Lesson video

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

By the end of this lesson, you should be able to interpret and sketch graphs of longitudinal waves.

And to be able to do that, we first need to be able to describe longitudinal waves and really understand how the graphs of those waves are linked to what's happening in the waves themselves.

And whilst we're looking at longitudinal waves, we first of all obviously need to describe what a longitudinal wave is and understand that properly.

And to do that, we need to introduce some new terms. We need to think about what a compression in a longitudinal wave is and what a rarefaction is in a longitudinal wave.

And we also need to be able to identify what the wavelength of a longitudinal wave is and what the amplitude is.

These last two terms are a little bit more complicated for longitudinal waves than they were for transverse waves, but we'll have a look at that in this lesson today.

So the lesson is split into two halves.

We're going to look at sound waves first of all, because they're the sort of longitudinal waves that you are most familiar with, and we can use those to actually be able to describe what a longitudinal wave is and how it works.

And then in the second half of the lesson, we'll look a little bit more detail at how we represent longitudinal waves, and in particular by looking at some longitudinal wave graphs.

So first of all, let's have a look at sound waves.

Up until now, we've been looking at transverse waves mostly, but today we're going to shift our gaze and have a look a little bit more closely at longitudinal waves.

And as well as sound waves, the type of longitudinal wave you're most likely to be familiar with is an ultrasound wave.

You might even have an ultrasound scan of yourself when you are a baby growing inside your mother.

What ultrasound is, it's a very high frequency sound wave that can pass through the human body without causing any harm.

And the ultrasound reflects off different surfaces to generate a real time image of what's going on inside the body with the help of computer that generates the image from the detection of those reflections.

And then another example of longitudinal waves are the P waves in earthquakes.

Earthquakes create both longitudinal and transverse waves.

So sound waves are longitudinal waves as all of those examples we gave before.

Sound waves can travel through air and they can also travel through other gases and through solids and liquids, but what they can't do is travel through empty space, which we also call a vacuum.

And this is because they need a wave medium to travel through.

They need something to vibrate and oscillate backwards and forwards to carry the wave as they move.

Now, if we think about air, air contains, air is a gas and air contains particles that we've learnt move very, very quickly in all directions and they're bouncing off each other all the time.

So you can always picture the gas particles around us bouncing off in all directions in three dimensions.

How do they carry a sound wave as well as all that motion? When a loud speaker starts working, what it actually does is it adds an extra movement to the sound wave.

So they're still whizzing around in all directions, but what we've got here is a picture of the extra movement that is added to those particles to form the sound wave.

And we're going to concentrate on this extra movement when we're thinking about sound waves bash backwards and forwards, but overall, the loud speaking isn't making the air particles move further forwards than they would normally move then.

They're normally, as we said before, whizzing around randomly in all directions and staying roughly in the same sort of places they were on average as they were before.

So what's happening here is the loudspeaker pushes the particles of air close to it forwards.

They bash into particles a little bit ahead of them and push them forwards, but when they collide with those particles, they bounce backwards again and stay roughly where they were, and they keep bashing into other particles nearby and stay in roughly the same place.

And as the particles pass that vibration on, the sound waves moves forward.

On this picture, you can see little arrows on three of the particles showing how they're vibrating backwards and forwards, are being made to vibrate backwards and forwards by the loud speaker's motion, and the loudspeakers moving in exactly the same way as these other particles.

So this is the same picture again with the loudspeaker at one end of the tube moving backwards and forwards to create the sounds.

It's vibrating backwards and forwards to create the sound.

And as it's going forwards, it's pushing it into the air particles close to it and it's pushing them in one direction.

And as it moves backwards, those first particles will be bumping into the particles in front of them and bouncing backwards off them into the space left by the loudspeaker.

And each time those particles of air push into the ones in front of them, they set those particles moving forward and bumping into the ones in front of them.

