 Hi, I'm Zor. Welcome to a new Zor education. We continue talking about waves. We have learned about longitudinal waves, which basically are like waves of the sound in the air, when the molecules of air are moving in the same direction where the waves are propagated. So the waves are actually the differences in pressure of air in this particular case. So the molecules are going this way back and forth and the sound is propagating the same way. So the movements of carriers of the oscillations are exactly in the same direction as oscillations themselves. Today we will learn about different kinds of waves, maybe the waves which you're kind of more accustomed to. These are waves when the molecules of the carrier of the waves are moving not in the direction but perpendicular to the direction. So it's like a rope and you are shaking up and down one of the ends of the rope. The waves are going along the rope and the movements of the molecules of the rope are across that direction. So this is called the transverse waves. So today we will talk about transverse waves. That's the waves when the carrier is moving perpendicular to the direction of the waves. Okay, now the usual introduction. This lecture is part of the course called Physics for Teens presented at Unisor.com. I would suggest you to watch this lecture from the website. So you go to Unisor.com, there is a menu, you choose Physics for Teens, that's the course. And within that course there is a rather large chapter waves and you click the waves and then you will get again the next menu where you will find transverse waves and this is the first lecture in that particular chapter. There is also a prerequisite course called Mass for Teens on the same website, which I suggest you to be familiar with because physics without mass, basically it's not really the real physics. Now the website is completely free. There are no advertisements, there are no strings attached, you don't have to pay anything. You don't even have to sign on if you don't want to. For self-study without any kind of supervision, et cetera, you don't really have to sign in. There are exams on the site which you can just test yourself or your supervisors can test you. Back to business. So again, we know about longitudinal waves and again the best example is sound waves in the air when the molecules are oscillating along the direction where the propagation of the waves is. And now we will talk about a typical example is a rope. So let's just consider what happens if you have a rope and then somebody takes this particular end of the rope and starts moving it up and down. Well, again, we know from experience that the rope will take something like this shape and the waves will go that way. This will be the direction of the propagation of waves. What my purpose in this lecture is to explain why it happens. It's not easy. I was myself actually struggling with understanding why the waves are actually propagating. And today we will talk about a qualitative picture, so to speak. We don't really go too much into any kind of formulas. We will try to explain how the propagation of waves actually is happening based on certain model. So the first thing which we need, we have to model this rope in such a way that we can later on apply certain mathematical foundation to this. So what comes to mind? Well, we know that rope contains little pieces. Little pieces contain even little smaller pieces down to molecule level. So something probably is related to molecules and relationship between molecules, etc. So I'm suggesting as a model actually to consider the rope as consisting of small objects. Maybe it's molecules. Maybe it's just small segments of the rope connected with certain rods. So every piece has certain mass. There is a certain distance, let's call it r. This is also m, this is also m. So this is like a necklace, basically. Necklace of beads which are connected with, let's consider weightless but rigid rods, diamond necklace. So these are diamonds and they have certain weight. And these are connection between these diamonds which we consider to be weightless. But they are rigid so they can actually go up and down independently from each other but they are at fixed lengths from each other. All this. So how can we analyze analytically this particular necklace? Well, it's not easy. Let's simplify it even more. Here is the simplification which I suggest. That's it. Just two beads of certain mass connected with a rigid rod but they can actually change the direction relative to each other. Okay, now the bead on the left would be the beginning of my rope. That's where I'm actually moving, forcing this particular left bead to move up and down. Now what I would like to see is how my bead on the right would actually move in this particular case. And again, this is not easy. Okay, now before going into these analytics I would like to finish something which is really a formal part which everybody, all the textbooks and all the teachers are basically explaining as the most important part of the transverse waves which I don't consider to be the most important. It's basically a terminology. So this is called a crest of the wave and this is a crest and this is a crest. This is the part where the wave goes to the uppermost level. This is a trough. This is a trough. Now this is distance from this to this. It's amplitude. I will use letter A for amplitude. The distance between one crest and another is wavelengths, usually Greek letter lambda. Well, the whole thing resembles a sinusoid, right? Well, it's not really a sinusoid, but very close to it. It's obviously periodic and it's kind of moving up and down so it really looks like a sinusoid. That's all right. Now, if you measure the time between one particular piece of the rope, this one moves, let's say, from the top position down to the bottom position and then to the top, that is called period. It's a time period. Time between the same particle or the