A lightweight piece of rope is attached at one end to a wall. The opposite end of the rope is repeatedly moved up and down, producing some waves along the rope. Figure one shows the waves. Which arrow in figure one shows the wavelength of a wave on the rope? Tick one box. J, K, L.
Okay, so let’s first take a look at figure one. We can see in figure one that we’ve got a wall and the ground. We’ve got a piece of rope attached to the wall. And at the other end, the rope is being moved up and down and this is producing waves on the rope. Now, the first part of this question is asking us to find the letter J, K, or L which represents the distance that’s the wavelength of the wave.
Now, the wavelength of a wave is the distance between two equivalent points on the wave, in other words the distance travelled in completing an entire cycle of the wave. So normally, we measure it from peak to peak. So for example, here is a peak and here is the next adjacent peak. Well, the wavelength is this distance here or we can measure it from trough to trough as well. So here is a trough and here is the next adjacent trough. The wavelength is this distance here and by the way this distance and this distance are exactly the same because they’re both the wavelength of the wave.
And we can see, for example, for the peak-to-peak case, when measuring the distance between this point and this point and in between those two points, we’ve got an entire cycle of the wave. And to be honest, we can pick any point on the wave that we want and we can measure the wavelength from it to the next point that’s equivalent to it.
In other words, let’s say we picked this point here. Well then, the next adjacent equivalent point to the right will be the point after which an entire cycle of the wave has been completed.
So let’s see what the entire cycle looks like. Now, we start along the wave and go this way. We might be tempted to think that this point here is the next adjacent equivalent point. But actually, it’s not because even though these two points are at the same height, the wave in this case is going down, whereas the wave in this case is going up.
So we need to continue on with our wave until we get to this point here. Now, this is the point that’s equivalent to our very first point — this one here — because an entire cycle of the wave has been completed between this point and this point. And so the distance between these two points is the wavelength which by the way happens to be the same as this distance or this distance.
So anyway, upon closer inspection, we can see that all the letters J, K, and L are measured from the same point; that’s this point over here. But they’re measured this time to the left and they’re measured different distances. So we need to find the letter J, K, or L that represents one wavelength.
And so if we’re gonna start here, we need to move one entire wave before we get to our wavelength. Now, at this point, we’ve reached the dotted line representing L. However, we’ve not reached the complete cycle of the wave yet. So we keep going. And then, we arrive at the dotted line. This dotted line represents K, but we’re still not one wavelength across. So we keep going until we get to this point over here.
Now, this point represents the distance being measured as J. And it’s the distance between this point and this point which are equivalent points that are separated by one cycle of the wave. Therefore, the arrow that shows the wavelength of a wave on the rope is J.
Moving on then, we’ve talked about the wavelength of a wave. So let’s consider its amplitude.
The distance that the rope is moved up and down is increased. How does the amplitude of the waves along the rope change? Tick one box. Amplitude increases. Amplitude decreases. Amplitude does not change.
Okay, so first of all, in this question, we’ve been told that the distance that the rope has moved up and down is increased. In other words, initially, we were moving the rope up and down by this much. But now, we increase that distance. So we move the rope up and down slightly more.
And so the new wave will look something like this. And by the way, we’ll only draw this half so that we can compare it with the old wave which is on the right half of the diagram.
So we’re being asked to find what happens to the amplitude of the waves on the rope. To answer this question, we first need to recall what amplitude actually means. The amplitude of a wave is defined as the maximum displacement of a wave relative to its rest position.
So what this means is the following. If the wave was not being moved up and down, then we’d just have a flat rope going like this. And so therefore, that is its rest position because that’s how the rope would be if we weren’t moving it up and down. And the amplitude is defined as the maximum displacement compared to that position.
In other words, the amplitude is this distance here which is the maximum displacement up or this distance here which is the maximum displacement down. And of course, both of these distances are the same. But one of them is in the upward direction and the other one is in the downward direction.
So what happens to this distance when the distance that we move the rope up and down is increased? Well, we can see that in the wave beforehand; this would have been the amplitude. However, now the amplitude is this: the amplitude is larger. And so we tick the box next to amplitude increases because that’s what happens.
Alright, so we’ve now talked about wavelength and amplitude. Let’s discuss the speed of the wave on a rope.
Describe how the speed of the wave along the rope can be measured accurately.
Okay, so if we wanna measure this, we need to recall that the speed of an object is defined as the distance travelled by the object divided by the time taken for the object to travel that specific distance. Now, in our case, we’re trying to measure the speed of the wave along the rope.
So let’s say once again that we have the same setup as before, except that we’re not moving the rope up and down yet. So here is the wall that the rope is attached to and here is the point that the rope is attached to the wall. Here is the floor and here is the other end of the rope. Now, very soon, we’re going to set off a wave moving along the rope.
But before we do, we want to measure the distance that the wave is going to travel. So let’s say that we’re going to have our wave travelling the entire length of the rope. So we need to measure this distance, which we’ll call 𝑙. And so we can start by saying measure 𝑙; that’s the length of the rope.
Then, once we’ve done that, what we can do is to set off a wave along the rope. We can do this by moving the free end of the rope up or down slightly. And as soon as we do that, we need to start a stopwatch because of course we want to time how long it takes for our wave to travel along the length of the rope.
