 All right, okay. So in Shaduli, you had any exam in NPS? And Prasim, no exams. UT is going on, is it? Oh, 13 last year, that's what I was saying. I thought 30th is the last year. Okay, so still not many have joined in. I hope all of you are aware that the KVPY is postponed. Are you guys aware about that or not? KVPY is not on the coming Sunday, all right? All right, so I guess we can start now. Others will join meanwhile, fine. So last session, what we did was we completed the chapter oscillations, okay? All right, somebody's saying Olympiats also got postponed. Yeah, these Corona God knows when it will get over. It is making our life tough, okay? And most probably the Omicron will be gone in a month's time and hopefully no other variant come. Otherwise, maybe 12th also will become like 11th. Then it'll be continuously three years, okay? But anyways, so I hope all of you are vaccinated, right? Vaccination is done in the school nowadays. Not yet, done. Tejasani is done, Siddharth is also done, good. Get yourself vaccinated, all right? It is important for your own good and then you can be a little bit relaxed about Corona. Anyways, so in the last session, we completed the chapter oscillations, fine? So in the oscillations chapter, we had taken two sessions, all right? And now in the today's session, we are going to start the last chapter of physics, which is waves, fine? So in your school, I think in NPSHSR, the syllabus is already over, right? And NPS Kormangala, what is going on? In NPS Kormangala, which chapter is going on? Shaduli, what do you mean by, I think, waves only? You don't attend the classes, is it? Okay, so oscillation is almost done. So it is waves only, which is happening in your school. So good that we are completely in sync with your school. And just that NPSHSR has run up a little faster. But most of the schools, they are doing this chapter called waves, all right? So now, first of all, we need to understand that this is a standalone topic or standalone chapter in grade 11th, which is completely different from all the chapters which has come before this, okay? Because before this chapter waves, all the chapters were talking about study of matter. How the mass moves, what is the property of a matter? Okay, only these two things were focused. In thermodynamics also, thermal properties of matter only was there, okay? In mechanics, it is all about how the mass is moving. Even the oscillation, the previous chapter to the waves, even there our focus was how mass is moving. This is the only chapter in grade 11 in which we do not talk about the movement of the mass as such, okay? This is not mass, this is something else, okay? This is waves. So everything in this world, in this universe will either behave like a particle or a mass or it can behave like a wave, okay? We'll discuss what is a wave, but right now I'm just setting the context here. So have you heard about dual nature, duality and stuff like that? Have you heard this phrase dual nature of light and, right? So dual nature means what? Dual nature means that the same thing can behave like a mass or a particle and the same thing can behave like a wave, okay? So depending on the situation, something can behave like a particle, depending on the situation, same thing can behave like a wave also. Now, it may sound very strange to you on the face of it, but just look at yourself. For example, let's say you are with your parents. You behave in one way, when you are with your friends, you behave in another way, right? When you're happy, you behave in one way. When you're angry or sad, you behave in a different manner, right? So we humans have 100 types of behavior. Similarly, the matter also can behave at least two ways, particle, way the particle is and the way the wave says, fine? So what I'm trying to tell you is that wave and particle, they are two different universes in the physics. You can study the same thing as a particle and same thing can be studied as a wave, okay? In this chapter, we are going to exclusively focus on waves, the property of the waves and how we can analyze the wave motion, okay? And understand how it moves, when it moves, what is the velocity of the wave and stuff like that. So you can see in front of you, this is a wave, all right? There is a water body, on the water body, I have just put a stone, right? And the waves have been generated, all right? So first let us try to understand what is a wave and then our discussion can move forward. So write down waves, all right? So can you tell me or let's say, what do you understand by waves? Can you tell me your understanding first? Don't worry about right or wrong. Try to tell me what is a wave. Anyone, what is a wave, okay? Good, others, all right? Everyone, okay, you may not like to define what is a wave. Tell me some examples of the waves. Give me some examples that this is a wave, this is the wave and stuff like that, okay? I'll just write down one by one. Somebody is saying the sea wave, then ripples on the water, then sound waves, okay? Microwave, all right? So EM wave, microwave is an EM wave, all right? Sunray is again electromagnetic wave only, fine? These are some of the waves. Radio wave is also electromagnetic wave. It is an electromagnetic wave, all right? Fine, so we will try to understand little bit about the waves by taking an example of ripples on the water because this is the only example where you can actually visualize better. You can't see the EM wave, you can't see the sound wave, sea wave, I mean, of course you can visualize a little bit but ripples on the water is the best possible one which we can understand in greater detail to get a greater insight about the waves, all right? So this thing, the front page itself is the example of ripples on the water, right? So what has happened here? Something has happened or not? How you create a water ripple, everyone? You might have thrown the stone over here. When you throw the stone, there will be some disturbance that has been created over here, okay? When you throw the stone here, exactly at that moment, does the particle over here know anything about you throwing stone there? The particle has no idea that you have thrown stone at the center but still after some time, even this particle start some sort of movement, right? You can see that the circle has reached here, okay? So what is happening is that when you see the water ripple, when you see the water ripple, you may feel like that the water is moving like this and this thing is traveling. Some sort of disturbance is moving away from the disturbance, fine? Now, have you seen Diwali lights in which a lot of lights are there and these lights, they only do on and off. When the lights are doing on and off, you feel as if light is traveling from one place to the other place. Have you seen such kind of thing? Everyone, right? So exactly that is happening over here. Do you think that the water is going this way? Do you think that water is traveling like this? Everyone, what do you think? Is water traveling like that? If water start traveling like this, if you throw a stone here, slowly entire water will just reside, slowly and slowly, entire water will just go away, but that doesn't happen. So it is not that water is going forward. It, you feel as if water is moving forward, just like the Diwali lights. What water is doing is, what the water is doing is that it is only moving up and down, okay? So these particles, these particles which are there on this circle, these particles which are there on this circle. So what happens is that these things, these entire particle on the circle, they'll together move up, they'll together go down, okay? But one circle when it moves up, other goes down. So like this, they continue and you feel as if the water is moving, but the reality is it is not the water that is moving. It is the disturbance that is moving, okay? The circular disturbance is spreading. Water is there only, okay? So if somebody asks you what is that something which is moving like this, that is not water, that is a disturbance. Of course, it is moving with the help of water. If water would not have been there, the disturbance would not have moved forward. So it is like, you know, when you travel, you're using vehicle, you're using car to travel. So similarly, the wave is using water to travel, all right? So I hope this thing is clear, all right? What is clear? That the wave is nothing but a disturbance that is created at one location and that travels from that location to another location. And in this case, it may use a medium. Use a medium. Shabili, you note down the, make your notes, okay? We'll discuss it, don't worry. It may use medium like water to move. The wave uses medium to move forward, but the medium doesn't travel along the disturbance. It is not the medium that is traveling along the disturbance, although medium is moving. It is not that the medium doesn't move at all. Look at this water, what is moving up and down like this but the wave is going like this. Disturbance is going perpendicular to the movement of the water. So I'm just saying that the water doesn't travel with the wave, okay? So now why are we studying this chapter? Once we are clear that what is a wave, then the next question that should come in your mind that why we are studying it? The first answer to that is because it is in your curriculum and they're going to ask questions from it. But beyond that, if you ask that, okay, fine, that is okay, that we are getting marks, but what else? The other very important aspect of the wave is that you cannot imagine your day-to-day life without waves. Okay, just look at the kind of examples that you have given. Someone has given example of microwave. So if you do not know anything about the waves, you can never create microwaves. If you do not know about the waves, you cannot create antennas to catch any kind of information from the style. Let's say you are catching the television signal. What is that? It is a wave only, isn't it? When you're sitting here attending the class, your laptop is