same little piece of the rope makes the full cycle from top to top or from bottom to bottom from middle point to middle point, whatever you want to do. So this is the period. Well, basically that's it about terminology. So we've finished this and this is as much as everybody knows about the waves. Whatever I'm talking about right now is much more involved and in my particular viewpoint it's more important because I would like to analyze the forces which are acting in this kind of a movement. So let me get rid of this. So we've finished this formal part of this lecture. Now it's more involved, I would say. So we are talking about a movement of two particles, one and two. This is my vertical direction and this is the rod between them. Let's call this alpha and this is beta. Both have mass, m and m and there is a fixed distance between them, r. We are forcing moving alpha up and down from minus a to plus a. This is amplitude, right? Remember this amplitude. Now initially, let me put this a little higher. Okay, initially let's consider we start from this position and this will be alpha and this will be beta. So they are lying down. What kind of forces are basically acting? Well, let's just think about the movement. As I'm moving alpha from position minus a to plus a through the zero what actually happens with alpha? Well, if I'm doing movements like that that means that I'm speeding up first and then slowing down to speed zero, right? And then go back again, speeding up and slowing down. That's basically my movement. Otherwise I will not be able to stop here and here. So I begin my movement and I end my movement which means I have to speed up and then slow down. As alpha is moving this way how is beta moving? Well, let's just think about it. From this position to this position this is my rod. I'm moving only this part. What happens with beta? Well, if I'm moving to this position, let's say well, beta is supposed to move this direction because this is supposed to be the same length as this one. Not that, something like this. So this length is supposed to be equal to this length. This length is the distance R between these two particles, these two objects. And this moves up and this still here which means this is kind of delaying movement and that's why it will not move vertically up because we don't have any force which moves up. We have the force only here which means that basically stretching the rod. The rod becomes tense and it's a tension force which in the beginning it always acting along the rod because we are basically pulling by this rod. So the tension begins here but as this thing is going up the tension becomes... the tension is at angle and there is a vertical and horizontal component. Vertical is very small in the beginning mostly it's horizontal. That's why it's moving this way. But movement again, as I'm moving upwards my position is something like this. So my beta object, this is beta is moving closer and closer to the vertical along which alpha is moving and it also goes upwards following the alpha because alpha is basically hooked to it by the fixed rod. So as alpha moves this way beta moves this way. So this is just qualitative picture. Now I would like to analyze all the forces which are acting in this particular case. So, getting back to our picture level, bigger scale. So this is alpha, this is beta. So what kind of forces? Well, obviously there is a fixed force which is not very interesting. That's the weight. Since we have a mass, we have p alpha equals to mg. We are assuming that masses are the same, right? So this is the rod between them. Now, the only force besides the weight which acts on the beta object is whenever we are pulling up, we are talking about the movements up the first quarter, let's say from the bottom minus a level to zero when we are speeding up. So we are speeding up the alpha which goes to zero. This is zero level and this is minus alpha level. So from here we are moving to this. We are speeding up alpha. There is an acceleration which means there is a force, obviously. So there is something which drives alpha upwards. Now the force, the only force which acts on beta is tension of the rod which pulls beta basically along the direction. So the force of tension always directed along the rod itself. So the tension should be here. This is function, tension, beta of time. Oh, it depends on time, obviously. Well, first of all, direction depends because in the beginning these two were in this position, horizontal position. So the tension force was horizontal. Then as alpha moves up, beta is supposed to follow it so the tension force changes direction and most likely changes the value as well. Now, the same tension force acts on alpha but in a different direction. So the force of the tension which acts on alpha should be directed opposite. So if alpha pulls beta, the way how tension force pulls beta up, it pulls alpha along the same direction, along the same rod but in opposite direction and they are actually equal to each other in absolute value at any given moment of time. So this is how rod pulls beta up and kind of holds down alpha and that's basically the function of the rod to keep the distance the same. Now, we are moving alpha straight up and we have to have some kind of force and there is an acceleration. I mean, we can always just think about movement of alpha as being sinusoidal up and down. So the y coordinate of alpha can be something like minus a cosine omega t. y I put minus a, so at t is equal to 0, this would be 1 and would be coordinate minus a. So at time t is equal to 0, y coordinate would be at minus a. Then at certain moment of time, t which is equal to t over 4, my coordinate would be 0. Then at time t is equal to t by 2, my coordinate would be a. It goes to the top level, a. This is 0. So from 0 to t over 4, I am going from minus a to 0. Then from t 4 to t 2 to t over 2, I will go to plus a. Then from t over 2 to 3 quarters of a