 Hi, I'm Zor. Welcome to a new Zor education. The previous lecture was about introduction into transversal oscillation. And basically we were talking about complexity of real transversal oscillations. That let's just talk about rope. If you are have some kind of an infinite rope but you have one end of it and you are making harmonic oscillations up and down. The waves will go along the stretch of the rope but every segment of the rope will actually move also up and down, relatively speaking repeating the movement of the end which we are using to drive the whole oscillations. Well, it was much more complex than just up and down movement of every segment. And I did explain in the previous lecture that it's actually some kind of a combination of vertical and a very small horizontal displacement as well. Now this lecture will further simplify our model and we are talking about model. We are not talking about reality. Reality is always much more complex. So we will consider a model of this ideal rope which does not have this small horizontal movement. It has only up and down movement and each segment basically repeats the segment of the driving end of the oscillation of the rope. So this is a simplified model of transversal oscillations using this ideal rope as an example. And obviously it's a continuation of the previous lecture where I was just explaining the complexity of the problem and this is a simplified model. Now this lecture is part of the course called Physics for Teens presented at Unisor.com I do suggest you to use the website Unisor.com to basically go through the whole course because this is the course Physics for Teens is a course and it has obviously lots of different lectures. They are logically connected to each other. Now there is a prerequisite course on the same website. It's called Mass for Teens. You can't study Physics without knowing Mass and Calculus is a must. Now the website is completely free. There are no advertisement and no strings attached. So pure knowledge. Right, so let's just go into this simplified model of ideal rope. So basically it looks like this. Whenever we are moving up and down this end of the rope, and let's just consider the rope is infinite, now what we are saying is that every segment, every point on this rope makes the same movement up and down movement as the driving end which we are actually using to initiate the whole oscillation process. Right, so no horizontal displacement only up and down exactly the same way as this one. Which is not what's happening in reality. In reality the real movement is some kind of elliptical movement but we are not talking about this. We are just completely ignoring the horizontal, and very small by the way, horizontal displacement and we are talking about the vertical displacement of every point considering it's repeating the movement of the end. Now yes it is repeating however there is a delay obviously because whenever you start here every other point repeats obviously the movement of this but with a certain time delay. And this time delay depends basically on the wave propagation, how fast these waves propagate through the rope. And it depends on many factors primarily on physical properties of this ideal rope which we are talking about. There are certain materials maybe the rope can be made, maybe it's made of steel, maybe it's made of some kind of fabric or whatever it is it's made of. And it's always different because it depends on its mass per let's say unit of length and maybe some kind of flexibility. I don't know. There are lots of physical characteristics which should be analyzed to find what exactly is the speed of wave propagations. Now we did talk about at the previous lecture about the propagation of the waves that we were considering a small segment with a rigid rope in between. So if you are making up and down movement of this one and there is no friction let's say and this is a solid rope this one will follow this but with a delay so it will be something like this if you start this it will be movement of this type and that basically means that again the movement will be repeated but with a certain delay which depends on the speed of propagation of the wave in this particular rope. So this speed is actually a physical characteristic and it should be given. What else should be given to find out exactly how our segments each point actually in a rope, how it behaves. Well very simple. First of all we will introduce system of coordinates. So the vertical displacement will be a function y. If this is the point so it will be a function of how far this point is from the beginning and obviously of time. So we have to define this function to find out exactly how our point is moving in the vertical direction and again our assumption is it's an ideal rope, it's ideal conditions and simplified model when this particular point is repeating exactly the movement of this but with a time delay. Okay now time delay what is the time delay I told you just now that the speed of propagation of the waves is some kind of a characteristic which should be given but we can't really find it out from other properties. So let's say v is the speed of propagation. So if x is the distance in the lengths of the rope from the point which we are interested in in the beginning and the wave is propagating with certain speed v that means that if you will divide x by v you will have a time delay between this movement and this movement. Okay that actually is sufficient to basically find out what exactly the oscillation of this point is. If we know this one so let's say what is the oscillation of this one? This is y of 0 time comma t. So 0 means 0 distance from the