 So coming back to superposition, suppose the two waves have different amplitudes, write down y1 is equal to a1 sin omega t, y2 is equal to a2 kx, kx minus omega t plus pi, okay. I will try to, see y2 you can write it as a2 sin of this cos of pi, you need to open it up plus, see otherwise you cannot deal with this. Now when you do y1 plus y2, this will come out to be a1 plus a2 cos of pi plus a2 sin of pi. Now do you remember this expression, a sin theta plus b cos theta, do you remember how to further make it compact, when you multiply and divide by root over a square plus b square, similarly over here your a is this and your b is this, getting it? So you are going to get it as root over a square plus b square sin of kx minus omega t plus theta, where this is cos and this is sin, so tan of theta is a2 sin of pi divided by a1 plus a2 cos of pi we have done something like this last class okay, then what is this? Is this the new amplitude? How much is that? Find out, substitute this as a and this as b, so resultant amplitude root over a1 square a2 square plus 2 a1 a2 cos of pi, good. So resultant amplitude square is equal to a1 square plus a2 square plus 2 a1 a2 cos of pi, can I write it in terms of intensity? Resultant amplitude square is i resultant, this is equal to i1 plus i2 plus 2 root i1 cos of pi okay, this is what will happen if their amplitudes, the waves amplitudes are different okay, fine in and out, now similar to the previous this thing, find out the maximum or maximum and minimum intensities, i1 and i2 are given to you, you can change the value of pi, then pi is 0, so what is the maximum possible intensity? When it will be maximum cos of pi will be 10, 1, so i1 plus i2 plus 2 root i1 i2, so this is the root of i1 plus root i2. So root i1 plus root i2 whole square is the maximum amplitude intensity, sorry, minimum intensity will be happening when cos of pi is minus 1, okay, so root i1 minus root i2 the whole square, so maximum intensity divided by the minimum intensity, see this is not in your school curriculum, okay, not part of your school curriculum, but whatever you have done earlier was in school curriculum, this is there in J curriculum, is root over i1 plus i2 divided by root over this whole square, now I can write it in terms of the amplitude also, so this can be written as a1 plus a2 divided by a1 minus a2 the whole square, so if the amplitudes are equal minimum intensity is 0, but if they are not equal minimum is not 0, go out, okay, so there are actually no questions in your textbook from this topic, but we will take up, let us talk about from where and how this phase difference come, because you can see everything depends on phi, what is the value of phi, okay, we take up one simple scenario, there are many more scenarios that you will be learning in wave optics in class 12th, okay, now consider that scenario, please write down two waves, two waves start with zero phase difference, two waves starts with zero phase difference, one wave travels more distance than the first wave, the second wave travels more distance than the first wave, will there be a phase difference, initially there is a scenario, okay, from here one wave goes like this, here the other wave goes, when they start there is no phase difference, but is there a guarantee that when they mean there will not be any phase difference, there is no such guarantee, see it is like this, if this is a wave and you start like this, if suppose this point meets with that point they are not in the same phase, because this wave has travelled more distance, getting it, so just because this wave has travelled an extra distance there will be a phase difference automatically generated, the reason is just because one wave has travelled an extra distance, okay, so initially the phase difference is zero, so I need to find out how to get the value of the phase difference, okay, so let us try to see that, now can you tell me that if I talk about one wavelength, let us say this is lambda, for one wavelength lambda what is the phase change, 2 pi, right, so for lambda, for lambda 2 pi is the phase, after every lambda you add 2 pi in the phase, yes or no, that is the way we have defined or we have found out the wavelength lambda, okay, for lambda distance 2 pi is the phase, so for distance of x1 the phase will be 2 pi by lambda times x1, yes or no, phase per minute lambda divided by phase per minute distance multiplied by the distance, okay, any doubt, so let us say this travels at distance of x1, so between, let us suppose this is zero phase, this is zero phase, so at this point this will be the phase of the first wave, let us say this distance is x2, so this is phi 1, so for x2 the phase difference will be 2 pi by lambda times x2, this is phi 2, so difference, the phase difference between this wave and that wave will be how much, 2 pi by lambda x2 minus x1, this is the phase difference, yes or no, right, so if this phase difference is an even multiple of pi, what will happen, constructive difference, so if this happens intensity at this location will be 4 I naught, how to find that location, the location will be such that x2 minus x1 will be n times wavelength, the difference in the path travel, if it is equal to the integer times the wavelength, there will be a constructive difference happening there, okay, now if the phase difference 2 pi by lambda x2 minus x1, this is equal to 2 n minus 1 times pi by 2 destructive difference or x2 minus x1 is let us say an odd multiple of lambda by 2, it will be a destructive interference, so I do not want to stretch it more because already we are beyond your curriculum, but I have to tell this because just telling theory is not sufficient, so how to visualize the phase difference is also important, in doubts quickly tell me, it should be lambda by 4, what, it should be lambda by 4, sorry it is odd multiple of pi for destructive, isn't it, that is what we have done it, in chemistry we had that and the waves were like this, it is constructive, but they are going opposite side, so at the end point, in chemistry about the shortages, what is the molecule of it, it is same thing, you are treating electron as a wave, particle wave, so it is saying that they are going opposite side also, it goes constructive and then it goes that way, like if they are going this and then they go construct, it goes up, they go constructive, it goes up means what, so like probability of finding electron becomes higher, if two waves constructively meet, so the amplitude increases, amplitude means the density of electron there will be higher, when they constructively meet, when they meet destructively electron vanishes because electron is a wave there, so what, it does not matter, you are talking about kx minus omega d meeting kx plus omega d, see if you take your wherever they are meeting as x equal to 0, put your origin there, so kx your x is fixed, so k into x is just another constant, it adds to the phase phi, one is phi plus omega d, other is