Introduction to mechanical waves
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Properties of periodic waves
In the last video we talked about the idea that if I start with some type of a string there, and if I were to take the left end of the string-- I could just have equally have done the right, but if I take the left end of the string and jerk it up, then all the way down, and then back to its resting position, it'll generate this disturbance in the string. And the disturbance might initially look like this after I've done that jerking up and down once. And that disturbance is going to propagate down the string. It's going to move down the string like that. Let me color this in black. So this is right after I do that first cycle-- that first jerking up and down. The string might look something like that. And if we wait a little while, the string might look something like this, assuming that I only did that once. The string might look something like this, where that pulse has actually propagated down the string. That pulse has propagated down the string. And in the last video, we said, hey, this disturbance that's propagating down the string, or propagating down this medium-- although it doesn't necessarily have to have a medium- we called this a wave. And in particular, this wave right here, this is a pulse. This is a pulse wave because we only have, essentially, one perturbation of the string. Now if I kept doing that-- if I kept going up and down, and up and down, essentially, if I periodically did it at regular intervals, then my string would look something like this. Doing my best to draw it neatly. It might look something like this, where once again, the perturbations are going towards-- the disturbances are going to move to-- the right. They're going to move to the right with some velocity. And what I want to do in this video is really focus on this type of wave. This type of wave right here, which you can imagine, since I'm periodically moving this left side up and down, up and down, and creating these periodic movements in the wave, we call this a periodic wave. This is a periodic wave. The motion is repeated over and over again. So what I want to talk about is some of the properties of a periodic wave. Now, the first thing you might say is, hey, how far are you jerking it up and down? How far are these movements from rest? So if this is the resting position right there, how far are these movements above the resting position and below the resting position? And we call that the amplitude of the wave. So that distance right there- I'll do it in magenta-- that distance right there is the amplitude. Sometimes mariners will have an idea of wave height. Wave height, they normally refer to from the bottom-- from the trough-- of a wave to its peak. Amplitude, we're talking about from the resting position to it's peak. So let me label peak. I think you know what peak means. Peak is the highest point on the wave. That's the peak. And that's the trough. If you're in a fishing boat and you wanted to see how big a wave is, you'd probably care about the wave height-- not so much if your boat's sitting down here, you have to care about this whole distance. But anyway, we won't talk too much about that. So that's the first interesting idea behind a wave. And not all waves are being generated by Sal jiggling a string on the left-hand side. But I think you get the idea that these waves can represent many different-- this graph can represent many different types of wave forms. And this, essentially, displacement, from the resting position, or from the zero position, that is your amplitude. Now the next question you might ask is, OK, I know how far you're jiggling this string up and down, but how quickly are you doing it? So how long does it take for you to go all the way up, all the way down, and back again? So how long for each cycle? Cycle is me going up, down, back again. How long for each cycle? Or you might say how long for each period? We're saying this is periodic. Each period is each repetition of the wave. So this idea of how long for each cycle, we call that the period. And this is going to be a unit of time. Maybe I'm doing it every two seconds. It takes me two seconds to go up, down, back again. Up, down, back again. That's going to be two seconds. A very related term is, how many cycles am I doing per second? So in other words, you could say, how many seconds for each cycle? We could even write that. So for example, a period might look like something like 5 seconds per cycle. Or maybe it's 2 seconds per cycle. But what if we're asked how many cycles per second? So we're asking the opposite question. It's not how long, how many seconds does it take for me to go up, down, and back again. We're saying in each second, how many times am I going up, down, back again? So how many cycles per second? That's the inverse of period. So period, the notation is normally a big capital T for period. This is frequency. It's normally denoted by an F. And this, you're going to say cycles per second. So if you're going 5 seconds per cycle, that means you're doing 1/5 of a cycle or, 1/5 of a cycle per second. All I did is invert this right there. And that make sense. Because the period and the frequency are inversions of each other. This is how many seconds per cycle. How long does one up, down, back again take? And this is how many up, down, back agains are there in a second? So they are inverses of each other. So we could say that frequency is equal to 1 over the period. Or you could say that period is equal to 1 over the frequency. So if I told you that I'm vibrating the left end of this rope at 10 cycles per second-- and, by the way, the unit of cycles per second, this is a hertz, so I could have also written this down as 10 hertz, which you've probably heard before. 