Lesson video

Hello there and welcome to this lesson from the Measuring Waves unit.

The lesson is all about the wave equation, which is an equation linking together some of the key properties of waves.

My name's Mr. Forbes, and I'll be leading you through the lesson.

By the end of it, you should be able to describe how a wavelength of a wave and the frequency of a wave are connected together when it's moving through different materials.

So if you're ready, let's start.

Here are some of the keywords that you'll see throughout the lesson that we need to understand.

Wave speed, frequency, wavelength, lambda, which is that symbol inside the brackets there, and the wave equation.

Let's go through some of the meanings of those, but you'll find out more as we go through the lesson.

Wave speed is basically the speed of a wave as it travels through some sort of material or medium.

So just the speed of the wave.

Frequency is the number of waves produced every second.

A frequency is how often something happens.

So for waves, that's how many waves there are every second.

Wavelength is the length of a wave, so that's a measurement of the distance from one part of a wave to the next part that's the same.

We normally measure from wave crests to the next wave crest.

That symbol lambda, shown in the brackets again, is used to represent the wavelength in some calculations.

And the wave equation itself is what we're going to come up with during the lesson.

I'll mention it now.

It's wave speed = frequency x wavelength, and we're going to see exactly how we come up with that equation.

The lesson is in three parts.

First part is all about the wave medium and what that is.

The second part will be the wave equation and how we've generated that.

And the third part will be a closer look at how we can link together the frequency of a wave and the wavelength and do some calculations based on that as well.

So let's make a start and look at what the wave medium is.

A medium is basically a material that waves will travel through.

So a medium could be a wide range of things depending on the type of wave.

I've got a few examples to help you through that.

The first are water waves.

Water waves like ripples travel through the water, so the water's acting as the medium here.

The second, a mechanical wave sort of like the waves in a spring.

And in the spring, the medium will be the coils of that spring because they move around.

A third example, sound.

Sound could travel through water or solids and the medium is whatever it's going through.

Most commonly you'll have sound travelling through air, so that would be the medium.

Okay, a first check for you to see if you understand what a medium is.

Which of these is used to describe what a wave travels through? Is it wave structure, wave material, or wave medium? Pause the video and make your selection and then restart when you're happy.

Well done.

It was wave medium.

Medium is how we describe the material a wave passes through.

A second check for you here.

I want you to be able to identify a particular medium.

So shaking the end of a slinky spring produces a wave that travels along it.

Which of these is the wave medium? Is it the air around the spring? Is it the coils in the spring? Or is it the size of the wave? Pause the video and select and then restart when you're happy.

That's right, it was the coils in the spring.

The coils are the things that are actually moving and passing the wave along.

So they're the medium.

Now the medium's important because it's the medium that determines the speed of a wave.

When a wave is travelling through a medium, it's at a fixed speed, but if it changes to another one, its speed will change as well.

Let's have a look at a few example media: air, glass, and water.

Now if I've got a sound wave passing through those three different things, it'll actually travel at a different speed in each one.

Here's the typical speeds.

So the speed of sound in air is about 330 metres per second, but in water it's much, much faster.

It's 1,500 metres a second or so, and in glass it's faster still.

So you can see there that the wave medium is what makes the difference to the speed of sound.

But another important thing is, the wave speed depends on the type of wave as well.

Light is much, much faster than sound in media.

I'll give you the example so you don't have to remember these numbers, but you can see they're very much larger.

300 million metres per second is the speed of light in air.

That's a lot faster than the speed of sound.

As I mentioned earlier, the speed of sound through a particular medium is constant, it doesn't change.

So sounds travels at about 330 metres per second in air, but that's assuming that the medium itself doesn't change.

You can actually make a change to a medium.

You can change the properties of air and that will have an effect on the speed of sound in it.

So changing the medium can change the speed of the wave passing through it.

Some examples here.

If you change the depth of water that a water wave is going across, that will change the speed of a wave.

So making it shallower or making it deeper will make the wave speed up or slow down, and you may have performed experiments based on that idea earlier.

If you change the temperature of air, you're actually changing the properties of the medium.

So that will have an effect on the speed of sound.

So when I said earlier that speed of sound is 330 metres a second, that's for a very particular conditions, very particular pressure, very particular temperature.

And if I alter those two things, I'll actually change the speed that sound travels through it.

