Lesson video

Hello, my name is Mr. Fairhurst, and this lesson is about representing transverse waves.

In this lesson, what we're aiming for is at the end of the lesson for you to be able to interpret and sketch what we call displacement-time graphs of transverse waves.

These are different from the displacement-distance graphs and we need to think about a few other things first before we get to them.

During this unit, we'll be using a number of keywords.

We'll be looking at what we mean by wave medium and we'll be using this idea that we've seen before about displacement and applying that to new situations, and we'll be thinking about the period of a wave and the frequency of a wave, which are both key terms that apply to the speed or the rate of vibration in the wave.

We'll start by looking at what we mean by frequency and period.

These are the terms related to the rate of vibration of the wave, how quickly the wave oscillates up and down, if you like.

And once we've done that, we can look at how we calculate the frequency and period of a wave, and then we're going to use those ideas together to represent transverse waves as the displacement-time graphs, which is a new way of displaying these graphs.

So first of all, let's have a look at frequency and period.

Let's just remind ourselves what we mean by a transverse wave.

In this picture, we've got a transverse wave moving forwards from left to right along the rope, and it's a transverse wave because the rope's been shaken at right angles to the direction the wave is travelling.

In this case, it's been shaken up and down.

Now, we call the rope the wave medium because the wave is moving through the rope, and without the rope there wouldn't be a wave.

In this example, the wave medium is the water because water is what the wave is moving forwards through, and the water as the wave moves forward is oscillating or vibrating up and down at right angles to the direction that the wave is moving.

So let's think about this wave moving through the water.

The frequency of the wave is the number of waves passing a point each second.

And frequency is measured in units called hertz, or Hz for short.

And a frequency of one hertz, as this wave is shown here, means that the source is producing one wave every second, or one complete oscillation in each second.

In this ripple tank, the waves being made by the bar vibrating up and down is moving at right angles to the direction that the wave is moving.

It's making the water move up and down, but the wave is moving forwards away from it.

So that's a transverse wave.

And in this instant it's making eight waves each second.

That means that the frequency of the wave is eight hertz.

So let's have a look at an example of how we can work out the frequency.

A ripple tank makes 24 waves in four seconds.

So what's the frequency of the wave? Well, the frequency is the number of waves in each second.

So what we need to do is to divide the total number of waves, 24, by the number of seconds to find out how many in each second.

So the sum we need to do is 24 divided by four, which will give us six hertz.

Have a look at this example and have a go at this yourself.

Just pause the video and have a go.

So how did you get on? In this case, we had 36 waves in total over a period of nine seconds.

What's the frequency? How many waves in one second? All we need to do is to divide the 36 by the number of seconds that we've got and we have this sum, 36 divided by nine, which gives us four hertz.

And we can think about this backwards, if one wave takes four, so if the frequency is four hertz, that's four ways in every second, we've got nine seconds.

So nine lots of four, that will give us 36 waves in total.

So it works both ways.

What about this example? This one just gives us a more awkward answer and maybe it takes a little bit longer to think about.

A ripple tank makes two waves in four seconds.

What's the frequency of the wave? It's exactly the same as before.

We want to know how many waves in one second.

So you've got two waves divided by the four seconds, and that gives us half a wave in each second.

So if we count four seconds, half a wave, one wave, one and a half waves, two waves, that's taken us four seconds to get a total of two waves.

So even though the frequency is less than one, it's still the right answer.

Have a go at this example yourself and just pause the video whilst you do so.

So how did you get on? Following the same process, we want to know how many waves in one second? Which is the frequency of the wave.

We've got six waves in total over a period of 24 seconds.

So in one second, we want six divided by 24.

You can perhaps do this on your calculator.

60 divided by 24 gives us a quarter, or 0.

25 hertz.

So as well as the frequency, we need to think about the period of the wave.

This is a time taken for one complete oscillation.

So if you like, it's almost the opposite of frequency.

On this rope, two waves have been made every second.

So on there you can see two complete waves in a period of one second.

Those waves moving forwards on the rope.

And that means that each wave takes half a second to be made.

The frequency is two because it's two waves per second, but the period, the time for one wave is half a second.

So in other words, the period is 0.

5 seconds.

The symbol for period is a capital T.

