 So write down all of you the principle of superposition of the wave. So like I told you at the start superposition is what when two waves meet okay so we need to understand what will happen if two waves meet fine. So it is a straight forward thing so if let's say y1 is equal to ff1 x,t and y2 is equal to f2 x,t these two waves if they meet they will give you y is equal to y1 plus y2 as simple as that you need to just algebraically add them up fine nothing else you have to do is add them. So feel free to use sin a plus sin b is equal to 2 sin a plus b by 2 cos a minus b by 2 so all those things are valid okay so feel free to do all of that okay. So basically this is what when two waves meet will happen now since we are only focusing on the sinusoidal wave primarily so let us discuss what exactly will happen with the sinusoidal wave okay so we will take a special kind of situation the assumptions or the situation is this the situation is that we are taking two waves having same frequency same wavelength so we will take two cases inside this situation case number one amplitude is same for both and there is a phase difference I will discuss what is the phase difference and case number two amplitudes are different and so are phase, phases also are different from each other okay so first we will take a simple scenario case number one okay so how we can write down two equations for two waves that are meeting I can write it as let us take amplitude as small a small a sin of kx minus omega t one of the wave second wave is let us say this a sin of kx minus omega t why because frequency and wavelength are same but there is a phase difference so I will add phi to it okay so phase difference is what between these two phase difference is phi so phase difference phi does not depend upon time it is fixed okay this scenario is also called superposition of two coherent waves what are coherent waves for which the phase difference does not depend on time it will happen only when frequencies are equal if frequencies are not equal let us say if you have kx minus omega one t one phase minus if you have omega 2t plus phi you can see the phase difference depends on time okay phase difference depends on time it will be independent of time only when the two frequencies are equal okay but anyways take these two waves and superpose them these two waves are meeting each other so all of you tell me what you get y as equal to have you simplified it you can use sine a plus sine b formula let me know once you're done type in once you're done just type in that it's done sine a plus sine b is 2 sine a plus b by 2 cos a minus b by 2 use that are you getting this to a cos phi by 2 into sine of kx minus omega t plus phi by 2 how many we're getting this so now does this look like a wave equation is this a wave equation yes or no if this is the wave equation what is the amplitude what is the amplitude of the wave good so new amplitude of the wave is 2 a cos of phi by 2 now one thing you need to see here amplitude depends on phi okay what if is equal to 0 what if it is equal to 0 for some value of phi what does it mean if this is equal to 0 if amplitude becomes 0 for some value of phi there will be some phase difference between the two waves phi for which 2 a cos phi by 2 is 0 let's say phi is pi phi is let's say 180 degree so these two waves will meet and it will become 0 what does it mean it means that the two waves when they meet they destroy each other okay so let's say this is one of the waves when this wave meet this one when they superpose they will yield nothing this is what it means this never happens with the masses when two masses meet but it can happen that one wave meet other wave and nothing comes out okay for what values of phi this will be equal to 0 for the values for which cos of phi by 2 is equal to 0 so phi by 2 should be equal to what 2n plus 1 pi by 2 for an odd multiple of pi by 2 cos of phi by 2 will become 0 okay so phi should be equal to 2n plus 1 times pi n can be anything n is an integer n belongs to integer it can be 0 1 2 3 minus 1 minus 2 could be anything if phi is equal to this then these two waves will meet and they will destroy each other are you getting it this is called destructive interference destructive interference or you can say destructive superposition anything all right they are not oppositely moving they're not oppositely moving you can see that this wave is moving in positive x direction kx minus omega t this also moves in the positive x direction will they destroy each other at any value no just now I told you for this value of phi if this phi is equal to pi or 2 pi not not 2 pi sorry pi an odd multiple of pi pi 3 pi 5 pi 7 pi 9 pi like that if phi is that then these two waves will meet and destroy each other is it clear now how it comes the amplitude you equate it to 0 and see what is the value of phi for which amplitude will become 0 and you can see that this is the value of phi that comes up anyone has any other doubt clear now let us see case number two the last thing that we are doing today is case number two what was the case number two amplitudes are different and there's a phase difference so y is equal to y1 is equal to a1 sin of kx minus omega t and y2 this is not from your m crt this is this you can treat like a numerical okay rather than a derivation because it is beyond n crt this is a situation okay now I want to add them up okay so y is equal to y1 sorry plus y2 is trigonometry done in mathematics for you trigonometry is done okay great so it's here is one of the trigonometry question which you have to solve so how will you add these two trigonometric equation so that I want to add these two trigonometric function so that it looks like a wave equation I want it to look like wave equation like a sin kx minus omega t plus phi there are two sin functions I want only one look like a wave function wave equation try doing it spend one or two minutes it'll be interesting trigonometry question should I give you a hint should I give you a hint the hint is expand this expand this