So that's why I can only give an approximation.

Okay, a check for you here.

True or false? All waves travel at the same speed.

I'll be asking you to justify this in a bit as well.

So pause the video, make your selection, and then restart.

Well done.

The correct answer was false there, but why is it false? I want you to select from one of these two to explain why the speed is different.

Well done again.

The speed of a wave depends on whatever medium it's travelling through, not the properties of the wave.

So if you've got a different medium, you're going to have different wave speeds.

Okay, I've got the first practise activity here to check that you've understood about what a medium is and how waves pass through it.

I've got a wave being produced in a slinky spring here by vibrating one end of it side to side.

So it makes a wave that travels towards the right here.

I want you to suggest two ways of changing the speed of that wave travelling through the slinky and then I want you to suggest to me one way that you could change this process but that wouldn't have an effect on the wave speed.

So pause the video for a while, write down your answers, and then resume when you're ready.

Okay, the first part of that was to suggest two ways of changing the speed of the wave.

Now any change to the medium will affect the speed of the wave, so your suggestions should be based around that.

So making a physical change to the wave medium could affect the wave speed.

Some possible things we could do with the slinky are these.

We could use thicker or thinner coils on the spring.

If you've got big heavy coils, a big stiff spring, that will make an effect.

The second thing we could do is we could stretch it.

That will change the physical properties of the spring as well.

So stretching it out longer will change the wave speed.

Okay, the second part of the task was to give me a suggestion for something that wouldn't have an effect on the wave speed.

You should have realised that anything that doesn't change the properties in a medium won't change the wave speed.

So if I try to change the wave instead of changing the medium, I'll still get the same wave speed so there won't be any change.

Okay, so the wave speed's not changed by changing the wave itself.

Some suggestions that you could have given me.

Shaking the spring up and down faster and slower.

That changes the wave, it changes its frequency, but it doesn't change the wave speed.

Shaking the spring harder and softer would change the amplitude of the wave, but again, it shouldn't change the wave speed.

So well done if you've got either of those suggestions.

There's probably a few more you might have come up with.

Okay, we're going to move on to the second part of the lesson now and that is looking at the wave equation and where it comes from.

We're going to try and find a connection between the wavelength, the frequency, and the wave speed of a wave.

And to be able to do that, we're going to have to imagine a wave moving.

So we're going to have a fixed marker, a position the wave moves past, something like this, and we're going to have waves moving past it like this.

So the waves move across the marker and we're going to count the number of waves that go through.

That's going to be connected to the frequency, because the frequency is how many waves pass something per second.

So we're going to use that marker to help us count the waves go past each second.

Okay, and here's our first example.

There's the marker positioned and we're going to have a wave move past it towards the right.

There it goes.

Okay, so let's imagine that wave has moved four metres in a second.

That gives us a wave speed of four metres per second.

So we've got the speed of the wave.

Now counting the number of waves that have gone past, you can see there's two complete cycles of the waves.

So two waves have gone past in that one second, giving us a frequency of two hertz, two waves per second.

And finally, we can see how big a distance those two waves fit in.

There's two wavelengths there, there's two whole waves, so that's two wavelengths.

That fits in four metres.

So if we've got the length of one wave, one wavelength, that should be two metres.

Here's a second example.

We've got a different wave passing the marker.

Here it comes.

And this wave as well was travelling at four metres per second, so it's gone four metres in that one second it's been travelling.

We've got a wave speed of four metres per second.

This time though you can see there's more waves there, more complete cycles.

If you count those waves, there's 1, 2, 3, 4.

So we've got four waves that have passed that marker, giving us a frequency of four waves per second, four hertz.

And we can again try and find the wavelength from that diagram.

If you look, there's four wavelengths inside four metres, and that gives us a wavelength of one, one metre per wave.

A third wave now.

This third wave passes the marker, it's quite different than the others.

I've got the wave moving at a different speed this time, it's moving at six metres per second.

So in that one second it travelled six metres, so a wave speed of six metres per second.

Now if you want to count the waves you can pause and see, but I'm going to tell you there's 10 full waves have passed there and that again gives us a frequency, a frequency of 10 hertz.

And we can calculate the wavelength this time.

You've got six metres and you've got 10 waves.

So 10 wavelengths in six metres gives us a wavelength of 0.

