 Alright guys so let us start this chapter waves so we will quickly have a small discussion on what is waves and so on and so forth then we can start the topic right down waves. Now when you hear the word waves what comes in your mind honestly what is the first thing that comes ocean ocean up and down sound okay sound comes in your mind okay fine so these are the things that come in our mind so let us take examples of let us take an example of water ripple do you think water ripple is a wave do you think water ripple is a wave water ripple is an example of a wave so if water ripple is an example of a wave then how this created we what we do how the water ripple is created can can you tell me how you create water ripples throw a pebble throw a stone okay so what does stone does what does stone doing when you throw it on the water disturb the water okay so basically it doesn't matter what if you throw a stone or from your hand only you do it there should be some disturbance okay disturbance is created at one location and what is happening to the disturbance then when you create a disturbance in one of the location then that disturbance propagates isn't it suppose right now disturbance is here only this location has no idea that there is a disturbance over there but this disturbance will travel and reach here also okay so this is a way all right so if we crudely you know tell what is a wave wave is nothing but wave is a disturbance that travels from one location to the other okay so this is the crude definition of what is a wave now if it is a wave if it is sorry if it is a disturbance will this disturbance have some sort of energy or not everyone a disturbance has energy or not disturbance will have energy okay so many times when I say that disturbance is what is traveling from one location to other the importance is not that clear okay so rather than calling it a disturbance we could call it energy so energy is transferred from one location to the other fine and this energy can have information within it okay I'll talk about it in some time this energy can create force fine so by this disturbance which has energy is getting transmitted from one location to the other then I mean how it is different from a particle going from point a to point b how it is different from that can anyone answer that this object has gone from a to b so the kinetic energy from here has gone there so what is the difference between this situation and that situation all of you tell me when a particle is going from a to b and a wave is going from a to b is there any difference is there any difference the water ripples when they are moving what is happening concentric circles are there and they are expanding like that okay when concentric circles are expanding do you think water is residing do you think water is moving away and going further away from the disturbance is that is what happening what do you think when you disturb the water and you see that the disturbance created in a circle and circle expands when circle expands is water is also moving residing water is moving or not you know understanding my question okay my question is that when you throw stone you see these concentric circles getting created and they are expanding so when these concentric circles are expanding is water also moving like this that is my question is water also going like that okay if you say yes with some of you have said then if I keep on throwing stone if I keep on throwing stone the wherever I am throwing stone the water will be gone and all the water will be at the corners but that doesn't happen ever okay so water is not moving get that water is not moving with the wave all right please understand that wave can use a medium like water to move forward but the medium doesn't move with wave okay so you need to get this very very clearly that the medium is just like you know a means to transport the energy or the disturbance from one location to the other okay have you seen the Diwali lights or any New Year lights wherein there are a lot of LED bulbs like that if you put LED bulbs and sometimes you feel as if the light is you know traveling like that have you ever seen that what LED does the LED only does on and off and you feel as if the light is traveling like that have you have you ever seen that right so exactly something like that is happening what is happening the medium the particles of the medium they are just oscillating they're oscillating up and down so you'll feel as if water is moving forward but it is not water is it it is a disturbance that is moving forward okay so keep that thing very very clearly we will come back to it again okay but let me first classify a few things that there are waves that require medium and some waves do not require a medium all right have you done the waves introduction earlier ever in class 8 9th somewhere this is the first time in 8th okay see the thing is that you you might have seen in your class 11th curriculum this is the last chapter okay before this chapter can you recognize that everything was about matter everything mechanics is about how mass moves how particle moves mechanical properties of matter rigid body motion simple harmonic motion of a mass then properties of a mass mechanical property of solid mechanical properties of liquid thermal property of the matter so everything is about mass in class 11th okay so you may not know but in class 12 everything is about charge but there are few chapters which are disconnected for example in this chapter this chapter is not about mass getting it so do not try to relate whatever you have learned already directly in this chapter because whatever things you have learned is valid for the masses what I'm telling you is that when let's say two particles are coming and hitting like this you what you do you conserve momentum right you have coefficient of restitution and whatever you have you conserve momentum when two particles come and hit like this but when one wave meet the other wave like this you can't conserve momentum okay so you need to have something else some other let's say principles you should have some other way of dealing of how what will happen if one wave meets the other wave okay so that is why I call that this chapter is different from your entire physics you have been learning till now in class 11th okay so do not force yourself to apply things directly on this chapter okay doesn't the mass only collide as in yeah Mars will collide so when mass collide you when wave meet when see meeting when you can say when two masses meet okay meeting and collision just the English word okay so when two wave meet you have a different way of dealing with it and when two masses meet you have two different ways of dealing with it you have a different way of dealing with it okay that is what I'm saying here okay so now waves that just the way waves move there are two types of waves okay the wave which require medium to move this require medium to move this one doesn't require medium okay doesn't require medium what is the above one called which require medium this one is called this is the mechanical waves correct this is a mechanical wave and this is what electromagnetic waves so this is what classification based on based on how wave travel alright so this EM wave you will learn it in class 12th mechanical wave is there in 11th okay but of course there are few basics and few fundamentals which are valid for EM wave as well as mechanical waves so that is why the name of the chapter in class 11th is waves so that we cover the basics which is valid for both and then we will talk about mechanical waves thereafter okay now there is another way of classifying the waves write down classification based on how medium oscillates just now we discussed that the medium won't move with the wave but medium is oscillating okay so the way medium oscillates you can discuss or you can classify based on that also okay so waves can be of two types with respect to how the medium is behaving can you guess anyone you have already learned it in class 8th or 9th correct correct so one is transverse wave okay the other one is longitudinal wave so in transverse wave the medium oscillates perpendicular to the wave velocity okay in longitudinal waves medium oscillates parallel to wave velocity fine so EM waves by the way