10 hertz just means 10 cycles per second. If my frequency is 10 cycles per second, my period is going to be 1 over that. So 1 over 10 seconds per cycle, which makes sense. If in 10 times, I can go up and down, a whole up, down, back again, if I can do that 10 times in a second, it's going to take me 1/10 of a second to do it each time. Now another question we might ask ourselves is, how quickly is this wave moving, in this case, to the right? Since I'm jiggling the left end of the string. How quickly is it moving to the right? So the velocity. So to do that, we need to figure out how far did the wave go after one cycle? Or after one period? So after I jiggled this once, how far did the wave go? What is this distance from this resting point to this resting point there? And we call that a wavelength. And there's a lot of different ways you can define a wavelength. You could view a wavelength as how far the initial pulse went after completing exactly one cycle. Or you could view it as the distance from one peak to another peak. That is also going to be the wavelength. Or you could view it as a distance from one trough to the other trough. That's also the wavelength. Or in general, you could view the wavelength as one exactly equal point on the wave. From that distance to that distance. That is also one wavelength. Where you're completing, between that point and that point, you're completing one entire cycle to get exactly back to that same point. And when I say exactly back to that same point, this point doesn't count. Because this point, although we're in the same position, we're now going down. We want to go to the point where we're in the same position. And notice over here, we're going up. We want to be going up again. So distance is not one wavelength. To go one wavelength, we have to go back to the same position. And we're moving in the same direction. So this is also one wavelength. So if we know how far we've travelled after one period-- let me write it this way; wavelength is equal to how far the wave has traveled after one period. Or you could say after one cycle. Because remember, a period is how long does it take to complete one cycle. One to complete up, down, and back again notion. So if we know how far we've traveled, and we know how long it took us, it took us one period, how can we figure out the velocity? Well, the velocity is equal to distance divided by time. For a wave, your velocity-- and I could write it as a vector, but I think you get the general idea. Your velocity-- what's the distance you travel in a period? Well, the distance you travel in a period is your wavelength after one up, down, back again. The wave pulse would have traveled exactly that far. That would be my wavelength. So I've traveled the distance of a wavelength, and how long did it take me to travel that distance? Well, it took me a period to travel that distance. So it's wavelength divided by period. Now I just said that 1 over the period is the same thing as the frequency. So I could rewrite this as wavelength. And actually, I should be clear here. The notation for wavelength tends to be the Greek letter lambda. So we could say velocity is equal to wavelength over period. Which is the same thing as wavelength times 1 over my period. And we just said that 1 over the period, this is the same thing is your frequency. So velocity is equal to wavelength times your frequency. And if you know this, you can pretty much solve all of the basic problems that you might encounter in waves. So for example, if someone tells you that I have a velocity of-- I don't know-- 100 meters per second to the right, so in that direction-- velocity you have to give a direction-- and they were to tell you that my frequency is equal to-- let's say my frequency is 20 cycles per second, which is the same thing as 20 hertz. So literally, if you had a little window where you're only able to observe this part of your wave, you'd only observe that part of my string. If we're talking about 20 hertz, then in 1 second, you would see this go up and down twenty times. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. In exactly one second, you would see this go up and down 20 times. That's what we mean by the frequency being 20 hertz, or 20 cycles per second. So, they gave you the velocity. They gave you the frequency. What's the wavelength here? So the wavelength, in this situation-- you would say the velocity-- your velocity is equal to your wavelength times your frequency. Divide both sides by 20. And actually, let me make sure I get the units right. So this is meters per second, is equal to lambda times 20 cycles per second. So if you divide both sides by 20 cycles per second, you get 100 meters per second times 1/20 seconds per cycle. And then this becomes a 5. This becomes a 1. So you get 5, and then the seconds cancel out. So you get 5 meters per cycle. So this is equal to 5 meters per cycle, which would be your wavelength in this situation. So it's 5 meters. You could say 5 meters per cycle, but wavelength implies that you're talking about the distance per cycle. So in this situation, if this is moving to the right at 100 meters per second and this frequency-- I see this moving up and down 20 times in a second-- then this distance, right here, must be 5 meters. Likewise, we can figure out the period very easily. The period here is just going to be 1 over the frequency. It's going to be 1/20 seconds per cycle. So using these formulas-- and I don't want you to memorize a formula, it should be intuitive for you. And hopefully, this video made it a little bit intuitive. But using this, you can really answer almost any question if you're given two of these variables and you need to solve for the third. Anyway, hopefully you found that helpful.