So it's a bit like time but it's a particular time, so we give it a capital letter, and that's 0.

5 seconds.

On this rope, there's a fixed marker and the waves moving forward past the marker, and we've counted four ways passing that marker in every second.

That means that each separate wave passes in a quarter of a second.

So a quarter, a half, three quarters, one, four waves over a second, but each wave taking a quarter of a second to pass that point.

That means that the period of the wave is a quarter of a second, or 0.

25 seconds.

So let's have a look at an example.

Two waves are made on the rope in eight seconds.

What's the period of the wave? Well, this time we need to do the sum back the other way around than when we did for frequency.

We want to know how long one wave takes.

We've had eight seconds in total and it took eight seconds to make two waves, or for two waves to pass.

So that's four seconds per wave, and the sum we did was eight seconds divided by the number of waves, which is four seconds per wave, or the period is four seconds.

And we can put the T instead of writing the word period.

Have a look at this example yourself and see what you can work out as the period.

Just pause the video again whilst you do so.

Okay, what did you get? The time taken was two seconds and we had eight waves.

So how long does it take for each wave? This time we divide the time, two seconds, by eight to get the time for each wave.

And two divided by eight gives us 0.

25 seconds, or a quarter of a second.

So the time period is 0.

25 seconds.

What about this example? Eight waves pass a fixed marker in four seconds.

What's the period? So the waves moving forward pass the marker.

We've counted eight in four seconds.

So the time for each wave to pass is four seconds divided by eight, which gives us 0.

5 seconds.

So the period again is 0.

5 seconds.

And finally, for now, just try this example and again, pause the video whilst you do so.

Okay, so this time we've got two waves passing a marker in 10 seconds.

So the time for each wave to pass is the total time, 10 seconds, divided by the number of waves passing, which is 10 divided by two.

And that gives us a time of five seconds, which is the period for one wave.

Okay, here's a set of questions for you to practise to see if you've understood what we've talked about in this section of the lesson.

Just pause the video whilst you have a go at those.

Okay, so let's have a look at the answers.

First of all, a wave's got a frequency of five hertz.

How many waves does that make in one second? Well, it's five because the frequency means how many waves in one second.

How many waves in five seconds? Well, five in one second times five gives us 25 waves.

And the period of the wave is the time it takes for one wave.

We've got five waves in one second, so we can do one second divided by five, which gives us 0.

2 seconds for the period.

You might notice it also looks exactly the same way, if you've got 25 waves in five seconds, we could do five divided by 25 and we get exactly the same time period because each wave is taking the same amount of time each time.

Okay, what about part b? A wave has now got a period of 0.

1 second.

How many waves in one second? Well, the number of waves is one second divided by 0.

1, which is the time for one wave, which gives us 10 waves.

You might also think about this as if the time period of the wave is a 10th of a second, that's 10 per second.

And the frequency, as we said, it's 10 waves in one second, the frequency will also be 10, this time 10 hertz.

You must use the correct units for frequency.

And part c now, one water wave passes the marker every two seconds, what's the frequency? Now, the most common mistake here is to say two because you're not thinking it through properly, but hopefully you have done one wave in two seconds.

The frequency is the number of waves one divided by the number of seconds two, which gives us half a hertz, half a wave every second.

The period of the wave, if it's one wave every two seconds, that's the period, it's one wave in two seconds.

So the period is two seconds.

So hopefully you got most of those right, if not all of them, and well done if you did.

In this part of the lesson, we'll be looking at how we can calculate the frequency and period now that we understand what the two terms mean.

So we've just seen in the last part of the lesson how we can work out the frequency and period of a wave.

It turns out that the frequency and period of the wave are connected in a different way as well.

And that way is that if we, you may have spotted this, the shorter the period of the wave, the higher its frequency.

And it turns out that if we half the period of the wave, the frequency is twice as high.

So say the period of the wave was half a second, the number of waves in one second gives us two.

So the frequency is two.

If we make the period of the wave shorter, maybe make it a quarter of a second, that means we've got four waves per second, and that means that the frequency is now four hertz.

So what we've just seen there is if we half the period of the wave, we double the frequency and we can put that together in an equation.

And the equation, if you like, it's just a shorthand way of saying that.