beginning. So if this function is given and v is given then you can find out the time delay as a function of distance you can always divide distance by speed, get a time delay and I can say and this is very important that function y of x t now what I'm saying is it's supposed to behave exactly the same as function y of 0 and t which is the oscillation vertical oscillation of the beginning of the ending or beginning whatever of this rope. Driving end let's put it this way. So it's supposed to be exactly like this but with a time delay. Now we actually know from mass scores or just from general logical observation that if one function is different from another function only by shift of the argument the whole graph is shifted which means that in this particular case it's supposed to be equal to 0 time comma t minus x divided by v minus time delay. So if I will put into this function what was its value time delay back then I will have what is value of this function at this point now. So if t is now t minus x divided by v is how this one because it's 0 how this one behaved x divided by v seconds ago and after x divided by v seconds from this the movement actually was repeated at point x. Now this is a very important consideration and I would like you to completely understand it. Again if you would like to know what exactly the value of y at point x now what was the value of displacement at this point 0 x divided by v seconds ago because by this x by v seconds this movement would be propagated to this point so that's very important consideration from which we will derive basically this function if we know this now what is this function well it again it's supposed to be given because that's exactly how we are oscillating this rope up and down now if we assume that this is some kind of harmonic oscillation which usually people do we can say that y of 0 comma t is equal to a now in this case let's start with sine it can be sine it all depends on initial conditions so let's say it's sine of omega t now why I put sine well because let's consider that at time t we do not have any displacement from this so the point is exactly this if my point at time 0 is let's say a maximum so it will be a cosine but it will be still the same oscillations it all depends on initial conditions ok so we know this function but if we know this function we know this function so y of x comma t is equal to a comma sine of omega t minus x divided by v basically that's it everything else is just transformation of this equation which basically defines completely defines the motion of this point at any distance x at any time t however what people usually do they introduce certain other characteristics of oscillations and they are just replacing one variable with another so let's just talk about different variables which characterize the oscillations so what kind of variables well obviously we have a time variable now what else what's interesting is that this is periodic function now the periodic function and what is the period in this case but from 0 let's say to this this is one period no from 0 to 0 I'm sorry from 0 to 0 so this is one period this is another period etc so the function has periodic characteristic and obviously there is something which is called the wave lengths so the wavelength lambda is this distance where this particular point is exactly the same position as the previous one well the shortest distance obviously whenever we are completely repeating the whole movement and then the movement repeats again so this is this piece from from this to this is lambda and this to this is another lambda so lambda is the wavelength the shorter the wavelengths the more frequently obviously our our oscillations are happening so lambda is wavelength now what's also is interesting if we have lambda as a wavelength we have the period t this is the time period so it's the time which takes the wave to go up and down and down again so if it goes up down and then down and then up and that's the end of it well the time during which any particular point is making these movements which is a complete cycle it goes from let's say from initial position of 0 it goes up then goes down to 0 then down and then go back to up the time is called the period now obviously in this particular case if we know the period if we know the wavelengths and we know the period it's just shorter to period then immediately from here the speed of wave propagation is obviously down to divided by t the lengths divided by time during which it happens that's basically the end the game this depends on different properties of the of the rope the propagation even if it's the same rope but we will start oscillating it much faster we will reduce the wavelengths and the t the period will also be reduced but the propagation which depends only on the characteristic of the rope will probably be the same well at least approximately now what else do we have well obviously we have amplitude that's the maximum deviation up or down from the middle point now and amplitude is participating here now in this particular formula we have this omega which is angular frequency now what is angular frequency well if you will take the regular sinusoid then this period is 2 pi right so this is the sine of t or in fact it doesn't really matter now if you will have function sine of 2t this is t and this is y now what will be the graph in this particular case well if the period of this is 2pi period of this would be 2t should be equal to 2pi which means period will be for t it will be only pi since it will be like this if this is 2pi and this is pi so the period will be twice as small right and basically if you will have