Waves. Attnom Academy

S4.10.1 Chapter 1

Kinds of waves

Mei flicks one end of a long rope, and a bump runs to the other end. What actually moved along the rope?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Pieces of rope, which travelled from her hand to the far end.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The bump. Each piece of rope only moved up and down a little.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Nothing, because the rope stayed where it was.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Compare transverse and longitudinal waves with a rope and a spring
Explain that waves carry energy, not matter

A wave is a disturbance that travels from one place to another and carries energy with it. A rope, a spring, the surface of a pond and the air can all carry waves.

Tie a ribbon to the middle of a long rope. Mei holds one end and flicks it up and down once. A bump runs along the rope. When the bump reaches the ribbon, the ribbon rises. Then it comes down again.

After the bump has passed, the ribbon is where it was before. The bump went on to the far end. The rope did not.

Draw it first

A bump runs past a ribbon

1 At 0 s the bump is near Mei and the ribbon is at rest.
2 At 1 s the bump is under the ribbon, and the ribbon is at its highest.
3 At 2 s the bump has gone on. The ribbon is back at rest, on the same dashed line. It has not moved along the rope.

The bump carried energy from Mei's hand to the far end. At the far end it can shake a hand tied there. So a wave carries energy from place to place. It does not carry the material it travels through.

Now let Mei keep flicking, up and down, again and again. A steady train of bumps and dips runs along the rope. The highest point of a wave is a crest. The lowest point is a trough. The dashed line is where the rope lay at rest.

A wave on a rope. The picture is a snapshot: each piece of rope is at its own height.

A piece of the rope moves up and down. The wave travels along the rope. These two directions are at right angles, which is 90°. A wave like this is a transverse wave.

In a transverse wave the vibration is across the way the wave travels.

Light is a transverse wave. So are the ripples on a pond, to a good first approximation. Strictly, each bit of water moves in a small circle. We treat ripples as transverse in this book.

Now take a long soft spring and push one end sharply towards the other end, along its length. The coils bunch up, and the bunch runs along the spring. Where coils are pressed together there is a compression. Where they are spread apart there is a rarefaction.

A longitudinal wave on a spring. The compressions move to the right. The thick layer only shakes to and fro.

Look at the thick layer. It moves a little to the right and a little to the left, along the way the wave travels. This is a longitudinal wave. The vibration is along the direction of travel.

In a longitudinal wave the vibration is along the way the wave travels.

Sound in air is a longitudinal wave. A loudspeaker pushes the air layers together and pulls them apart. The compressions and rarefactions run out through the air. In the air, as on the spring, each layer only shakes to and fro.

See it three ways

As a picture

In words

Across means at right angles to the way the wave travels. Along means on the same line as the way the wave travels.

In symbols

Transverse: vibration at 90° to travel. Longitudinal: vibration at 0° to travel, along the same line.

Examples of the two kinds. The pushed row is the row of buttons in the second Try it.

A rope, a spring, water and air are all a medium: the material a wave travels through. Waves on a rope or a spring, ripples and sound all need a medium. Light does not. Sunlight crosses empty space to reach us, so light needs no medium.

Now think of a cork floating on a pond. Ripples pass it. The cork bobs up and down, and when the ripples have gone it is about where it was. The ripples carried energy past it. They did not carry the water or the cork along with them.

A floating cork bobs as a ripple passes and stays where it is. Energy goes by, water does not.

How fast does a wave travel? The speed of a wave is the distance its pattern covers divided by the time it takes. You met this rule for moving things in the speed book: speed = distance ÷ time.

A bump that runs 10 m along a rope in 4 s has a speed of 10 m ÷ 4 s = 2.5 m/s. Waves in different materials have different speeds. The later chapters measure some.

Try it

Send a bump down a rope

You need: a long rope or string, about 5 m, a ribbon, a scrap of cloth or a piece of paper, a friend who can hold the far end, a post or a chair to tie the far end to, if you have no friend.

Be safe: Do this on flat ground, away from people walking past. Flick the rope sideways and low, not towards anyone's face.

Tie the ribbon to the middle of the rope with a loose knot. Lay the rope on a flat floor, or hold it just off the ground.
Hold one end. Your friend holds the other end, or ties it to the post. Pull the rope almost straight, not tight.
Flick your end sideways once, quickly. Watch the bump run along the rope.
Watch the ribbon as the bump reaches it. Then watch where the ribbon is after the bump has gone.
Now flick your end again and again, at a steady pace. Watch how the ribbon moves.

What you will see: The ribbon moves sideways as the bump passes and then goes back to its place. It does not travel to the far end. The bump did.

If you cannot try it, this is what happens.

Try it

A pushed row of buttons

You need: seven flat buttons of the same size, a smooth table.

Be safe: Keep the buttons away from young children, who can swallow them. Slide them along the table, not towards anyone.

Put six buttons in a straight line on the table, touching each other.
Slide the seventh button along the table so that it hits the end of the line, not hard.
Watch which buttons move.

What you will see: The button at the far end of the line jumps away. The buttons in the middle hardly move. The push went along the row and carried energy to the last button. This is a longitudinal pulse.

If you cannot try it, this is what happens.

A wrong idea many people have

A wave carries water, or rope, or air, from one place to another.

Why it feels right: You see a bump run along a rope and ripples spread over a pond. It looks as if the stuff itself is moving.

The test that breaks it: Float a cork or tie a ribbon on. After the wave has gone by, the cork or ribbon is where it was. A wave that carried the water along would take it too.

What is true: A wave carries energy and a pattern from place to place. The material only moves a little to and fro, or up and down, and stays near where it started.

A wrong idea many people have

Sound is a wave like a rope wave, with the air moving up and down.

Why it feels right: Pictures of sound waves are drawn as a wavy line, which looks just like a rope wave.

The test that breaks it: Look at the spring. The layers move along the line of travel, not across it. Air layers do the same, so sound is not a rope type wave.

What is true: Sound is a longitudinal wave: the air shakes to and fro along the way the sound travels. The wavy line in a picture is a graph, not the path of the air.

A wrong idea many people have

Every kind of wave needs a material to travel through.

Why it feels right: Rope waves, ripples and sound all need rope, water or air, so it seems to be a rule for all waves.

The test that breaks it: The Sun is far off across empty space, and its light reaches us. If light needed a material, it could not cross the empty space between the Sun and the Earth.

What is true: Rope waves, ripples and sound need a medium. Light does not. Light crosses a vacuum, which is why we can see the Sun and the stars.

Key words

wave: a disturbance that travels and carries energy.
crest: the highest point of a wave.
trough: the lowest point of a wave.
transverse: a wave whose vibration is at right angles to the way it travels.
compression: a place in a longitudinal wave where the material is pressed together.
rarefaction: a place in a longitudinal wave where the material is spread apart.
longitudinal: a wave whose vibration is along the way it travels.

Back to your guess

The answer is B: The bump. Each piece of rope only moved up and down a little.

A wave is a disturbance that travels. Each piece of the rope moved up and down and stayed near its place. What travelled from Mei to the far end was the bump, and with it energy. The rope itself did not travel.

Keep it

Copy this into your exercise book.

A wave carries energy, not material. Transverse: vibration at 90° to travel (rope, light). Longitudinal: vibration along travel (sound, spring). Light needs no medium.

Draw: A rope with a ribbon, a bump in three positions, and the ribbon staying on one dashed line. Beside it, a spring with one compression.

S4.10.2 Chapter 2

Measuring waves

Kofi sends a wave along a rope. Ana asks how big the wave is. Where does Kofi put his ruler?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A From the lowest point to the highest point.
You picked A. Read on. At the end, the chapter comes back to your guess.
B From one crest to the next crest.
You picked B. Read on. At the end, the chapter comes back to your guess.
C From the rest line up to a crest.
You picked C. Read on. At the end, the chapter comes back to your guess.
D Along the whole length of the rope.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Label amplitude, wavelength and period on a wave diagram
Calculate wave speed from frequency and wavelength

To describe a wave with numbers, we need four measures. They are the amplitude, the wavelength, the period and the frequency. Each one has a unit, and each one is measured in its own way.

Start with the picture. We draw a wave on a graph. The up and down axis shows the displacement. This is how far a piece of the rope has moved from its rest place, in a unit such as cm. The dashed line through 0 is the rest line, where the rope lies when no wave is on it.

A graph of displacement against distance is a photograph of the rope at one moment. It shows the shape of the wave along the rope.

Draw it first

A rope wave on a distance graph

1 Draw the wave. The dashed line through 0 is the rest line.
2 The amplitude is the greatest displacement. Measure it from the rest line up to a crest. Here it is 2 cm. A trough is 2 cm below the rest line, so it gives the same amplitude.
3 The wavelength is the distance between two matching points on neighbouring waves, such as crest to crest. Here it is 1.6 m. Its symbol is λ, the Greek letter lambda.

Amplitude is a distance from the rest line, so it is never negative. A bigger amplitude means more energy in the wave. For sound, a bigger amplitude makes a louder note. For a ripple, it makes a taller ripple.