phi minus omega d, but then if they are like that, then they will be totally out of, they are not totally out of phase, one is not, you are saying that y is equal to a sin kx naught plus omega t and y is equal to a sin kx naught minus omega t, how can they constructively interfere, that is what you are trying to say, they will constructively interfere, if the phase difference between them is this 2 and phi, now how this happens, how this phase difference comes out to be 2 and phi, if they are meeting like that, that is a different debate, but this has to be there, now the electrons meet in a different way, the way 2 electron waves they meet, sometime they create s orbital, sometime they create sp, hybridized orbital sp2, sp3 depending on how the waves are, how the electron waves are meeting and usually what happens is that standing wave get generated, then only the orbitals sp2, sp3, sp3, r2 and all that, they are actually standing waves, so one wave has to come from this side, other has to come in the opposite direction, because 2 waves when they come in the opposite direction, then only standing wave get generated, so standing wave we have not yet done here, we have only talked about the waves when they are moving in the same direction, so if they are moving in the same direction, they will create a travelling wave, not a standing wave, so let us talk about standing wave, then it will be better, fine, so please write down reflection, reflection is a very very common phenomena in the waves, so there was, once there was a dog who was walking near the fence and it was constructed this thing wall, so every time dog used to bark, the sound used to get reflected and he hears his own sound, thought that there is another dog that side, so continuously for 3-4 hours he will bark, so it can fool you at times, so we are talking about reflection of the wave, so when we talk about reflection of wave, the way the reflection happens in transverse is different from the way it happens in the longitude, not only that the kind of reflection whether it is hard reflection or soft reflection even that has a bearing of what will happen after reflection, so let us take it one by one, we are talking about reflection on a transverse wave, first is the hard reflection, basically hard reflection I will tell you how it is defined, so when a wave is travelling from mu 1 to mu 2 and mu 1 is less than mu 2, so what happens is let us say this is a string, it is tied with a heavier mass density string, so a wave get generated from here, it touches this end, so part of it will get reflected, part of it will get transmitted, so I am talking about this reflected one, so this reflected one is getting reflected from mu 2 earlier the wave was in mu 1, mu 2 is more than mu 1, so the thing is that wave is getting reflected from a denser medium, very crude way you can say that, so when the reflection happens from a denser medium it is called hard reflection, got it? In your textbook a very simple example is given in which a string is tied to a wall let us say and there is a disturbance that is sent from here, the smallest unit of disturbance is called pulse, so whatever happens to this pulse it will happen to the entire wave because it is the pulse will be attached to create an entire wave, so building block of wave is pulse, so this wave is travelling like this, so now this point is fixed on the wall, so after sometime this pulse will reach the end, now reflection will happen, so what this point 1 is trying to do, tell me next if suppose wall was not there, so this point 1 will try to do what, talking about point 1, point 1 is in the medium and try to move up, transfer wave particle move up and down, so if this disturbance have to move forward then the point should move up, but what you are doing you are not letting it move up, it is applying a force on the wall in upward direction, wall does not move because it has infinite inertia, but what wall will do equal in opposite force on the string, string does not have infinite inertia, so by the moment wall applies a force in opposite direction and inverted pulse get generated like this and it will travel backwards, everybody understood? So if incoming wave equation is this, incident wave equation is let us say A sin kx minus omega t, what will be the equation of the reflected wave, assuming there is no energy loss in the reflection, otherwise if energy loss happens amplitude will change, amplitude is same, so what will be the reflected wave equation, that is all, A sin kx plus omega t you said, so what is the difference between it getting inverted or like this only it travels back. There will be an additional phase of pi, see it is this point is suddenly replaced by that the pulse 1 2 3 after reflection 2 suddenly coincided with 1 as if the wave got shifted this way, so a phase of pi has increased, so because of reflection a phase of pi will get introduced in the hard reflection, any other doubt? So why is the minus become plus in omega t? It is travelling in opposite direction, negative x direction, positive x I am taking like that, so this is the hard reflection, let us talk about soft reflection of transverse wave, soft reflection the same scenario mu 1 is more than mu 2 now, the wave was there in the denser row and it is getting reflected from the rarer medium, so basically reflection from the rarer medium, so in your textbook a simple example is given, there is a ring attached to one end of the string and it is passing through a pole like that, this ring is mass less, now this pulse is travelling, now what will happen if this pulse reaches here? It will pull the ring up, ring will go up or not, ring will go up, so ring goes up with the pulse it goes up and then it will come down also, so it goes down the rope has little bit of inertia, so it will not stop there it will go down, so as if somebody has taken this end and then like that, up and down like that, so because of that up and down movement itself there will be a reflection coming out like that, but now it will not be inverted, it will be exactly this way only, but just that velocity is changed, so if incident wave is a sin kx minus omega t, the reflected will be a sin kx plus omega t, so these are the two scenarios of the reflection for the transverse wave, please write down the longitudinal wave, we have one minute we can finish that, or we can do the reflection, let us not finish quick, fine so this is what we have done today, next class we are going to do the reflection of the longitudinal wave and then we are going to talk about the standing waves, after that we are going to talk about the top loss effect, next entire class will be on this chapter only, after that we will be doing some problem practice, and that your syllabus is over, once your syllabus gets over, we can do some problem practice on some specific chapters, we can revise in fact, we have six classes to revise entire class 11, or we can do optics, we will discuss after finishing.