6 metres, it's 0.

6 metres per wave.

Now we're going to put that data from that experiment into a table and see if we can find a pattern in it.

Here's what we saw.

You had a frequency of two hertz, wavelength of four metres and a wave speed of eight.

Looking at those two numbers, if you multiply the frequency by the wavelength, it gives you the wave speed.

If you look at the second example, we had a frequency of four hertz, a wavelength of one metre, and that's given us a wavelength of four metres per second.

Four times one is four.

So again, the same pattern there.

Third example, we had a frequency of 10 hertz, a wavelength of 0.

6, and that give us a wave speed of six metres per second.

In each example you can see the wave speed is simply the wavelength and the frequency multiplied together, so a general pattern there.

The wave speed is equal to the wavelength multiplied by the frequency, and that's the wave equation.

If we write down what we've just seen mathematically, we get an equation.

An equation like this.

Wave speed is equal to the frequency times the wavelength.

We can write that in symbols and we'll try a bit of practise with that later.

In symbols it would be v = f x lambda, where v is the wave speed, f is the frequency, and lambda is the wavelength and I've got the units there as well, metres per second, hertz, and metres for all those three.

Let's try an example calculation based upon what we've seen in that wave equation.

So I've got a question here.

The end of a spring vibrates with a frequency of 2.

0 hertz, producing a wave of wavelength 1.

5 metres.

Calculate the wave speed.

So we can do this in simple steps.

The first thing you should do with any calculation in physics is write out an equation.

So here's the wave equation that we saw earlier.

Wave speed = frequency x wavelength.

The second thing we do is we look back to the question and see can we find the frequency in the wavelength and substitute the numbers in.

Reading that, we've got a frequency of two hertz, so we can put that in, and a wavelength of 1.

5 metres.

So writing out those values gives us this, wave speed = 2 hertz x 1.

5 metres.

And the final step is just simply multiplying those two values together.

Wave speed is three, and because it's a speed, we're measuring it in metres per second, so it's three metres per second.

I've got this question for you now.

What I'd like you to do is to pause the video, follow the same steps as I did and see if you can come up with an answer.

Welcome back.

So hopefully you followed this procedure to be able to get to the answer.

We first of all write down the equation, wave speed = frequency x wavelength.

Then we try and spot those two values in the question.

And looking carefully, frequency of 2.

4 hertz, wavelength of 0.

1 metre.

We write those two values down.

Wave speed = 2.

4 x 0.

1.

And entering those into a calculator or doing it in your head should give you a value of 0.

24 again metres per second because it's a speed.

Well done if you've got that answer.

So we're going to move on to the second task of the lesson.

Here it is.

We've got three groups of students producing a series of water waves in trays and they're measuring frequency and wavelength and recording the results in that table at the bottom there.

And I'm asking you to calculate the wave speed for each group.

So that's three separate calculations.

And when you see your answers, they'll be slightly different.

So I'm going to ask you to suggest a reason for the differences in the speeds.

I should point out here that the wavelength is in centimetres, so that's going to give us an answer for wave speed in centimetres per second.

I'd like you to pause the video, try and solve it and then restart.

I'll be back with you soon.

Welcome back.

Let's have a look at a solution for that task.

The first part of it was to calculate the wave speed for each group.

You should have come up with answers like this.

Okay, remember, the process is just using the wave equation and that's multiplying the frequency by the wavelength.

So for group A, the frequency of 5.

0 times a wavelength of 1.

2 gives us 6.

0 centimetres a second.

For group, B we've got 4.

0 x 1.

4, that's 5.

6.

And for group C we've got 4.

5 x 1.

4, which is 6.

3.

And as you can see, there's slightly different wave speeds for each group there as well.

The second part of the question was about why they might be getting different wave speeds, and the reason you've got a different wave speed is there's something different about the medium.

So you could have suggested things like this: different depth of water.

The students might have had deeper or shallower water and that would affect the wave speed.

It could even have been a temperature difference, that will make a slight difference to wave speed.

It's possible even that they might have used different water.

Salt water and fresh water will have different wave speeds because they're a slightly different medium.

So well done if you've selected any of those.

Okay, we're going to move on to the final part of the lesson.

We're going to look at the link between frequency and wavelength.

