 you know, if you fix x, if you put x equal to 0, you will get a equation of S A j m or not. So, let us talk about that as well. At a particular x, at a particular x equal to x naught, let us say, the equation will become what? a sin k x naught minus omega t plus phi. So, I can write it like this, a sin minus omega t plus theta or I can write it as minus of a sin omega t minus theta. Theta is what? Theta is my k x naught plus phi. So, this looks like an equation of S A j m, right? So, wherever you go at whatever x, you find that all the particles are performing S A j m. Now, tell me how will I find the velocity of the particle? I have found out the velocity of wave till now, okay? Velocity of wave is omega by k. This is what we have derived till now and we have found out that all particles, they perform a sin j m along y axis, okay? Now, tell me how will I find velocity of particle? V p, how will I get that? For velocity of particle at a particular x, I am asking. So, if I differentiate y, will I get velocity of particle? Is dy by dt velocity of particle or not? Right? Particle is moving along y axis. So, dy by dt keeping x constant because particles velocity you will find at a particular x. This will be A minus of A omega cos of k x minus omega t velocity of particle as expected at a particular x, the velocity of particle is having the equation of the S H m. You can see that this you can write down as omega under root of A square minus A square sin square k x minus omega t plus phi and this can be written as omega under root A square minus y square. So, this is the velocity of particle which looks exactly like the S H m equation, okay? And there is a mean position at y equal to 0. For which particle? For all the particles. Now, just to tell you some relation because it is not systematic. So, it comes in bits and pieces that, okay, this is this. So, do you know this? Now, this. So, there is no correcting story as such. So, that is why I have to move like that. So, can you tell me what is the slope of the shape of the wave? So, this is a wave, okay? So, suppose at x comma t, I am trying to find out what is the slope of the wave. Are you getting? So, basically dy by dx at a particular time is how much? This is equal to what? Slope. Slope of the shape of the wave at a particular x and t. Some of these things are not in your answer on t. Anybody got it? a k cos of k x minus omega t plus phi, okay? So, if I multiply and divide the slope and the velocity of particle is minus of k omega cos of k x minus omega t plus phi. So, looking at these two equations, what comes in your mind? These two equations, what comes in your mind? Divide it, get rid of sin, sorry, cos, this, these two gone. So, velocity of particle divided by the slope is equal to minus omega by k. What is omega by k? Velocity of wave. So, velocity of particle is minus of velocity of wave into the slope at that location. Slope of the wave at that location. This relation you can use probably for graphical questions. So, at a particular location and at a particular time, if you have the shape of the wave, find out the slope of that wave at that location, multiply that slope with the velocity of wave, you will get the magnitude of velocity of the particle. I am coming to that. There is some significance of k and omega that we are going to discuss next. But right now, there are so many things, one by one we are discussing. So, we have discussed velocity of wave. We have discussed that all particles perform SHM. Velocity of particle is dy by dt and there is a relation between velocity of particle and velocity of wave. Now, this is a wave. So, if we are studying a wave, there will be something like frequency of wave and the wavelength of the wave. Yes, a is the amplitude. You can see the wave here a comes. Write down wavelength, wavelength is represented by little lambda. The definition of wavelength, anyone? So, it is the distance between any two equivalent, difference in distance between any two points which have the same phase. Or distance between two crests or two troughs, like that you have studied. So, there are multiple crests and troughs. So, which one you will take? Adjacent one. So, please write down the shortest distance, the shortest distance between the two points in phase. The shortest distance between the two points in phase is the wavelength. What does it mean in phase? We have the exact same void. We have just discussed what is phase. This is the phase. Yes or no? This inside this is phase. Now, this phase repeats after what angle? There is a sign function, sign repeats after 2 pi. So, the phase will repeat after 2 pi. So, if two adjacent ones are some distance apart and they are in phase, there will be in terms of angle, they will be at a distance of 2 pi because after 2 pi phase repeats. So, phase repeats. Now, you can say that they are in phase. So, let us say the particle 1 is this a sin kx minus omega t plus phi after let us say wavelength lambda it comes out to be the same phase. So, how will I write y 2 as a sin of what? kx plus lambda kx plus lambda minus omega t plus phi. So, with the t is different. No, I have frozen the wave at a particular time I have taken a snapshot. You cannot deal with wavelength of a moving wave. By the time let us say this is a wave. So, when you are masing the descent into crests by the time you go from here to here wave should not move. You should measure the distance at a same instant. Yes or no? I am saying that after lambda same thing repeats. So, if at x equal to x this thing is happening at x equal to x plus lambda also same thing will happen because after lambda it is repeating lambda is a wavelength. So, instead of x I am putting x plus lambda getting it. So, this can be written as a sin of kx minus omega t plus phi plus k lambda got it. Now, k lambda should be equal to what? 2 pi right because these two things are same y 2 and y 1 if they have to be same k lambda should be equal to 2 pi because after 2 pi only sin will repeat itself this thing is exactly same as that this thing is extra adding up if you add a minimum 2 pi here same y 1 will come out. So, that is the reason why k lambda should be equal to 2 pi and from here we are going to get lambda as 2 pi by k write down y 2 is what is y 2? See y 1 and y 2 are the two points on the wave have the same phase. So, this is let us say y 1 and this one is y 2 y 1 and y 2 all right. So, we have derived the wavelength to be 2 pi by k k is also called wave number write down time period what is the time period? Time period to come back to the same phase correct. Please write down time period is the minimum time, time period is a minimum time taken by a particle it is a minimum time taken by a particle to come back to the same phase. Now, can you derive how much will be the time period? It is the derivation exactly same as 2 pi by omega. So, let us say after time capital T it will come out to be in the same phase. So, a sin k x minus omega t plus capital T plus phi this will come out to be a sin k x minus omega t plus phi minus omega capital T. So, omega capital T should be 2 pi. 2 pi for it to repeat. So, T will be equal to 2 pi by omega. So, frequency of the wave will be omega by 2 pi. So, these are some straight forward relations from the wave equation that we have assumed. You assume it is sinusoidal ok any doubts anything. So, there will be lot of numericals only on this equation that will be kinematics of wave. So, sinusoidal wave is any wave that repeats the pattern. Sinusoidal has a sine wave shape. So, if it has like sharp turns. That is not sinusoidal that you can say triangular wave. Sir can you have an accelerating wave? Accelerating here it can be possible. Velocity of the wave depends on the property of medium. If property of medium changes the wave accelerates. But at a particular location x the property of medium will be constant. So, velocity at that x will be fixed ok.