t, I go back to 0. At moment t is equal to t, 3 quarters of a t, my coordinate would be again 0 and t is equal to t, my coordinate would be as before minus a. That's what we started. It would be minus a if t is equal to 0. So this is what this formula gives me. So I know the movement of alpha. I would like to know what would be the movement of beta. And again, right now we are talking about only from here to here. So from t is equal to 0 to t is equal to t over 4, where my coordinate would be 0. This is acceleration. Then would be deceleration. Okay, so acceleration. What must be my force acting on alpha to give the straight up acceleration? Well, I cannot really put this force straight up. Why? Well, because this is at angle, right? Which means there is a horizontal and vertical composition of this, right? This is vertical, which actually pulls down, and this is horizontal, which moves to the right. Which means my force should neutralize this horizontal component and this vertical component and add something more to accelerate the movement up. So the force should be something like this. Now, when we are moving a rope up and down, well, we do not think about this. We think about we are moving only vertically. But we are not thinking that we actually have to have some kind of angle because we are pulling the rope, well, the next bead, let's put it this way, we are pulling the next bead up and it moves towards the vertical direction. It's a minor movement. Don't take me wrong. Most important movement, major movement is obviously up and down. But there must be some because if this is the distance between two molecules, obviously it's a small distance, but it's supposed to be from this to this, which means we are moving to the left. And that's what actually makes the force, if we are moving strictly vertical, the alpha bead, then the force must have certain angle. Now, what we can do actually, we can put some kind of a railing here. We will move it up and down, but we have to really think about that the result is the reaction of the railing which doesn't let the alpha to move left or right. So we will move it up and down, but there is also some kind of a reaction of the railing which keeps it on this vertical direction. So the resulting force in this particular quarter of a period, the resulting force will be at slight angle. That's very important actually. It's not very important to really shape the rope, but it's very important to understand all the forces which are acting in this particular case. It's an extremely complex movement, by the way. We just think about this, we are making waves in the rope and it's really simple. It's not simple. And don't forget that all these forces are changing with time. Primarily the direction is changing. The rope used to be this way and at the end of this it will be almost vertical, right? So it changes. Okay, now, will beta ever reach the same vertical where alpha is moving? No, it will move closer. But as soon as alpha reaches the level zero, which means that's the maximum speed of alpha, then it will start decelerate. Alpha will start decelerate. Now beta is moving in both directions, vertical and horizontal, vertical here and horizontal here as a result of this tension force. And though it will not move in a vertical direction with the same speed as alpha, but it will follow. So it will also increase its vertical component. Let's not talk about how fast it will go towards the left, towards this vertical. It will go with certain speed. But since most important direction of the alpha is vertical, the beta will also increase its speed vertically to a certain degree. Okay, so that's what happens during the first quarter of a period. I would say that the direction, the trajectory, let's put this, the trajectory from this level would be something like this. It would be closer to, if I will continue with movement of alpha indefinitely up with a constant speed or accelerating, it doesn't really matter, what's important is we do not decelerate if I'm just moving up all the time. The beta would move closer and closer to the vertical so that would be some kind of asymptotic movement of beta. As alpha moves up, beta would move closer and closer and follow alpha. Eventually it will be very, very close and vertical component of beta will be almost the same as vertical component of alpha. But also it will have always horizontal, weaker and weaker, so it's slower and slower moving towards vertical. The horizontal movement would be faster here and much slower here, and again it's asymptotic movement. There is a curve called tractrice, which actually resembles this. That's when you're just pulling with a constant speed you're pulling on a chord some kind of an object, you're going this way and the object is going this way. So this is called tractrice. So it resembles it, but not exactly. And in our case our movement is much more complex. We are not moving up with a constant speed, we are accelerating and then we're decelerating. And now we are approaching the second quarter of the movement from 0 to up, to the top amplitude, to the A. Okay, what happens there? Okay, let's draw another picture. Again, I'm talking today only about pictures of how basically the whole thing might move almost without any quantitative characteristics. Well, except I just had a formula for movement of the alpha. I assume this is the formula, but again it's just a model. Okay, now we're talking about from 0, somewhere here is minus A, but this is plus A. Now in the beginning of this second quarter my, by the way, in this particular case t period is equal to 2 pi over omega. Omega is angular speed or angular frequency. So now we're talking about the second quarter from c over 4 to c over 2. Now at this moment, since we are accelerating all the time my beta object is moving up and closer. So probably it's somewhere here, approximately. And this is my alpha. And again, let's talk about forces