they are transverse EM wave are never longitudinal fine so all the EM waves are transverse there are few mechanical waves that are longitudinal do you remember which one is the most popular longitudinal wave which one sound no no what the rebel is not longitudinal sound okay this is a very good question if it is a EM wave then what oscillates medium is not there you say the medium is transverse then what is oscillating medium is not there at all so how will you know that EM wave is a transverse wave in electromagnetic wave there is electric field that oscillates okay electric field is oscillating and wave is wave is what traveling like this there's a magnetic field also I'm just oversimplifying it so wave is going forward like that and electric field is oscillating like this understood electric field is doing up and down like that who has that question understood okay great so this is the classification so you now you have good amount of understanding what is a basically a wave and what kind of wave exists there's one very important thing about the wave you can never pinpoint the location of a wave getting it the wave will be spread in an area always for example if a particle is moving with velocity you can exactly pinpoint that here is a particle here is this mass 1 comma 0 2 comma 3 particle is going from 1 comma 0 2 comma 3 but you can't say something like that for wave wave is present in entire medium okay when water ripples are created it is not that water ripples got created only at one location it is spread in a zone fine so you might have heard about Heisenberg uncertainty principle have you heard of it from where that comes why it is why it is there because the electrons behave like a wave okay so if electrons start behaving like a wave you cannot pinpoint the location of electron you will have electron density basically what is electron density it is nothing but a wave you say that in this zone wave exist or in this zone electron exist fine so anyway we will discuss it more in class 12 right now our focus is 11th curriculum I said by the way did you know that class 10th board exam will happen in the month of May or June I think June and class 12th board will also happen in the month of June May or June okay so and what about your exams class 11th exam does that got postponed as of now March okay anyways so let us first see how the chapter is organized okay so you know I might be going little slowly but then you you know initially we go very slow once everything for in place our pace will increase right now we are just introducing the chapter write down waves so waves topic can be this chapter you can have multiple classifications in the chapter there will be four broad classifications first one is kinematics of waves just like in simple harmonic motion right same way here also we have a section that talks about kinematics okay so a wave equation will be given to you you will be asked to find out the wavelength frequency time period amplitude and all those things that is the kinematics of the waves and we do not care we do not care about how the wave get got created in which medium it is moving and what is the effect of the medium on the wave we don't care about all those things over here okay then we have we have the velocity of wave in a medium okay so just to give you an analogy just like when you travel in a car car is your medium so your velocity depends on the velocity of car not your own this thing similarly when a wave is moving in a medium the velocity of the wave depends on the medium only it depends on what kind of property the medium has okay so we will be discussing about how the medium affects the velocity of the wave and what is the equation so you can say that second part is like laws of motion where you try to get into the cause what causes the wave to move which is the medium okay it's like why do you think water ripples got created in the water only when you throw the stone on the field there are no ripples created okay so medium has a big role then third part is the major section which is super position of the wave so this is something big super position of waves wherein you will learn about what will happen if two waves meet and both of the waves it is like those it is like the analog of the collision between the two masses similarly you can say collision between the two waves okay so and one interesting thing is that when two waves are traveling the same direction when they meet they will behave differently when two waves traveling in opposite direction meet they will they will behave differently when two waves having different frequency when they meet they will behave differently okay so there is a wide spectrum of behavior that you need to understand so that is why there are three things here first is when wave both the waves traveling in same direction okay when both the wave travel in the same direction a third wave gets created all right then the wave space is less waves traveling in opposite directions if wave is traveling in opposite direction then something called standing waves gets created okay standing wave is a very common phenomena every musical instrument works on the standing wave only okay so we will see how it gets created then you over here our assumption is that frequency is same which need not be true but then we are studying that only over there then waves having different frequencies when they meet what will happen okay so something called beats will get created so we will talk about these things under superposition of the waves okay then the fourth section of the chapter is about Doppler's effect is anyone of your school is done with this chapter ypr or hsr or dps Doppler's effect so do anyone of you remember what is the Doppler's effect I think it was a small part was there in class 8th Doppler's effect is an apparent change in the frequency the radar sonar okay these equipments which you might have heard of radar and sonar they work on the Doppler's effect only so what will happen is that when you throw the wave forward and it hits an object and reflects back to you there will be change in frequency suppose you have thrown 200 hertz frequency forward it hits the object and comes to you the frequency will be now 198 hertz and the change in frequency depends upon the object's velocity your own velocity so if you know your own velocity which might be at rest also you will understand with what velocity the object is moving from where the frequency has hit and come to you okay so Doppler's effect is very useful in radar and sonar so this is what we are going to learn under Doppler's effect fine now you guys now might have an fair bit of idea that this is not a very small chapter it is but then it is not as big as rbd also you can say that this is slightly bigger than the simple harmonic motion okay and I have what I have seen over the years that people like to ask a lot of questions from Doppler's effect there are a lot of questions Doppler's effect then superposition of the wave comes next fine so these two topics a lot of questions are asked in in community exams but having said that any point in time they can surprise you with the questions from the other topics as well okay anyway so this is a chapter let us now start the chapter only okay so you might have a fair bit of idea how important the wave is in your day-to-day life right I don't need to emphasize is enough because the the the sheer fact that we are having this class is because of wave okay Wi-Fi is what a radio wave which propagates and reaches your laptop or desktop or mobile phone you are able to talk on the mobile phone because of the wave only which get transmitted from the tower to your mobile phone okay so wave is there everywhere every wireless technology works on the wave okay so you cannot ignore the importance of it fine okay so enough of introduction we will now get down to the mathematics of the waves so just like the simple harmonic motion you remember how we started we tried to find out what could be the equation which governs the simple harmonic motion right similarly we will try to see what is the equation that govern the wave equation okay but in SHM the oscillation the periodic motion everything was very straightforward we don't need to visualize too much okay but here in waves I feel that you need to