 common example of reflection. Very common. Okay. That example of dog that I have given right? This one. But like some useful district which are regularly, mirror? Sonar, correct. Give me an example which is useful as in super position is what is happening. Reflected wave and the incoming wave, they superpose the meeting. No such example coming in your mind. You can say that but in reality become very difficult. It doesn't happen like that. But then all musical instruments, whenever you play a musical instrument, there is a wave. Suppose take an example of guitar string. You plug the string, the wave travel wherever it is hinged from there, it reflects off and it then meets the incoming this thing and creates a standing wave. That is what we are going to discuss. All the musical instruments including tabla. Okay. They have this phenomena of reflection and that reflected wave coming and meeting the incoming wave. Okay. So let's say the incoming wave is this is right down incoming wave is a sign Kx minus omega t and the reflected wave two ways. One is telling in positive direction. Other is telling in negative direction. Right. So please add them. These two ways when superpose see what will happen. Correct. I am taking a soft reflection right now. That can be hard reflection as well. But that will not change much. Laplace and everything we discussed. Right. Did you add? I didn't know what to write. 2a sin a plus b by 2, which is sin of Kx cos omega t. Now tell me is this a wave equation? This is not a wave equation. The two waves get added up. This doesn't give you a traveling wave. That is what you can say. Okay. It is not a traveling wave, but it is an outcome of two ways meeting. Okay. Now let's try to analyze what is happening actually. So if you see that this is the term which depends on x. That is a term that depends on time. Okay. Now don't speak ahead of me. Okay. So tell me are there any points which are always at rest? Whatever time? Yeah. If sin of Kx becomes 0, y will be 0 for all the time. It doesn't matter what is cos omega t. Yes or no? Right. So please write down if sin of Kx becomes equal to 0 and will be at rest always. Any locations of these points? Can you tell me location of this point? Try to do it. Whenever I ask you something, don't just blabber really. Think and then speak. Think what it is. Solve it properly. Write a term using pen and paper. Every time you speak something wrong, you are getting negative mark in exam or whatever. You are not learning anything. Tell me x is equal to what? 0, 0, 0. Nobody got it? Where I have given you lambda here? I don't see. N pi by k. N pi by k. N pi by k. So if Kx is equal to 0, Kx should be equal to N pi. So x should be equal to N pi by k. So you can put the value of N as 0, 1, 2 like that. You keep on putting these values. You have different locations of x which will be always at rest. So these are the points which don't move at all. Does that thing happen in the travelling wave? That doesn't happen. So these are the special points. These points are called nodes. Now tell me if I put x equal to 1 centimeter. Basically what I am saying is I am just looking at a particular location. I am just looking at what is happening at x equal to 1 centimeter. Can you describe what is happening at x equal to 1 centimeter here looking at the equation? It will be a thing which is doing SHM with that amplitude. So if you take this bracket term as some constant, does it look like an SHM equation? Right? So that location x, let's say x equal to x naught. Location x equal to x naught is doing SHM with an amplitude of 2A sin of kx naught. So this is the amplitude of the SHM of that point which is at a distance of x naught. Are you feeling it? But if kx naught happens to be N pi, amplitude will become 0. So this is the amplitude of the SHM of that point. Now does that thing happen in a travelling wave? In travelling wave the amplitude of all the point is A. It goes maximum A, minimum it goes to 0. But in this case when one wave is coming like this, other wave comes like that. There will be some point which will be always at rest. The particles are doing SHM with different different amplitudes. So have you ever seen such thing? Suppose a string is tied on two ends. You just plucked at the center and you see some, you know, the ites sort of structure. It vibrates. Have you seen such thing? Have you seen? So this is the standing wave. These two points are nodes. Particles simultaneously go up, simultaneously go down. Time period of SHM doesn't depend on amplitude. Time period of all the particles are same. They together move up, together go down. Someone move at higher amplitude, someone goes to the lower amplitude. So over here you can see that the distance between those two points is half of the wave length. It has to be equal to pi by k. Minimum distance between the two nodes has to be pi by k. So in this specific case, will it be half of the wave length? It has to be. That's what I'm telling you so many times. It has to be pi by k. Yes. Okay? So basically you cannot have a frequency whose wavelength is less than which supports this over there. Does that make sense? See here, the first distance of node is x equal to 0. Second distance of the node is pi by k. So the distance between these two points cannot be less than pi by k. You cannot have that kind of standing wave there. It will not vibrate like that. Are you getting it? It will vibrate. Now have you seen structure like this? This kind of thing you might not have seen. The two points, one and two, they are fixed. But if you vibrate with very high frequency, there will be natural nodes that get created. So total five nodes are there now. You have seen this? You can do something similar on a guitar. Okay, you know right then. Anybody else have seen something like this? Basically all the particles, they will vibrate like that together. So they are moving up and down together. But now the situation is different. There are total five nodes. And distance between the two nodes cannot be less than pi by k. Getting it? But now the frequency is higher. The distance between these two points is lesser than these two points. So this is what we are going to study. Standing wave is a hard reflection but the frequency is increased and you continuously vibrate. Continuously you are sending wave and continuously the wave is getting reflected and coming towards you. The wave doesn't stop coming. You have to do it continuously. I will show you that there are some videos but right now I can't take time to set up. Sir, also the distance between two nodes should always be pi by k. Yes, that is what we have defined. Sine k should be equal to 0 for the nodes and x should be equal to n pi by k. So when you put n equal to 0, you get first node. If you are taking your coordinate system such that first node is at x equal to 0, second node must be at pi by k. Third node must be at 2 pi by k. It may happen that 2 pi by k is more than the length of the string. Mathematically you can go to infinity. Actually only few nodes are possible because length is fixed. So between a node there will always be a point which performs an SHM of amplitude A. Aptitude A plus A and minus A. It has to go up and down. So I was coming to that only. So these are the points which are at rest always. There are few points which vibrate with maximum amplitude are called antinodes. The points which vibrate or the points which perform SHM with maximum amplitude are called antinodes. So can you tell me where are those points similarly you can derive? Sine kx should be equal to what? Sine kx should be equal to 1. Sorry, the amplitude will be 2a. Solve it properly. Can it also be minus 1? Sine of kx. Correct. So it should be plus minus 1. See amplitude can be negative also. Negative just signifies which direction it is moving. So it should take plus minus 1. So kx should be equal to what? Sine kx is plus minus 1. 2n minus 1 pi by 2. 2n minus 1 pi by 2. Minus 1 of plus 1 does not matter. kx will be equal to 2n minus 1 pi by 2k. So basically antinode lies between the nodes. There will be one node between two antinodes exactly in between. There will be one antinode between the two nodes always. And you can also see that if suppose you take this standing wave. Total wavelength is how much? If you draw the total wave, it will be this. From here to here is total one wavelength. So 1i is like half of the wavelength. This distance should be lambda by 2. So like that it will help you to visualize in terms of lambda also what is the distance between the nodes. From here to here the distance is lambda by 4. In terms of lambda you can talk or in terms of k also. Mathematically. Any doubts?