catching the wave from the router. The router is emitting the radio waves, your laptop catches the radio waves, and then only you are able to create, you are able to have internet and you are able to attend the class. So like this, there are so many examples. Your mobile phone cannot run if we do not understand about the wave. And the primitive example, if you talk about something which is very, very important, the sound. Sound is a wave. Without knowing that sound is a wave, you cannot do anything with it. So it is a very important aspect of our life that the wave is around us everywhere. So when I say wave is a disturbance, disturbance doesn't mean that it is something negative and you don't want it. Don't attach your emotions with it. Disturbance can be a very nice looking sine curve also. Disturbance simply means that something which was not there earlier, suddenly it has come up. Like water was like this, plane and symbol, you thrown stone, you disturbed it. So disturbance can be something which is very important to us. Disturbance can carry energy in which there are information. So what is Wi-Fi? Wi-Fi is having a lot of information about the internet. So that is what it is. So everything that is wireless, you cannot imagine without a wave. And that's the reason why it is there in our curriculum. So with a wave, you can transport energy from one place to another without moving the mass to that place. So it is a wireless means of transporting the energy. So the energy can be in any shape. It can be electrical energy. It can be magnetic energy. So I hope things are clear. So now once we are very clear about what is wave, how important it is, we need to now understand one thing that wave is an ocean. What I mean to say is that wave has lot many details into it. And I hope all of you know that the electron has a dual nature. And when it comes to chemical bonding, the electrons wave natures matter more. Because you're talking about orbitals, you're talking about hybridization. Hybridization is what? SP3, hybridization is what? The S wave is mixing up with P waves and creating SP3 orbitals or SP2 orbital and stuff like that. So that is the reason why we have already, have accepted that the wave is there in every aspect of our life. So there are different kinds of waves that are there. So it is better to classify them. For example, when you study the matter in chemistry or elements in chemistry, what is the first thing you do? You classify them. There are metals, nonmetal, group 1A, group 2A, 2B, 3A, 3B, like that. You create groups that is called periodic table. Similarly, when we are studying, let's say waves, we want to classify them, then only it makes sense to study them one by one. And when we talk about classification itself, classification can be based out of anything, any logical thing. For example, let's say if I classify your classroom on basis of height, someone who is less than 5 feet 5 inch and more than 5 feet 5 inch, one type of classification. Another classification could be based on gender, male and female classification. So let us try to see what are the classification that can be done with the waves, which are logical. So first type of classification is classification based on how wave travels. Okay. Wave that requires medium to move. Okay. Then there is this wave that doesn't require medium to move. All right. Okay. Like I think I've already discussed the wave that doesn't require medium to move is electromagnetic waves. Like X-rays, sunlight, radio waves and all those kind of electromagnetic radiations. They're all EM waves. Okay. Then the wave that requires medium to travel. For example, sound waves. Okay. They are the mechanical waves. Okay. So the mechanical waves, they require some sort of medium to travel. If medium is not present, they cannot move. Okay. So like I told you, sound requires medium to travel. Right now, what is the medium between like if I'm talking to someone between me and that other person, the medium is air, atmospheric air. All right. So when I speak, I create disturbance near my mouth. Okay. And that disturbance oscillates by using the air molecules and that reaches the ears. If the air is not present, no matter what I do, you will not able to hear anything. I also may not be able to hear my own voice because what is sound sound is disturbance to the air. Okay. And you might have heard about people changing the medium and their sound completely changing. Have you heard about someone inhaling helium gas and then speaking stuff like that? Have you heard about something? Right. I'll just show you if you change the medium, then how the sound can change. Okay. So I think I showed it to the last batch. William, look at this. This is the air. Down the balloon and you'll sound like Daffy Daffy every time. My cracker is my yard. Pick a ball. Okay. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. Focus. you taking the example of the water ripples that the medium what it is doing it is moving up and down so it is oscillating okay so there can be classification based on that how it is moving medium all right and by the way I forgot to mention here that the study of EM wave will be done in grade 12 good news the bad news is that mechanical wave is there in grade 11 okay so in this chapter our only focus is mechanical waves but since we are talking about the waves for the first time we are going to discuss basics about the waves which is applicable everywhere even in the EM wave that will be applicable okay only the basics all right so classification based on how the wave or how the medium oscillates so this wave you can classify them as transverse waves have you heard of transverse and longitudinal but ever before it was there in grade 9th right I think yes transverse and longitudinal in the transverse wave the medium is oscillating perpendicular to the velocity of the wave okay in longitudinal the medium is oscillating parallel to the velocity of the wave so the best possible example of a longitudinal wave is sound and transverse wave like water ripple is there uh the the wave on a string is there all right I'll just show you one example of longitudinal which you can see sound is a longitudinal wave but you can't see the sound wave right so let me just show you look at this hope ad will not come look at this this is the disturbance this disturbance is moving forward okay all right so this is the longitudinal wave you might be wondering that what is the use of this thing I mean this is just an example just something so that you can notice that such you know that longitudinal wave exists the longitudinal wave on slinky may not be useful but longitudinal wave as sound once you understand how it works you can see from your eyes now you can study the sound waves in a better manner so that is the that is the idea okay so these are the classifications of the waves now coming to the chapter in this chapter what are the things that we are going to study so let us talk about that also right so you know that in oscillations we had what in oscillation we have divided the chapter like kinematics of the SHM or kinematics of the oscillations then we talked about the causes of the oscillation then we talked about the energy of the oscillation or non-ideal scenarios of the oscillations right so something like that we can classify for this chapter also so write down waves waves is slightly bigger chapter than the oscillation see I have come to know that many of the schools nowadays they are skipping lot of topics at least in the grade 11 okay so of course that it may be comfortable for you writing your school exams but then that will affect you badly when you write the computer exams so make sure that even if let's say a topic is skipped in the school exams so you study when you study for the school exam you don't study that but don't skip that topic completely okay because when it comes to an exam like J, KVPY or CET, Bitside, VIP, whatever exam you write later on all these things will come okay they are not skipped from every exam just because they are skipped in your school exams right so the waves you can classify it into four different sections okay the first section is pretty obvious kinematics of the SHM not SHM I think kinematics of the waves under this topic we are just going to create some sort of mathematical formula for the wave saying that this is a wave equation and after that we will analyze it okay given this is the wave equation what will be the wavelength frequency speed and stuff like that but we never will get into that what is causing that speed from how the wave is getting generated okay we are assuming that wave already exists just like any other kinematics chapter we are not getting into why this is happening we are just assuming that this is happening let us study it like that okay so the next thing is velocity of the wave now the speed of the wave or velocity of the wave depends on the medium like for example when you travel in the car it is the car that determines how fast you are moving simply here also the medium determines how fast you are moving okay so we are going to get into that what are those properties of the medium that make sure that velocity of the wave is this much and we will try to create some equations from which we can find out the velocity of the wave okay then this section the third section is a little bigger third section is the super position waves so you must understand that when we were studying the mass we had how many chapters we had like kinematics motion 1d motion in 2d then laws of motion work by energy rigid body motion okay oscillation so many chapters gravitation but entire study of the wave is done only in one chapter okay so here you will learn everything in one chapter itself right but not as much detail manner as you have done the study of mass okay but anyways you will see many things are there in the single chapter so in when we studied the mass the particles we also studied what will happen if let's say two masses will come and hit each other collision happen then what will come