A stormy sea is one of the most awesome demonstrations of the power of Nature. Waves crash against the coast with enough energy to turn over cars and destroy buildings. But how is that energy transferred from the deep water where the waves are generated?

Rather less violent, but equally inspiring is the rainbow, which is sometimes left behind, when the storm has passed.

Both of these natural phenomena are examples of wave motion and that's what this video's all about. To understand wave motion, I'm going to have to develop a mathematical description of an idealised wave, which can then be applied to all sorts of real-world situations.

So what exactly is a wave?

Well, a wave is really a mechanism by which energy gets transferred from one place to another. And that idea can be seen here, in the wave pool at a holiday resort. A moving paddle, behind the wall, generates waves which travel away from the wall and are eventually dissipated on the sloping beach.

Well, here's a scaled-down and simplified version of the wave pool. It's called a ripple tank, and I'm going to use it to demonstrate the key features of waves. Waves are generated at this end. And, if I turn it on, you can see that the cylindrical paddle oscillates, and that drives the surface of the water up and down and creates parallel waves that travel along the tank in this direction. These sloping beaches round the edge are just to stop reflections and allow us to concentrate on the parallel wave crests that move along. But to actually look at a single wave crest as it moves along the tank is rather difficult, so to do that, we've got a diagram that shows it.

The wavy line represents the surface of the water, as we look from the side, and the height of the waves has been exaggerated. This is the profile of the wave and it's moving to the right at a constant speed. If we freeze the motion, you can see three wave crests with troughs between them. The amplitude of the wave is defined as the maximum displacement of the water surface from its mean position. It's equal to half the peak-to-trough displacement of the wave, and it's usually denoted by a capital A.

We can also use this static picture of the wave to define its wavelength. This is equal to the horizontal distance between two corresponding points on the profile - these might be two adjacent crests, but two troughs would do just as well, or any other pair of corresponding points. The wavelength of a wave is represented by the Greek letter lambda.

In fact you can think of this snapshot of the wave as a graph, showing the vertical displacement of the wave, as a function of distance, at a particular instant of time.

Let's start the wave moving again.

In the ripple tank, it's rather difficult to see the waves from the side like this. So to improve things, we've put a strong directional source of light up there which shines through the surface of the water, projects down onto this mirror here, and throws an image forward onto the screen. And here, you can see the parallel wave crests moving down the screen.

If we freeze the picture here for a moment, you can see that the wavelength is the distance between two adjacent wave crests.

Let's start things moving again, and think about the idea of the speed of the wave.

How would you measure the speed if you were sitting beside the tank, watching the wave pass a particular point?

Well, here's the graphical representation we had before, showing a side-view of the wave. For the moment, I want you to concentrate on the behaviour of the water surface at one particular point. As the wave passes by, the water surface at this point is simply moving up and down, periodically.

Let's forget about the distance axis, and consider the periodic up and down motion of this point on its own. In order to see how the displacement of this spot varies with time, we can plot a graph of displacement against time. This curve looks very much like the static wave profile you saw earlier, but it's actually a plot of the motion of one particular position on the water's surface, against time. It still shows the amplitude of the wave, which has the same definition as before. But now, the separation between two crests of the wave profile is an interval of time. This time interval is known as the period of the wave, and it's usually represented by a capital T - this is the time it takes the spot to do one complete cycle.

So what's the relationship between the period of a wave and its wavelength? Well, here's the displacement versus distance graph again. Let's watch exactly one cycle, . . . and then freeze the action.

During a time interval equal to one period, the wave has travelled forward a distance equal to one wavelength. So the speed of the wave, which is defined as the distance travelled divided by the time taken, is simply the wavelength divided by the period.

Okay, so what do you think would happen if the paddle were moving up and down at a different rate? Would it change the speed of the waves? Well, I can alter the rate of the paddle using this control here. So, first of all, if I slow it down... you can now see that the wave crests are further apart - the wavelength is longer. But they're still travelling at the same speed along the tank.