visualize it a bit more so I am what I'm going to do is a small exercise with you and then we'll get back to the equation of the waves okay so suppose there is a wave there is a wave like this wave can be of any shape okay it can be of any shape any random shape fine this wave is moving forward with velocity V okay and this wave is moving on a string have you seen a wave on a string everyone have you seen a wave on a string what do you have to do just take a string tie at one end and move this up and down this pulse will get created start moving forward so this is a disturbance that is moving forward okay same thing so what you need to tell me this wave which is moving forward tell me after after a small time delta T draw the shape of the wave how the shape of the wave looks like on this string only after time delta T draw the wave in delta T time then all of you okay are you here I thought you're starting to join send me a message now so what will happen to the wave can anyone guess what will happen to the wave wave as it is will move forward as it is understood the blue one is after time T after time delta T are you guys able to understand this everyone disturbance is moving forward as it is whatever the shape of the disturbance entire disturbance will just proceed forward okay now tell me this point this is the string is taking the shape of the wave right the string is taking shape of the wave so the point on the wave is point on the string also this is how the string was earlier this is how the string is later on so point number one point number two point number three four and five these are the five points on the string initially tell me where are these point after time delta T on the blue wave where are they let me know once you're done so I think many of you have understood I'll use this pink color point one will go down point two will go down point three moves up particles are moving up and down only it reaches here point four reaches there point three reaches there point two goes down point one goes down so all the points they have moved and took the shape of the blue curve so the the points on the string they're moving up and down only but it appears as if the disturbance has moved forward I hope it is clear to everyone no one has any doubt type in is it clear okay great fine now if this thing is clear we can talk about the wave equation so this is my y-axis and this is my let's say x-axis okay so if I tell you if I tell you at any point in time and at a location of x if I tell you what is a y coordinate what is the y coordinate if I tell you every x every time t does that describe the entire wave will you know that will you know everything about the wave if I tell you what is why y coordinate at every x and every t is that sufficient or you need to know something else that is sufficient right so if you know you are effectively telling me where the particle of the medium will be at every coordinate and at every time getting it so this will be the wave equation so a wave equation is basically nothing but y coordinate as a function of x comma t okay anyone has any doubt quickly type in otherwise I'll proceed further this should be the wave equation right y coordinate if I tell you as a function of x and t how does it play out if if you have to find out how the y coordinate changes at a particular location let's say x equal to 2 put x equal to 2 and then for different different times you know what is a y coordinate of that x coordinate okay so like that you can freeze t also you can see that at t equal to 0 at t equal to 0 how y changes with x okay so there is no other variable there's only y x and t okay now I'm going to ask you a very let's say critical question here so think about it I want you to tell me that let's say this is the diagram I'll just clear it a bit so that doesn't look better so now let's say this is the y this is the y from here to here this is the y this is x from here to there it is x coordinate all right so this is the y coordinate at this location x and a particular time t getting it so this is the y coordinate of this location whose x coordinate is x and time is t now I'm asking you that at t equals to 0 okay at t equal to 0 where that red line was okay velocity of the wave was velocity of the wave is given to us v now tell me tell me the x coordinate what is the x coordinate of that red line at t equal to 0 it's x coordinate is x at t equal to t let me know if you have not understood the question where and got it aditya got it the question is that right now it's x coordinate is x so what would be the x coordinate of that red line at t equal to 0 in time t how much the wave has traveled in time t the wave has traveled a distance of v into t so this red line has moved what distance the red line has moved a distance of v into t okay the exact thing as it is moves forward now when I say x comma t I mean all the x coordinates all of it everything has moved by a distance of v t okay so the x coordinate of a particular location if it is x okay so at t equal to 0 the point was at a location of x minus vt because vt is a distance traveled in time t so before t seconds it was behind by a distance of vt okay so at t equal to 0 f of x t was actually f of x minus vt comma 0 which is your y so y was f of x minus vt comma 0 everybody understood this type in now 0 is not a variable 0 is just a constant so you can say it is f of x minus vt fine so the wave equation although it is a function of x and t it cannot be any function of x and t it has to be a function of a combination of x and t which is function of x minus vt so what does it mean let's quickly discuss that so we have what I am telling you is f of x minus vt function is the wave equation so if you look at this ex by t square this is a function of x comma t but not a wave equation why because this is not a function of x minus vt whereas if I tell you e to the power x minus 2t divided by 2x minus 4t whole square this is this can be written as x minus 2t divided by four times x minus 2t whole square this is a function of x minus 2t getting it getting a difference so this is the wave equation above one is the wave equation and velocity of the wave is what what is the velocity of the wave for this one velocity is two meter per second over here two units correct fine clear so we have what we have understood is that the wave equation can be of any it can look like anything as long as it is a function of x minus vt it cannot be any function of x comma t does we need to be constant no we need not be a constant okay we can change depending on the medium if medium changes for example do you know that the light when it travels inside the glass its velocity changes you know this light is a wave right so its velocity will change right of course it can change as a function of time so medium should change as a function of time it depends on the medium aditya you need to understand that the velocity doesn't depend on the wave doesn't matter what shape and size of the wave it is you need to understand what is the property of medium if medium keeps on changing after every few seconds correspondingly your velocity will also keep on changing but do not correlate it with mechanics okay fine so basically your this equation function of x minus vt is a wave that is traveling in positive x direction okay then similarly there is let's say if you write x plus vt this is the wave that travel in negative x direction okay so if it is x plus vt it goes in the negative x direction minus x minus vt it is positive x direction then shouldn't a sound wave reach any place what do you mean by that I didn't get you like if I shout oh okay okay understood understood what they're saying hold on all right so basically the kind of wave which we have studied till now is one dimensional wave okay wave can be of three types depending on how they are moving 1d 2d and 3d okay because we are learning things for the first time we have taken the most simplistic kind 1d wave let's say on a string there's a string on a straight line wave is on the string only it is 1d 2d wave can you give example of 2d wave everyone a wave travels in two dimension what are the pulse correct okay 3d wave sound sound moves in three dimension okay so the equation of wave you know the equation of wave would be it not only just depends on x