Wavelength is a length along the way the wave travels, so its unit is the metre, m. You can measure it from crest to crest, trough to trough, or between any two matching points on neighbouring waves.

Now tie a ribbon to the rope and watch it. Its displacement changes over time: up, back to the rest line, down, and up again. A graph of displacement against time shows this. It is not a photograph. It is a record of one place, as time goes by.

The ribbon on the rope. The time from one crest to the next is the period.

The period T is the time for one complete wave to pass a point. It is also the time for one complete vibration. Its unit is the second, s. On the graph, it is the time between two crests.

The frequency f is the number of complete waves that pass a point each second. Its unit is the hertz, Hz. One hertz is one wave each second. A source that makes 5 waves each second has a frequency of 5 Hz. One kilohertz, kHz, is 1 000 Hz.

Period and frequency are two ways to say the same thing. If 5 waves pass each second, then one wave takes 1 ÷ 5 = 0.2 s. So f = 1 ÷ T and T = 1 ÷ f. A short period means a high frequency.

Same frequency, different amplitude. The taller wave is the louder sound.

Same amplitude, different frequency. More waves each second is a higher note.

Now put the measures together. In one period T, a wave moves forward by one wavelength λ. So its speed is the distance divided by the time: v = λ ÷ T. Since 1 ÷ T is the frequency f, this becomes v = fλ.

The wave equation is v = fλ. Speed v is in metres a second, m/s. Frequency f is in hertz, Hz. Wavelength λ is in metres, m. The units agree: m × (1 ÷ s) = m/s.

See it three ways

As a picture

In words

The speed of a wave is its frequency times its wavelength. The frequency is one divided by the period.

In symbols

v = fλ, f = 1 ÷ T, T = 1 ÷ f. Rearranged: λ = v ÷ f and f = v ÷ λ.

The rope wave of the next worked example. Its wavelength is 0.6 m.

Worked example

Waves on a rope have a frequency of 2.5 Hz and a wavelength of 0.6 m. Find their speed.

Write the equation: v = fλ.
Put in the numbers: v = 2.5 Hz × 0.6 m.
v = 1.5 m/s.

Answer: 1.5 m/s

Sound in air travels at about 340 m/s, whatever the note. So a note of higher frequency must have a shorter wavelength. Rearrange the equation to find the wavelength: λ = v ÷ f.

A sound wave in air. Its wavelength is the distance from one compression to the next.

Worked example

Sound travels at 340 m/s in air. A note has a frequency of 0.8 kHz. What is its wavelength?

Change the frequency to hertz: 0.8 kHz = 0.8 × 1 000 = 800 Hz.
Rearrange v = fλ to λ = v ÷ f.
Put in the numbers: λ = 340 m/s ÷ 800 Hz = 0.425 m.

Answer: 0.425 m

Both graphs together give the speed. The first graph, taken at one moment, shows the wavelength of the rope wave: 1.6 m. The second graph, taken at one place, shows the period: the time between two crests.

Worked example

The rope of the first graph has a wavelength of 1.6 m. The ribbon graph shows its period. Find the speed of the wave.

From the ribbon graph, the first crest is at 0.5 s and the next crest is at 2.5 s. So T = 2.5 − 0.5 = 2 s.
Find the frequency: f = 1 ÷ T = 1 ÷ 2 = 0.5 Hz.
The wavelength is λ = 1.6 m.
v = fλ = 0.5 Hz × 1.6 m = 0.8 m/s.

Answer: 0.8 m/s

A wave's speed is fixed by the medium it travels through. Change the frequency and the wavelength changes, but the speed does not. Raise the pitch of a note in air and the wavelength shortens. The sound still travels at about 340 m/s.

The arrow from trough to crest is 4 cm, twice the amplitude. Measure from the rest line.

The short arrow runs from a crest to a trough. It is only half a wavelength. A wavelength is crest to crest.

Try it

The period of a swinging stone

You need: a small stone, about the size of an egg, about 1 m of string, a high hook, a nail in a beam or a branch to tie it to, a clock or phone that shows seconds.

Be safe: Hang the stone clear of faces and windows, and keep other people away from its path. Use a small stone and tie it well, so that it cannot fly off. Do not tie it where you must climb.

Tie the stone firmly to the string. Tie the other end of the string up high so the stone hangs in the air.
Pull the stone a short way to one side and let go. A swing to and fro and back to the start is one full swing.
Start the clock when the stone passes its lowest point. Count 10 full swings and stop the clock.
Work out the period: T = time for 10 swings ÷ 10.
Work out the frequency: f = 1 ÷ T. Then try a shorter string, about 25 cm, and compare.

What you will see: The stone takes about 2 s for each swing on a 1 m string, so its frequency is about 0.5 Hz. A shorter string swings faster: its period is shorter and its frequency is higher. Timing 10 swings gives a better answer than timing one.

If you cannot try it, this is what happens.

A wrong idea many people have

The amplitude is the height from the trough to the crest.

Why it feels right: The whole height of a wave is the first thing you see. A ruler held beside the wave reads from the bottom to the top.

The test that breaks it: Look at the graph with the two arrows. The distance from the rest line to the crest is 2 cm. The distance from trough to crest is 4 cm. These are not the same, and the first is the amplitude.

What is true: Amplitude is measured from the rest line to a crest, or from the rest line to a trough. Crest to trough is twice the amplitude.

A wrong idea many people have

The wavelength is the distance from a crest to the next trough.

Why it feels right: A crest and a trough are the two obvious points of a wave, and they are next to each other.

The test that breaks it: Measure crest to trough on the graph: 0.8 m. Now go on to the next crest: 1.6 m in all, and the wave has repeated. A full wave needs both a crest and a trough.

What is true: Wavelength is the distance between two matching points: crest to crest or trough to trough. Crest to trough is half a wavelength.

A wrong idea many people have

A wave of higher frequency travels faster.

Why it feels right: Frequency and speed both sound like "how fast", and a high note feels quick.

The test that breaks it: Sound in air travels at about 340 m/s for every note. A high note and a low note from the same drum reach you together. If higher frequency meant a higher speed, they would arrive at different times.

What is true: Frequency says how many waves pass each second. Speed says how far one wave goes each second. The medium sets the speed. With v = fλ, a higher frequency then means a shorter wavelength.

Key words

displacement: how far a piece of the medium is from its rest place, with a direction. It has a unit such as cm.
rest line: the line on a wave graph where the medium lies when no wave is on it.
amplitude: the greatest displacement, measured from the rest line to a crest.
wavelength: the distance between two matching points on neighbouring waves, in m. Its symbol is λ.
period: the time for one complete wave to pass a point, in s. Its symbol is T.
frequency: the number of waves that pass a point each second. Its symbol is f. f = 1 ÷ T.
hertz: the unit of frequency, written Hz. One hertz is one wave each second.
wave equation: v = fλ. The speed of a wave is its frequency times its wavelength.

Back to your guess

The answer is C: From the rest line up to a crest.

The size of a wave, its amplitude, is measured from the rest line to a crest. The distance from lowest to highest is twice that. The distance from crest to crest is the wavelength, which is a different measure.

Keep it

Copy this into your exercise book.

Amplitude: rest line to crest. Wavelength λ: crest to crest. Period T and frequency f = 1 ÷ T. Wave equation v = fλ. The medium sets v.

Draw: A wave on axes with the rest line. Add an arrow for the amplitude and an arrow for the wavelength. Write v = fλ beside it.

S4.10.3 Chapter 3

Waves on water

Priya's ripples run from deep water onto a shallow part of the tray. What happens to them?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Ripples slow down and bunch up as they run onto shallow water.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Ripples speed up and spread out as they run onto shallow water.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Ripples stop where the water gets shallow.
You picked C. Read on. At the end, the chapter comes back to your guess.
D Ripples do not change when the depth changes.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Measure the speed of ripples in a tray of water
Explain how water waves slow down and bend in shallow water

Drop a pebble into a pond and rings spread out from the place it fell. Each ring is a line of crests. Scientists study water waves in a shallow tray, seen from above, called a ripple tank.

Seen from above, a line that joins the crests is a wavefront. The wave travels at right angles to its fronts. A tap on the water makes circular wavefronts. A straight bar moved up and down makes straight ones.

Ripples from one tap. The rings are equally far apart. That gap is the wavelength.

The gap between two neighbouring rings, or two neighbouring straight fronts, is one wavelength. To measure it well, measure across several waves and divide. Do not measure a single gap, because the reading error is a bigger share of a small length.

Five fronts have four gaps between them. Four gaps are four wavelengths.

Count the gaps, not the fronts. Five fronts leave four gaps, so the distance from the first front to the last is four wavelengths. A common slip is to divide by five.

To find the speed, use the wave equation from the last chapter: v = fλ. The frequency f is how many taps you make each second. The wavelength λ is the length you measure. Keep λ and v in the same length unit, such as cm and cm/s.

Worked example

Taps at 4 Hz make five fronts with 28 cm from the first to the last. Find the wavelength and the speed.