Alternatively, if I increase the rate at which the paddle moves up and down... now the wavelength's much shorter, but again, the waves are travelling at the same speed down the tank. So the speed of the waves is independent for any particular tank of water. Let's take a look at this, using our graphical representation.

This is the original displacement against distance graph of the water surface. With a digital clock, you can see that the wave takes two seconds to travel a distance of one wavelength. So the period of this wave is two seconds. This means that only half a cycle of the wave passes by a fixed point in one second.

If the rate of the paddle, and hence the rate at which the water moves up and down is now reduced - by a factor of two, say - the wavelength becomes longer, as we saw in the tank. Since the speed of the wave is unaffected, it now takes four seconds to travel a distance of one wavelength - that's only one quarter of a complete cycle, passing by, each second.

On the other hand, if the paddle moves up and down more rapidly, the wavelength becomes shorter. In this case, the wave takes only one second to travel a distance of one wavelength. So, with the paddle oscillating like this, one complete cycle goes past every second.

The number of cycles passing any point per second is known as the frequency, and it's equal to the reciprocal of the period: f equals one over T. If the period is 2 seconds, the frequency must be half a cycle per second and so on.

So you now have two relationships that are true for all waves: speed equals wavelength over period, and frequency equals one over period. Expressing the right hand side of the first equation as 'one over T times lambda' and substituting 'f' for 'one over T' gives the fundamental equation of wave propagation. Speed equals frequency times wavelength.

Frequency - the number of complete cycles that pass a fixed point in one second - is measured in hertz. And at the moment, the paddle is oscillating about 6 times a second, so that's a frequency of 6 hertz. Now, if I double that frequency to about 12 Hz, that means that the frequency of the water waves is about 12 Hz, but the wavelength has halved - so, the speed remains the same. And if I were to decrease the frequency again by a factor of two, then the wavelength would double, and once again the speed would remain the same.

This illustrates the important result that the frequency of a wave multiplied by its wavelength gives a constant number - the speed of the wave. So a long wavelength corresponds to a low frequency, while a short wavelength corresponds to a high frequency.

Although the speed stays the same at different driving frequencies, it does depend on things like the geometry of the tank, and the nature of the fluid through which the wave travels.

To make sure you've got the hang of these various definitions of wave parameters, try the following exercise.

Here are two representations of the same wave. The upper graph is the displacement of the wave plotted against distance - this graph is a snapshot of the wave at a fixed instant of time. And the lower graph shows the displacement plotted against time and this shows the wave behaviour as a function of time, at a fixed point in space. Pause the video now and carry out Task 1.

The next thing I want to look at is what happens to a water wave when it passes through an aperture. Well, I've fitted an aperture here, but it's rather too wide for what I need at the moment. So if I fit these sliding doors, I can make an aperture that's about - well - about 3 cm across. And if you now look at the screen, you can see that as the waves pass though the aperture, they spread out.

This spreading out of a wave as it passes through an aperture is known as diffraction, and it's a property of all types of waves. So, what do you think controls how much the wave spreads out? Well, one possibility is the width of the aperture. And what do you think would happen if I made the aperture smaller?

Well, you might be surprised to learn that a narrower aperture leads to more spreading out of the wave - more diffraction.

If, instead, I make the aperture wider . . . then after it's settled down, you can see that the wave becomes less spread out - the diffraction is less pronounced. And, if I were to make the aperture very wide indeed, then there'd be hardly any diffraction.

Well, I'm now going to return the aperture to its original width, and take a look at the effect of changing the wavelength of the wave. Now, remember, I can change the wavelength by altering the frequency of the paddle, using this control here.

If I make the wavelength longer, you can see that the waves become more spread out beyond the aperture - more diffraction. And if I make the wavelength smaller again - here's where we started - then even shorter wavelength - then there's less diffraction.

So here are the two separate factors that affect the diffraction of a wave passing through an aperture. The wave spreads out more if the aperture is made narrower or if the wavelength of the wave is increased. And conversely, it spreads out less with a wider aperture or a shorter wavelength. In fact, all kinds of wave - not just water waves - exhibit diffraction at an aperture, as long as the aperture's not too big.

