 So, let us proceed further, we are going to have one class only for the problems, but the problem is that in this chapter, there are many questions that combines multiple concepts together. So, we cannot have questions on single concept, write down, principle of superposition, just write down superposition, this is the biggest topic of this chapter and the most important topic, if question from this chapter comes, 90 percent from this, 90 percent chance, it will be from this topic, that is, now superposition means what, it is like a collision, ok, two mass when they collide, that situation you have discussed, momentum and all get conserved, but we have not yet learned what will happen if two ways collide, that is what we are going to learn, ok. Now, when the two waves meet, there will be different, different things that will happen and that makes this scenario very unique, when mass comes and hit the other mass, there is nothing other than force, ok, so momentum get conserved and you are able to write the momentum equation, but over here there will be multiple scenarios, ok, so, just first write down the scenarios, so if two waves, I am talking about superposition of two waves, because if you know how two waves superpose, after superposition you can superpose a third wave also, ok, so 1 plus 2 is 1, then third one becomes a second one, so it is always two at a time, so two waves, two waves traveling in same direction, when meet creates a third wave or creates another wave, that is the combined, that is what two waves traveling in the same direction, when meet, oh, we are talking about a third wave get generated, only one wave get generated, two waves they meet and a third wave is the outcome of the two waves meeting, two waves knock on, second two waves traveling in opposite direction, having same wavelength and frequency create standing wave, two waves having different frequencies, we are going to discuss it anyways, just I give you a small example, you might have tried playing guitar, that is all, so when you play the guitar, you will see that this kind of wave will get generated, if you plug the string, you will see this kind of thing get generated, when it vibrates the string when it vibrates, this is also wave, but what is happening is that these two points are fixed, one where the two ends where the string is held, this is wave which is not moving anywhere, what happens is one wave get generated when you plug it, then the wave get reflected from here, then this reflected wave travels in opposite direction, it meets the wave coming from this direction, when they meet, they create standing wave like this, we are going to discuss it in detail, two waves having different frequencies, when they meet can generate beats, this beat is not your song or a beat, as I gave beat or song, which one, beats is something else, I am going to show you actually, I will play a video for this, just write down three scenarios distinctly, because when we are going to take it one by one, you should not wonder that what is this, what is something new, these are all the three scenarios that come up, when two waves meet, in the first two scenarios, I am assuming that frequency and the wavelength is same, in the third scenario frequency is different, there is a fourth thing also, but that is not in the syllabus, so now you might have got an idea that two waves meeting is not as simple as two masses colliding, so according to super position principle, y is equal to simply y1 plus y2, y1 and y2 are the wave equations of the two waves, just write down, according to super position principle, y is equal to y1 plus y2, when the two waves meet, simply you have to take the algebraic sum of the two, going to discuss it, y1 is not a, y1 is a sin kx minus omega t, getting it, so when you add it, you have to add the sin, according to super position principle, they can be n number of a's, y1 plus y2 plus y3 plus y4, so keep on adding it, so let us take y1 as, write down in your textbook, they have taken cos of sin, sometimes they take cos, sometimes they take sin, they have taken sin, sin, they have taken, a sin kx minus, when they talk about beats, they have taken it as cos, y2 is equal to a sin kx minus omega t plus phi, they are the two, can you describe what is the similarity of the two, velocity is same, what else, amplitude is same then, time grid is same, wavelength is the same, what is the difference, phase difference is there, I could have written here phi1 and this is phi2, phi1, phi2 then phi2 minus phi1 will be the phase difference, so it is similar, now please write down y1 plus y2, and tell me what equation you are getting, y1 plus y2, what you will get here, you sin a plus sin b formula, y1 plus y2 will be 2 a sin a plus b by 2, then cos a minus b by 2, so cos of this and sin of kx minus omega t plus phi by 2, does it look like a wave, does it look like a wave equation, this is your amplitude, if I rather than this bracket, if I write capital A, suddenly it will look like a wave equation, so when two waves like this meet, it will create a third wave of what amplitude, 2 a cos of phi by 2, and when it comes to the wave, when we talk about the energy contained in the wave, we talk about intensity, intensity is a better representation of energy in terms of the wave, because there is a continuous transmission of energy, so it is better to talk in terms of power delivered for an area, so intensity do you know proposal to what, amplitude square do you know this, you do not know, now you know, please write down intensity is proposal to amplitude square, as you mean frequency constant, if you compare the two waves having the same frequency, intensity will be proposal to amplitude square, see the right guy, so it is straight, straight, yes sir, 90 degree, not like this, that is more than 