out of it so we learned about the conservation of momentum and coefficient of restitution and all those things we have understood and that is the way we have analyzed it now under the super position of the waves exactly same thing is happening one wave coming from here another coming from here they will meet so once two waves have met what will happen you can't use conventional momentum because mass is nothing mass is not there it is just that one disturbance from here under disturbance from here they meet so what happens to the disturbance that is what we are going to find out and unlike the mass what happens after these two disturbances meet depends on how the wave was traveling before the disturbance have met okay so there are three aspects okay what will happen when the two waves that are traveling in the same direction meet traveling in same direction when they meet what will happen when two waves they travel in the opposite direction and meet okay this first and second in these two cases we assume one thing that the frequency and wavelengths are same okay in the third aspect of this section we are going to talk about what will happen when the two waves they are traveling with different frequencies when they will meet okay so that is what we are going to study in the first case I'll just quickly tell you first case you'll get another wave which is traveling second case you'll get standing wave third case you'll get beats beats is a phenomena that you will see later on when we study and the last part of the chapter talks about Doppler's effect so I hope all of you were there I mean all of you were at some point in your life where they're in the railway station and have you ever observed that when train is approaching to you with a siren that is on the sound of the train is the sound of the siren is one type and when it passes you and goes away from you then somehow the siren sound appears to have changed have you observed that or not when train is approaching and it passes you two different kind of noise you see from the siren or not right so what happens is that what happens is that when when someone who is emitting this sound be train or anything when that something is moving then the person let's say I'm hearing that sound then I will not hear the same frequency whatever the train is emitting because train is moving while it is emitting that frequency I will feel some other frequency so the frequency what I feel depends on how you are moving and in which direction you are moving at the same time the frequency also changes if train is stationary but I start to move okay so there is an apparent change in the frequency and this thing is used by the radar and sonar okay the radar and sonar they are the two equipment that uses the Doppler's effect to find out the velocity and the direction of the movement of the enemy aircraft or whatever it is fine so without the understanding the waves you cannot hope to create radar or sonar also fine so this is the overall construct of the entire chapter now let us talk about the waves okay let us start the chapter wave that is what I mean okay first thing is what kinematics of the wave now if I have to study kinematics of the wave I need to know equation of the wave some sort of equation right you remember in oscillations how we started the kinematics of the oscillation we have we we had observed something and we have created an equation of the oscillation that a is equal to minus omega square x raise to power n you remember that n belongs to odd integer and when n is equal to 1 it is shm equation all right so something like that we have to observe for the wave come up with some sort of equation and then only our study can start god hasn't come and told us that listen this is the equation of the wave we we have to create equations of the wave right so let us observe few things about the wave and then we can proceed so let's say this is the waves so what is this it is a one-dimensional wave this is a y-axis this is the x-axis and this is a disturbance that is going in that direction with certain velocity v you can say that it is a string wave on a string so the string has taken this shape and this is a wave and this wave is moving forward okay so this is how it is all right after time t okay this is at t equal to 0 after time t the entire wave as it is will move forward the disturbance has moved forward so i hope you are getting this the same disturbance i'm talking about the idealistic case of course in real world when the wave is moving forward the amplitude will decrease and it will die down but in ideal world exactly as it is the wave will move forward this is all of you understood or not this is the wave at t equal to t this is the wave at t equal to 0 seconds everyone type it is it clear now if if i have to tell you the equation of the wave if i have to tell you the equation of wave then i should tell you an equation i should tell you one equation which will help you to find out the shape of the wave at any time right so how will you define the shape of the wave you if i tell you that i want what is a y coordinate if i ask you what is the shape of the wave you should be able to tell me the y coordinate of any point which is located at a x coordinate x at a time t if you can tell me a function of x and t okay what will be the y coordinate where the particle will be then you have already given the wave equation okay so this is what we are trying to achieve we are trying to find out what is that function which is a function of x and t that will give me the y coordinate all right and right now i am focusing only on the one dimensional wave there can be many kinds of wave for example this one here the this wave is two dimensional it is spreading in 2d okay when i'm speaking when i'm speaking whatever i'm saying it is spreading everywhere it is going top it is going front it is going back and forth everywhere it is going so the sound is a three dimensional wave similarly when we we have a string and we move it up and down the wave that get generated it moves along the string in that straight line so it is a one dimensional wave okay so just to keep my life simple i am assuming one dimensional wave to study first okay this is an example of one d wave like on a string this is the wave that is moving now will any function of x versus t will be the wave equation the answer is no it has to be some combination of x and t okay will only sinusoidal wave possible no the disturbance can be of any shape maybe this is the shape of the disturbance this disturbance is moving forward like that is this sine function no cosine function no it is some random thing that is moving forward so disturbance can take any shape need not be sinusoidal only but yes there has to be some combination of x and t let us try to correct that so i am going to show you something let's say if i tell you that this is the x coordinate of this point this point x comma t okay this is this is the point let's say point a can you tell me at t equal to t its x coordinate is x all right add t equal to zero what will be its x coordinate what will be its x coordinate at t equal to zero the wave is telling the velocity v every point has moved a distance of velocity to time okay what will be the what was the x coordinate of that point on the wave at t equal to zero at t equal to t its x coordinate was x so don't you think that it was somewhere here and this distance this point which was here let's say a dash has traveled a distance of v into t all of you agree type in entire wave has traveled v into t even this distance is v into t from here to here that distance is v into t all the points have traveled v into t distance okay so the x coordinate of this point at t equal to zero was x minus v t so it has traveled v t distance forward so to take it backwards by v t at t equal to zero that was its location okay so the y which is a function of x comma t if i substitute x as x minus v t will i get the same y or not this y coordinate will be same right so at t equal to zero x coordinate was x minus v t t equal to zero so the function x comma t should be a function of x minus v t okay so any mathematical function which is a function of x minus v t is a wave equation for example if i say y is equal to e to the power x minus two t divided by x minus two t whole square this is the function of x minus v t the velocity of the wave is two but if you write y is equal to e to the power x minus two t divided by x cube is this a function of x minus v t or not second one is it a function of x minus v t second one it is not there has to be x minus two t in the denominator also so it cannot be any function of x and t it has to be a combination of x and t in such a manner that x minus v t should appear together as a single variable fine so when you make the coefficient of x as one the coefficient of t is the velocity of the wave i hope this is clear to everyone anyone has any doubts i'm discussing in a little bit more depth about the waves these things are not discussed in the school or anywhere else so but then this you can understand okay e to the power x minus v t i wrote you can write anything right so this is the wave here we assume that the wave is traveling in the positive x direction if the wave is traveling in negative x direction it will be a function of x plus v t wave traveling in negative x direction if it is x plus v t function okay so this is the basic principle behind the equation of the wave and luckily we have in our grade 11 curriculum only the sinusoidal wave in detail okay so i'm now going to shift our attention towards the sine wave equation but you must be very much sure that only sinusoidal wave need not be there it can be taking any shape of the wave need not be sinusoidal so even this is a wave equation only okay sinusoidal wave is a special form of the equation the wave that's it so all of you let us talk about sine wave equation this will make your life a little simpler all right so what is sinusoidal wave sinusoidal wave is the wave that follows sine or cosine curve