We've already said that the wave speed of a particular wave in a particular medium is constant and we've looked also at this wave equation, wave speed = frequency x wavelength, but what would happen if you changed the frequency of a wave when it was moving through a medium? Well, if the frequency changes and the wave speed doesn't change, there must be a change to the wavelength.

What we're going to do is watch a short video of a simulation to show the relationship between frequency and wavelength and what happens if I increase the frequency of a wave in an object and how that affects the wavelength.

This is a simulation of a wave on a rope.

We're going to use it to explore the relationship between the frequency and wavelength.

I've set it to produce waves of frequency of one hertz.

So let's get it running to see the wave pattern.

As you can see, the waves travel along the rope from one end to the other.

We're going to pause it and measure the wavelength.

So we're using the ruler up here.

A little bit further on.

So we'll get the full wave.

You should be able to see that the wavelength is about six centimetres just there behind that window.

Let's get it going again and see what happens to the wavelength when I increase the frequency using this control down the bottom.

I'll gradually increase the frequency to two hertz.

And you should be able to see the wavelength changing.

It's much shorter now.

Let's get to two hertz exactly.

And pause it again so we can see the wavelength.

Position the ruler here.

And you can see the wavelength now is about three centimetres.

So doubling the frequency has halved the wavelength.

Let's continue with that and increase the wavelength to three hertz and see what happens.

Again, the wavelength is getting shorter as we go.

And there we have three hertz.

And pausing again.

Now the wavelength of the wave is decreased down to two centimetres.

Okay, let's see what we learned from that video.

This is a true or false question.

So, increasing the frequency of a wave increases its wave speed.

Is that true or is it false? After this, I'm going to ask you to justify your answer as well.

So pause the video and make a selection and then restart.

Excellent, it's false, but why is that? So again, I'm going to show you two options.

There they are.

The wave speed is proportional to the frequency, or the speed of a wave depends only on the medium it is in.

So why was that statement false? Excellent.

It's because the wave speed doesn't change.

It only depends on the medium it travels through.

So only changing the medium changes the wave speed, nothing to do with altering the frequency.

Okay, thinking back to that simulation, I've got a diagram here representing it again.

I'm shaking a rope up and down with a frequency of one hertz and I'm producing a wave there.

You can see there's one full wavelength between me and the wall.

What would happen if I altered the frequency of the vibration, if I shook it up and down faster? Well, the wave speed doesn't change, so what does? Shaking it up and down faster will cause the wavelength to change.

I've doubled the frequency and you should see now there's two full wavelengths inside that length.

So there's a frequency of two hertz and the wavelength has halved.

Doubling the frequency has halved the wavelength.

Mathematically what's happening here is something called inverse proportionality.

When I double one thing, I halve another thing so that the relationship still works.

It's written out like this.

You can see there the wavelength symbol, the lambda, and then another symbol between, that's the proportionality symbol.

And then one over f, that's one divided by the frequency.

That's an inverse proportionality relationship.

So what it's basically saying is something like this.

If I've got a frequency of two hertz and I've got a wavelength of six metres, doubling the frequency to four hertz will change the wavelength by half.

So doubling the frequency halves the wavelength.

So as you'd expect, you get a wavelength of three metres.

If I did the same thing again and increased the frequency from four hertz to eight hertz, or doubling the frequency again, I'd halve the wavelength again and I'd get a wavelength of one and a half metres.

Okay, let's do a quick check whether you've understood what happens when you change the frequency.

If a frequency of a sound wave in air halves, then the wavelength must have double.

Is that true or false? Pause and select and then restart.

That's true.

Well done.

Can you explain your answer though? Can I have a justification for it? I've got two possibilities here.

I'd like you to select from them.

The wavelength is inversely proportional to the frequency, or the wave speed changes to account for the change in wavelength.

Which of those? Well done.

It was the wavelength being inversely proportional to the frequency.

If I'm changing the wavelength, I change the frequency.

If I change the frequency, I change the wavelength.

They're inversely proportional to each other.

The wave speed doesn't change at all, unless I change the medium.

That relationship is true for all waves.

Whenever I change the frequency, I'll be changing the wavelength.

So I have got something like a slinky spring here.

Vibrating at low frequency might give you a long wavelength, and then vibrating at higher frequency will give me a shorter wavelength.

So shaking slinky at different frequencies will give different wavelengths.