90, come front little bit, straight, spine straight, okay, so intensity is proposal to amplitude square, if frequency is constant, if frequency is not constant, as if you are comparing two waves having different frequencies, then intensity is proposal to frequency square, amplitude square, so intensity is power by time, power per unit area, power is energy per unit time, okay, fine, so here usually we are comparing the waves having the same frequencies, so intensity is proposal to amplitude square, so initial intensity is, initial intensity is some constant into a square, final intensity is square of this, 4 a square cos square phi by 2, yes or no, so can I say that intensity is equal to 4 times i naught cos square phi by 2, all of you understood this, right, so intensity is 4 times the individual intensity multiplied by the cos square of phase difference phi 2, okay, now what is the maximum possible intensity, if I can change the phase difference, what is the maximum possible intensity, 4 i naught, where, no k is not equal to 1, intensity is here also k will be there, right, constant of proportionality, no, k into a square is i naught, so both the k's should be, because frequency, wavelength everything else is same, so i max is 4 i naught, what is the minimum value of intensity, minimum is 0, now this is something very unique, very unique can't be together, right, there is something unique about the waves, so two waves can meet and they can destroy each other, intensity will be gone, that doesn't happen when two masses can meet, it's like one wave this, if this wave meets that wave, nothing is left, fine, so it can happen, similarly, one wave having intensity i naught meets the other wave having intensity i naught and i naught and i naught creates 4 i naught, so basically what is happening is that the energy is getting concentrated and energy can become 0 also when they meet, in terms of when we talk about the waves, now tell me what should be the value of phi for which the maximum possible intensity comes, so for that cos square phi by 2 should be 1, so cos of phi by 2 should be plus minus 1, so phi by 2 should be n pi, so phi should be equal to 2 n pi, so if phase difference between the two waves is 2 n pi then you say that the maximum intensity will get generated, you also say it constructive interference happens, this is for constructive interference between the two waves, reading it, see I will just tell you the day to day life example of constructive and destructive difference, have you ever seen oil spilled on the surface of water, when you move past it the colour changes, colour of the film changes, you don't know you haven't seen oil spilled on the water, the weird rainbow thing, yeah rainbow sort of thing, when you move past the colour continuously changes and there is a location like that, location becomes completely dark and there is a location when you move past suddenly becomes very bright, so when it becomes suddenly bright constructive difference is happening, when it has become completely dark there is a destructive difference that is happening there, but there are many other examples that we will be discussing later on not now, so this is for light, this is for the light and that is EM wave that you are going to learn in class 12th, okay so tell me what is the value of phi for minimum intensity, cos square phi by 2 should be equal to 0, so phi by 2 should be equal to what, 2n plus 1 times phi by 2, so the value of phi should be 2n plus 1 pi, so if it is an odd multiple of pi destructive interference happens and even multiple of pi constructive interference happens, understood any doubts, keep it to yourself, no doubts, see this phase difference determines everything, because of only difference between the two is the phase difference, everything else is same, so what if everything else was different, yeah that is I mean slowly and slowly we will do it, now I will change the amplitude, okay is this clear to all of you, everything is different because they interact, they interact now why not, if you force them to interact they will interact, what to do, right if I force you to interact with him even though you do not like him, you will talk right now why I hate you, he loves me okay I hate you that is the kind of interaction you will have, right, so see right now also in this room there are multiple ways, light is coming from here, here everywhere else light is coming, but the problem is that these light sources they are not having same phase difference, when k and omega are different, the difference between these two phases change with time, yes or no, when you do cos a minus b by 2 kx minus omega t disappears, when you do cos a minus b by 2 omega 1 and omega t both will be there, k1 and k2 will also be there, so with time the amplitude changes and the frequency of change is very fast, 50 times in a second, so you cannot see it, you can see only when the phase difference between the two waves is constant, so that is why we are studying this, sir can em waves interact with mechanical, no, no it is like the aliens interacting with you, good question, em wave interacting with mechanical waves, em wave can generate momentum and it can transfer energy, so in that way some interaction is possible, but that is, sir you have to do something. But I am not talking about that kind of interaction, we are talking about this kind of interaction, they will not be interference between em wave and mechanical wave, by the way since he has pointed out that velocity we have found out for the sound is gamma p by rho and p is equal to rho i t by m, you know this or not, so p by rho is i t by m, so gamma i t by m is also the speed of sound, so velocity can be written as root of gamma i t by m and hence velocity is proportional to root over temperature of the air.