while moving okay and in our textbook they have assumed that this we can treat as a sine wave a sine kx minus omega t plus phi okay how is different from the sine function of that shm equation you remember shm equation was a sine omega t plus phi in shm equation this kx term was not there that is the only difference okay all right so this is the sine wave and if this is a sine wave equation can you tell me what is the velocity of the wave to find out the velocity of the wave you need to find out how it is a function of x minus vt try to tell me what is the velocity of a wave if sinusoidal wave equation is given in front of you so on no one make the coefficient of x as one the coefficient of t will be the velocity so you can write it like this a sine take k common will become x minus omega by kt plus phi so you can see that it looks like a function of x minus vt isn't it so the velocity of the wave becomes omega by k is this clear to everyone okay okay in this equation a is referred as amplitude of the wave okay and sir how did sine equation come sine equation we are assuming we are saying that let us study sine wave equation it can be anything okay even tan x minus omega t whole cube divided by cos cube x minus omega t even this is a wave equation you can study anything but we are choosing to study sine wave is it clear it can be any function as long as this x minus vt is there and you will understand one more thing soon let me first write down few things this kx minus omega t plus phi this entire thing inside the sine this is called as phase of the wave the phase a is the amplitude kx minus omega t plus phi is the phase okay now tell me in this wave equation in this wave equation if i put some value of x let's say i'm just looking at what is happening at x equal to five meters x is not a variable i'm putting x as some number two two meters five meter whatever it is then does this equation becomes shm equation or not if this thing becomes a constant if i put x equal to two meters will it become shm equation or not it becomes shm equation because shm equation is what omega t plus phi so phi would be phi plus kx now new phi if x is a constant if you're not changing x all right so that is one more reason why we are studying the sinusoidal wave so that you can correlate with oscillations all right so let us study it a little bit more all of you write down y is equal to a sine kx minus omega t plus phi okay now tell me if you differentiate y with respect to time at a particular location of x let's say x you're keeping fixed okay what do you get what is dy by dt what do you think well also the wave is omega by k so time derivative y is y is what who is moving in the y direction who is moving in the y direction wave is moving in the x direction along the y direction what we move others correct the particles they are moving okay particles are moving in the y direction so when you differentiate y with respect to time you will get velocity of the particle you will not get velocity of the wave velocity of the wave is omega by k you already found that so dy by dt is velocity of the particle this is equal to minus of a omega cos of this x i'm treating constant because i'm differentiating at a fixed location of x okay so vp is equal to now look at this a omega root over 1 minus sine square plus of omega root over a square i'm taking a inside so this thing is y square right that thing is y square y is a sine kx minus omega t the velocity of the particle is minus of omega root over a square minus y square now does it remind you anything what does it remind does it look familiar this equation velocity of the form velocity formula in shm it is a velocity formula in the shm right don't worry about the minus sign it is just telling the direction nothing else okay so the particles are doing shm you can prove it like this also so this is the shm shm with y is equal to zero the mean position okay and the maximum possible velocity of the particle is at the mean position which is y is equal to zero that is omega times a fine so now i think it is even more clear why we are studying the sinusoidal wave so that whatever you're studying the oscillation you can use it again okay particles are moving up and down they are doing simple harmonic motion and the wave sinusoidal wave is traveling or in another way manner you can say that if if particles are doing simple harmonic motion then the wave is sinusoidal wave okay now let me quickly talk about few more things about this equation y is equal to a sign kx minus omega t plus five right so when you're differentiating it with respect to time keeping x constant you get minus a omega cos of kx minus omega t plus five and when you're differentiating this with respect to x keeping t constant keeping t constant means what at a particular moment you have taken a snapshot of the wave you're not letting the wave change with the time you've just taken a snap of the wave which was like that and that is like keeping t constant and then you're differentiating that curve with respect to x what you will get when you do do it divide by dx what you get for a curve you get slope all of you right divide by dx is the slope at some location so it will be ak cos of kx minus omega t plus five here you kept x constant there you kept t constant so this is what velocity of the particle this is what the slope so when you divide it you will get velocity of the particle divided by the slope at any moment this is equal to minus of omega by k minus of omega by k is the velocity of the wave so the velocity of the particle is equal to is equal to minus of slope times velocity of the waves so this relation if this relation holds good it is a sinusoidal wave and you can use this relation in maybe some graphical questions for example if i tell you that this is the sine wave and if i tell you the velocity of the wave is three meter per second and this particle if you draw the tangent over here this angle is five if i tell you that and then if i ask you what is the velocity of that point p another velocity of particle so it will be simply equal to minus of three tan of five tan of five is a slope of this point divide by dx so like that you can you know maybe if you encounter such numericals you can do that fine so what are we doing we are just trying to understand this equation more and more okay this is not part of theory we are just discussing that okay this is a wave equation then what all things we can keep in our mind when we solve numericals so these are the few things there are many more things we'll discuss it let me show you one more thing let's say this is the wave okay now this wave has moved forward like that okay you need to tell me i'll just mark few points here so i'll mark few points over here point one two three these are the points which are there on the wave at t equal to zero t equal to zero one two three four five six so after this wave has moved forward the white line the white curve became the yellow curve right so at t equal to t it has moved forward in a yellow curve if you tell me where are these points one two three four five six where are they on the yellow line or yellow curve what happened to those points let me know once you're done we redirect you forward on the yellow curve so are you telling me are you telling me that this point is moving forward is this point moving forward i think you are confusing with the previous example where i have told you this thing right here i asked you what about the disturbance you're looking at you're not looking at the points of the medium you're in this equation of wave you're just looking at the wave this point on the wave where it was over here i am asking you about the particles not the wave wave has moved forward that is all fine i am asking you that this point let's say this is a string on this string the wave is there okay wave has moved forward let's say like this so are you saying that the points on the string they are moving forward no right it is not that all the points have moved forward now tell me everyone nobody else till now okay so like what we discussed the points are moving up and down right so point number one will come down there is no other location it has to be on the yellow line point number two goes up point number three goes up four goes down five goes down six moves up so like that they are moving moving one comes here one dash two goes there two dash three reaches three dash four goes four dash five is five dash six is six dash so that is what points are doing they're moving up and down they make you feel as a wave is traveling forward is this clear to all of you typing quick okay now tell me which are those two points which are moving similar uh similar kind of movement where are you seeing actually there there is there is there are two points which are similar to each other tell me which are those two points as this white wave moves forward becomes yellow wave there are two particles that are moving together what are they one and five okay somebody saying one and five let's look at there's one there's five are you saying that they are moving similar way sorry this is one that is five one has traveled this much distance in the same time five has traveled a lot more distance they are not similar correct two and six look at the two and six they are similar two and six one is called crest or not one is the crest all of you agree crest of the wave one right so if i put a point over here seven the seven will travel like one this will be seven dash okay so you can say that one and seven two and six they are similar their movements are similar okay when their movements are similar we say that they are in phase one and seven in phase with each other two and six in phase with each other okay so why i'm discussing all of this because i want to define something called as wavelength okay in your grade ninth the wavelength was probably defined as distance between two consecutive crests or troughs which is an incomplete definition okay why they made you remember something which is incomplete because