And so what we're seeing here is that push from the loudspeaker being moved from one set of particles to the next as it's going forward, as a wave is moving forwards.

But each of the particles in turn is staying roughly in the same position.

If I add on here some arrows to show how the individual air particles are moving, I've just highlighted three of them here with the arrows.

As the air particles behind these ones I've highlighted push them, it will push 'em forwards, they'll bump into the particles in front of them and be bounced backwards, and they'll oscillate about a fixed point.

So in the air, as the sound is moving forwards, the particles are being caused to move backwards and forward to oscillate about their average position.

But the sound wave itself is moving forwards and it's carrying energy with it, it's transferring energy with it, but without transferring any of the air.

If you are at the other end of a sound wave listening with your eardrum, what you'll notice is that the energy makes your air drum vibrate, so you hear the sound, but you don't get an air being blown full of air with the waves coming forwards.

In fact, you don't feel any breeze in your ear at all because the air is not moving forwards with the wave.

Now, if the air particles are being caused to move backwards and forwards by the speaker and bouncing back when they hit the particles in front of them, then the distance each air particles moving backwards and forwards and vibrating backwards and forwards is going to be the same distance that the speaker is moving.

And if you've seen a loud speak without the cover on, you'll see that it doesn't actually move very far at all most of the time.

So the distance that the air particles are actually being caused to move by the sound wave is quite small.

And just the same as for the transverse waves earlier, the maximum distance of each air particles is made to move either forwards or backwards from where it was before is its amplitude.

And we can mark on the amplitude of some of these air particles motion here.

So each of those particles with the arrow has been moved forwards or backwards by the maximum distance, or its amplitude.

Now, we can describe this pattern of particles with some key terms that we mentioned right at the start of the lesson.

Places where the air particles have been squashed together are called compressions.

And so on this particular picture, you can see three compressions where they're being compressed or pushed together.

It's got the word press in the middle, which reminds me anyway that they've been pressed together.

And places where they are most spaced out are called rarefactions.

It's got the word rare in it, which sort of means that there's rarely very many particles in this space.

It doesn't mean that, but it's what I imagine it means.

So we've got compressions and rarefactions that we use to describe the points on the longitudinal wave.

And you'll notice that the rarefactions are halfway between each pair of compressions, and each compression is halfway between two rarefactions and they're all equally spaced along the wave.

If we want to measure the wavelength of a sound wave, we measure as a distance between two sets of compressions like so.

Now, if you remember, the actual air particles themselves are not moving backwards and forwards very much, so the air particles don't move along the whole of the wave, they just move a tiny fraction.

But it's the pattern formed by the air particles in the longitudinal wave, in the sound wave, that cause these compressions that we measure the wavelengths from.

So again, the wavelength is a distance between two compressions, or in the top right (indistinct), also the distance between two rarefactions, because in our previous diagram, if we just go back to that briefly, we'll see that they are equally spaced.

So it's the same wavelength between each of those.

Okay, let's check to see if you've understood what we've been talking about.

It's quite complicated, so it's worth just pausing there and having a real think about what's going on.

Here's three more statements.

Here's three statements about the amplitude and wavelength of the wave that's shown in the diagram.

And there's a distance marked x on there.

Just pause the video and write in the boxes what you think about each statement, whether you think they're right or wrong.

And just mention or just tick the box to say whether you are absolutely sure it's right or wrong, or whether you are a little bit uncertain, you think it's right or wrong, but you're not absolutely certain.

It's a tiny distance and the length of a sound wave can be two, three, four metres long.

So it's certainly not that.

What about statement C? The wavelength is equal to the distance x.

Well, x is marking the distance between two compressions, which is what we gave as our definition of a wavelength of a longitudinal wave.

So that last statement is the only correct one.

Okay, so that's quite a lot of information to take in.

I think we just need to have a stop there and have a practise at seeing if we can put those ideas into practise and apply them to a new situation.