it will depend on y and z also okay so sound is a 3d wave but we are not talking about it right now understood enough now what is the next thing huh sorry so y is equal to f of x minus vt is the wave equation all right so basically there is a there is a sinusoidal wave what is the sinusoidal wave a sinusoidal wave is a wave that follows sine curve while moving it can be sine it can be cosine because sine and cosine are exactly same the shape of the curve okay going forward our attention is this sinusoidal wave okay so this equation represent a sinusoidal wave equation a sine of write down kx minus omega t plus phi this is the equation of a sinusoidal wave now looking at this wave equation can you tell me what is the velocity of the wave everyone what should be the velocity of the wave this should be function of x minus vt so what is the velocity of the wave everyone okay so basically how will you write it like this x minus vt format how will you write it like this you take omega you take a k common okay you have to make the coefficient of x as 1 how do you do it like this straightforward take k like this it becomes x minus omega by k times t plus phi right so now this is what it is x minus vt so the velocity of velocity of the wave is omega by k i hope it is clear to everyone how to find the velocity of the wave okay a is called amplitude write down a is the amplitude the value kx minus omega t plus phi this is the phase okay so three things we have already found out amplitude phase and velocity okay so now let's study this equation little bit more because there are many things to find out from there if you differentiate y with time what what exactly you will get dy by dt will give you what at a particular location x as in x is fixed you will get it as minus of a omega cos of what is this if velocity of wave is omega by k then if i differentiate what i get what it is at a particular location i'm keeping x fixed i'm not differentiating x my question is if i differentiate y with time what i get usually i get velocity right when i differentiate you remember shm equation x is equal to a sin omega t plus phi when you differentiate that you get the velocity of the shm but here velocity of wave is already omega by k you're found out what you get when you differentiate that what is this who is moving in the y direction is is the is the wave moving in the y direction who is moving in the y direction particles particles are moving in the y direction okay so this will give you basically velocity of the particle okay this is basically velocity particle only now a very interesting thing if you find it interesting vp is equal to minus of a omega root over 1 minus sin square i'm writing cos theta as 1 minus sin square theta nothing great about it this fine now i'm okay yes correct now i'm putting a inside this square root so i get minus omega root over a square minus a square sin square kx minus omega t plus phi what is a sin kx minus omega t what is this why this is why so velocity of the particle is minus of omega root over a square minus y square so can you recognize that this is an shm equation okay this is an shm equation so basically what particles are doing particles are performing shm if it is a sinusoidal wave okay so write down if if it is a sinusoidal wave particles perform shm and where's the mean position look at the equation can you tell me at what y the mean position like what is the y coordinate of the mean position with mean position as y is equal to zero so all the particle it is not just that just one particle all the particles are doing shm like that okay and the mean position is the common this thing a straight line what does negative sign mean in vp see negative sign means that as the t increases rather than moving up it first goes down that is what it means simple as that it's like you know you can see that there's a sine wave it's a sinusoidal wave let's say starts like this as it moves forward how the particle how the wave will move wave goes like this okay wave goes like this so this particle which was here it went down rather than moving up okay so that is a minus sign okay so don't read too much on it it's normal it's okay clear all right so now we have understood few more things we have understood that particles are doing shm and the mean position is at y is equal to zero let us find out few more things so if I again differentiate it I'll get acceleration of the particle and I don't need to differentiate it again and again because if it is an shm acceleration of the particle would be omega square omega square y because oscillation is happening on the y axis okay it can move up also depending on what is the value of phi okay don't worry about it this is a wave sine wave moves forward so where this particle go this has to go down no in order to lie on the blue line okay there are all these particles they move down all this particle move up okay these particle move down okay so we have learned about the fact that the particles are doing shm we have learned how to find the velocity of the particle we have learned how to find the velocity of the wave we know what is the amplitude and what else we knew something we know what is the phase okay so now let's see if we can find out something else also again write down this a sine kx minus omega t plus phi now when we you know hear about the wave what are the basic factors that come in our mind basic things for example when you hear shm basic thing that should come in your head is what is the time period of shm what is the frequency of the shm so when you hear about the wave what is the first thing that should come in your mind correct right so there are few things that should come in your mind and wavelength is definitely yeah wavelength is definitely one of the things which will strike in your head okay so let us first see what is wavelength wavelength means what can anyone describe what does wavelength mean I will draw a sinusoidal wave here what is a wavelength everyone should answer I'm not asking a very difficult question do I what is the wavelength distance between two consecutive crafts troughs and crusts two consecutive waves okay time taken time taken particle to complete a section it is a length it is not time distance between the two particles okay great okay now till now we have been learning about the wavelength as distance between two consecutive crests and troughs like this is the wavelength or this is the wavelength fine so this is what we have learned that wavelength is this now why they are telling you this because it is easy for you to gaze what is a wavelength like this but this is not a definition of the wavelength okay this is how you measure the wavelength okay the definition of wavelength is this it is the minimum distance right down minimum distance along the line of velocity velocity of wave minimum distance between two points that are in phase now can anyone tell me what does in phase means in phase means what particles are doing SHM in phase means what same time interval okay three all the particles are moving up and down both have the same displacement why at the same instant okay so can you tell me are these in same phase this point and this point are they in same phase these two point number one two three four which one of them are in phase it appears as if one and two are in phase but they are not in phase means in phase means moving together very layman words moving together like you see two particles they are simultaneously at the same level moving up and down together like this so if I look at the movement of the wave wave move forward okay this particle goes down two is moving up so one and two are not in phase okay similarly you can see that three is moving down so three and one could be in phase four is moving up so two and four are in phase one and three are in phase okay so distance between one and three is the wavelength distance between two and four is a wavelength okay and of course if you talk about the crests okay both the crests they will move down only so both are in phase itself so it makes sense to tell you when you were in a lower standard that distance between two consecutive crests and or two consecutive troughs if you look at the trough they'll move up that they can't go down further because they're already at the lower most okay