So what about light? Does light spread out after passing through an aperture? Well, here's an aperture. And if I put this piece of card up here, and we look at the shadow cast by this aperture, then it doesn't seem to be causing much diffraction of the light.

This nice sharp outline seems to imply that light travels in straight lines. So does it propagate like a wave or not? Well, light does propagate like a wave, and it does get diffracted. However, the wavelength is very small, and this aperture is far too large to cause any appreciable diffraction. Remember how the diffraction of the water waves became less apparent as I made the aperture much bigger than the wavelength. To demonstrate diffraction with light waves, I need to use much smaller apertures. I also need to use a different light source - one with a very narrow beam and a very narrow range of wavelengths. And in order to achieve that, I'll be using this laser beam and, of course, I'll be taking care not to look directly into the beam.

Now this slide contains the apertures I'll be using, and in fact the first one is just 60 micrometres wide, or 60 microns for short. That's about one sixteenth of a millimetre. So, if I place this slide in front of the laser beam, you'll see the diffraction pattern on the screen over there.

The aperture's aligned vertically, and that sharp spot of light has clearly been spread out, or diffracted, in a horizontal direction by the small aperture. You can see that the diffraction pattern consists of a series of dark and bright regions spaced regularly either side of the centre. The image appears bright in regions of so-called constructive interference, where light waves from different parts of the aperture combine to reinforce each other, and dark in regions of destructive interference where light waves combine in such a way that they cancel each other out. The double-headed arrow indicates the width of the central bright region.

This spreading of the laser beam shows that light does indeed propagate like a wave. Now, if you remember with the water waves, you saw that changing the aperture changed the amount of diffraction. So, again here, if I use a smaller aperture, you should see that the diffraction pattern spreads out more. This next aperture's just 30 microns wide.

Again, the double-headed arrow indicates the separation of the first dark regions on each side of the centre. And if I swap again to an aperture that's 120 microns wide, you can see that the diffraction pattern becomes less spread out. So this is exactly the same effect we saw with the water waves - the spread of the diffraction pattern increases with a narrower aperture, and the decreases with a wider one.

Now I'm going to use a different laser. This one produces green laser light, and I've already put the 60-micron aperture in front of it, so let's take a look at the diffraction pattern produced in this case.

Well, if you compare this pattern with the one produced by red light passing through the same aperture, you should be able to see that the green pattern is less spread out than the red one. So, how can you explain this result?

Well, since I've used the same aperture, it must be something to do with the light. And, in fact, it's the wavelength. Now, if you remember, there was less spreading of the water waves by the aperture when their wavelength was reduced. And so, the conclusion is that the green laser light must have a smaller wavelength than the red laser light.

The different colours of visible light are simply the way our eyes interpret the different wavelengths. And outside the range of visible light are other forms of electromagnetic radiation which also show all the properties of waves. Gamma-rays, X-rays and ultraviolet light all have shorter wavelengths than visible light, which occupies this very narrow band of wavelengths.

Notice how increasing wavelength corresponds to decreasing frequency, right across the spectrum.

Infrared radiation, microwaves and radio waves all have longer wavelengths than visible light. And within the visible range - the familiar rainbow of colours - red light has the longest wavelength, and blue light the shortest.

Now I'm going to carry out a diffraction experiment using hundreds of narrow apertures at the same time. This will give a much clearer pattern on the screen over there, and enable-you to take fairly accurate measurements from your TV screen. In order to do that, I'm going to be using a device called a diffraction grating - that consists of hundreds of narrow lines spaced very close together. And, in fact, this slide contains three separate diffraction gratings.

The first one I'll be using has 300 lines per millimetre - that's like 300 apertures packed into every millimetre across it. So the spacing between the individual lines is one three-hundredth of a millimetre - or about 3.3 microns.