Five fronts have four gaps. So four wavelengths together are 28 cm.
One wavelength is λ = 28 cm ÷ 4 = 7 cm.
The speed is v = fλ = 4 Hz × 7 cm = 28 cm/s.

Answer: 7 cm and 28 cm/s

Try it

Find the speed of ripples in a tray

You need: a flat tray or a wide dish, with water about 1.5 cm deep, a ruler, a pencil or a stick, a torch, a clock or phone that shows seconds, a phone camera to take a photo from above, if you have one.

Be safe: Keep the tray and the water away from sockets, wires and anything that runs on electricity. Wipe up spills at once, because a wet floor is slippery. Do not shine the torch into anyone's eyes.

Put the tray on a table or the floor, on a dark cloth. Pour in water about 1.5 cm deep. Wait until the water is still.
Lay the ruler flat on the edge of the tray where it shows along the water. Shine the torch low from the side, so the crests show as bright and dark lines.
Tap the water with the tip of the pencil at one end, at a steady beat. Count your taps for 10 s and divide by 10. This is the frequency f in Hz. Try for about 4 taps each second.
Look from straight above, or take a photo. Measure the distance across 5 fronts, which is 4 gaps. Divide by 4. This is the wavelength λ.
Work out v = fλ.

What you will see: The ripples are a few centimetres apart and travel several tens of centimetres each second. Your numbers will differ with the depth of the water and how fast you tap. Measuring four gaps together is far better than measuring one.

If you cannot try it, this is what happens.

Now change the depth. Once the water is shallower than about half a wavelength, water waves travel more slowly where it is shallower. Ripples in a tray 1.5 cm deep are in this range. This is also why ocean waves slow, shorten and rise as they reach a beach.

A flat plate in the tray makes part of the water shallow.

Send straight ripples across the tray from the deep end to the plate. Here is what happens at the edge of the shallow part. The frequency does not change. The source makes the same number of waves each second, and the same number cross the edge each second.

The speed falls in the shallow water. The wave equation v = fλ then tells us the wavelength must fall too, because f stays the same. So the fronts bunch up in the shallow water. The new wavelength is λ = v ÷ f.

Next, let the ripples meet the edge at a slant. Each part of a front slows down as it reaches the shallow part. The part that arrives first slows first, so the front turns. It is like a cart whose wheel meets sand first and swings round.

To describe the turn, draw the normal: the line at 90° to the edge. The ripples travel along arrows at right angles to the fronts. In the slower, shallow water, the ripples turn towards the normal. This bending is refraction.

Ripples cross from deep to shallow water. The fronts are closer together, and the way they travel turns towards the normal. It is drawn with the speed ratio 0.6 of the worked example below. In a real tray the bend is often smaller, because the edge of the plate is not sharp.

Look at the edge. Every front that reaches the edge carries on into the shallow water without a break. So the fronts meet the edge equally far apart on both sides. That joining is why the direction must turn when the wavelength changes.

If the waves run from shallow water into deep water, they speed up and the wavelength grows. They then turn away from the normal. Along the normal itself, waves do not turn. Their wavelength and speed still change.

Worked example

In the deep part of a tray the wavelength is 5 cm. Ripples with a frequency of 4 Hz then cross to the shallow part, where the wavelength is 3 cm. Find the speed in each part.

In the deep part v = fλ = 4 Hz × 5 cm = 20 cm/s.
The frequency stays 4 Hz in the shallow part, because the source does not change.
In the shallow part v = fλ = 4 Hz × 3 cm = 12 cm/s.

Answer: 20 cm/s and 12 cm/s

The ratio of the two speeds is 12 ÷ 20 = 0.6, which is the same as the ratio of the wavelengths, 3 ÷ 5. This works because the frequency is the same in both parts. The picture above is drawn with this ratio of 0.6.

Going further: the sea shows the same slowing on a huge scale. For a wave much longer than the water is deep, the speed is v = √(g × h). Here h is the depth in m, and g is 9.8 m/s². This rule is for tides and tsunamis. It is not for small ripples in a tray.

The speed of a long wave in the sea. 198 m/s is about 710 km/h, nearly as fast as a jet airliner. The shallow sea near a coast slows it to about 36 km/h.

Worked example

A long wave crosses a sea that is 90 m deep. Find its speed.

The rule for a long wave is v = √(g × h).
Put in g = 9.8 m/s² and h = 90 m: g × h = 9.8 × 90 = 882.
The square root is √882 = 29.7.
So the speed is about 30 m/s.

Answer: About 30 m/s

As such a wave nears a coast, the water gets shallower and the wave slows. The energy the wave carries is squeezed into less water, so the wave grows taller. This is why a tsunami is small far out at sea and tall near the shore.

Try it

Ripples over a plate

You need: the tray and the water from before, about 1.5 cm deep, a flat tile or flat piece of board about 1 cm thick, with its top just under the surface, a pencil, a ruler and a torch.

Be safe: Keep the tray away from electrical things, and wipe up spills. Set the tile down gently, because it can crack the tray.

Lay the tile flat at one end of the tray, so that the water over it is about 0.5 cm deep.
Tap the water at the other end at a steady beat. Watch the ripples cross onto the tile.
Look at how far apart the crests are over the tile compared with the deep part. Measure both, across four gaps.
Try to tap so that the ripples meet the edge of the tile at a slant. Watch the way they turn.

What you will see: The crests are closer together over the tile. That is a shorter wavelength at the same frequency, so the waves are slower there. At a slant the ripples turn towards the normal on the tile. The effect is small, so look carefully, or take a photo from above.

If you cannot try it, this is what happens.

Five fronts, with 20 cm across them. Count the gaps between them: one, two, three, four.

A wrong idea many people have

To find the wavelength, divide the distance across the fronts by the number of fronts.

Why it feels right: We count things to share a total. Five fronts look like five waves, so it seems right to divide by five.

The test that breaks it: Look at the picture: 20 cm across four gaps, not five. Divide 20 cm by 5 and you get 4 cm. The real gaps are 20 ÷ 4 = 5 cm.

What is true: Divide by the number of gaps, which is one less than the number of fronts. Six fronts have five gaps.

A wrong idea many people have

The frequency of ripples changes when they pass into shallow water.

Why it feels right: The ripples look different in the shallow part, with the crests closer together. It seems that the number of waves must have changed.

The test that breaks it: Count crests that pass a point in the deep part in 10 s, then in the shallow part in 10 s. The counts are the same. If one more arrived than left, the water at the edge would pile up forever.

What is true: The source sets the frequency, and it does not change when the wave crosses an edge. The speed and the wavelength change. The wavelength is v ÷ f.

A wrong idea many people have

Water waves speed up in shallow water.

Why it feels right: A wave on a beach seems to rush up the sand, and it looks fast as it breaks.

The test that breaks it: The ripples over the tile have crests closer together at the same frequency. With v = fλ, a smaller λ and the same f give a smaller v.

What is true: Waves slow down as the water gets shallower. A breaking wave looks quick because it is rising and tipping over, not because its speed has gone up.

Key words

ripple tank: a shallow tray of water, seen from above, for studying water waves.
wavefront: a line joining the crests of a wave, seen from above. The wave travels at right angles to it.
normal: a line at right angles (90°) to an edge, used to measure angles.
refraction: the turning of a wave when its speed changes at a slanted edge.

Back to your guess

The answer is A: Ripples slow down and bunch up as they run onto shallow water.

Water waves travel more slowly in shallower water. Their frequency stays the same, so the wavelength gets shorter and the crests bunch up. If they meet the shallow part at a slant, they also bend.

Keep it

Copy this into your exercise book.

v = fλ. Measure λ across several gaps: fronts − 1. Shallow water slows waves. f stays the same, so λ shortens and the wave bends towards the normal.

Draw: Straight wavefronts crossing an edge at a slant, closer together in the shallow part, with the normal and two arrows.

S4.10.4 Chapter 4

Sound and ultrasound

Tariq claps in front of a tall wall 85 m away and hears an echo. How long after the clap does the echo reach him?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A About 0.25 s, because the sound only has to reach the wall.
You picked A. Read on. At the end, the chapter comes back to your guess.
B About 0.5 s, because the sound goes to the wall and back.
You picked B. Read on. At the end, the chapter comes back to your guess.
C About 2 s, because sound is slow.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Measure the speed of sound using an echo from a wall
Explain how ultrasound is used to scan unborn babies and find sea depths

Sound is a longitudinal wave. Layers of the medium are pushed together and pulled apart, and the pattern of compressions and rarefactions runs outwards from the source.

Sound needs a medium to travel through. It can travel through a gas, a liquid or a solid. It is usually slowest in a gas, faster in a liquid and fastest in a solid. In a solid the particles are close and spring back quickly.

Sound is about 340 m/s in air at room temperature. It is a little slower when the air is colder and a little faster when it is warmer.

In a vacuum there are no particles, so there is nothing to push together. Sound cannot travel in a vacuum. Scientists show this with a bell in a jar. They cannot do it at home, because it needs a pump that takes the air out.

With air in the jar you hear the bell.