And what we've said is that the frequency is one divided by the period.

Let's just put those numbers into the same equation and see what we get.

So if I remember right, we said that the period that the wave starts is half a second.

So one divided by half a second on the right of this equation, one divided by a half gives us two, which was the frequency.

If we half the period to make it a quarter of a second, how many quarters in one? It's one divided by a quarter.

How many quarters in the whole one is four, the frequency is equal to four.

So if we half the period, the frequency is doubled, and this equation is a shorthand way of saying that.

And we can make it even shorter by using symbols for frequency and for period.

What this relationship is showing is that the frequency is inversely proportional to the period of the wave.

Inversely proportional means that if one thing doubles, the other value halves.

If one triples, the other will be a third as big, and so on.

We can see that if we try a few examples.

So here's the first example.

What happens if the period of the wave is doubled to the frequency? Well, I've illustrated that the time period of the wave is doubling by making the T symbol for period twice as big.

And the one divided by something that's bigger gives you a smaller fraction.

So that means that the frequency would be smaller, and in this case, if the period is doubled, the frequency is halved.

Have a look at this example and see if you can work out what happens to the frequency.

Pause the video if you need a little bit of time to do this.

Okay, so in this case, I've redrawn, rewritten the equation, but this time with the time period four times smaller, which seems to suggest that the right hand side of this equation is much bigger and the frequency must therefore be bigger.

So if the period is four times smaller, the frequency is four times higher, it's four times greater than it was before.

Let's look at this other example, but this time we're changing the frequency.

What happens if we double the frequency? What happens to the period? Well, if the left hand side, the frequency is twice as big, the right hand side of this equation needs to be twice as big, and the way to do that is to make the period smaller.

So one divided by something smaller will give you a bigger value, and in this case the period needs to be twice as small.

It needs to be halved.

Have a go at this equation, or this example and see what you think.

Just pause the video again if you need to.

So how did you get on? Once again I've written out the equation, and this time I've made the frequency half the size, 'cause the frequency has been halved.

And if we think about what we need to do to the right hand side to make that half as big as well, we need to double the size of the period, the big T there.

So we need to double the period in order to half the frequency.

Here's some questions for you to have a practise.

Have a look through those, write down your answers, and just pause the video whilst you're doing so.

So how did you get on? Part a, a water wave's got a period of five seconds.

What's its frequency? Well, frequency is one divided by the period, it's one divided by five, or 0.

2 hertz.

So we can use that equation to get the answer very quickly.

Part b, a wave on the rope's got a period of.

5 seconds.

What's its frequency? Frequency is one divided by the period, one divided by 0.

5 gives you two hertz.

Part c, a wave with a period of.

25 seconds.

How many waves pass a marker in one second? We're asking for a frequency, number of waves a second, so frequency is one divided by the period, and one divided by 0.

25 gives us four, four waves per second, or four hertz.

Part d, a transverse wave with a frequency of eight hertz.

It's period T is going to be eight times smaller so its period is going to be one eighth, or 0.

125 seconds.

To use the equation, you may have had to rearrange that, which is fine.

And part e, a wave on the string has got a frequency of 25 hertz.

How long is each full vibration? Well, frequency is one over the period, or period is one over the frequency.

It's one 25th of a second for each one.

And if you think about that, if it's got a frequency of 25 hertz, that's 25 in a second, or each one is a 25th of a second.

In decimals, that's 0.

04 seconds.

And now that you understand about frequency and period and are able to calculate them, we can now move on to looking at displacement-time graphs.

We can actually represent a wave by two different sorts of graph, a displacement-distance graph, or a displacement-time graph.

As you can see here, we've got both graphs and they both look very, very similar, but there are some very important differences between them.

And it's important first of all to check what sort of graph you have and what you are looking at.

And the only way to be really certain is to look at the scales of a graph.

That's always the first thing you should look at.

When you look at a graph, look at the scales.

So the right hand graph has got time on the horizontal axis rather than distance, and that is a displacement-time graph.

Again, a displacement-time graph is not a picture of a wave drawn to scale.

It's nothing at all to do with that.

What it shows is how the displacement at one point changes as the wave passes.

So the graph on the right hand side shows how point x on the ripple tank goes up and down as the wave moves past it.