So first of all, I'll shine the red laser beam through the grating with 300 lines per mm. Instead of smoothly changing dark and bright regions, the diffraction pattern this time consists of a series of isolated spots on the screen. But the image is created by the same process of interference between light waves coming from different positions on the grating. These spots are referred to as diffraction orders, and they come in pairs either side of the centre. The central spot is called the zero order, the two outside that are the first order, then come the second order, and so on.

If I now swap the grating for one that has only 100 lines per mm - that's a spacing of one hundredth of a millimetre or ten microns - the diffraction pattern becomes less spread out. So larger, more widely-spaced apertures give rise to less diffraction.

And finally, using a grating with 600 lines per millimetre, a spacing of about one point seven microns, you can see that the pattern becomes very spread out indeed. So just as you saw with the single apertures, narrower apertures give rise to more diffraction. Here are all three diffraction patterns displayed together from the three different gratings. Pause the video now and carry out tasks 2 and 3.

Now, I'm going to illustrate how a different wavelength alters the diffraction pattern. I've put the original 300 lines per millimetre grating back, but now I'm going to use the green laser beam. So let's look at the diffraction pattern now.

Here are the red and green diffraction patterns displayed together for comparison. As you've seen before, a shorter wavelength gives rise to less spread in the diffraction pattern.

At this stage, I just want to summarise the main points you should have picked up so far.

To start with, I used the ripple tank and graphic sequences to define various basic parameters of a wave, and I found the important relationship speed equals frequency times wavelength.

And in Task 1, you tested your own understanding of these terms by making some measurements on two representations of the same wave, from your television screen.

I then illustrated the property of diffraction of a water wave passing through an aperture, and showed that the spread of the wave could be increased both by using narrower apertures and by using longer wavelengths (and vice-versa).

Light also propagates as a wave, but the wavelengths are very much smaller. So much narrower apertures are needed to demonstrate diffraction.

Finally, I used a diffraction grating, rather than a single aperture, so that you could make quantitative measurements of the spread of the various diffraction patterns.

You should have concluded, firstly, that the distance of the nth spot from the centre, s-subscript-n is proportional to the order of the pattern, n.

And, secondly, that the distance is inversely proportional to the spacing between the lines on the grating, d.

Finally, the demonstration with the green laser light means we can suggest a third relationship, namely that the distance is also proportional to the wavelength of the light, lambda.

Combining these three relationships gives us a first approximation for an important expression that describes the diffraction pattern produced by a grating.

As you saw in tasks 2 and 3, the proportionality seems to deviate at large distances from the centre of the pattern - at high orders - and the reason for this will be explained in the book, where the idea of diffraction is developed further.

To finish off, I want to give you a quick preview of the experiment you'll be doing later. This grating is similar to the one that you've been sent - it contains 300 lines per millimetre - and you'll be using it to look at the spectra of household light bulbs. Just so that you're sure of what you'll be looking at, I've set up this demonstration here. These two lamps contain different bulbs. This one is a conventional tungsten filament bulb, whereas this one is a modern energy saver bulb. To the naked eye, both appear similar - they both produce a nice bright white light. But, of course, white light is a mixture of all colours, and this grating will allow you to see just what that mix of colours is.

Now, in the experiment, you'll be asked to produce a narrow source of light like this. It's important to realise that this is not a diffraction slit - it's far too wide to diffract the light coming from the lamp - it's merely there to provide a well-defined source. And if I now look through the grating towards that, I can see the spectrum of the tungsten light bulb. The spectrum shows a continuous distribution of colours, indicating that a whole range of wavelengths is present.

If instead, I look towards the energy saver bulb, I see something quite different. Now, at each order of the diffraction pattern, I can see a spectrum that contains discrete lines of particular colours - corresponding to certain wavelengths only. In fact the view you have on your TV screen is limited by the capabilities of the television camera. Your eye is a much more sensitive optical detector when it comes to discriminating between different shades of colour, so when you come to do this experiment for yourself, you'll see a much clearer difference between the two spectra. And later, in book 7, you'll see that line spectra such as these also provide a profound insight into the quantum world of atoms.