they don't want to make things complex all right because you take any two points on the crest one and seven there they will move in a same manner okay if you take two crests or two troughs they will move in a similar manner okay so wavelength is actually what wavelength is the minimum distance along the velocity of wave between the two points that are in phase that are moving together okay so one and seven they are crest only so they move in a similar manner so distance between one and seven you can call it as a wavelength one and seven is wavelength and distance in two and six is also wavelength it's the same distance okay but then if you look at this point which appears to be similar to two on the first look but if you look at its movement it actually goes down two goes up and that point goes down so this point is not in phase with two that is not the wavelength you to find out a point which is moving exactly like how two is moving so that is in phase okay so that is a definition of the wavelength clear okay so we have defined the wavelength let us try to find out the value of the wavelength mathematically how we can find it out so y is equal to a sin kx minus omega t plus five this is the equation of and if wavelength is lambda then if we put x as if we substitute instead of x x plus lambda if we substitute that will we get the same value of y or not everyone will we get the same value of y or not right we get the same value of y so using that information get the value of wavelength what is the wavelength in terms of what is given anyone okay so what we do we substitute x as x plus lambda equate it to a sin of because the value of y should be same but if we equate it like this we will get the value of lambda to be zero okay so what we do is that we add two pi that is a minimum time period of the sine function okay now when you equate these two you will get k times lambda to be equal to two pi so from here wavelength will be two pi by k okay clear so this is a wavelength now similarly there is something called as time period of the oscillation okay so you know that particles are performing the shm with angular frequency omega right every particle is moving up and down so if you have to find out the time period we know it will be equal to two pi by omega okay and if you want to derive it what you can do is that you can put the value of t to be equal to t plus capital T exactly like same derivation you will get the time period as two pi by omega okay and one thing you might have noticed here lambda divided by time period is equal to omega by k which is the velocity of the wave and one by time period is frequency so lambda into frequency is the velocity of the wave so for every wave lambda into frequency is the velocity of the wave only that is the relation which you have learned in ninth also right have you seen this relation before in chemistry all right so enough of the analysis let us try to solve some of the questions one by one you can answer first part what is the answer pretty simple 0.005 second part you don't need to do the calculation you can directly tell the expression itself what is the correct the wavelength is 2 pi by lambda lambda is coefficient of x so pi by 40 meters time period of oscillation what it is two pi by omega omega is three frequency is one by time period so three by two pi parts fine now do the deep part wavelength the formula is two pi by lambda right sorry two pi by k is the formula k is the coefficient of x do the deep part everyone you just need to put the value of x and value of t okay i think all of you are getting this thing 0.005 sin of 36 is this what everyone is getting but what is the value of sin 36 roughly what it is 36 is in radiance okay and sin of 36 will it be equal to sin of 36 minus 2 pi yes or no 2 pi the time period so i can subtract it as many times as possible okay so pi is 3.14 2 pi is what 6.28 so if i subtract the 2 pi 3 5 times that is 36 minus 10 pi so you will get sin of 36 minus 3.14 right so 3 4 what is this sin of 4.6 okay sin of 4.6 is nothing but sin of pi plus pi by 2 pi is 3.14 and pi by 2 is 3.14 divided by 2 1.5 something so 4.6 so this is roughly equal to 1 only the sin of 36 is 1 so the answer is minus of 0.005 itself no no sin sin of pi plus pi by 2 is minus 1 this is plus 0.005 is this clear to all of you how did we get the value of sin 36 these are all mathematics here okay they want you to do such mathematics in the school level questions itself okay this is from your textbook 14.2 clear to everyone can i move ahead type in quick let us take one more numerical tell me for this so what is the answer can answer be a everyone can it be a without calculating itself you can reject a how can you reject a why a cannot be the answer without you calculating anything which direction the wave is traveling plus x direction right so is this is this wave traveling in the x direction no it is traveling in the negative x direction there's also negative x direction it is plus in between it can be either c or d right in option c omega is 12 pi and k is um pi by 30 so the velocity is equal to 12 pi divided by pi by 30 360 meters per second so c appears to be the correct option so i hope the thought process is clear how we have navigated our answer it is clear to all of you type in how they calculate phi who who calculated phi how to calculate phi okay so good question very good this is the equation v is 316 the second option also but there is a plus sign here this is the wave that is traveling in the negative x direction it is given that the wave is traveling in the positive x direction in the fourth option wavelength should be 60 right fourth option also the velocity is 360 meters per second but then yes i missed one check also wavelength should be 60 meters what is wavelength in terms of k 2 pi by k k is pi by 30 here so it comes out to be 60 it will not come out to be 60 in this space okay so we need to check for the wavelength also sorry about that clear now all right no so somebody was asking what is phi phi depends on what is the value of y at x equal to 0 and t equal to 0 they should tell you some initial condition because when you're you know for example let's say this this is a wave that is traveling okay somebody say okay t equal to zero is this somebody will start the watch little later by the time the wave has reached there it's the same wave the velocity is same wavelength is same frequency is same everything is same but your t equal to zero and my t equal to zero are different so phi depends on that at t equal to zero and x equal to zero what is y that is how you find out all right so next topic next topic that we should be doing is the speed of the traveling wave wave is traveling we need to find its speed so that we understand how fast our information is traveling have you ever let's say seen a movie in which some war movie where the enemy is approaching with horses and the the other person is putting his or her ear on the ground to hear whether the enemy is approaching them or not seen those movies right okay so basically why it is happening like that because it is a sound only but sound travels through the air or it can travel through the ground also but when it travels through the ground it travels a lot faster because the medium is such so when you put your ears on the ground you may be able to hear something which is happening very far at the far at much earlier manner okay so that's the reason why you have to do that so when we talk about the speed of the traveling wave like I told you earlier the medium matters through which medium the sound is traveling or through which medium the wave is traveling and also what medium how the medium is moving it is all about medium so how the medium is moving means whether the wave is transverse or longitudinal so we are going to discuss it one by one like I told you it depends on the medium and also it depends on whether it is a transverse or longitudinal fine so we need to be very specific about the velocity of wave as in different formulas will be there for different mediums and different formulas will be there for whether it is transverse or longitudinal because it depends on that right so we will first we will take the example of transverse wave on a string all right so write down so there is a full-fledged derivation of transverse wave on a string how to find the velocity we are not getting into the detailed derivation of the velocity of the wave but whatever is there in our curriculum in our textbook they have done dimensional analysis to find out the expression of the velocity of the wave so we are going to do that only okay so every medium will have two properties one is elastic property other one is the inertial property on those two property only the speed of the wave depends upon okay now don't worry that it sounds a little weird I mean this derivation itself is assuming that you should know that it depends on inertial and elastic property okay so it just need to let's say remember it okay there is it like it is like you already know that velocity depends on inertial elastic property because the way we are deriving is not the true derivation it is just like say I'll say that it is the proof of something which we know already like that we are deriving okay so the if it comes in your school exam this is the way you have to write the way I'm writing okay so transverse wave on a string the speed depends on the inertial property which is mass per unit length this is new let's say okay so inertial property has to do with mass all right now you can't say that inertial property of a medium is total mass of the string because string can be like one kilometer long one meter long five meter long so ultimately the same wave can travel on different different lengths right so it is not the total mass that matters it matters how the mass is distributed okay so inertial property you can take it as mass per unit length and elastic property is represented by the tension now you'll say that okay why tension is elastic property the answer to that is tension is