which are acting on this. Alpha stopped accelerating. It starts decelerating. So alpha is decelerating. Now beta, what kind of forces are acting right now? Well, beta continues moving with the speed alpha has reached at this particular moment. Up and horizontally. Now it will probably be faster now than alpha. So alpha is breaking down the speed, but beta is still moving with the same speed alpha was at point 0. So what does it mean? It means that the tension would be different instead of rod pulling beta up. Now beta would push the rod up. So the tension would be directed this way. So this object would push the rod. Instead of rod pulling the object, object would push the rod. So that would be my t beta of t of time. Well, obviously there is a weight. So these are basically interesting forces. Now onto beta would only, its weight would actually do. And then the beta would actually push the rod. And this pressure would be applied to alpha. Now alpha is still supposed to move on this particular distance. So what actually would be forces which are acting in this particular case in this subject? Well, my force obviously should be independent of this, but let's talk about what kind of force I should really. Since our force should actually break down the whole system, it should resist. So the alpha would push the rod in opposite direction. Because the rod is supposed to be rigid, right? Now obviously there is a weight. And then what does it mean that what kind of force should be actually acting on alpha? Well, we have to break it down. So the force should be down. But again, not exactly vertically down. Since this force will have horizontal and vertical composition, now we should really somehow neutralize. So the force would be at this particular angle. Not exactly vertical. So what's interesting is that again, the force which we are applying onto the object whenever we are pulling it, we are thinking we are pulling it up and down, the real force would be again at slight angle from the vertical. On the way up it was angle towards this direction, on the acceleration, on the deceleration, the force should really be directed against the movement, right? To decelerate. So whenever we are moving up, we are accelerating and then decelerating. So decelerating means we are applying the force, which actually going back in an opposite direction to a movement, which means down. And again, at slight angle, because we really have to apply some kind of a pressure onto the road. Now, what happens in this particular case? Well, if I am applying this pressure towards the road, this pressure also has certain force applied against these beta, because the length is supposed to be the same. And what does it mean? What happens with beta when alpha is decelerating? Beta is trying to move as it was because of inertia, but the fixed length of the road prevents it. If this goes down, down, down, this is also supposed to decelerate somehow. But since the road is the same, it will start moving this direction, slightly outside. So the angle, this angle would be smaller, I mean bigger, sorry, bigger here, and then bigger again here, because the beta would go with the same, the beta will have certain inertia. We are breaking down on the alpha movement, but the beta will continue moving with the same inertia as before, but its direction is restricted because we have the fixed length. If all of a sudden I just momentarily stop alpha, what would be with the beta? Well, since it was moving up in this direction, now we are stopping alpha, it would continue, upwards movement will give you, basically, it will go on the circle, but it doesn't go on the circle because alpha is not stopping, alpha is just slowing down, which means the beta will start also moving slower, but again it will be on the further and further distance. So if before my movement was like this, now my movement would be like this, and eventually almost horizontal. Now I'm kind of extrapolating these thoughts, so as alpha is moving vertically up and down, beta is supposed to move on some kind of a curve, it would be further in the beginning and at the very end of this half a period and closer somewhere in the middle. Now what happens somewhere here? Well, it depends, for instance, if the speed is not very high at this middle point, which means it will reach certain position like this one, and then we are decelerating. Well, what happens, it might actually go this way and then go this way along the same trajectory, or it might actually make some kind of a little loop, what if the speed is so significant for whatever reason, that after this position it will go even more around, this will stop, and this will go even by inertia a little bit more, then goes down and that's supposed to follow, so maybe it will be something like this direction. I'm not sure myself, it depends actually on many different things. The length of the road, the masses of these, and on the amplitude, so these and the speed of movement up and down, or if you wish, the angular speed of this movement. These are all analytical things which we will explain in some details, not in all details, because it's a very complicated movement actually. But my purpose was to basically talk about forces which are acting and the direction, the trajectory of the second bead, second bead on this necklace, if the first bead is going vertically up and down, so this direction would be again kind of a very complicated curve, and then add to this the third bead, for instance, the third bead would follow the second one, right, also in the same length, and it would be even more complicated curve. Well, again, the most important is obviously direction up and down. The direction left and right would not be as visible as in this particular case, but you should understand that this is complicated and this kind of a movement would be definitely not exactly straightforward. And just imagine it's quite simple if you have a rope