so this is the way the wavelength is defined a definition is one thing finding out is another thing so if I tell you that this is the equation a sign of kx minus omega t plus five by the way one more thing we talked about the wavelength being the distance between two consecutive uh points having same uh phase tell me that with time things are changed okay so with time it is not that um what I'm trying to say is that uh suppose you take two crests and two troughs so this point will no longer remain crests after let's say few moments this will go down this will go down so how will you pinpoint one location you pinpoint a location by assuming what you're doing is you are taking at some fixed time you take a snap of the wave what I'm basically trying to say is you take a picture of the wave at a particular instant then on the piece of the paper you look at the two points and find out the distance it should not happen that while you're finding the distance wave is already moving forward understood at a particular instant you need to freeze the wave and then look at it I hope I've made myself clear here so can you find out the wavelength from this equation find out the wavelength okay Aditya got something others that's okay let us try to do this suppose wavelength is lambda suppose wavelength is lambda then whatever is the location why whatever is a location why same location should be there at k times x plus lambda because after every lambda the phase should repeat itself do you all understand say these two should be equal yes or no type in yeah what I'm saying that I'm saying lambda is a wavelength so whatever is happening at x same thing should happen at x plus lambda also because the two points are in phase so rather than x I can write x plus lambda as well so that is what I did fine so I need to equate this to this now this I can write as a sign kx minus omega t plus phi plus 2 pi I can add otherwise lambda will come out to be 0 and I can add 2 pi any number of times sign function is repetitive so I will add minimum repetitive cycle which is 2 pi I can add 4 pi also but 4 pi is like taking this crest and the other crest you get 2 times lambda okay I want minimum distance so 2 pi I'm adding so when you equate these two you'll get k lambda to be equal to 2 pi so from here lambda will give you 2 pi by k so if you write the wave equation like the way it was written the wavelength would be 2 pi by k fine k is by the way referred as wave number all right so we have got the wavelength also now the only thing left is frequency in order to find frequency we can find the time period and then take the inverse of it so all of you take this and find out the time period what is the definition of time period by the way what is the time period definition it is the minimum time minimum time taken by the particle to complete one oscillation can I repeat what I said I have written everything whatever I said I was defining time period okay Pranav got it others am I going very slowly I somehow feel like I'm going a little slow tell me is the pace fine should I increase the pace okay so now now if time period is capital T then whatever was happening at t equal to small t same thing should happen at small t plus capital T so this a sign kx minus omega t plus capital T plus phi okay these two are equal if I directly equate it I'll get capital T to be zero so what I'll do I'll add 2 pi and then equate a sign kx minus omega t plus phi plus 2 pi so from there you get omega into capital T to be equal to 2 pi so t is equal to 2 pi by omega all right so from here the frequency is omega by 2 pi hertz okay minus see minus you can add and subtract 2 pi and time period cannot be minus right so even if you're getting minus take a mod so I can as well as subtract 2 pi now ha now frequency is omega by 2 pi and wavelength is uh 2 pi by k meters so you can see that frequency into wavelength is equal to the velocity of the wave okay have you heard of this wave equation for every wave frequency into wavelength should give you the velocity of the wave everyone have you heard of it that frequency into wavelength should give you the velocity of the wave okay so this is perfectly matching these are the kinematics details of the wave let us uh let us now take one example from your textbook only okay what we will do is that probably will our syllabus gets over one or two classes sooner so we'll have a separate problem solving session of main or advanced level for the waves right now the focus is cool okay do it yourself don't open the book totally defeats the purpose else do this done I think couple of parts are straightforward you can tell me a amplitude is 0.005 meters okay anybody got wavelength lambda is 2 pi by k k is what k is 80 here so you get pi by 40 okay meters c um period t 2 pi by omega which is 2 pi by three seconds okay then uh frequency is 1 by time period so 3 by 2 pi hertz okay what else they're asking amplitude period wavelength what is the speed of the wave speed of the wave is what omega by k right so 3 by 80 meter per second this is the speed of the wave now um this one the last part try doing it yourself try doing it yourself okay no not in terms of sign of an angle you have to tell me the exact answer why is what 0.005 sign of x equal to 0.3 0.3 into 80 is 24 minus 60 so this is 0.005 sign of minus 36 what is sign of minus 36 minus 36 radiance is how much can you find out without using calculators it is in radiance okay it's in radiance so is 36 is too big an angle or it's a small angle 36 radiance is a huge angle okay it's a big angle very big so what you have to do is bring it down between 0 to pi at least how you bring it down between 0 and 2 pi 0 and 2 pi what you do to bring it down sign has a period of 2 pi so you can add 2 pi multiple times you can add 2 pi multiple times so you can keep on adding 2 pi multiple times 2 pi is what 2 pi is 2 into 3.14 so you can multiply 6 into 3.106 actually 12 12 into 3.14 how much it is you can add 12 pi how much is 12 into 3.14 36.68 36.68 minus 36 becomes 0.68 I hope this calculation is correct of yours 0.68 0.68 is how many times pi are you sure this is 36.68 I don't think so 37.68 37.68 so this is 1.68 1.68 is roughly pi by 2 yes or no so this is 0.005 sign of pi by 2 which is approximately 0.005 all of you understood this calculation this calculation part is important is it clear to everyone sir should we add 2 pi it 2 pi he added yeah you have added 2 pi six times that is what 12 into pi you added 2 pi six times you can add as many times as possible type in is it clear to everyone other is it clear Arjun fine so let us move little bit more so we will now talk about see kinematics again kinematics of the wave is very similar to kinematics of the SHM okay so we'll be taking few more questions but let me finish the second part of the chapter also that talks about the velocity of the wave then probably we can take a lot more questions right down the speed traveling wave why we are calling it as a traveling wave because there can be standing wave also what is a standing wave have you seen in the guitar when you pluck the string it just vibrates like that okay that is a standing wave it doesn't go anywhere the speed of a traveling wave depends on first of all medium and second it also actually depends on whether it is transfers or longitudinal okay these two things it depends upon so let us try to see how it is related to the medium okay once you tell me that it is a transverse wave then it only depends on medium right so now on which properties of the medium should the velocity of the wave depends upon what do you think what is the how the medium should be how the medium should be to support the propagation of wave to support the wave propagation how the medium should be medium should be dense okay so what the medium should do medium should do what shm medium should do shm okay in order to do the shm what do you need at the mean position force is zero when it is oscillating but it should still go down and move up even if force is zero so inertia is required if inertia is not there then the particle won't do shm because as soon as force is zero the particle stops because of inertia only even though at the mean position force is zero it keeps on going down okay so inertial property of the medium will support the wave