like a stress that comes because of the strain okay and strain means that the string has elasticity okay all right so if I say that the velocity of the string depends on mu and t so I can say v is proportional to mu to the power a t to the power b now can you do dimensional analysis to find a and b try doing it yourself get the value of a and b anyone are you getting anything all right so I can see some of you got it good so dimension of the velocity should be equal to dimension of mu to power a dimension of tension to the power b okay dimension of velocity is length t minus one this is equal to dimension of mu this mass per unit length so m l minus one to the power a tension is force forces mass times acceleration so this will be m to the power a plus b minus a plus b t minus 2b so when you correlate minus 2b is equal to minus one so b comes out to be half coefficient of t coefficient of m should be zero a plus b is zero so b is half so a should be minus half and minus a plus b should be equal to one so you can see that when you put it over there you get one okay so the velocity is proportional to mu to the power minus half t to the power half and luckily or surprisingly the proportionality constant is one so velocity is equal to under root t by mu this is your textbook derivation of the velocity of the wave dimensional analysis we have used okay so the velocity the transverse wave on a string the velocity of that will be under root t by mu on a string all right so let us take some numericals it is simple try doing it done get a rough answer calculation might be little tough t is given to us directly 60 neutrons right and mu is mass per unit length so total mass all of you are done like this only but you don't want to calculate so this is roughly 93 meters per second okay good so let me let me solve a couple of questions with you in which there is no calculations so the excuse of not solving it because of calculation will go away this is mass m all right point a point b the distance between point a and b is l all right so we we have created some sort of disturbance over here and this disturbance is moving this way okay the capital M is very large compared to mass of mass per unit length is given as mu okay the entire system is accelerating with acceleration a upwards okay with acceleration a this entire thing is in some lift and this lift is accelerating with attrition a upwards we need to find time taken for a pulse to go from let me name it as one and two from one to two try solving it pressure got something sharduli got something siddharth got something everyone is getting something something okay can you tell me what is the tension t is what look at this let's say t is a tension gravity force is mg acceleration is a upwards net force equal to mass m acceleration right so t minus mg is net force in the direction of acceleration this is equal to m into a under t by mg plus a divided by mu and time taken will be length divided by the velocity which is this the answer okay so i can see many of you got it some of you might have done some silly error refer it all right so one last question before the break is this now when you are using this formula under t by mu as the velocity of the wave although mu is mass per unit length of the rope but many times we ignore the mass of the rope right but we need not be ignoring it because that mass of the rope itself can create tension for example let's say a rope is hanging from the top like this imagine you yourself are hanging like that you'll feel the tension or not similarly a rope when it is hanging like this total mass is m distributed along length is l you need to find out this pulse how much time it will take to reach the top okay try doing it step number one is get tension as a function of x what is x you can imagine over here at a distance x from the bottom what is the value of tension over there yeah what you did not understand which part of the problem the rope is hanging vertically what is there to understand over there the rope is hanging under the gravity like this anybody got the value of tension as a function of x okay so what we will do is that we will draw a free body diagram of this much until as you draw free body diagram of that much tension will not appear in the fbd because it is an internal force this is t how much is the mass of this rope what is the mass there's much portion of the rope total mass is capital N for a length l so for a length x the mass is m by l times x that into g everything's at rest so t minus mg is 0 so t is equal to m by l x g okay so the velocity at any moment where tension is t is under root t by mu mu is m by l so it comes out to be root over x g this is the velocity and you can see that velocity changes at every distance x it is maximum at the top where x is maximum and tension is also maximum all right now we need to find time how do you get time velocity is not constant otherwise you could have done distance by velocity so what you can do is that you can write velocity dx by dt yes integration your favorite topic do it like this dx by x integral is what everyone what is dx by x integral dx by root x integral what it is 2x dx x to the power minus half dx what is the integral x ratio power minus half plus 1 divided by minus half plus 1 2 root x to l root g t so from here you'll get time taken to be equal to 2 times root l by g this is the answer you want to take a break or should we continue break all right so let us take a break now we will meet soon all right am i audible everyone okay so let us continue we have studied till now about the uh transverse wave so let us talk a little bit about the longitudinal waves also because uh the sound wave is a very common example of a wave and it happens to be a longitudinal wave all right so let's quickly discuss about it so now um in case of transverse wave we were tracking the y coordinate right y coordinate was a sin kx minus omega t plus phi our transverse wave was like that but in case of longitudinal wave the problem is there is no y coordinate everything is happening on the x axis right the particles are oscillating back and forth in the line of the velocity they're going like this they're oscillating like that the y coordinate is zero so how will we write the equation for the longitudinal wave all right so what we do is that in case of the transverse wave what is the y coordinate y coordinate is a distance from the mean position similarly here also if distance from the mean position of any particle is let's say s so we will track s and everything else remains the same okay so make sure that you understand this thing that ultimately the mathematical where the expression is exactly the same so we don't need to study the kinematics of the longitudinal wave separately because the equations are same the kinematics of the longitudinal wave will be same as the kinematics of the transverse waves okay all right so we know that the sound is sound is a longitudinal wave and one more thing that is very common with the longitudinal wave is that the longitudinal wave is inside the medium for example when I'm speaking the sound is inside the air okay another example is let's say let's say water is there okay so on the surface of the water transverse wave is possible but inside the water only longitudinal waves are possible fine so usually usually longitudinal waves are within or immersed inside a medium fine okay now we will try to establish some relation with velocity and the property of the medium for the longitudinal wave all right so in case of longitudinal wave which is inside a medium it depends on the elastic and the inertial property so elastic property of the medium is represented by the bulk modulus okay and the inertial property what do you think is the best representation of inertial property if let's say the sound is traveling inside this room inside the air so what do you think is a best representation of the inertial property everyone correct others answer inertial property in the case of string it was mass per unit length right here you can talk about density as the inertial property fine the mass density the volume density okay rho the density now do you guys remember the formula for the bulk modulus it was there in the properties of the solid what are the formula for the bulk modulus tell me b is equal to what no one remembers b is equal to correct correct delta p divided by the volumetric strain not pressure extra pressure pressure is anyway there the atmospheric pressure is there right everywhere the atmospheric pressure is there you need to create extra pressure to change the volume and things this is a bulk modulus in a differential form you can write bulk modulus as this the same thing okay now if I tell you that the velocity depends on bulk modulus and the density can you find out the value of a and b using dimensional analysis try doing it yourself this is a derivation from your textbook anybody about to get the answer dimension of the velocity should be equal to dimension of the density to the power a to dimension the bulk modulus to the power b dimension of the density is k g per meter q so m l minus 3 the power a bulk modulus what is the dimension everyone what is the dimension for the bulk modulus look at the formula for the bulk modulus delta v by v in the denominator is dimensionless it has a dimension of the pressure only okay pressure is what force per unit area force is mass into acceleration divided by the area so which is just a second the screen appears to have paused so m l minus 1 t minus 2 b they just only got the answer already m a plus b l minus 3 a minus b and t to the power minus 2 b okay now when you correlate this with that you'll get minus 2 b is equal to minus 1 so b is equal to half a plus b should be zero so a is equal to minus half you can check here substituting whether you are getting one or not you will get it okay so the velocity is proportional to rho to the power minus half b to the power half and luckily and surprisingly the constant of proportionality is one so velocity is equal