a function sine of omega t the period will be 2pi divided by omega so this is the function which basically combines angular frequency and the period now using this particular equation which is basically images the following from the definition of period and angular frequency and using another thing which is kind of obvious we can transform this formula to many different waves many different waves for example I can say that this is equal to a sine of omega t minus ok what I can do is instead of omega times x divided by v I will use omega from here which is 2pi divided by t right so what will I have I will have omega x so omega is this 2pi x divided by t and divided by v ok now what is v times t from here that's lambda so I can put instead of this I can put lambda and now let me introduce one more characteristic 2pi divided by lambda is called wave number pure artificial concept to make it easier and write it down as a sine of omega t minus kx now what does it mean actually this is a very important actually form of the same equation and the form it says that the motion that is buoyant at distance x from the beginning is exactly the same law as the initial oscillations and the initial is a sine of omega t in this particular case shifted in phase by distance multiplied by wave number it's a little bit more convenient representation of the same equation so in the notes for this lecture on this website and resort.com and I have notes for every lecture by the way which is basically like a textbook I can put not only this but also some other forms of the same equation just basically substituting one after another so now we will use this type of substitution but we can use many others so there are many different forms of the same equation but all of them are basically the same in their major sense major form which is actually that equation which represents the movement of point at distance x is almost the same as the one which represents the beginning of the rope or driving end of the rope but I recall shifted by phase something which depends obviously on different characteristics one of the characteristics is obviously the length the phase shift in phase is proportional to the length which is kind of obvious and all we basically need to know is the coefficient of proportionality which in this case is called wave number now I started with a sign because I was assuming that in the beginning we have in the middle point and then I go up and down I can start from the upper point and then go all the way down from A to minus A in which case what's changed well the only thing which actually changes instead of sign I can actually use cosine and that would give me the initial cosine of omega t will give me a t is equal to zero co-ordinate as A so basically that's the upper point everything else remains exactly the same and I will have the cosine here so sine and cosine are actually again it's almost the same the difference between these functions is only a phase shift remember the graph of cosine is something like this graph of cosine is this so one of them is just shifted relative to another so that's why cosine or sine doesn't really matter it all depends on the initial condition very exactly my rope and then everything else follows from it okay what else here I think we are completed right so this equation and all similar equations which can be derived from this one are determining the movement of any point on the rope assuming that this is some kind of an ideal situation and all the points on the rope will certainly delay the motions of the the driving end of the rope now other than this very very simplifying I would say idea hypothesis whatever you call it everything else is basically straight forward very simple mass and it all depends on the phase shift of the function by the way if our initial oscillations are not harmonic it's just any function but basically the same consideration will be valid so y of x comma t would be equal to y of 0 t minus x divided by v so this formula still stays and depending on what exactly this function y of 0 something it can be anything not necessarily harmonic but whatever it is the others the other points will repeat the movement of the rope shifted in phase by certain particular value which depends on how far it is and how fast the wave propagation happens on this rope but obviously if it's not wave like almost harmonic or something like this movement of the initial point there will be no propagation so to speak so we don't really have to talk about any kind of weird movement of the initial driving point we are usually talking about harmonic oscillations of the driving end of the rope and that would result in more or less to a certain degree of precision the repetition using this particular formula or any other incarnation of this ok so I suggest you to read the notes for this lecture you have to go to unison.com choose physics 14 course and then the part of the course is called waves and then there is a transverse waves and that's where we will have this lecture called wave equation 2 wave equation 1 was related to waves in media and their propagation that's one thing about I mean there are many different wave equations so that's why it's 1, 2 etc so that one was more for about longitudinal oscillations of media like for instance air with sound waves but this particular is about transverse oscillations so again I do suggest you to read this lecture and don't forget that most chapters most parts of the course they have exams which you can just take yourself as many times as you want just to make sure that you really are mastering the contents of this ok that's it thank you very much and good luck