With the air pumped out, the bell still moves, but you hear almost nothing.

As the air is pumped out, the ringing gets fainter. When almost no air is left, it is almost silent. The bell hangs on a thin thread, so that the sound cannot run along something solid to the glass. You can still see the bell move, because light does cross a vacuum.

Light is far faster than sound. In a vacuum, light travels at 3 × 10⁸ m/s. That is nearly a million times the speed of sound in air. This is why a flash of lightning is seen long before the thunder is heard.

Now use an echo to measure the speed of sound. Stand in front of a large wall and clap. The sound travels to the wall and bounces back. You hear the echo a short time later.

Sound goes to the wall and comes back. So the distance it travels is twice the distance d to the wall. The speed is distance divided by time: v = 2d ÷ t.

The sound covers the 85 m twice, there and back.

Worked example

A person stands 85 m from a wall. An echo comes back 0.5 s after a clap. Find the speed of sound.

The sound goes to the wall and back, so it travels 2 × 85 m = 170 m.
The speed is v = distance ÷ time = 170 m ÷ 0.5 s = 340 m/s.

Answer: 340 m/s

A hand held stopwatch cannot time 0.5 s well, because your reaction time is about 0.2 s. The trick is to clap again and again, so that each clap comes with the echo of the one before. Then the time between claps is the echo time. Count many claps and divide.

Try it

Measure the speed of sound with an echo

You need: a large flat wall, such as the side of a tall building, at least 50 m away across open ground, a long string or tape, or a known length of pace, to find the distance, a clock or phone that shows seconds, two flat pieces of wood to clap together.

Be safe: Stay well away from roads and from the edges of cliffs, roofs or deep water. Do not clap close to anyone's ears. Use a wall that belongs to nobody who would mind the noise.

Find the distance d from where you stand to the wall. Walk it and count paces, after you have measured the length of one pace on a string.
Clap the two pieces together. Listen for the echo.
Clap again so that your clap comes just as the echo of the last clap arrives. Keep a steady rhythm.
When the rhythm is steady, count 20 claps and time them. Divide the time by 20. This is the echo time t.
Work out v = 2d ÷ t.

What you will see: Your answer will be close to 340 m/s, but not exactly. Timing 20 claps instead of one clap makes the answer much better. The speed of sound also changes a little with the temperature of the air.

If you cannot try it, this is what happens.

The same idea gives the distance to a reflector. If you know the speed of sound and the echo time, then v × t is the whole way there and back. Half of that is the distance.

An echo gives the distance to a cliff when the speed and the time are known.

Worked example

A shout echoes from a cliff and the echo returns 1.2 s later. Take the speed of sound as 340 m/s. How far away is the cliff?

The sound goes there and back in 1.2 s.
The whole way is v × t = 340 m/s × 1.2 s = 408 m.
The cliff is half of the whole way: 408 ÷ 2 = 204 m.

Answer: 204 m

Ordinary sound is used in this way, but sound that we can hear is a nuisance for it. For many jobs, scientists use sound that is too high for us to hear. Sound with a frequency above 20 000 Hz is called ultrasound.

Humans hear from about 20 Hz to about 20 000 Hz. The top of the range falls as people grow older. Sound below 20 Hz is called infrasound. Elephants use it. Many bats and dolphins use ultrasound.

The scale goes up in tens: each tick is ten times the one before. The top of the human range is 1 000 times the bottom.

Ultrasound is the same kind of wave as ordinary sound. It is longitudinal, and in a given medium it travels at the same speed as ordinary sound. It differs only in having a higher frequency, and so a shorter wavelength.

A ship uses ultrasound to find the depth of the sea. This is sonar. The ship sends a short pulse of ultrasound down. It reflects from the sea floor, or from a shoal of fish, and comes back up. The ship times the echo.

Sonar. The pulse goes down and the echo comes up, so the ship halves the whole way.

Sound in sea water travels at about 1 500 m/s. The speed changes a little with temperature, salt and depth, so the ship uses a value from measurements. The depth is half the distance there and back: d = v × t ÷ 2.

Worked example

A ship's sonar pulse returns 0.8 s after it was sent. Sound travels at 1 500 m/s in the sea. Find the depth.

The echo sounder times the pulse as it goes down and comes back.
Going down alone takes half the time: 0.8 s ÷ 2 = 0.4 s.
The depth is speed × time = 1 500 m/s × 0.4 s = 600 m.
In kilometres: 600 m ÷ 1 000 = 0.6 km.

Answer: 600 m, which is 0.6 km

Doctors use ultrasound to look inside the body without cutting it. The scanner has a probe that sends in pulses of ultrasound, with frequencies of several million hertz. A clear gel on the skin keeps air from lying between the probe and the skin. Air reflects almost all the ultrasound.

Echoes come back from every boundary where one kind of tissue meets another: skin, fluid, the baby and bone. The scanner times each echo, and notes how strong it is. A computer turns these into a picture. Sound in soft tissue travels at about 1 540 m/s, so the echoes come back in about a tenth of a millisecond.

A scan. The echo from the baby comes back to the probe, and the time says how deep it is.

Ultrasound does not ionise, which means it does not knock electrons off atoms. At the low powers used for scans, no harm to the mother or the baby has been found. This is why it is used to look at unborn babies, in place of X-rays. Scans are still only done when they are useful.

A wrong idea many people have

The echo time is the time the sound takes to reach the wall.

Why it feels right: We think of the wall as the place the sound is going to, and the echo as the end of the journey.

The test that breaks it: Take a wall 85 m away and the speed 340 m/s. Reaching the wall alone takes 85 ÷ 340 = 0.25 s. The echo reaches you after 0.5 s, twice as long, because it must come back.

What is true: The echo time covers the way there and the way back. The distance to the wall is half of v × t.

A wrong idea many people have

Light and sound travel at about the same speed.

Why it feels right: A clap seems to be seen and heard together. A flash and a bang from a nearby firework seem to come at once.

The test that breaks it: Watch a firework across a wide field or a far storm. The flash comes first, and the bang follows seconds later. At 1 km, light takes about 3 millionths of a second, and sound takes about 3 s.

What is true: Light is nearly a million times faster than sound. Sound is slow, about 340 m/s in air, and it can never be faster than light.

A wrong idea many people have

Ultrasound travels faster than ordinary sound, because it is higher.

Why it feels right: Higher frequency sounds like "more" and "quicker", and we call the sound "ultra".

The test that breaks it: In air at 340 m/s, a note of 400 Hz and an ultrasound of 40 000 Hz both travel at 340 m/s. The ultrasound has 100 times the frequency, so with v = fλ it has one hundredth of the wavelength.

What is true: The speed is set by the medium. Ultrasound has the same speed as ordinary sound in the same medium. Only its frequency and wavelength differ.

Key words

vacuum: a space with no particles in it. Sound cannot cross it, and light can.
echo: sound that reflects from a surface and returns. It travels the distance there and back.
ultrasound: sound with a frequency above 20 000 Hz, too high for people to hear.
infrasound: sound with a frequency below 20 Hz, too low for people to hear.
sonar: a way to find depths and objects in water by timing the echo of a pulse of sound.

Back to your guess

The answer is B: About 0.5 s, because the sound goes to the wall and back.

The sound has to go to the wall and come back, a total of 170 m. At about 340 m/s that takes 0.5 s. An echo always covers the distance twice.

Keep it

Copy this into your exercise book.

Sound needs a medium and cannot cross a vacuum. Echo: v = 2d ÷ t. Depth = v × t ÷ 2. Ultrasound is above 20 000 Hz. Light is far faster than sound.

Draw: A ship, the sea and the sea floor. Draw a pulse arrow going down and an echo arrow coming up. Mark the depth.

S4.10.5 Chapter 5

Reflection and refraction

A ray of light meets a mirror and makes an angle of 20° with its surface. At what angle was the ray arriving, in the way scientists measure it?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A 20°, because that is the angle the ray makes.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 70°, because scientists measure from the line at right angles to the mirror.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 110°, because the ray bends back by the same amount.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw ray diagrams for reflection and refraction at a boundary
Explain total internal reflection and its use in optical fibres

Light travels in straight lines called rays. A ray is drawn as a straight line with an arrow that shows the way the light goes. This chapter follows rays when they meet a mirror, and when they meet the edge of glass or water.

Every angle is measured from the normal. The normal is a dashed line at 90° to the surface, drawn at the point where the ray meets it. The ray that arrives is the incident ray. The angle of incidence is the angle between the incident ray and the normal. The ray that leaves a mirror is the reflected ray. The angle of reflection is the angle between that ray and the normal.

The law of reflection: the angle of incidence equals the angle of reflection. Both are measured from the normal.

The law of reflection says two things. The angle of incidence equals the angle of reflection. And the incident ray, the normal and the reflected ray all lie in one flat plane. On a diagram, that plane is the page.

A very common slip is to measure the angle from the mirror, not from the normal. If a question says a ray makes 30° with the mirror, the angle of incidence is not 30°. It is 90° − 30° = 60°. Always draw the normal first.