The wave speed is staying the same in all those examples because I'm not changing the medium in any way.

Only the wavelength is changing.

High frequency gives shorter wavelength, and low frequency gives longer wavelengths.

Another check here.

Which of these happen when the frequency of a water wave increases from 0.

2 hertz to 0.

4 hertz? Does the wavelength double, the wavelength halve, the wave speed double, or the wave speed halve? Pause and select and then we'll go through the answer.

Excellent.

The wavelength halves.

If you are doubling the frequency, which is what we're doing from 0.

2 hertz to 0.

4 hertz, then we're halving the wavelength.

Okay, we've reached the final task of the lesson which brings all of the concepts together.

It's got quite a few stages, so I'll go through it in detail.

A sound wave with a frequency of 500 hertz and a wavelength of 0.

66 metres is produced by a loud speaker.

Firstly, I'd like you to calculate the wave speed.

So you're going back and using the wave equation here again.

So calculate the wave speed of that sound for me.

Second thing I'd like you to do is to describe what happens to the wave speed when the frequency of the sound is doubled to 1000 hertz.

Next, I'd like you to describe what happens to the wavelength when the frequency is increased to 1000 hertz.

And the final part of the task, I'd like you to describe what happens to the wavelength if the frequency is reduced from 500 hertz to 100 hertz.

And that's a fairly tricky one there at the end.

So I'm going to ask you to pause the video and spend a bit of time trying to answer those four parts for me and restart when you're ready, and we'll go through them.

Welcome back.

Let's go through the answers to that question.

First thing we are going to do is calculate the wave speed of the sound.

To calculate the wave speed, we're going to use the wave equation and we're going to extract some data from that question at the top there.

So we'll write down the wave speed equation and we need to find the frequency and the wavelength to solve this.

So the frequency is 500 hertz and the wavelength is 0.

66 metres.

So we put those values in.

Wave speed = 500 hertz x 0.

66 metres.

And we just calculate by multiplying those two together to give us 330 metres per second, which is what we saw the speed of sound was earlier in the lesson.

Next, it's describe what happens to the wave speed when the frequency of the sound is doubled to 1000 hertz.

So if we're doubling the frequency, does that have an effect on the wave speed? Well, no it doesn't.

It doesn't have any effect at all because the medium hasn't changed, we haven't changed the air in any way.

So there is actually no effect there.

So well done if you spotted that.

Moving on to part three of the question, we're describing what happens to the wavelength when the frequency was increased to 1000 hertz.

Now in this case, we do know that the frequency and the wavelength are connected.

If you change the frequency, you change the wavelength.

If you change the wavelength, you change the frequency.

So there is a change here.

We know that the frequency is doubled, and we know the wavelength is inversely proportional to that.

So if you double the frequency, you must halve the wavelength.

So there we go.

We've got a wavelength halved from 0.

66 metres to 0.

33 metres.

The final part of the question was the trickiest of them.

Describe what happens to the wavelength if the frequency is reduced from 500 hertz to 100 hertz.

So what I'm doing there is I'm decreasing it to a fifth of its original value.

So if I'm decreasing the wavelength to a fifth, I must be increasing the wavelength by a factor of five.

So what I'm going to get is a frequency five times smaller, so a wavelength that's five times larger.

So 5 x 0.

66 metres is 3.

30 metres.

If you got that, very well done.

That was by far the trickiest of the tasks so far.

Well, we've reached the end of the lesson now, so I'll just go through a quick summary.

The first thing that you need to remember is that the wave speed depends upon the medium it's passing through.

It doesn't depend on the properties of the wave, it depends on the properties of the medium.

So you can make the wave speed up or slow down only by altering that medium.

Then we looked at the wave equation, which connects frequency, wavelength, and wave speed.

And that is wave speed = frequency x wavelength or in symbols, v = f lambda.

And in the final phase of the lesson, we looked at the relationship between the wavelength and the frequency, and found that wavelength is inversely proportional to the frequency.

If you double the frequency, you cause the wavelength to halve.

And if you halve the frequency, you cause the wavelength double.

And we've got a few little diagrams at the bottom to try and represent that.

Low frequency gives long wavelength, and a high frequency produces a short wavelength.

So congratulations for reaching the end of the lesson.

I'll see you again in some future ones.