propagation what else what else this is an important property of the medium another property of medium disturbance is let's say here I have started the disturbance here and somehow the wave has propagated and entire thing is started oscillating how is that possible how is it possible that this point got disturbed because of that point point number one and point number two two got disturbed because of one the reason is elasticity good so elasticity elastic property and inertial property these are the two properties on which the speed of the sound sorry I'm saying sound again and again the speed of a wave will depend upon there is no other thing okay so let's take an example of transverse wave on a string okay so if I take an example of transverse wave on a string then I need to take let's say I have I can choose anything right I can say that mass will represent inertial property and strain will represent elastic property or Young's modulus or bulk modulus will represent elastic property anything okay so it is up to me what to take so it makes sense to take the inertial property of a string you should not take the total mass of a string because your string can be practically very big so infinite mass all right so it it's not the entire mass that matters it is about how it is distributed what is the density of the mass mass per unit length okay represents the inertial property write down inertial property is represented by represented by mass per unit length which you can say is mu okay mu I mean if you see this mu coefficient of friction will come in your mind but then we take mu here as well now for elastic property when we talk about the string what is the first thing which will come in your mind with respect to mechanics with respect to mechanics what is the thing tension t okay so elastic property is represented by tension t now some of you will be like how do you relate to elastic property from where this comes from the fact is that we say that the string is uh inextensible and all those things but if a string is inextensible tension will never come okay because there is a strain in the string that is why there is a stress in the string stress in the string is represented by the tension only okay but that small strain is negligible so I can very well say that tension can represent the elastic property okay so now the velocity of the wave depends on mu and t I hope till now everything is clear I want you to use dimensional analysis please use dimensional analysis and come up with the equation do the dimension analysis you know right how to do the dimensional analysis okay pranodharma gadi got something did you get your physics ut marks from the school how was it okay so velocity is proportional to how you do it mu to the power a and t to the power b like this we do dimensional analysis right I can say that velocity depends on mu and t like this so dimensionally I can write down this dimensional equation l t minus one this is equal to dimension of mu is what mass per unit length ml minus one raised to power a tension is force mass acceleration so ml t minus two so this can be written as m a plus b l b minus a t to the power minus two b okay so if I compare it minus two b is equal to minus one from here b is equal to half and coefficient of mass a plus b should be zero so a should be equal to minus b which is minus half okay so then you can check coefficient of l is b minus a minus one okay b minus a is not minus oh sorry I don't have to check that that is coefficient of t t is minus two b that we have already equated b minus a should be equal to one and yes it is one so it matches so b is equal to half and a is equal to minus half so v is proportional to mu to the power minus half and t to the power half so from here we'll get velocity being proportional to root over t by mu okay and you will get with proportionality constant c under root t by mu and experimentally and surprisingly it is found out that c is equal to one so velocity is root over t by mu is this clear to everyone anyone has any doubt quickly type in no doubts okay let's take few questions on it so I have let's say let's first take the school textbook thing find out the answer okay all of you do this this is from your school textbook a direct formula substitution you have to do it okay you cannot be lazy to not to do it okay Pranav got something others all of you should get the answer roughly you can tell the answer if not exactly you can tell between this and that the answer Pranav if four options are there between 84 and 96 they can a big range okay I think it's a direct this thing I don't need to discuss much on it mass is given a steel wire of this length is there so first what you do you find out the mass per unit length that is 5 into 10 to the power minus 3 divided by 0.72 okay this is mu tension is 60 newtons okay so the velocity is root over t by mu direct substitution t by mu and a divided 0.72 will go in the numerator now don't expect me to tell you how to multiply and divide I'll directly write down the answer which is 93 meter per second okay let's take couple of more numericals so there is a string okay the string has mass per unit length as mu okay mu is the mass per unit length this length of the string is l this length is l mass per unit length is mu and capital M is actually very large compared to the mass of the wire so don't worry about taking this mass including that mass it is negligible okay now what is given to you is that this entire thing is accelerating upwards with acceleration of a you are basically inside the lift this entire set up is there inside the lift and lift is accelerating with a upwards what you do is that you disturb the wire here little bit you just plug it like a guitar string and this disturbance starts moving this disturbance is moving from this end to that end you need to find out how much time the disturbance will take to go from here point a to point b find out time taken by the disturbance or the pulse from a to b yeah this has mass I am telling you to ignore this mass because mass which is hanging is very big okay Aditya got something did you get the value of tension on the string how much is the tension on the string draw the free bar diagram here this is t this is mg correct mg this mass is actually accelerating it is accelerating with expression a right so I can write down t minus mg is equal to m into a I am getting t as m times g plus a okay this is the tension t and the velocity of the wave is root over t by mu t by mu this is velocity that's all I mean once you get the velocity of the wave the time taken will be l divided by velocity so this is the answer this I think came sir what is what is mass not what do we find t what if mass this mass if you don't ignore you find out tension here okay nothing different it would be this mass is let's say small m then your tension over there will be equal to n plus a small m into g plus a that's all that's the only change we can take tension constant yes tension is constant why not tension is a constant see there is no variation of tension due to gravity on this horizontal string gravity is vertical okay any other doubt anyone has there's no variation the see the pulse will go from here to here only I'm not one I'm I don't care about movement of the disturbance here okay no no no pronoun if you consider the pulse moving on this then yes tension changes and velocity will also change as you go from here to there okay but throughout this string even if you don't ignore mass tension will be constant you're assuming the string to be horizontal which it cannot be if there is a mass the string cannot remain horizontal but hypothetically we are assuming clear let us take one example of variation the velocity like more I can see that some of you are constantly asking about that thing so let's take this final example before we take a break there is a string which is hanging like this okay a uniform mass density string total mass of the string is m and length of the string is l okay it is hand from one of its ends like this what you do is that you create a small disturbance at the bottom this disturbance will move up okay you need to find out how much time it will take for the disturbance time taken by the disturbance to go from bottom to the top most point find out to go from bottom to the