Physics: Wave on a Rope: Frequency and Wavelength

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What's the simplest kind of wave you can think of? You might think it's just a little pulse, traveling down a rope. But it turns out that mathematically, there's an even simpler wave, although it doesn't look like it at first. And that's when you take a wave and you wiggle it continuously, so you produce a sinusoidal wave, rather than a single pulse. Here's a picture of somebody wiggling a rope and they've started this wave train traveling down, and if they keep on going and if they wiggle it so that this is up here, a sinusoidal function, then this is really very easy to describe mathematically and turns out to be incredibly useful in physics, because there's an awful lot of practical situations where waves really do look like this - a traveling a sinusoidal wave, traveling in a certain direction. So if we want to describe it, let's try to write down the mathematics, kind of like kinematics, just describing this traveling wave.
To begin with, let me try to imagine it physically. Supposing there's somebody over here who is wiggling a rope and the rope is just in front of me in space. So there's this wave traveling by me this-a-way, to your right. And what do I see here? Let me define a coordinate system. So this will be the x-axis, and this way will be the y-axis. And what am I seeing? I'm seeing a wave going by on the rope. I see this sinusoidal pattern that's traveling to your right. And focus your attention on a single chunk of rope. Rope isn't moving that way. It's the wave that's moving that way. Rope is moving up and down. If you had a little spot painted on the rope, you'd see the spot going up and down. In fact, it would look like simple harmonic motion for any given spot.
So let met draw a graph for y as a function of time. That means the sideways displacement. And I'm going to pick one particular spot - let's say x = 0. So right here, I'm going to watch the spot. And it's going to be going up and down, and here's a graph of y versus time. And it goes up to some maximum, which we would call the amplitude of the wave. It goes up to A and then down to minus A, up to A and down to minus A. And that's a perfectly understandable graph. The formula that I would write down is y - I'm sitting at x = 0, but I'm letting time go by - is just Acos(t). It's just the same old formula as you would write down for simple harmonic motion. And what's omega mean here? There's no angular motion. Omega is just defined to be 2 divided by the period - the amount it takes for the wave to make a complete cycle.
So if you look at this picture, how do I read off the period? It's the amount of time to go from one peak, say, to the next peak, or from one trough to the next trough. It's just one full wave. How long does it take? Capital T. Omega is 2 over period, and of course, as usual, if there's a rope going up and down and it takes a time T, you can also talk about just the plain old frequency, f. It's . Frequency is the frequency of oscillation of the rope measured in cycles per second. And omega is, as usual, it's a frequency, kind of. It's an angular frequency. It's radians per second. Nothing's going in a circle. It's just the argument of a cosine function, so it's in radians.
So this is the formula for y at x = 0 as a function of time. That's not the complete description of the wave. It's just looking at this little spot. So let me think of the wave in a different way. Remember there's this wave traveling by me, and if take a snapshot, what do I see? I see a sine wave. So let me examine that snapshot. In other words, let me look at T = 0 and ask, what's the shape? Well, here's a graph. If it's a sinusoidal wave, it's a sinusoidal curve. And at x = 0 and t = 0, we were at the maximum amplitude. So this vertical height here is exactly the same in the two graphs. It's the amplitude of the wave. I'm seeing this wave here and the top is displaced by the amplitude away from the equilibrium center line.
The horizontal axis here is something totally different. It's x. It's position along the rope. So as I walk along the rope, at some spots the rope is up, at other spots, the rope is down. So this is essentially that snapshot that I was talking about. And how do you characterize this? We already know the amplitude. The wavelength is the name that we attribute to the description of how far, in distance, is it from one peak to the next one. It's a natural name, the wavelength. And I could go from peak to peak or trough to trough. It wouldn't make any difference. So the wavelength of a wave is called lambda (). And lambda can also be written as, if you like, . That's in analogy to period is equal to . This is just a definition. k is called the wave number, and it's just sometimes convenient, just like sometimes it's convenient to define omega instead of period.
