 of motion the particles are doing when it comes to transfer itself. Particles of medium, this particle came down so after that what will happen to this particle? So overall what it is doing? Is it SHM? The particle of medium will go from here to down and then will further move up. Is it the kind of movement it will do? Right it will just do an oscillation it need not be SHM. It will be SHM for a sinusoidal wave but if I talk about in general it will be an oscillation with mean or the mean position as this line. All the particles mean position will be this line. Okay so the particles are doing the oscillation with mean position as the x-axis getting it? Okay what about here in the longitudinal wave what the particles are doing? Again it is a periodic motion or oscillation fine. So basically the particles are doing oscillation only whether you talk about transverse or longitudinal the kind of motion the analysis is exactly same just that in this case it is vertical over here it is horizontal. Okay so going forward whenever we are going to introduce a concept we are going to study just transverse and after studying that concept in transverse we are going to just make small adjustment for the longitudinal. We do not have to derive everything again for the longitudinal because essentially at the basic level both are similar. Getting it? Okay now tell me what happens when you speak in the air what happens what kind of wave it is by the way? Longitudinal wave right so what exactly happens when I speak in the medium? Air gets compressed. Air gets compressed and then it gets you know so compression and rare fraction okay so basically it's a pressure wave sound is a pressure wave so when the air gets compressed pressure increases there's a rare fraction pressure decreases are you getting it? Okay so this is just definition of what is wave now till now we have only known what is wave in a qualitative manner we need to jot down some equation and say that this is the equation of wave okay so right now our purpose is to find out what will be the equation of the wave okay so write down topic equation of wave once I get what is the equation of wave half the job is over because everything that can be defined for the wave is inside the equation of wave. Let's take an example like this this is my x-axis and this is my y-axis got it? I have drawn a portion of the wave okay isn't here equation of wave will be what what kind of equation it will be just tell me function of x, y, t like that what will be function of I will first tell me what I am trying to find equation of wave so what it is actually like what I am trying to put in the equation equation of what equation of wave and wave is what this this is the wave so I am trying to give you a function which will plot this at any time at any location x if you tell me what is x and t I will tell you what is y that will be the equation of wave yes or no so I am trying to find out what is this function y is equal to f x, t is there any doubt here this is what I am trying to find out which function of x, t definitely it can't be any function of x and t so x by t square is not a wave equation even though it is a function of x and t but it is not a wave equation so I need to find out what is a combination of x and t that will generalize the equation of wave so I am trying to find out what is that function x comma t okay so let's say that x is this okay okay so this is let's say at this point x is the coordinate and t is a time okay now can you tell me this is this is by the way the value of y right from here till here okay now tell me if velocity of wave is v it is let's say triangle like this v is the velocity okay so now you have to tell me that at what time same y was there at x equal to 0 and t equal to t y is y over here so at what time this y was at x equal to 0 see I am asking you this y which is there at x equal to x and t equal to t I am asking you at x equal to 0 when the same y was there what was the time t minus x by v x is the distance it has taken a time of how much x by v to travel from here to here if x is a distance right this entire wave as it is moved forward so the distance x from here to here it will take a time of x by v so same y was there at x equal to 0 and t equal to t minus x by v edit out okay so function of 0 comma t minus x by v is nothing but function of t minus x by v or you can write it as a function of vt minus x by v now v is constant so function of constant doesn't mean anything so this is function of vt minus x okay so if there is a wave equation it has to be a function of vt minus x or x minus vt same thing a function of t minus x by v give me one example which is a function of x minus vt give me equation of wave okay I tell you e to the power minus of x minus 2t q divided by x minus 2t raised to power 2 by 3 is this a equation of wave that you put a minus over here this is the equation of wave is this a function of x minus vt yes the function of x minus vt so what is the velocity of wave 2 2 meter per second in positive x direction and this is the equation of wave getting it it is a function of x comma t but more than that it is a function of a combination of x and t which is x minus vt are you getting it okay so but then this this is not a wave equation e to the power t cube divided by x minus t whole square this is not a wave equation so but v can be okay because here t is sitting alone it has to be x minus vt together as a single variable getting it so if that is that is treated as a single variable in an equation then that will be a wave equation so this is a wave equation okay and here the here when we are deriving it when we are deriving it we are assuming what we are assuming the wave is traveling in positive x direction please write down a wave traveling in positive x direction has this equation any function of x minus vt positive x direction can you tell me wave traveling in negative x direction what will be the function function of x plus vt this is negative x direction okay is it clear which one i see random random it could be anything okay it could be x minus 3t here and 3t there but it can't be 2 here and 3 here okay it could be like this you know that it is it's like this e to the power minus x minus 2t whole cube divided by 4x minus a t it can be like this i can take 4 common it will be x minus 2t so you can rearrange the term to check whether it is a function of x minus vt this is a wave traveling in positive x this is in the negative direction any doubt any other doubt quickly ask you just started the chapter it is a cos wave or sino wave just there is a phase difference of pi by 2 i'll not write it how does it repeat the function if you like function of x minus vt does it repeat multiple times what do you mean repeat like the same or the same thing keeps coming back huh it is just the function of x minus vt here so that is the sole reason why we have defined it see guys listen here anyways so equation of wave we have discussed a generic equation now we are going to make up a specific equation okay the specific equation sinusoidal equation i am writing it as a sin see at times we confuse the motion of wave with the shm okay it is why doesn't need to you know you're dragging the wave so even when you say that it has to do shm oscillation you need to think before you know that what i'm trying to ask okay so basically if you keep on increasing time okay so the why location should repeat after every time because why is the tracking of the particle okay and the way we have derived the function what was that we have already assumed that somewhere something is happening at x equal to x let's say something is happening same thing should have been happening at x equal to 0 also the the way we have derived it itself automatically assumes that okay and this is a special kind of wave sine wave and this is also called simple harmonic wave why it is called simple harmonic wave because the particles in this case are doing simple harmonic motion and you tell me what is the velocity of wave quickly tell me omega minus v t is it a function of x minus v t this is a function of x minus v t or not so where is v over here omega divided by k omega by k you have to see here coefficient of x should be 1 take k common everything will be divided by k then x minus omega by k t so the velocity is omega by k positive x direction negative x direction positive x direction so write down velocity of wave and it is omega by k positive x direction function yeah function is it just omega by k velocity is constant omega by k okay now why we have written like this k x minus omega t because it becomes simpler that we are going to learn about it later on but whatever we write it has to match with this that we have just derived now this term inside the sine term is called a phase of the wave so write down k x minus omega t plus phi represents the phase the location the y coordinate will be same for the same phase it is unknown so another way to find out the velocity of wave is differentiate the phase so basically what is happening is let us say this is a wave okay this wave move forward all I have to do is see where this line goes this line goes here right so I have to see with what velocity is line is moving so I have to see the constant phase with this point and that point are in phase okay so the constant phase point with what velocity it is moving that is what I have to find out right so phase is constant and if differentiated I will get dx by dt minus omega i is equal to zero so dx by dt gives me the velocity of that constant phase point which is omega by k okay so another way to find out the velocity of wave is to differentiate the phase whatever is there inside the phase so why is the why is the derivative of the phase it's a constant phase right partial derivative this is partial derivative actually phase physically same thing like what in shm that there was a phase right similar case in case of the wave also in case of shm there was only time factor because I don't need to bother about you know the particle is same here I am tracking all the particles so even x comes in the equation anything else