top most point tension is constant or it keeps on changing everyone tension constant how it is constant when you when you hand yourself from hand like this on let's say on a rod you feel maximum tension where on legs or on hands where you feel the maximum tension on hands so tension is changing it is not constant okay so if tension is changing what you can do is that you can find out tension as a function of x distance from here is let's say x at a distance of x can you tell me what is the tension over there all of you how much is the tension over there at a distance of x Aditya got something others okay just do the free boy diagram right take that much piece of the row and show the forces that's all how much is the mass of this much x length mass is how much m by l into x this is the mass of this portion that into g and this is t it is in equilibrium there is no acceleration anywhere so you can write down t minus m by l x g is equal to 0 so t is equal to m by l g times x do you all understand this this is t all of you understand this type in so velocity is what velocity is root over t by mu so at this location velocity is t by mu so m by l g x mu is how much what is mu mu is m by l only so good thing is m by l m by l gone so velocity is root over g x okay so then how you find time taken for this pulse to go from this point to the top most point what to do next we can write v as dx by dt dx by dt is root over g x fine so dx divided by root x is equal to root g into dt what is the integral of dx by root x dx by root x what is the integral two root x or one by two root x what is this integral x to the power minus half dx integral is x to the power minus half plus one divided by minus half plus one so it is two root x okay don't make such errors so two root x when you put the limits it become two root l this is equal to root of g t so time taken will be two times of root of l by g so this is how you can deal with the change in the velocity and there is no point finding the acceleration of the wave why there is no point because wave is not a mass for a mass if you find acceleration you can apply Newton's second law and say force is equal to mass and acceleration but over here even if you find acceleration of a wave there is no mass you can't say mass and acceleration is something there is no physical meaning of the acceleration of the wave so we don't focus on it we focus on the velocity for a wave fine so let's take a break now we will take a break and we will meet huh wave has acceleration why not this do don't you think this is accelerated v is equal to root g x is it constant velocity it can actually but acceleration has no physical significance so that's why we don't care about it all right so all of you we will meet at 6 27 pm come back in time fine so let us move forward with something new we we have discussed about the transverse wave on the string all right now let us discuss about the longitudinal wave okay longitudinal wave little bit of discussion is required first on the equation of it okay because whatever discussion we had we always assume that we are talking about the transverse only longitudinal wave slightly different from the transverse wave because in the transverse wave the oscillation happens on the y-axis and the wave travels on the x-axis like this okay oscillations are happening like that so this is transverse in the longitudinal what happens is that these are the particles let's say particles the medium they are oscillating like this okay the oscillations happen on the x-axis itself and the wave also propagates on the x-axis these are velocity wave also oscillations also happen like that and velocity also like this so i'll just show you a small the demo of the longitudinal 1d longitudinal wave it'll be clearer than so spring see look at this you can see that all of you how wave is traveling disturbance is moving in the same direction in the direction of the oscillation of the particle particle here is this spring this was slingy right fine so this is the small demo of how longitudinal 1d longitudinal wave will be okay so the equation of the transverse wave when we write we like write like this a sin of kx minus omega t plus five how do you think you can write down equation for the longitudinal wave there is nothing on the y-axis so you can't write y is equal to something for longitudinal then what you write instead of y instead of y you write x but x is here x is the x coordinate what is y here here can can you tell what is y here you can say y coordinate okay fine that is y coordinate but physically what is what does y represent physically distance from the correct distance from the mean position mean position now with with the transverse wave the good thing is that mean position lie on the y is equal to 0 for all the particles mean position is y is equal to 0 okay over here the mean positions for the longitudinal wave will be at different different locations so these are the mean positions x coordinate is different and oscillation is happening on the x coordinate only so here also there should be something that represents distance from the mean position it cannot be x coordinate because the mean position of the x coordinate itself is different different so we say that s is a variable which tells us distance from the mean position for that particle so s is equal to a sin kx minus omega t plus phi okay now if we plot rather than plotting the wave like this if we plot on the y-axis s rather than the y coordinate if you plot s and this is x you will see that it looks exactly same as the transverse wave okay but then this is not y-axis it is distance from the mean position all right so mathematically both the equations are same both the equations mathematically they are same so whatever we do going forward with respect to the transverse wave same thing is valid for the longitudinal also keeping in mind that we are taking s instead of y is this clear to everyone type in is it clear to everyone okay so so you know the most common form of the longitudinal wave is the sound wave only okay so we can find out the speed of sound as well which is of our interest okay but before we do that first try to let's try to find out what is the equation for the longitudinal wave and you know the basic difference between longitudinal and transverse wave is that transverse wave usually I mean most of the time is on the surface okay because the particle have to move up and down all right at the boundary this transverse wave comes longitudinal wave is usually inside the medium one is on the medium other is inside the medium okay so it's like if you talk about the surface of the water water level on the surface there can be transverse wave water ripples will be there but inside the water water ripples cannot be there but if wave exists it will be longitudinal wave okay similarly when I'm speaking let's say I'm talking to someone okay so I will disturb the particles which are near my mouth when I speak I disturb those particles and that disturbance will propagate and then it reaches your ears and you hear that okay so that is the physics related to the how you hear it once it goes inside your ear it is by biology I will not discuss that so depending on which medium you are your sound quality will also differ if you are in a vacuum you'll not be able to talk because there are no particles to propagate the sound waves it needs medium to propagate okay if you inhale helium and then try to speak you'll see that your voice will be completely changed okay so depending on medium your sound will be different how longitudinal wave converted to transverse wave it's a disturbance only it can get it will disturb the longitudinal wave can reach the surface of the water and on the surface same energy will get converted into the transverse wave as well okay okay so now let us talk about we have seen the transverse wave equation on a string okay let us see the longitudinal wave what is the velocity of it on what it depends on so one thing which we all agree on is that longitudinal wave is inside the medium you are immersed inside like I am immersed in the in the room which has gas oxygen nitrogen some dust particles so inside the medium I am and then I am speaking so that's how longitudinal wave works so like we already discussed the velocity depends on the