Let's just write down the mathematical formula for this curve right here. So what is it? It's y = Acos(kx) And this is really y of x at time t = 0, if you really want to be precise. So let me write it like that. And why is it kx? Well, that's why I defined k the way I did. At x = 0, we're at cosine of zero at max. And when are you at max the next? When kx is equal to 2 - the cosine of 2 is 1 again - so when x = , in other words, when x = . So this is really the correct mathematical formula for this function.
Now once again, let's stop and think about this wave traveling by. In one period of time T, every chunk of rope has undergone one complete oscillation. And in that same chunk of time T - capital T, the period - what's happened? Remember the wave is traveling. This was at a high point, and then it was heading down. Her was the next high point, one wavelength away. And this chunk of rope went down and back up again, and in that period of time, that one peak has worked its way over from here - a total distance, .
So the wave has traveled a horizontal distance, , in a time, T. So how fast was the wave going - the velocity of the wave? I'm not talking about the velocity of the rope. I'm talking about the velocity of the wave. It's gone a distance in a time T. It's lambda times frequency. That's a really important formula. It tells you, given wavelength and frequency, what the speed is - or more commonly, given the speed on the rope and you know one of these, you can deduce the other.
When you look at this formula, you think to yourself, supposing that the person over there, who has been patiently jiggling away with frequency f, producing this traveling wave, suppose they double their frequency all of the sudden so they're wiggling the rope up and down faster and faster? What am I going to see here?
Well, this formula might lead you to believe that I see a wave going by twice as fast, but I don't. You really don't. The velocity of a wave on a string is a property of the string, or the rope. It's a function of the density of the rope and the tension in the rope. It's got nothing to do with the wave that you're putting on it. I haven't proven that and we need to. That will come in a future tutorial, but it's true. So what happens if they speed up their frequency and the velocity isn't changing? Well, the formula says the wavelengths get smaller, and that's exactly right. The waves look to me bunched together closer. The rope is going up and down more rapidly in time, and so they're squished together in space. And the whole thing is still moving. It's a new sine wave with a smaller wavelength. It's got a larger frequency. It's going faster. And the whole thing is moving sideways with the same old velocity v.
So that's the speed of a wave. It's a very useful formula. You know, I wrote down y at time 0 and y at position 0, but I never wrote down y, the displacement of the rope at general position and general time t. And the formula is really very simple. You can almost guess it from what we've written down. It's just cos(kx - t).
So let's just look at this formula and think about it for a second. When time equals to zero, it's my old formula, Acos(kx). And when x = 0, is that minus sign a worry? No, it's okay. It's Acos(-t) is mathematically the same as cos(t) It's just a property of the cosine. So in the limiting cases, this formula works.
Now is there any way of understanding why I wrote this down? Sure, it's just a consequence of what I had here. Rewrite lambda in terms of k, and rewrite f in terms of omega. Remember those were just definitions. And you will discover that v is also equal to . This is just the definitions of omega and k in terms of lambda and f and a little bit of algebra.
So let me look at this expression, Acos, or in other words, v times t. And all of this is what is inside the cosine. And if you stare at that function, think about what it's telling you. At time t and x = 0, the argument is zero. We're at a peak. And now let's follow that peak through time. So as time is advancing, the peak is where this thing is staying zero. In other words, x = vt is following that peak. And that means the location of the peak x is just traveling along horizontally with velocity v.
So when I look at the formula this way, I realize that this is just the way of mathematically describing a sinusoidal wave, which is traveling. And it's traveling in a specific direction, because x is growing with time. If you flip the minus sign to a plus sign, that's a traveling wave going the other way - to the left.
So traveling sinusoidal waves are really fairly straightforward to describe. If I tell you wavelength and period and amplitude, we really know the whole story. I know at any position along the rope at any time, where is the rope? What's the sideways displacement y?
It's really not the whole story of waves. There's lots of other properties of waves. For instance, I haven't calculated v yet. I haven't said a word about the energy being transferred by this wave. I certainly haven't talked yet about what happens when one wave and another wave come together and interact. So all of that is still to come.
Waves
The Basics of Waves
A Wave on a Rope: Frequency and Wavelength Page [3 of 3]