So this situation here, this is a model that some students have made to try to explain how a sound wave moves through air.

Now, a model isn't exactly what's going on, but it's a representation of what's going on to try to help us to understand more clearly what's happening.

And what these students have done is they've made a model with ping pong balls in front of a loudspeaker to try to represent what happens when a sound wave moves forward.

So what I'd like you to do is to have a go at those four parts of the question, put down as much detail as you can, and then just pause the video whilst you do that.

Okay.

How did you get on? What do you think the ping pong-balls represent? Well, hopefully you said it's the air particles because that's the obvious thing.

The ping-pong balls represent the air particles or the wave medium that the wave is travelling through.

And what happens to them when the loudspeaker vibrates? So let's just imagine the ping-pong balls hanging by threads.

The loudspeaker starts to move backwards and forwards.

It, first of all, as it moves forwards, it hits the first ping-pong ball, which then is pushed forward, batches into the next one.

That one batches into the next one and so on.

So the vibration passes all the way along the ping-pong balls to make the one on the far right hand side push forwards.

Whilst that's been happening, the loud speakers move backwards.

So the first ping-pong ball's been allowed to swing back, backwards from where it was to start with, and in turn, all of the other ping-pong balls can swing backwards as well, and then it will swing back forwards and keep doing so whilst the loud speakers pushing it and then giving it space to swing backwards.

And we've summed that up on this text down here.

How is that similar to how a sound wave moves? Within a sound wave, the air particles are doing pretty much what the ping-pong balls are doing.

They're moving backwards and forwards, being pushed forwards by the loudspeaker, bashing into the air particles in front of them, and being pushed backwards again.

And if the loudspeakers move backwards, they've got space to move into until the loudspeaker moves forwards and pushed 'em forwards again.

So all of those air particles are moving backwards and forward and passing on the vibrations added to their movement by the loudspeaker.

So those statements, again, describe what's happening.

You can always just pause the video and have a look at these in more detail if you'd like to.

And then the final part of the question was asking, how is the model different to how a sound wave moves? Remember, a model isn't exactly what's going on.

A model is what we've constructed in our minds to help us think about what's happening.

It's a close approximation to reality, but it isn't exactly what's going on.

So in this model here, the ping-pong balls are hanging on threads, whereas in the air, the air particles are actually whizzing around very, very quickly all over the place.

And there's many, many times more air particles than there are ping-pong balls handing on threads.

There's in the odds of 10,000 trillion air particles, which would take up the same space that these ping-pong balls are taking up here.

That's far more air particles than you can imagine.

And they're so small you can't even see them.

And the other thing about air particles is they're already whizzing around all over the place.

The movement that we're describing here, the ping-pong balls passing on the vibration one to the next is the added movement caused by the loudspeaker to the air particles.

It's the sound wave.

And I suppose with that model, we're ignoring all of the other movements and just think about the movement caused by the sound wave that we actually hear.

And again, we've just jotted those down here.

And if you'd like to read them in more detail, just pause the video and have a look at those for a few moments.

So now you understand what a longitudinal wave is.

We're going to move on to see how we can represent longitudinal waves in the form of a graph.

This part of lesson is quite challenging and it's really here just to give you a really good idea of how the longitudinal graphs link to the movement of the particles in the waves.

It's unlikely that you'll be asked very detailed questions of this in the exam, but it will really put clearly into your mind what's going on.

It will make those longitudinal wave graphs make a lot more sense once you've done this, what I'm going to do now is I'm going to try to explain to you how the particles change their positions because of a longitudinal wave and how we can draw a graph of that movement in the same way we did this for transverse waves.

So I'm starting here with a set of particles which are understood.

These are particles before the wave passes by.

And if we imagine a loudspeaker, maybe if these were sound air particles, for example, if there's a loudspeaker making them move forwards and backwards bashing into each of them, bump backwards as we've described before, how might they be moving? Well then the arrows here show how the particles are caused to move forwards and backwards in order to make the longitudinal wave.