The law holds for every ray on every surface. A rough surface, such as paper, is made of tiny patches that face different ways. Each ray obeys the law at its own patch, but the patches point in many directions, so the light scatters. A smooth mirror keeps parallel rays parallel, and you see an image.

Now look at your own image in a flat mirror. Rays from your face reflect from the mirror and reach your eye. Your eye follows each ray back in a straight line, to a point behind the mirror, where the image seems to be.

The image is as far behind the mirror as the object is in front. The rays only seem to come from it.

The image in a flat mirror is the same size as the object. It is as far behind the mirror as the object is in front. It is upright. It is left and right reversed. It is a virtual image: light does not really pass through it, so you cannot catch it on a screen.

Now turn to refraction. Light travels at 3 × 10⁸ m/s in a vacuum and almost as fast in air. In glass and in water it is slower. The refractive index n of a material is the speed of light in a vacuum divided by the speed in the material.

Refractive index and the speed of light in m/s. Ordinary glass is about 1.5. Other kinds of glass differ a little.

When a ray goes into a material where light is slower, it bends towards the normal. When it goes into a material where light is faster, it bends away from the normal. A ray that travels along the normal does not bend, but it still changes speed.

Air to glass: the light slows down and bends towards the normal.

Glass to air: the light speeds up and bends away from the normal.

How much does a ray bend? Snell's law gives the answer. For light going from air into a material of index n, sin i ÷ sin r = n. Here i is the angle of incidence and r is the angle of refraction, both from the normal. The sine of an angle is the ratio of two sides of a right angled triangle.

In general, light goes from a material of index n₁ into one of index n₂. Then n₁ × sin i = n₂ × sin r. In every case the ray bends towards the normal when n goes up, and away when n goes down.

A ray enters water at 45° to the normal.

Worked example

A ray goes from air into water, whose refractive index is 1.33. The angle of incidence is 45°. Find the angle of refraction.

Use sin i ÷ sin r = n, so sin r = sin i ÷ n.
sin 45° = 0.7071, so sin r = 0.7071 ÷ 1.33 = 0.5317.
The angle whose sine is 0.5317 is r = 32.1°.

Answer: 32.1°

A rectangular block of glass shows both bends. The ray bends towards the normal as it goes in, and away from the normal as it comes out. The two bends are equal and opposite, so the ray that leaves is parallel to the ray that went in. It is moved sideways.

Through a glass block, the ray leaves parallel to the way it came in. It is at 40° to the normal again, but shifted sideways.

Now go the other way. A ray inside glass or water meets the edge with air. It bends away from the normal. As the angle of incidence grows, the bent ray swings closer to the edge. At one angle it runs along the edge. That angle is the critical angle, c.

At the critical angle, the angle of refraction is 90°. So sin c = 1 ÷ n for light going from a material of index n into air. For glass of index 1.5, c is about 42°.

Past the critical angle, no light can leave. All of it is reflected back inside, by the law of reflection. This is total internal reflection.

Three rays in glass. Under the critical angle the light goes out. At it, the light runs along the edge. Beyond it, all the light is reflected.

Total internal reflection has two conditions. The light must go from a higher n to a lower n, such as glass to air. And the angle of incidence must be greater than the critical angle. Light going from air into glass can never be totally reflected.

Even when light does cross an edge, a little of it is also reflected. The diagrams leave that weak reflection out. Total internal reflection is different: there, all the light is reflected.

A ray in water at 60° to the normal is past the critical angle. All of it is reflected.

Worked example

Water has a refractive index of 1.33. What is the critical angle for light going from water into air?

At the critical angle, sin c = 1 ÷ n.
sin c = 1 ÷ 1.33 = 0.7519.
The angle whose sine is 0.7519 is c = 48.8°.

Answer: 48.8°

Diamond has a very high index. Its critical angle is small, so light that enters a cut diamond is reflected inside again and again before it comes out. This is part of why diamonds sparkle.

An optical fibre is a thin, flexible thread of glass. Light that enters one end meets the wall at an angle greater than the critical angle. It is totally reflected, again and again, and follows the fibre round bends to the far end.

A ray follows a bent fibre by total internal reflection. Every meeting with the wall is past the critical angle.

A real fibre has a core of glass inside a coating of glass of slightly lower index. The total internal reflection happens where the core meets the coating. A fibre cannot be bent too sharply. The light would then meet the wall at less than the critical angle and leak out.

Fibres carry telephone calls and the internet as pulses of light, which you meet in the last chapter. Doctors also use bundles of fibres in a thin tube, an endoscope, to look inside the body without a large cut.

Try it

Test the law of reflection

You need: a small flat mirror, a torch, two cards held with a narrow gap between them, a sheet of white paper, a pencil and a ruler, something to hold the mirror upright, such as a lump of clay or a book.

Be safe: Never shine the torch into anyone's eyes, and never use the Sun or a mirror in sunlight for this. Tape the edges of a glass mirror, which can cut. Keep the mirror and the clay away from young children.

Draw a straight line across the paper. Stand the mirror upright on the line, with its shiny face towards you. Draw a dashed line at right angles to the mirror, as the normal. Mark where it meets the mirror.
Lay the torch behind the two cards. A thin beam should come through the gap. It should cross the paper at a slant and meet the mirror at the foot of the normal. Keep the beam low and flat on the paper.
Mark two small dots on the beam that goes to the mirror, and two on the beam that leaves it. Take away the mirror and the torch, and draw lines through the dots.
Mark a point on each line exactly 10 cm from the mirror point. Measure how far each of the two points is from the normal.
If the two distances are equal, the two angles are equal.

What you will see: The two distances come out almost the same, within a millimetre or so. A point 10 cm along a ray is 10 cm times the sine of the angle away from the normal. So equal distances mean equal angles.

If you cannot try it, this is what happens.

Try it

The silver surface of water

You need: a clear glass or jar, filled with water, a table.

Be safe: Use a glass or jar that cannot tip over, away from the edge of the table. If you have a clear plastic cup, use that. Wipe up any spills.

Stand the glass of water on a table, away from the edge.
Crouch so that your eye is a little lower than the top of the water. Look up at the underside of the water surface, through the side of the glass.
Look steeply upwards. Then look at a slant, nearly along the underside of the surface.

What you will see: Looking steeply up, you see through the surface to the ceiling. At a shallow slant, the surface looks like a silver mirror. You see the bottom of the glass or the objects in the water. This is total internal reflection.

If you cannot try it, this is what happens.

A wrong idea many people have

A ray that makes 30° with a mirror has an angle of incidence of 30°.

Why it feels right: The angle you can see is the one between the ray and the mirror. It is the first angle we think to measure on a flat surface.

The test that breaks it: Draw the normal first. The ray makes 30° with the mirror, and the mirror makes 90° with the normal. So the ray makes 90° − 30° = 60° with the normal. The law of reflection uses the angle from the normal.

What is true: Angles of incidence, reflection and refraction are always measured from the normal, never from the surface.

A wrong idea many people have

Light always bends towards the normal when it changes material.

Why it feels right: Our first picture of refraction is a ray going into water or glass, and that ray bends towards the normal.

The test that breaks it: Look at the ray going from glass into air. It bends away from the normal, and the angle in the air is bigger than the angle in the glass.

What is true: A ray bends towards the normal when it goes into a material where light is slower. It bends away from the normal when it goes into a material where light is faster.

A wrong idea many people have

Total internal reflection can happen when light goes from air into glass, if the angle is large enough.

Why it feels right: A large angle seems to be all it takes, and light reflects from a glass window at a slant.

The test that breaks it: From air into glass, sin r = sin i ÷ 1.5. This is always less than 1, so a refracted ray always exists. A window at a slant also reflects a little, but some light always goes in.

What is true: Total internal reflection needs two things. The light goes from a higher index to a lower index. The angle of incidence is greater than the critical angle.

Key words

angle of incidence: the angle between the incident ray and the normal.
angle of reflection: the angle between the reflected ray and the normal.
virtual image: an image made where rays seem to come from, so it cannot be caught on a screen.
refractive index: the speed of light in a vacuum divided by its speed in the material. It is written n.
critical angle: the angle of incidence in a denser material at which the refracted ray runs along the edge. sin c = 1 ÷ n.
total internal reflection: all the light reflected inside a material, when it meets the edge beyond the critical angle.
optical fibre: a thin glass thread that carries light along it by total internal reflection.

Back to your guess

The answer is B: 70°, because scientists measure from the line at right angles to the mirror.

Scientists measure the angle from the normal, the line at 90° to the mirror. A ray that makes 20° with the surface makes 90° − 20° = 70° with the normal. That is the angle of incidence.

Keep it

Copy this into your exercise book.

Angles are from the normal. Reflection: i = r. Slower material: bends towards the normal. Total internal reflection: denser to less dense, beyond the critical angle, sin c = 1 ÷ n.

Draw: A glass edge with three rays: one bending out, one along the edge, one reflected, and the normal as a dashed line.