So here is the wave that we create by moving the free end of the rope. And we can see it moving along the length of the rope. Now, it continues to do so and we keep the stopwatch running until the point that the wave reaches the fixed end of the rope. Then, we stop our stopwatch.
And so as a result of this, the time on the stopwatch will be the time taken for the wave to travel from this end of the rope to this end of the rope. And if you remember that’s the distance we measured earlier. So we’ve got the distance travelled and the time taken to travel that distance. So let’s recap.
After measuring the length of the rope, we moved the free end of the rope and started our stopwatch simultaneously. Then, we stopped our stopwatch when the wave reached the fixed end of the rope. Once we do that, we have all the information we need to calculate the speed of the wave.
So we start by measuring the length of the rope, then we move the free end of the rope and start our stopwatch, and then we stop our stopwatch when the wave reaches the fixed end of the rope.
Anyway moving on, we’ve looked at one way to measure the speed of the wave along the rope. Let’s now consider how the speed relates to the wavelength of the wave and the frequency of the wave.
The speed of a wave can be calculated using the following formula: wave speed is equal to frequency times wavelength. The waves along the rope have a speed of 60 centimeters per second and a frequency of 1.25 hertz. Find the wavelength of the waves. Wavelength is equal to blank centimeters.
Okay, so first of all, we’ve been told that wave speed is equal to frequency times wavelength. We can write this symbolically if we represent wave speed with the letter 𝑠, the frequency with 𝑓, and wavelength with the Greek letter 𝜆, in which case we can say that 𝑠 is equal to 𝑓 times 𝜆 or to put it more simply 𝑠 is equal to 𝑓𝜆.
Now, in this question, we’ve been told that the waves along the rope have a speed of 60 centimeters per second and a frequency of 1.25 hertz. Notice by the way that the speed has been given to us in centimeters per second and not meters per second, which is the standard unit of speed. However, this is not going to be a problem and we’ll see why in a second. But before we think about that, let’s rearrange this equation to solve for what we need to find.
We’ve been asked to find the wavelength of the waves. So we need to isolate 𝜆 on one side of this equation. So we start with 𝑠 is equal to 𝑓𝜆. And then, we can divide both sides of the equation by the frequency 𝑓. This way, the frequency on the right-hand side cancels out. And what we’re left with is that the wave speed divided by the frequency is equal to the wavelength.
Now, at this point, we can plug in the values that we’ve been given in the question. We can say that 60 centimeters per second — that’s the wave speed — divided by 1.25 hertz — that’s the frequency — is equal to 𝜆. However, the unit hertz which is a standard unit for frequency is actually the same thing as one divided by the unit of seconds which is the unit of time. And so we can replace hertz in this equation with one divided by seconds or one per second.
And then, we can see that we’ve got a per second in the numerator and a per second in the denominator. Those both cancel out and we’re just left with a unit of centimeters. So our wavelength is going to be in centimeters which is perfectly fine because that’s what we need to give our answer in.
And so at this point, we can say that 60 divided by 1.25 centimeters is equal to 𝜆. And when we plug that into our calculator, we find that this is equal to 48 centimeters. And hence, our final answer is that the wavelength we’ve been asked to calculate is 48 centimeters.
Now, we’ve just calculated the wavelength. Let’s compare this to the amplitude of the wave.
The peaks of the wave along the rope are 24 centimeters higher above the ground than the troughs of the wave are. How does the amplitude of the waves compare to their wavelength?
Okay, so here is our rope once again and here is the wall that the rope is connected to and here is the ground. Now, we’ve been told that the peaks of the wave are 24 centimeters high above the ground compared to the troughs of the wave.
In other words then, the distance between the peaks and the troughs is 24 centimeters because the troughs are a certain distance above ground, whatever that distance may be, and the peaks are another 24 centimeters higher above this distance. And so the vertical distance between the peaks and troughs is 24 centimeters.
Now, we’ve been asked to compare the amplitude of these waves to their wavelength. Well, we saw earlier that the amplitude was defined as the maximum displacement compared to the rope’s rest position. And in this case, the rest position is here. So the amplitude is this distance here or this distance; they’re both valid.
Basically, it’s the distance between the rest position of the rope if there were no waves on it and either a peak or a trough. And that distance happens to be half of this distance here. In other words, the amplitude of the wave is 12 centimeters. And this distance here is 12 centimeters as well, just in the downward direction. So let’s say that the amplitude, which we’ll call 𝐴, is equal to 12 centimeters.
Now, earlier on in the question, we calculated the wavelength of the waves to be 48 centimeters. And in this part of the question, we’ve been asked to compare these two quantities. One way we can compare them is to divide the wavelength by the amplitude. So we can say that 𝜆 divided by 𝐴 is equal to 48 centimeters divided by 12 centimeters. And in this case, the units of centimeters cancel out. So we’re just gonna be left with a number. And this number happens to be four.
Now, to get a little bit more meaning out of this, we can rearrange the equation. We can multiply both sides of the equation by the amplitude 𝐴. This way, the amplitude on the left-hand side cancels. And what we’re left with is that 𝜆 is equal to four times 𝐴. In other words, the wavelength is four times bigger than the amplitude.
And so we can say as our final answer that the wavelength is four times larger than the amplitude.