And as you can see here, we're starting to get at the bottom line what looks like a longitudinal wave with compressions and spaces called rarefactions between because of the movement of those particles.

And if you look carefully, you can see how the arrow shows how the particle at the top has moved forwards or moved backwards by a certain distance to reach their new position.

So the bottom diagram shows a wave moving along these particles and how the particles have moved in order to form the wave.

Okay? And there's the compressions and rarefactions added on just to show you where they are.

Rarefactions where the particles are most spread out and compressions where they're least spread out.

One thing to notice is the particle at a compression, or at a rarefaction is in its usual position.

It's not really moved at all from where it started from.

So now we've described how the particles are moving forwards and backwards in the longitudinal wave.

The question is how do we plot that in a graph? In a transverse wave, it's fairly straightforward.

As you can see in the top image here, the transverse wave particles are moving upwards or downwards and we can plot their displacement on the graph quite easy in an up and down direction.

But in the longitudinal waves, the particles are not moving up and down, they're moving backwards and forwards, and that's not easy to plot.

What we do instead is we think about the backwards and the forwards movement as a displacement, and what we say is that the forwards movement of the particles is a positive displacement, which we can go plot on a graph going upwards.

And if the particles are moving backwards from their usual position, we draw them on the graph below their usual position.

So the graph we've got here matches up to the movement of the particles above.

And they've had a few lines in there, you can see that the particles that have not moved away from their usual position are placed on the line of the graph at the bottom.

And the particles that are moving forwards are shown on the graph as lines going upwards.

And the size of the arrows on the graph, the displacement that we've shown there are the same size as the movement forwards, the displacement of the particles in the longitudinal wave.

So going all along the graph, all the particles that have been moved forwards from their usual position are above the line.

All the particles that have been moved backwards from their usual position are shown on the graph below the line, and all the particles that are in the same position that they normally are are drawn on the line.

So it's just like a transverse wave graph, but this time we're plotting forwards and backwards movements of the particles as up and down on the graph.

So let's check that you've understood that.

Here's a graph that represents a longitudinal wave.

Which point shows the maximum forward displacement of the particles on the wave? The maximum forwards displacement.

Pause the video and work out which is your best answer.

Okay, what do you think? The correct answer is D.

And if you look above the graph at the wave, you can see that it's A and C, which are plotted on the line.

They correspond to particles that have not moved from their usual position.

And at position B, the particles in the longitudinal wave have moved backwards.

So that shown below the line.

At D, the particle there has moved forward, and it's moved forward the maximum amount.

So that's the maximum forward displacement at part D.

Same graph, but this time, which point shows compression? If you need to, just pause the video and then start it again when you're ready.

Okay, what do you think? The correct answer is A.

You can see it's compression because it's the point on the longitudinal wave above the graph where all the particles are closest together.

It's not easy to tell whether point A or C is different on the graph, but you can see from the longitudinal wave that's not particularly important at this stage of your learning.

So just think about the compression in terms of the movement of the particles.

But the point I'm making here is that where there's a compression, the line on the graph is crossing the horizontal axis, because the particle there hasn't moved.

Again, the same graph, but I've added some arrows now.

This time, which arrow shows the amplitude of the wave? Again, pause the video while you think about this and then start it again when you are ready.

Okay, so this is a little bit more challenging because it's hard to actually think about what's happening with these arrows here.

The amplitude is the maximum displacement.

So we've got to find the point where the particle has moved the furthest distance from its average position.

And we need to draw a line on the graph, or show that on the graph as a distance it's moved.

And the correct answer here is B.

That particle above B, the particle above B has moved the furthest distance and it's moved backwards so it's drawn below the line.

And finally on this section, which arrow shows the wavelength of the wave? Again, pause a bit if you need to whilst you make your selection, what do you reckon? Well, the wavelength is a distance between two compressions or two rarefactions, or any two places on the wave that are the same, if you like, on two adjacent waves.