S4.10.6 Chapter 6

Lenses

Held far from a window, a magnifying glass puts a small, upside down picture of the window on a sheet of paper. Why is the picture upside down?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Rays from the top of the window cross after the lens and reach the bottom of the picture.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The lens turns the picture over, as you would turn a coin.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The lens bends every ray down.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw ray diagrams for a converging lens
Explain how lenses correct short sight and long sight

A lens is a clear piece of glass or plastic with curved faces. It bends light by refraction at each face. A converging lens is thicker in the middle and brings parallel rays together. A diverging lens is thinner in the middle and spreads them apart.

The principal focus, F, of a converging lens is the point where rays parallel to the axis meet after the lens. The axis is the line through the middle of the lens at right angles to its faces. The distance from the lens to F is the focal length, f.

Parallel rays meet at the principal focus. A fatter lens bends them more, so its focus is nearer.

A lens works both ways, so it has a focus on each side, at the same distance f. A fatter, more curved lens has a shorter focal length. A real lens bends light at both faces. To keep the picture clear, we draw one bend, at one line through the middle.

To find the image of an object, draw an arrow for the object. Then trace three rays from its tip. These three rules are enough.

Rule one: a ray that travels parallel to the axis bends and goes through the focus on the far side. Rule two: a ray that goes through the centre of the lens carries straight on. Rule three: a ray that passes through the focus on the near side leaves the lens parallel to the axis.

Where the rays meet is the image of the tip. If the rays really cross there, the image is a real image. You can catch it on a screen. It is upside down. The picture of the window is a real image.

An object beyond 2F gives a real, upside down, smaller image between F and 2F.

An object at 2F gives a real, upside down image of the same size, also at 2F.

An object between F and 2F gives a real, upside down, larger image beyond 2F.

Move the object closer, to a point inside F. The rays now leave the lens spreading out. They never cross on the far side. Look backwards along the rays, and they seem to come from one point on the object's side.

An object inside F gives an upright, larger image on the same side as the object. This is a magnifying glass.

This image is a virtual image. Light does not really pass through it, so it cannot be caught on a screen. Your eye sees it because the rays seem to come from it. A virtual image from a converging lens is upright and larger. This is how a magnifying glass works.

The image from a converging lens. A real image is upside down. A virtual image is upright. An object at F gives no image: the rays leave parallel.

Diagrams give the shape of the answer. A rule gives the numbers. For a thin lens, 1/f = 1/u + 1/v. Here u is the distance from the object to the lens and v is the distance from the lens to the image. All are in the same unit.

This book counts distances to a real image as positive, and a virtual image has a negative v. Other books use other sign rules and write the formula another way. The physics is the same. To find v, rearrange: 1/v = 1/f − 1/u.

The magnification is the height of the image divided by the height of the object. It also equals v ÷ u when you ignore the sign. A magnification of 2 means the image is twice as tall as the object. It has no unit.

Worked example

An object is 4 cm from a converging lens of focal length 3 cm. Find the image distance and the magnification.

For a real image, 1/v = 1/f − 1/u. Here f = 3 cm and u = 4 cm.
1/v = 1/3 − 1/4 = 4/12 − 3/12 = 1/12.
So v = 12 cm. The image is real and 12 cm from the lens.
The magnification is v ÷ u = 12 ÷ 4 = 3.

Answer: 12 cm and a magnification of 3

Any object between F and 2F gives a larger real image, as in the worked example.

Now the diverging lens. It spreads parallel rays so that they seem to come from a focus on the object's side. The three rays bend the other way. A ray parallel to the axis leaves as if from the near focus. A ray through the centre goes straight on. A ray aimed at the focus on the far side leaves parallel to the axis.

A diverging lens always gives an upright, smaller, virtual image between the lens and the object.

Now turn to the eye. The cornea and the lens of the eye act together as a converging lens. They make a real, upside down image of what you look at on the retina at the back of the eye. The brain turns the picture the right way up.

A muscle changes the shape of the eye lens. To look at something close, the lens is made fatter, so its focal length is shorter. To look at something far, it is made thinner. This focusing is why a clear eye sees both near and far.

In short sight, the eye bends light too much, or the eyeball is too long. Rays from a far object cross in front of the retina, and the image on the retina is blurred. Near things are clear. A diverging lens in front of the eye spreads the rays a little and moves the image back onto the retina.

Short sight without glasses. The rays from a far object meet in front of the retina, so the picture on the retina is blurred.

Short sight corrected with a diverging lens. The rays now meet on the retina.

In long sight, the eye bends light too little, or the eyeball is too short. Rays from a near object would meet behind the retina, so the image on the retina is blurred. Far things are clear. A converging lens in front of the eye helps to bend the rays and moves the image forward onto the retina.

Long sight corrected with a converging lens. The image is back on the retina. The object is drawn near, as a book would be.

Opticians give the strength of a lens as its power: P = 1 ÷ f, with f in metres. The unit is the dioptre, D. A converging lens has a positive power and a diverging lens a negative one. A short focal length is a high power.

A short sighted person sees clearly only as far as a distance called the far point. A diverging lens in front of the eye should make a faraway object seem to be at the far point. So its focal length equals the far point distance, with the sign of a diverging lens.

Worked example

A short sighted person sees clearly only up to 0.5 m. What power of lens do they need?

The lens must make faraway things look as if they are 0.5 m away. So it is a diverging lens with focal length 0.5 m.
The size of the power is 1 ÷ f = 1 ÷ 0.5 = 2 D.
A diverging lens has a negative power, so the lens is −2 D.

Answer: The lens is −2 D

Try it

Make a picture with a lens

You need: a lens from an old pair of reading glasses, or a magnifying glass, a sheet of white paper, a ruler, a room with a window or a lamp across the room.

Be safe: Never look at the Sun, and never point a lens at the Sun. A lens focuses sunlight into a hot spot that can burn skin, paper and eyes. Choose a window or a lamp, and keep the Sun out of the view. Do not look into a bright lamp through the lens.

Hold the paper up as a screen, with its back to the window. Hold the lens between the window and the paper.
Move the lens towards and away from the paper until a sharp small picture of the window appears on the paper.
Measure the distance from the lens to the paper. This is about the focal length f, because the window is so far away.
Work out the power: P = 1 ÷ f, with f in metres.

What you will see: The picture on the paper is upside down and smaller than the window. It is a real image. The window is far away, so the image forms very near the focus.

If you cannot try it, this is what happens.

A wrong idea many people have

A converging lens always makes an upside down image.

Why it feels right: The first picture we see, a window on a screen, is upside down. Most pictures of lenses show an upside down image.

The test that breaks it: Hold a magnifying glass close to some print. The print looks larger and the right way up. That image is virtual, and the object is inside the focal length.

What is true: A real image from a converging lens is upside down. When the object is inside F, the image is virtual, upright and larger.

A wrong idea many people have

A short sighted person needs a lens that magnifies, so a converging lens.

Why it feels right: We call a magnifying glass a lens that helps us see, and a short sighted person sees things badly.

The test that breaks it: In short sight, the rays from a far object cross in front of the retina. A converging lens would bring them together even sooner, and make it worse.

What is true: Short sight needs a diverging lens, to spread the rays. Long sight needs a converging lens, to bring them together sooner.

A wrong idea many people have

A lens has one focus.

Why it feels right: Pictures of parallel rays arriving from one side show only one point where the rays meet.

The test that breaks it: Turn the lens round and send the rays from the other side. They meet at a point the same distance from the lens on the other side.

What is true: A lens has a focus on each side, at the same distance f. Rule three uses the focus on the object's side.

Key words

principal focus: the point where rays that arrive parallel to the axis meet after a converging lens.
focal length: the distance from the lens to its principal focus. Its symbol is f.
real image: an image where rays really cross. It can be caught on a screen, and it is upside down.
virtual image: an image where the rays only seem to cross. It cannot be caught on a screen.
magnification: the height of the image divided by the height of the object.
short sight: a fault in which far things are blurred. A diverging lens corrects it.
power: the strength of a lens, P = 1 ÷ f with f in metres.
dioptre: the unit of the power of a lens, written D.

Back to your guess

The answer is A: Rays from the top of the window cross after the lens and reach the bottom of the picture.

Rays from the top of the window go through the lens, bend, and cross. After crossing they are below the axis, so they land at the bottom of the picture. Rays from the bottom land at the top. So the picture is upside down.

Keep it

Copy this into your exercise book.

Three rays: parallel through the far focus, through the centre straight, through the near focus out parallel. 1/f = 1/u + 1/v. Short sight: diverging. Long sight: converging. P = 1 ÷ f.

Draw: A converging lens with its axis and F marks, an object arrow, and the three rays meeting at an upside down image.

S4.10.7 Chapter 7

The electromagnetic spectrum

In empty space, a radio wave and a gamma ray leave a star at the same moment. Which reaches the Earth first?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A The gamma rays, because their frequency is higher.
You picked A. Read on. At the end, the chapter comes back to your guess.
B The radio waves, because their wavelength is longer.
You picked B. Read on. At the end, the chapter comes back to your guess.
C The radio waves, because they carry less energy.
You picked C. Read on. At the end, the chapter comes back to your guess.
D They arrive together.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

List electromagnetic waves in order of wavelength and frequency
Give a use of each kind of electromagnetic wave

Light is not the only kind of wave that crosses empty space. Radio waves, the heat from a fire, and the X-rays at a hospital are all waves of the same family. They are electromagnetic waves.