as sin of a plus b a is kx minus omega t and b is phi so let me expand it like all of you are stuck anyone got any this thing sin of a plus b sin a cos b plus cos a sin b okay so cos of kx minus omega t sin of phi now what to do what do you think we should do next why I have opened it like this what do you think good so you can take sin of kx minus omega t common you will have a1 plus a2 cos of phi plus you can take cos of kx minus omega t and you have a2 sin of phi common like this okay so you can see that it looks like a sin of kx minus omega t plus b cos of kx minus omega t capital a is this capital b is this now have you learned how to simplify this as a single trigonometric function a sin theta plus b cos theta how you simplify it or how you put it as a single trigonometry function do you know this what to do next tell me ideas I mean you could be wrong if I were in your place I would have been definitely wrong because it's not something which you can guess directly so at least you should think and answer something okay die three got it sin of phi okay what you do in these cases this is this is something which is very common okay so you need to learn this trick you if you are doing it for the first time it's okay but you should remember this trick for such situations in which you get this format a sin theta plus b cos theta whenever you get something like this and you want a single trigonometric function you can multiply and divide by root over a square plus b square it's a common trick don't worry now you know it root over a square plus b square sin of by the way can you guess why I have done like this at least that you can guess why I have multiplied divided by root over a square plus b square any guesses on that all of you why I've written like this I can take cos theta I can say that it is equal to this why I've divided by root over a square plus b square because this thing can never be more than one so I can say this is cos theta if this is cos theta what is this sin theta okay so it becomes what sin a cos theta plus cos a sin theta so it will become root over a square plus b square sin of kx minus omega t plus theta where theta is what theta is such that tan theta is b by a b by a what was b a2 sin of phi and a was a1 plus a2 cos of phi does it look look like a wave equation everyone is it a wave equation it's a wave equation okay and the amplitude is what amplitude is root over a square plus b square which is a1 plus a2 cos of phi whole square plus a2 sin of phi whole square so when you simplify it you will get under root of a1 square plus a2 square plus 2 a1 a2 cos of phi when you look at this what comes in your mind this is the amplitude if amplitudes are different vector addition come in your mind or not vector addition okay so this is what happens if two waves of different amplitude they meet their amplitude will result to this much now looking at this expression I want you to tell me what is the maximum amplitude and what is the minimum amplitude everyone maximum and minimum you can change phi maximum is when cos of phi is 1 so a max is root over a1 square plus a2 square plus 2 a1 a2 this will be root over a1 plus a2 whole square that will be a1 plus a2 this is maximum amplitude minimum is when what should be the value of cos of phi for minimum cos of phi should be minus 1 all right so a minimum is equal to root over a1 square plus a2 square minus of 2 a1 a2 which is what root over a1 minus a2 the whole square which is a1 minus a2 mod so this is minimum this is maximum okay so I have two minutes tell me anyone of you have any doubts from anywhere anyone any doubt from anywhere whatever we have done today please ask doubts I am not going to leave you early anyways how is it a vector sum see it is not a vector sum it looks like a vector sum okay vector sum when you add two vectors there some of the vector is isn't it a square plus b square plus 2 a b cos theta so it looks something like that this is what I was saying any of the doubt how a square was cos theta another expression becomes sin theta and how we get tan theta okay see I am assuming that if this is cos theta this is an assumption okay or this is an introduction I am saying let us assume this is cos theta so sin theta will be this how because sin theta is root over 1 minus cos square theta you can check that or you can see one more thing this is square plus that square is 1 so if this is cos theta this will be sin theta okay this you can do it your own sin theta is 1 minus cos square theta put cos theta is equal to this sin theta will come out to be this okay now tan theta how it is this you can divide sin theta and cos theta you will get tan theta equals to b by a and what is b b is this and what is a is that understood is it assume for some value of theta that is not decided yes I can assume it is some some theta for which tan theta is b by a okay why does it look like a vector see you may not be aware of something called phasor these are the rotating vectors okay which I don't want to talk right now I don't want to confuse you so we will discuss it in 12th this comes in 12th okay phasor right now you just assume that yes it looks like it looks like this because it comes out to be like that okay that's it any doubt alright guys so I am going to send you another assignment make sure you are struggle with the assignment and solve it this is the last this thing chapter make sure you do class 11 physics very very properly you know there are two reasons to it because if you do 11th physics properly 12th will be a cakewalk 12th is a very simple physics curriculum we will complete the physics curriculum by October first week itself next year and then we can get it down get down to the revision of things by problem solving so those who are doing it properly diligently will see that rapidly they will you know increase their capability at the end because everything will fall in place at the end all right okay so that's it from my side we will meet next week and next week we are going to solve a lot of problems on this chapter