medium and there should be some elastic property elastic property and there should be some inertial property of the medium what do you think inertia property should what parameter shoots best once you are let's say inside the medium which inertia property will be best suitable in a string it was mass per unit length what do you think here density right volume density rho it is volume density okay elastic property is taken by the bulk modulus the bulk modulus we are talking about flutes here okay so can you find out if velocity depends on the bulk modulus b to the power a and rho to the power b that use dimension analysis dimensional analysis find out the equation of the velocity of the wave all of you do you know what is the formula for the bulk modulus do you know this b is what delta p divided by delta v by v minus this is the bulk modulus formula okay prana got something see dimension of bulk modulus is a dimension of pressure only delta v by v dimension is zero nothing it is a ratio between the two volumes denominator anybody else velocity is l t minus one aditya also got it pressure is force per unit area so m l minus one t raised to power minus two power a density is mass per volume this will be m a plus b l minus a minus 3b you should have done it yourself instead of waiting for me to do this okay it is if you're just copying it it is as good as you're copying from a book it is i don't know it doesn't help okay so minus two a is equal to minus one so a is equal to half a is equal to half and a plus b should be equal to zero there is no m here so b should be equal to minus half now i have to check whether this is valid for l also is minus a minus 3b equals to one is minus a minus 3b is equal to one if it is then these are the correct ones you can see here minus a minus 3b so this is one only okay so it works so a is half and b is minus half so velocity is under root b by rho into some constant and surprisingly the constant of proportionality is one experimentally c is equal to one so velocity is root over b by rho okay now this is the velocity of the longitudinal wave inside a medium so using this can i find can i find speed of sound in our atmosphere the answer is of course yes that is why i have written this thing so now if it is yes then you know density of the air is 1.29 kg per meter cube on the surface of the earth approximately but the bulk modulus is not i'm given to you okay so can i find bulk modulus which is equal to minus of delta p by delta v by v so in a differential form it can be written as minus v dp by dv do you all understand what i'm trying to do here everyone have you understood till here whatever i am done so let us try to find out the bulk modulus of the of this process in which sound is traveling okay so there is one assumption we need to make what kind of process it is because for different processes bulk modulus is different okay for example in an ideal gas if you talk about that process of traveling of the sound is isothermal then v into dp by dv will be different if you say that it is adiabatic then v into dp by dv will be different so depending on the process bulk modulus will be different for the same gas please write down depending on the process v is different for same gas it is not a constant of a gas it is a constant of gas and the process which process it is so we need to assume something so Newton was studying it Newton is here also so Newton assumed that it's isothermal it's an isothermal process the propagating of the sound propagation of sound is isothermal process and it sounds very reasonable that of course it has to be isothermal the temperature should not change of the atmosphere so everything is happening at a constant temperature only so pressure into volume should be a constant c so if this is true if it is isothermal process p into v is a constant can you tell me what is the bulk modulus for an isothermal process minus v into dp by dv should be what find out the hint is differentiate it with respect to volume okay aditya got it others nobody else this could differentiate the sorry this you differentiate with respect to volume what is the derivative of a constant zero so right hand becomes zero only aditya got it okay so if you differentiate with respect to volume then I'll have p plus v dp by dv is equal to zero so from here minus of v dp by dv is equal to p so bulk modulus is the pressure itself if it is an isothermal all of you understood this is this clear that bulk modulus is p the atmospheric pressure type in is it clear all of you go through it differentiation for a minute differentiation is mechanical see p into v is a constant when you differentiate first you differentiate v what is the derivative of v dv by dv then you differentiate p so it's a chain rule right you have learned it in the bridge program this is equal to zero so dv by v is one p plus v dp by dv is zero that is what it is any other doubt is there a lag pranav is facing lag pranav always you face lag maybe some internet issue at your end maybe there's no lag all right so this is p now velocity of the longitudinal wave is under root of b by rho so velocity of sound is what in the atmosphere it is atmospheric pressure divided by density of the atmospheric gases so if you substitute the values atmospheric pressure is what 1.01 into 10 raise to power 5 divided by density of the gases around 1.29 this should be the velocity velocity of the sound okay and this comes out to be this comes out to be 280 meters per second all right this is what Newton calculated and during that point in time I guess there was no sophisticated instruments to exactly find out whether it is experimentally correct but given the reputation it was assumed that that this is correct but later on people found out that this is not the correct value of the speed of sound so there is something wrong and if there is something wrong what it would be the assumption would be wrong okay and what was the assumption that it is an isothermal process right the formula cannot be wrong formula is correct but when we write b is equal to p we are assuming it to be an isothermal process it is not so so Laplace came with its correction right down this is a Laplace correction he told that the travel of the sound travel of sound wave is a rapid process exchange heat and hence what is the process then hence the process is what kind of process it should be the process is closer to adiabatic okay and if it is an adiabatic process then p into v is not constant p into v raised to power gamma is a constant okay now I want you to do exactly the same exercise as just now and tell me what is a bulk modulus now find out bulk modulus is what now if this is true as in adiabatic process anyone okay so aditya got something others just differentiate it with respect to volume okay adiabatic got it anybody else only adiabatic and aditya got it some more time 30 seconds okay so now if I differentiate it I will get p into gamma into v to the power gamma minus 1 I'm applying chain rule okay differentiating v to the power gamma and then I will differentiate p so v to the power gamma dp by dv is equal to 0 okay so I can cancel out this from here so v will remain so v dp by dv is equal to minus of gamma p so minus of that is plus gamma p okay so bulk modulus is gamma times p fine so can you tell me gamma for the atmosphere gamma is what for the atmospheric gases everyone what kind of gas the atmosphere has largely diatomic so what is gamma for diatomic seven by five okay correct seven by five so now using this correction if I use the formula become gamma p by row bulk modulus is gamma p so this is under root of seven by five into 1.01 10 ratio power five divide by 1.29 okay this comes out to be 331 meter per second this is much closer to the actual value okay all of you have understood till now Laplace correction everybody understood are you guys facing difficulty in differentiating it I didn't get what kind of problem you guys are facing it's clear right all of you okay so this is the basic theory of the kinematics and the let's say the velocity of the wave how we you will find out okay of course we'll have a problem solving session in which we'll discuss a lot of problems right now since we have only let's say half an hour I'll just introduce to you our superposition okay little bit of that