An electromagnetic wave is a pattern of electric and magnetic fields that changes in step and moves through space. It is a transverse wave. It needs no medium, so it crosses a vacuum, and it carries energy.

All electromagnetic waves travel at the same speed in a vacuum: 3 × 10⁸ m/s, which is 300 000 km/s. Radio waves, light and gamma rays all travel at this speed. In air the speed is almost the same. In glass and water it is lower.

Every electromagnetic wave goes at the same speed in a vacuum.

What makes the waves differ is their frequency and wavelength. They are linked by the wave equation. For electromagnetic waves in a vacuum we write c for the speed: c = fλ. So the longer the wavelength, the lower the frequency.

The whole family is the electromagnetic spectrum. It has seven main kinds. In order of falling wavelength and rising frequency they are radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays.

The electromagnetic spectrum. The edges are not sharp, and different books put them a little differently. Some books count microwaves as the short end of radio. X-rays and gamma rays overlap.

The wavelengths run from kilometres to far smaller than an atom. A nanometre, nm, is 10⁻⁹ m, a billionth of a metre. The frequencies run in the opposite order, from a few kilohertz to more than 10¹⁹ Hz. Visible light is only a thin slice.

One megahertz, MHz, is 1 000 000 Hz, which is 10⁶ Hz. FM radio stations are often near 100 MHz. Mobile phones and Wi-Fi use frequencies from several hundred to several thousand megahertz.

Worked example

An FM radio station broadcasts at 100 MHz. Find the wavelength of its waves.

Change the frequency to hertz: 100 MHz = 100 × 10⁶ = 1 × 10⁸ Hz.
Rearrange c = fλ to λ = c ÷ f.
λ = 3 × 10⁸ ÷ (1 × 10⁸) = 3 m.

Answer: 3 m

Worked example

Blue green light has a wavelength of 500 nm. Find its frequency. Use 1 nm = 10⁻⁹ m.

Change the wavelength to metres: 500 nm = 500 × 10⁻⁹ = 5 × 10⁻⁷ m.
Rearrange c = fλ to f = c ÷ λ.
f = 3 × 10⁸ ÷ (5 × 10⁻⁷) = 0.6 × 10¹⁵ Hz.
In standard form, f = 6 × 10¹⁴ Hz.

Answer: 6 × 10¹⁴ Hz

Now go through the seven kinds, starting at the long wave end. Each has uses that come from how it passes through things and how things absorb it.

Radio waves have the longest wavelengths. They carry radio and television, signals to aircraft and ships, and the signals between spacecraft and Earth. Some stars and galaxies also send them out, and radio telescopes collect them.

Microwaves are shorter. A microwave oven uses them because water in food absorbs them and heats up. Mobile phones, satellites, Wi-Fi and radar use them too. The oven is shielded so that the microwaves stay inside.

Infrared waves are given out by every warm object. Heaters, toasters and your own body give them out. A television remote control sends a code in infrared. Cameras that sense infrared make pictures of heat, and they work in the dark.

Visible light is the only part we see. It lets us see, and it is used in lamps, cameras, lasers and optical fibres. Its wavelengths run from about 700 nm for red light to about 400 nm for violet light.

Typical wavelengths of the colours, rounded. The colours blend into each other, and indigo lies between blue and violet.

Ultraviolet is a little shorter than violet light, and our eyes cannot see it. It is used to sterilise water and surfaces, and to show security marks on notes and cards. Lamps that make some materials glow use it. Sunlight contains it.

X-rays pass through soft parts of the body and are blocked more by bone. Doctors and dentists use them to take pictures of bones and teeth. Airports use them to look inside bags.

Gamma rays have the shortest wavelengths and are very penetrating. They are used to kill cancer cells, to sterilise medical equipment, and to follow a tracer through the body. They come from the nuclei of atoms.

A use for each kind of electromagnetic wave. The next chapter looks at calls and at health.

X-rays and gamma rays overlap in wavelength. They differ in where they come from. X-rays come from fast moving electrons, and gamma rays from the nuclei of atoms. For the same wavelength they behave in the same way.

Not to scale. A gamma ray wave is far shorter than a radio wave. Both go at the same speed.

Try it

Feel infrared from a warm mug

You need: a mug of hot water, hot from a kettle but not boiling, poured by an adult, a card or a book, your hand.

Be safe: Use water that is hot but not boiling, and let an adult pour it. Stand the mug where it cannot be knocked over, and keep your hand to the side, never over the steam.

Stand the mug on a table. Hold the back of your hand beside the mug, a few centimetres from the side, at the height of the mug. Do not touch it.
Notice the warmth on your hand.
Slide the card in between your hand and the mug. Notice what happens to the warmth.
Take the card away, and notice the warmth come back.

What you will see: The warmth you feel at the side of the mug comes mostly from infrared. Warm air rises, so it does not reach your hand at the side. The card is opaque to infrared, so it stops it.

If you cannot try it, this is what happens.

Try it

See a remote control with a phone camera

You need: a television remote control or similar remote, a phone with a camera, in a dim room.

Be safe: The light from a remote control is weak and safe. Never look at the Sun, a laser or a very bright lamp through a camera or in any other way.

Point the end of the remote at your eye from a metre away and press a button. Look for a light.
Now look at the end of the remote through the phone camera screen, and press a button again.
Try the front camera and the back camera.

What you will see: To your eye there is nothing to see, because the remote sends infrared. Many phone cameras can see infrared, and show a flickering light. Some cameras filter it out, so yours may show nothing.

If you cannot try it, this is what happens.

A wrong idea many people have

Gamma rays travel faster than radio waves, because they have more energy.

Why it feels right: In daily life, things with more energy often go faster, and gamma rays sound like the most powerful waves.

The test that breaks it: A burst of gamma rays from a far galaxy can also give a flash of light. Both travel for billions of years, yet the flash can arrive within seconds of the gamma rays. If gamma rays were faster, they would arrive long before the light.

What is true: Every electromagnetic wave travels at 3 × 10⁸ m/s in a vacuum. The energy of a gamma ray goes with its frequency, not its speed.

A wrong idea many people have

Radio waves are sound waves.

Why it feels right: We hear a radio, so it seems that the radio waves must be sound that travels through the air.

The test that breaks it: Sound cannot cross a vacuum, but radio waves cross space to reach spacecraft far from the Earth. Radio waves are electromagnetic. The radio set turns them back into sound.

What is true: Radio waves are electromagnetic waves, like light. They are transverse and need no medium. The loudspeaker makes the sound.

A wrong idea many people have

A wave with a longer wavelength has a higher frequency.

Why it feels right: Both are measures of how big the wave is, and big things can seem to go together.

The test that breaks it: Use c = fλ with c fixed. A radio wave of 3 m has f = 3 × 10⁸ ÷ 3 = 1 × 10⁸ Hz. A wave of 1 m has f = 3 × 10⁸ Hz. The shorter wave has the higher frequency.

What is true: For electromagnetic waves in a vacuum, f and λ go opposite ways. A longer wavelength means a lower frequency.

Key words

electromagnetic waves: transverse waves of electric and magnetic fields. They need no medium and go at 3 × 10⁸ m/s in a vacuum.
electromagnetic spectrum: the family of electromagnetic waves, in order of wavelength.
radio waves: the longest wavelength electromagnetic waves.
microwaves: electromagnetic waves a little shorter than radio waves, used in ovens, phones and Wi-Fi.
infrared: electromagnetic waves given out by warm things, just longer than red light.
ultraviolet: electromagnetic waves just shorter than violet light.
X-rays: short wavelength electromagnetic waves that pass through soft parts of the body.
gamma rays: the shortest wavelength electromagnetic waves. They come from the nuclei of atoms.

Back to your guess

The answer is D: They arrive together.

All electromagnetic waves travel at the same speed in a vacuum, about 3 × 10⁸ m/s. Radio waves and gamma rays differ in frequency and wavelength, not in speed. They arrive together.

Keep it

Copy this into your exercise book.

Radio, microwave, infrared, visible, ultraviolet, X-ray, gamma: wavelength falls, frequency rises. All go at 3 × 10⁸ m/s in a vacuum. c = fλ.

Draw: Seven boxes in a row in order, with an arrow under them labelled frequency rises and one above labelled wavelength falls.

S4.10.8 Chapter 8

Waves in communication and health

Amina calls her cousin across the sea, and her voice arrives in a fraction of a second. What carries it?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Radio waves, microwaves and pulses of light in cables carry it.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Sound waves carry it all the way across the sea.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Electricity in copper wires carries it all the way.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Explain how radio, microwaves and light carry phone and internet signals
Explain why ultraviolet, X-rays and gamma rays can harm living cells

A wave can carry energy. By changing the wave in a pattern, it can also carry a message. The plain wave that is changed is the carrier wave. A radio station, a mobile phone and a fibre cable all work this way.

