talk about, how the air propagates inside this cylinder. Now, what I didn't really talk about is what happens if it reaches the end of this cylinder. Actually, it will reflect back, but we will talk about this in the next lecture. Also, another important factor, if you are, let's say, producing certain sound in the room, like I'm talking right now, it goes not along one direction, but in all three directions, which means energy is dissipating. So, that's why the air is, well, that's why the sound in the air is getting lower and lower the farther you are from the distance, because the energy of the oscillations are distributed among greater and greater sphere, so to speak. Okay, that's it for today. Thank you very much. I recommend you to read the notes for this lecture. It's basically the same thing as I was explaining. Plus, there are some pictures, much nicer than I'm doing right now. And that's it. Thanks and good luck.