And the gap between two crests on that graph is a time, because we're measuring time on the horizontal axis.

And if you think about how the wave is moving forwards as the wave goes up and down, we can think of that gap between two crests as equal to a period.

If you are watching the wave go past the point, when the crest of the wave goes past, you start timing.

When the next crest goes past you've measured the time for one wave to go past, or you've measured its period.

And that time period can be shown on a displacement-time graph like this, and it can be measured between two troughs, or between any two similar points on adjacent waves.

It's always the same amount of time.

Let's have a look at an example and something for you to have a go at.

In this example, a pupil makes a wave with a rope and her friend makes a graph to represent the same wave.

The graph she makes has time on the horizontal axis.

And what I'd like you to do is to have a think about these three statements and decide which ones you think are right and which one you think are wrong.

And in each case you decide whether you're absolutely certain you're right or wrong, or whether it is just the best guess by ticking the right box next to each statement.

Just pause the video and then have a look at these.

Okay, so let's have a look at the statements.

The first one, A, a snapshot of the wave on the rope.

Is that graph a snapshot a photograph, if you like, of the wave? And it's not, because it's not.

On the photograph, if you like, you would have distance along the horizontal axis.

Here we're measuring time.

What about statement B? The movement at one point on the rope.

Is that graph showing how one point on the graph moves? And yes, it is, it's showing how part of a rope at any one point is moving up and down as the wave moves past it.

And does the, statement C, does that graph show three complete wavelengths? Well, it does have three complete waves, but they're not wavelengths because we have time on the horizontal axis and not a distance.

So it's showing how the displacement of a point has changed over time, it's not shown the wavelengths of that wave.

It's got the distance between the crest, if you like, is a time and not a distance, so we can't measure wavelengths from that graph.

We can also use displacement-time graphs to measure the period accurately if we can plot the graphs out.

So on this graph, we're asked what the period of the wave is.

we've got time on horizontal axis.

So all we need to do is to pick two points on the graph that are the same point on opposite waves.

Here I've chosen the crests, and the difference between the crest is five seconds minus one second, which gives a period for the wave of four seconds.

Just pause the video and have a look at this graph and work out the period of the wave.

Okay, so what did we get? We could measure from crest to crest, as you've done here, and we could measure 45 seconds, take away 25 seconds to give a period of 20 seconds, or more easily, you could look to see where the graph crosses the zero point on the horizontal axis and you've got a complete wave between 20 and 40 seconds, and that will give you the same answer.

So you don't have to use the crest, you can use at any point and sometimes it's easier to use the zero line on the horizontal axis.

Now it's time for you to have a little practise.

Pause the video and see if you can answer these questions about this displacement-time graph.

Okay, how did you get on? First of all, you were asked to measure the time period, and we could measure that from crest to crest, but this scale is a little bit awkward there, so it's a lot easy to measure it from where the line on the graph crosses the horizontal axis.

So between two and zero seconds, it's the same point on the graph each place and we get a time period of two seconds.

We can measure the amplitude directed from the graph as well, which is the maximum displacement, and that comes in at seven centimetres.

However, for the frequency, we can't measure that directly.

And what we have to do is use our time period and the equation we saw earlier in the lesson, frequency is one divided by the period, which is one divided by two seconds, or 0.

5 hertz.

So just to finish the lesson, let's have a look at this summary.

What we've been doing is seeing how we can represent waves with a displacement-time graph.

There's a picture of a displacement-time graph and the time period mark with a capital T is the time for a complete wave to pass any point as the wave moves forward.

It's also the same as the time needed to make one complete wave, or one complete oscillation it's caused the wave.

And as we've just said, the gap between any two crests, or any two points that are the same on adjacent waves is equal to the period of a wave, and it's equal to that period because we're measuring time on the horizontal axis.

When we're thinking about the graph, we can also calculate the frequency from measurements of the graph from the measurements of the period.

Frequency is the number of waves produced in one second, it's measured in hertz, or Hz for short, it's got to be a capital H.

And we can use the equation, frequency is one divided by the period, to work it out, or in symbols, f equals one divided by a capital letter T for period.

I hope you've enjoyed the lesson and you've learned all you needed to know about interpreting displacement-time graphs for waves.