to under root b by rho but remember you are using this formula only for the longitudinal wave not for the transverse wave okay now we have been taking example of a sound wave and telling that it is the longitudinal wave fine so can we somehow get the velocity of the sound by using under root b by rho okay let's try to do that so we know the density of the air is 1.29 kg per meter cube bulk modulus is minus of v dp by dv this also you note down the atmospheric pressure is 1.01 10 to the power 5 Pascal so we want to basically find out that what will be the density of any longitudinal wave like sound wave in atmospheric pressure which is this having the density as that right so we know that the velocity formula is root over b by rho but the problem is b looks to be a little bit weird in a way the formula for b is this so can we use this formula to find out what is the value of bulk modulus for the atmospheric gases the answer is yes but you need to assume few things but Newton what he suggested that yeah Newton was here also so Newton assumed Newton assumed that the travel of sound wave is an isothermal process when sound wave that moves from one location to the other location Newton assumed that it is an isothermal process temperature remains constant and if temperature remains constant you can say that pressure into volume is a constant fine so can you differentiate this expression with respect to volume and tell me what you get bulk modulus as everyone try finding the bulk modulus take a derivative with respect to v and find out the bulk modulus okay others all right so we can use a chain rule here so p plus v dp by dv derivative constant is 0 so from here bulk modulus which is minus of v dp by dv is equal to the atmospheric pressure itself okay so we can use this directly and we know the value of p is this clear to all of you type in does it make sense okay so when it comes to the gas when it comes to the gas the bulk modulus depends process which process it is whether it is isothermal adiabatic and stuff like that so velocity comes out to be under root b by rho b is pressure the atmospheric pressure which is this divided by 1.29 which is the density under root b by rho and it comes out to be around 280 meter per second and by the way do you know actual velocity of sound in the air do you know how much it is any rough value you know no it is not treated actually it like I do not know if you have seen it earlier it is around 340 330 like that fine around 330 to 340 that is the speed of sound and 280 is completely off the mark right so this derivation is not correct so what newton did or what newton proposed was not correct but the formula of under root b by rho was correct so what newton did was newton assumed that it is isothermal process this assumption is wrong that it is an isothermal process it is not isothermal process because if there is let's say temperature difference between one location to the other will the sound wait for temperature to become uniform then sound will travel no even if there is a temperature difference sound will just travel through it so it is not an isothermal process what happens is that when sound travels the the the other parts they do not get time to exchange heat so it's a rapid process so if it is a rapid process others do not get time to exchange heat what do you think it is a what kind of process it is it is isothermal or what what could be the name of the process correct others right it is an adiabatic process so that is exactly what laplace did laplace did some correction to what newton had suggested it is called laplace correction okay so what laplace argument was that the travel of sound wave is a rapid process no time to exchange heat hence the process is closer to adiabatic got it it is closer to adiabatic process fine now for the adiabatic process it is not pv that is constant what is constant for adiabatic process everyone do you remember for adiabatic process what i can say pressure and volume is there any such expression like for for the isothermal p into v is constant for adiabatic what is the process equation correct we have p v raise to power gamma to be constant it is not pv constant pv raise to power gamma is constant now can you find out what is the bulk modulus minus of v into dp by dv how much it is everyone find out anyone what do you get bulk modulus as correct others differentiate it with respect to the volume again you'll get p gamma v to the power gamma minus one plus v to the power gamma into dp by dv this is equal to zero this will take away one minus of v dp by dv comes out to be gamma times p so the bulk modulus what Newton suggested was equal to p but Laplace came and told that bulk modulus is not p it is gamma times p now what kind of gases the atmosphere has it is monatomic diatomic polyatomic what is the majority of the gases diatomic for diatomic what is the value of gamma gamma is what for diatomic correct seven by five a factor of seven by five was missing in what Newton did so the velocity of the sound is under root b by rho which is gamma p by rho now when you substitute the values you will get around 331 meter percent which is very close to the real value all right so this is how you take care of the longitudinal wave and these derivations whatever we have done they are part of your school curriculum very important for you so make sure you understand them all right so right now you know we can actually do some numericals on finding the velocity wherein I can give you the different values of pressure and density but it'll look childish right the formula is there you keep on substituting values and getting the answer for the velocities so rather than that let us move forward and take some more topics and once other topics are over then you know whatever we have learned till now that will be any way part of the numericals that we are going to take a little later fine so let us move ahead so the next topic that we are going to discuss is the principle of superposition of the wave fine so here the principle of superposition says that if you have two two waves y is equal to f1 x comma t and y is equal to f2 x comma t there's a two functions are there two waves are there superposition is a technical word of telling that two waves are colliding like two masses they collide we'll say collision has happened when two waves they meet we say superposition has happened okay otherwise it is same as the collision only so what will happen when these two meet is addition of simple algebraic addition of these two functions nothing else okay for example let's say I can say these two waves sinusoidal wave if you consider this plus this this is what will happen now exactly what will you know this is what it is now after this it is mathematics right how you add them you'll use sine a plus sine b or some other thing you know if you want to simplify it as a sparse function make it look like a single trigonometric function whatever it is it is all mathematics beyond this fine so to keep the mathematics simple we are going to take only two cases that is in our curriculum fine so the situation that we will be talking about is same frequency I missed sine oh okay here okay so situation that we are going to talk about is same frequency let's say omega and same wavelength wavelength is 2 pi by k so wavelength is same k is also same okay under this situation only two cases will be taking case number one where the amplitudes are equal case number two where amplitudes are not equal okay so let us take the simpler case first where the amplitudes are equal anyone has any trouble you can let me know immediately y1 is let's say a sine kx minus omega t and y2 is a sine kx minus omega t plus 5 okay so you can see that phase of the first one is kx minus omega t phase of the other one is kx minus omega t plus 5 so the value 5 is referred as the difference in the phases phase difference okay that is the phase difference so all of you I want you to write y as equal to y1 plus y2 because guess what these two waves are meeting each other so what will happen after they meet how do you simplify it write down after this what we'll do what kind of formula can we use I want to see only one sine or cos function what should I do after this can I further simplify can I do something with it the answer is of course yes how can I use sine a plus sine b formula yes or no correct I can do that I can take a common so it becomes 2a sine of a plus b by 2 cos of a minus b by 2 so 2a cos 5 by 2 sine of kx minus omega t plus 5 by 2 how many of you got this right so does it look like a wave equation to you does it look like a wave equation everyone does it look like a wave equation yes or no if you call this as an amplitude now does it look like a wave equation if this is the amplitude right it does look like a wave equation having the same wavelength same frequency so two waves when they meet they create a third wave what is the condition they both are traveling in the same direction kx minus omega t kx minus omega t when two waves traveling in the same direction when they meet they create a third wave all right and whose amplitude is 2a cos of 5 by 2 this is the amplitude all right now tell me what if what if amplitude becomes 0 what does it mean it implies what if amplitude becomes 0 what does it mean does a wave exist then does a wave exist it doesn't exist wave is gone now that is something which happens only with the waves that two waves they can meet and destroy each other they both are gone it doesn't happen with the mass right like one mass coming from here one mass coming from there when they meet it is not that mass is disappeared but it can happen with the wave because wave is not mass so one wave that is like this can meet the other wave like that when they meet they become nothing they destroy each other that is called destructive interference so can you tell me one value of 5 for which destructive interference happens everyone what should i put value of 5 here so that amplitude becomes 0 pi is one such value okay what are there can you