And the correct answer here is A.

And if we think about it, if you look on the graph there on the horizontal axis, we've got distance measured and we've got the distance between two peaks of the wave that we've shown there.

And that, as you know from your earlier work, is the wavelength.

Okay, here's some practise for you.

We can also make longitudinal waves by pushing one end of a slinky spring backwards and forwards as we've shown here by the yellow arrow.

And we've got a slinky spring with a longitudinal wave on it, and we've got one below that with no wave.

When you're looking at how those coils on the spring have moved, which of those graphs do you think represent what's happening on that spring? Just pause the video and see if you can work that out.

Okay, have a look at this hint.

What I've done here is I've added some little lines to show how the coils, the turns of each coil of the spring have moved when there's a wave passing through them, whether they've moved forwards or whether they've moved backwards.

See if you can use that to help you check your answer to the first one or even come up with an answer if you didn't manage it before.

Again, pause the video whilst you have a think about it.

Okay, let's have a look at some answers.

So the answer for part A was graph B.

And underneath this graph, what you can see, I've put in here is the hint that shows how the particles, or how the turns of the coil when the longitudinal wave is passing have moved forwards or backwards.

And on the left hand half of the picture, you can see that all of the particles, all the parts of the turns of the coil have moved forwards.

So we've drawn the line going above the horizontal axis, and on the right hand side, all of those turns on the coil have moved a little bit backwards.

So we can draw the line going below the horizontal axis on the graph.

So that's the shape of the graph.

And if you look very closely at how far each of those turns of the coil has moved, you'll see that where the graph is the highest and the lowest, the turns of the coil have moved the furthest amount, they've got the biggest displacement.

So that matches up there.

And we can see that with these arrows that I've added on.

So this is just to show the displacement on the coil.

On the left hand side, they've moved forwards, on the right hand side, they've moved backwards and we can draw this by the line above or below the horizontal axis of the graph.

And we can apply the same idea when we sketch the graph in part B.

On the left hand side, starting at the left hand side of the graph, the particles have moved forwards.

So we can draw the line high up.

And then they cross over the line where the particles in their usual position haven't moved, which is represented by the black dots.

And then they move backwards.

So we draw the line underneath the graph, and then back up at the far end when they're moving forwards again.

So hopefully you got that graph sketched reasonably well.

What you'll be noticing by now is that all graphs form the same sort of shape, these single graphs here, it's called a sine wave.

So that's quite a complex set of ideas to get our heads around.

Let's try to summarise that and pick out the key ideas that we've learned from today's lesson.

So there's the picture of the sound wave with a loudspeaker vibrating at one end.

We identified compressions on the longitudinal wave where the particles are squashed most closely together, and rarefactions where they're most spread out.

We also identified the amplitude as the maximum displacement of each particle, the maximum distance it's moved forwards or backwards as the wave passes by it.

And that's the same distance that the loudspeaker moves forwards or backwards from its average position, 'cause it's the loudspeaker that's causing the movement in the particles.

And the wavelength defines the distance between one compression or rarefactions and the next and is a much bigger distance normally than the particles of air are moving in this sound wave.

We also noticed that the oscillations and the vibrations in the longitudinal wave are moving forwards and backwards in the same direction that the wave is travelling, that the displacement is the distance one part of the wave is moved forwards or backwards from its rest position.

And on our graph we show the forward displacement as a positive line above the graph line, and backward displacement is a negative distance below the graph line.

And we can therefore represent a longitudinal wave on a displacement distance graph, and it's shown like this.

Well, hopefully you've got your head around a lot of the ideas in that lesson.

As I mentioned earlier, you need to have a good idea of what's going on, but you're not going to be expected to reproduce the detailed thinking in an exam.

Although thinking it through and having an idea of what's going on is really going to help you.