of certain lengths and then you convert it into this type of things. This is a curve which is more, which is lengthier, so it cannot really be that every particle would move only vertically up and down. The length of the rope is fixed, it's this one, horizontally, so it should really compress a little bit, so the true picture would be the same rope. If this is the original length, then the length when it's actually moving up and down would end somewhere before that, because the length of this should be equal to this, which means that every particle of this rope actually moves left whenever we are making this type of things. And that's what this particular curve represents. It's squeezing, basically, and it's squeezing because this length is the same as this length, so the movement from here to here is not vertical, it's at the angle. And then, again, it goes here, not along the straight line, it will go along this curve. So, this is all the complications which are related to transverse movement, which means that transverse movement is not really up and down for every particular particle, as usually simplified in most of the textbooks. It's much more complicated. And whenever you are talking about, for instance, a string instrument like a violin, it's a perfect example of the waves, which are usually qualified as transverse. But if you think about it, this is your string, and then you're stretching the string. Now, what happens, actually? Well, you're stretching, which means you're stretching every rod, which connects two molecules, right? Every little rod has become... The story is even more complicated, because in this case, I was considering that the rod is actually a fixed length. In this case, to enable the vertical movement of each component of a string, we have to have these rods stretchable, so it's like a spring now. So every rod is a spring. And this gives me a new model of our transverse oscillations. The model, when this rod is not a fixed length, which gives this type of a wave, but the rod is stretchable, like a spring. So every rod between two molecules, every connection is a little spring. And another example, you have waves on the surface of the water. It's the same thing. It's much more complicated. It's not up and down, obviously. These are more like this one. So every molecule of the water goes along some kind of an ellipsoidal wave and then transfers to a different one. And what's most important is the purpose I wanted to explain all this is for you to understand why we have the propagation of waves. Very simply, because the maximum of the alpha point in this particular case is prior to the maximum of this one, because when alpha has already stopped, this one, the beta is still continuing to move up. And only at the very end, when it's already started moving down, only then the beta will finish this cycle and then it continues to follow. So alpha is always ahead of beta. And if it's always ahead of beta, it means there is a time difference between alpha reaching its crest at the top position and beta reaching its top. Because whenever alpha is here, beta is still here and then it goes around some circle or whatever it is, a curve, basically, which resembles a circle. And only then, when alpha starts moving down, only then beta reaches the top and then goes around the top and follows alpha. That's the delay between these two molecules, alpha and beta. And since there is a delay, that's basically what makes waves propagating. The alpha is reaching a certain maximum, a little bit later beta is reaching the same, let's say, maximum. And then the next object connected to beta, let's call it gamma, then gamma. So there is always a time delay, which depends on many different factors. As I was saying, it depends on the angular speed of the alpha oscillation, its amplitude, the weight and all other lengths of the rod and other physical quantities. But there is always a delay. And since there is a delay, we have this difference in time between alpha reaching the maximum and a little bit later beta reaching the maximum, then gamma reaching the maximum. And that's what makes the propagation of the wave. After reaching the maximum, they obviously follow the same direction. So again, it goes down and this goes down, but later on because it's on certain fixed lengths and then it only follows. Well, I think that's all I wanted to talk about today. No calculations, no big formulas, etc. It's a qualitative explanation of what kind of forces are acting on different beads, if you wish, to use this model, beads of necklace or every molecule of a rope, whenever the rope actually making these oscillations. And I was trying to explain that because of the time delay between alpha reaching the crest and beta reaching the crest, that's what actually makes the wave propagation. So that was the purpose of today's lecture. And I also hope that you understand how complex these things are. So whenever we are talking about certain relatively simple calculations which usually are presented in the textbook, it's a simplification. Reality is much more complex. And obviously we are researching our nature using the model. So we are simplifying the model, we are having a simplified view, which is not exactly what's going on, but to a certain degree it explains certain things. And during the next lecture, too, I will probably try to do the same. I will simplify as much as I can just to reach certain relatively simple formulas which assume, assumingly represent the reality to a certain degree. Okay? I suggest you to read the explanation which I have on the website. Every lecture has notes and the notes are basically the same thing as I am saying, plus a little bit better pictures maybe. So I do suggest you to go to Unisor.com, choose the physics routines, waves partition, waves part of this course, and the transverse waves as the topic we are talking about. Okay, that's it. Thank you very much and good luck.