think of any other value of 5 pi is one value then another one can i put 5 as 3 pi 3 pi also give me 0 or not 3 pi also gives me no 2 pi will not give you 0 if you put 5 as 2 pi 2 pi by 2 is pi so cos of pi is not 0 any odd multiple of pi okay so 5 should be equal to 2n plus 1 pi for destructive interference so if phase difference between first wave and the second wave is odd multiple of pi when they meet they destroy each other nothing will be remaining okay tell me one value of 5 so that amplitude is maximum 2a 2a or minus 2a it doesn't matter minus 2 it just means that it is on the downside what should be the one value of 5 that you can think of 5 is equal to 0 when you put 5 is equal to 0 and when you put 5 is equal to 2 pi right so any even multiple of pi i'll say constructive interference okay in destructive interference amplitude is minimum that is 0 in constructive interference amplitude is maximum that is 2a makes sense to everyone type in quick all right so this is case number one where two waves that are not only traveling in the same direction they have the same wavelength same frequency same amplitude just that the phase difference 5 is there okay now why don't you see such thing happening in air light is a wave right so why don't two light when they meet they destroy each other the answer to that is it does happen but we can't see it if the frequency wavelength they don't match if this omega and this omega they are not same the phase difference will depend on time with time within one second 50 times at the same place destructive interference happened and constructive interference happened our eyes can't see that so phase difference between the two should not depend on time the frequency should be same then only you'll be able to see destructive constructive interference okay i'll just give you one example have you ever gone or have you ever passed through let's say a place where the oil is spilled on the on the water and while you're walking through you'll see that the oil on the surface of the water sometime it shines very bright and suddenly turns black complete and then it shows you different colors red color green color yellow color has it ever happened as you're walking past through it oh no you haven't seen that okay so anyways so we are going to discuss about constructive and destructive interference with respect to light in grade 12th you have wave optics that will come in grade 12th there we'll discuss right now this much is enough for constructive and destructive interference discussion in this case the second case is when amplitudes are different rest all are same case number two amplitudes are not equal so let's say y1 is equal to a1 sine of kx minus omega t and y2 is a2 sine of kx minus omega t plus five the phase difference is five but the amplitudes are not equal so now mathematically you have to simplify it what comes in your mind I can you I cannot use sine a plus sine b formula here because I can't take a common a1 and a2 they are tell me one trick how can I simplify it further what comes in your mind okay so what we can do is we can expand it like sine a plus b then I can take sine kx minus omega t common it is something like this a1 it's mathematics okay just trigonometry sine a cos b plus cos a sine b so you can write it like that a1 plus a2 cos of phi sine kx minus omega t plus a2 sine of phi cos of kx minus omega t okay this is y after this put it here you can write it like this y is equal to a sine of kx minus omega t plus b cos of kx minus omega t where a is this and b is that so wait now tell me how can I further simplify this you have done this in trigonometry in max already a cos theta plus b sine theta how do you further simplify it to make a single trigonometric function remember that a cos theta b sine theta no you don't remember what you did do remember multiplying and dividing by a square plus b square under root anyone type in have you done it in max or not under trigonometry you have done it this now if I call this as cos of theta what this becomes what does this become if this is cos theta you can see that this is square plus that square is one so if this is cos theta this would be sine theta that is the reason why we have done this multiplying divided by root over a square plus b square okay so sine theta is this cos theta is this so now this become sine a cos b plus cos a sine b where tan of theta is b by a b is a to sine of phi does it look like something which you have done already this expression does it ring any bell no okay it will soon so y is equal to root over a square plus b square this is sine a cos b cos a sine b formula so it will become sine of k x minus omega t plus theta okay right they just we got it so now is this a traveling wave equation is this a wave equation type in right it is a wave equation it is a wave equation amplitude is root over a square plus b square amplitude is root over a square plus b square what was a a was a1 plus a2 let me just quickly check that a2 cos of phi b is a2 sine of phi like this so it becomes root over a1 square plus a2 square cos square phi plus 2 a1 a2 cos of phi plus a2 square sine square phi look at this plus that you can take a2 square common sine square theta plus cos square theta is 1 so it will come out to be root over a1 square plus a2 square plus 2 a1 a2 cos of phi does this ring any bell have you seen such expression before right you have seen it cosine form line vector exactly that way you have to find the amplitude when two waves when they meet having phase difference phi they will add like vectors the amplitude adds like a vector as if the angle between a1 a2 is phi so that way you can remember okay so in school you might have done the case in which amplitudes are equal when the two waves are meeting but then there's also you should be knowing because in computer exams there's no limit they can ask from anywhere okay so if you can change the value of phi what is the maximum value of amplitude that can happen in terms of a1 a2 type in it'll be a1 square plus a2 square plus 2 a1 a2 that is a1 plus a2 whole square under root so a1 plus a2 minimum what should be the value of cos of phi for amplitude to be minimum minus 1 minus of this it is mod of a1 minus a2 okay this is constructive this is destructive interference fine I hope it is clear to everyone type in all right so we have good 12 to 13 minutes we can solve some numericals I'm not going to teach any theory now let us take couple of numericals okay so let's say there are two waves two sin kx minus omega t and the second wave is 4 sin kx minus omega t plus phi by 3 okay these two waves they superimpose they meet you have to find resultant amplitude find out Trisham Siddharth got it others root over a square plus b square plus 2 a b cos of pi by 3 cos of pi by 3 is half so this will be root over 4 plus 16 20 plus 8 28 2 root 7 okay next you have y is equal to 4 sin of kx minus omega t and y is equal to 10 times sin of kx minus omega t plus phi these two waves are meeting this phi can change you have to find out what is the maximum possible amplitude when they meet divided by what is the max minimum possible amplitude when they meet what is the ratio for that correct pretty simple maximum possible amplitude is a plus b minimum is mod of a minus a plus a minus b so 10 minus 4 amplitude negative means nothing okay amplitude will always be positive maximum minimum minimum can be zero at the minimum value but it cannot be negative if amplitude turns out to be negative at any point in time when you calculate just take a mod of it like 4 minus 10 was negative take a mod 14 minus 6 7 is to 3 okay pretty simple straightforward numericals are pretty straightforward in this chapter but the derivations are not so straightforward okay so take it with a pinch of salt let me now since we have what we have couple of minutes we can take up some questions I'll take questions from anywhere okay now take this try this solve this question everyone you'll be much more confident if you towards the end you get some other questions correct yeah multiple options can be correct why not can be all of you agree that t2 is equal to 3.2 g that is 32 newtons and t1 is equal to 3.2 g plus 3.2 g how many if you got till here okay I have skipped some of the steps to get this okay but then you can drop the word diagram and get it easily so I'm not getting there the velocity in the first one is root over t by mu 1 10 is for minus 2 so that is 80 meter per second that's correct the first option is correct any other option v2 is what v2 is root over t2 32 divided by mu 2 8 into 10 is for minus 3 so this is root over 4000 right now root over 4000 is it 63 this is a 93618 6386 no it is not but then if you look at somewhere maybe in the internet or whatever it is they may tell you that option c is also sorry option b is also correct but then it doesn't appear okay it has to be exact fine and it may turn out to be exact if you take g as 9.8 all right so let's not mark b as correct it is only option one okay approximately it is 63 only but not exactly all right how much time we have oh two minutes we can do one question and here comes the question very very quick this one just one last question everyone okay said that got something others anybody else okay last minute people are not awake completely omega is coefficient of t so that is pi by 2 into 8 so 4 pi k is coefficient of x that is pi by 16 all right so the velocity is omega by k all right don't worry about like you know that it should be x minus omega t n like that so what you can do is that you can convert this take minus outside it'll become like this sine of pi by 2 x by 8 minus 80 okay taking plus or minus here it doesn't matter it just means that at t equal to I mean rather than starting like this the wave is starting like that it only means that nothing else okay so ignore minus sign v is omega by k omega is 4 pi divided by k pi by 16 so this is 64 centimeters per second is it clear to all of you typing clear all right so that's it from my side we will meet next week bye for now