In an analogue signal, the changes copy the message smoothly. In amplitude modulation the height of the carrier follows the message. In frequency modulation the frequency of the carrier follows it.

Amplitude modulation. The height of the fast wave follows the message.

Frequency modulation. The frequency of the fast wave follows the message.

Phone calls and the internet use digital signals. In a digital signal the message is turned into numbers, written with only two digits, 1 and 0. Each digit is a bit. A bit is sent as a pulse, high for 1 and low for 0.

A digital signal. Only two levels are used, so each bit can be told from the others.

Digital signals have an advantage on a long journey. A wave gets weaker and picks up noise on the way. A digital signal has only two levels. So even a weak, noisy pulse can be read as a 1 or a 0. It can be cleaned up and sent on as a fresh pulse. An analogue signal cannot be cleaned, because its noise looks like part of the message.

Many senders share the air. Each uses its own frequency. A radio set tunes to one frequency, which picks out one station and ignores the rest. Mobile phones also share the air by using different frequencies.

Now follow a call. The phone sends its code as microwaves to a mast. The mast passes it on, mostly as pulses of light in a glass optical fibre cable. The light is kept in by total internal reflection, which you met in the chapter on reflection and refraction. The fibre carries it over land and under the sea.

One way a call crosses a sea. The fibre between the masts runs over land and under the sea. The kind of wave changes along the way.

Light in a fibre is usually infrared, not visible light. A fibre can carry far more information than a copper wire, loses little energy, and is not disturbed by electrical interference. Most of the internet between cities and across oceans travels this way.

Microwaves pass through the atmosphere, so dishes on the ground can send them to a satellite and back. They travel in straight lines, so two dishes on hills need a clear view of each other. Wi-Fi also uses microwaves, to link a router to a phone or a computer.

A high layer of the air reflects some long radio waves. They can then travel round the curve of the Earth. This is why some radio stations can be heard from far away. Light and microwaves are not reflected in this way.

All of these signals travel at the speed of the wave. In a vacuum, microwaves and radio waves travel at 3 × 10⁸ m/s. Light in glass fibre is slower, about 2 × 10⁸ m/s. The time a signal takes is the distance divided by the speed.

Worked example

A signal crosses an undersea fibre cable 8 000 km long. Light in the fibre travels at 2 × 10⁸ m/s. How long does the signal take?

Change the distance to metres: 8 000 km = 8 000 000 m.
The time is t = d ÷ v = 8 000 000 ÷ 200 000 000 = 0.04 s.

Answer: 0.04 s

Two hilltop dishes linked by microwaves, 60 km apart.

Worked example

Two hilltop dishes are 60 km apart. Microwaves travel between them at 3 × 10⁸ m/s. How long does a signal take?

Change the distance to metres: 60 km = 60 000 m.
The time is t = d ÷ c = 60 000 ÷ 300 000 000 = 0.0002 s.
That is 0.2 ms, much shorter than a blink.

Answer: 0.0002 s

Now think about health. When an electromagnetic wave is absorbed, it hands over energy. It does this in tiny packets. The energy of one packet is higher for a higher frequency. A wave with twice the frequency has packets with twice the energy.

Radio waves, microwaves, infrared and visible light have packets with too little energy to knock electrons off atoms. Their main effect is heating. Strong microwaves heat water and can burn, so a microwave oven is shielded and its door has a safety switch. Strong infrared also burns.

Ultraviolet, X-rays and gamma rays have packets with much more energy. The shortest ultraviolet, and all X-rays and gamma rays, can knock electrons off atoms. This is called ionising. They are ionising radiation. Even the longer ultraviolet can harm a cell by changing its chemistry.

A ray with low energy packets only makes the atom shake. It warms.

A ray with high energy packets knocks an electron out. The atom becomes an ion.

Inside a living cell, the nucleus holds the DNA, which is the instruction for the cell. Ionising radiation can damage DNA. A cell can often repair small damage. A badly damaged cell dies. Sometimes a damaged cell lives on and copies its changed DNA. A change in DNA is a mutation.

If a mutation falls in the wrong place, the cell may grow and divide out of control. This is cancer. The more ionising radiation a body takes in, the dose, the greater the chance of harm. A very large dose can cause burns and radiation sickness, and can kill.

The boxed waves can damage living cells. The frequency, and so the energy of each packet, rises down the list.

Ultraviolet from the Sun causes sunburn, ages the skin and harms the eyes. It is the main cause of skin cancer. A little sunlight on the skin helps the body make vitamin D, but a short time is enough. Darker skin has more of the pigment that absorbs ultraviolet, which gives some protection but not complete protection.

To protect your skin, stay in the shade near midday. Cover up with clothes and a hat. Wear sunglasses that block ultraviolet, and use sun cream. Thin cloud does not stop ultraviolet, so you can burn on a cloudy day.

X-rays are useful. A doctor orders one only when the benefit is greater than the small risk. The dose from one dental X-ray is small. The risk is small, but it is not zero. So staff stand behind a screen or leave the room. A lead apron may cover the parts of the body that are not being pictured. A woman who might be pregnant must tell the staff first.

Gamma rays kill cells too, which is why a narrow beam aimed from several directions treats cancer. The tumour gets a large dose and the tissue around it gets much less. Strong sources are kept in thick lead or concrete, and workers limit their time near them.

For people near ionising radiation, three rules help. Spend less time near it, stay further away, and put something thick between you and it, such as lead or concrete. Each one lowers the dose.

What about mobile phones and Wi-Fi? Their microwaves cannot ionise atoms. Their main effect on the body is a very small warming. Large studies have not found that phones cause harm. A cancer research agency of the World Health Organization lists radio frequency fields as "possibly" cancer causing. It uses that group when the evidence is limited. Studies go on.

One more danger is not about packets but about heat. A very bright beam of visible light can burn the retina at the back of the eye. Never look at the Sun, and never look into a laser beam, even for a moment. The damage can be permanent.

Try it

Send a message with a torch

You need: a torch, a friend at the other side of a dim room, or across a dark yard, paper and a pencil for each of you.

Be safe: Never shine the torch into anyone's eyes, and never use a laser pointer for this. Do it where nobody can trip in the dim light, and not near roads.

Agree a code of two flashes for each letter. A is short short, B is short long, C is long short and D is long long. A short flash is a 0 and a long flash is a 1.
Write a message of four letters, using only A, B, C and D. Do not show it to your friend.
Send each letter by two flashes, with a pause between letters. Your friend writes down short or long for each flash.
Your friend uses the code to read the letters. Compare with your message.
Now hold a sheet of paper in front of the torch so that it is dimmer. Send the message again.

What you will see: The message is still read correctly when the torch is dim. Your friend only has to tell a short flash from a long one. A digital signal can be read even when it is weak.

If you cannot try it, this is what happens.

A wrong idea many people have

All electromagnetic waves are equally dangerous, or all are safe.

Why it feels right: The word "radiation" is used for all of them, and news stories treat it as one thing.

The test that breaks it: Infrared from a fire warms you, and visible light lets you see. Neither ionises atoms. A dental X-ray is a short exposure to a high energy wave. They are not the same kind of risk.

What is true: What a wave can do depends on its frequency and on the dose. Ultraviolet, X-rays and gamma rays can damage cells. Radio waves, microwaves, infrared and visible light cannot ionise, though strong ones can heat and burn.

A wrong idea many people have

A very small dose of X-rays does no harm at all.

Why it feels right: A dental or chest X-ray is safe enough for doctors to use, so it seems to have no risk.

The test that breaks it: Each packet of an X-ray can damage a molecule of DNA, and one change can start a cancer. The chance is very small for one X-ray, but it is not zero, and it grows with every exposure.

What is true: The risk from a small dose is small, not zero. Doctors use X-rays when the benefit is clearly greater, and they keep the dose as low as they can.

A wrong idea many people have

A strong enough radio signal would damage DNA like an X-ray.

Why it feels right: More power seems always to mean more danger, and "strong" sounds like "high energy".

The test that breaks it: Each radio packet has far too little energy to knock an electron off an atom. Making the beam stronger sends more packets, not bigger ones, so it cannot ionise.

What is true: A strong beam of radio waves or microwaves can heat and burn, but it cannot ionise. Only high frequency waves, with packets of high energy, ionise.

Key words

carrier wave: the plain wave that is changed to carry a message.
analogue signal: a signal in which the wave changes smoothly to copy the message.
digital signal: a signal sent as a code of 1s and 0s, as pulses.
bit: one binary digit, a 1 or a 0.
ionising radiation: waves or particles with enough energy to knock electrons off atoms. It can damage DNA.
mutation: a change in the DNA of a cell.
dose: how much ionising radiation a body takes in.

Back to your guess

The answer is A: Radio waves, microwaves and pulses of light in cables carry it.

The phone turns her voice into a code. Microwaves carry the code to a mast. Pulses of light in glass fibres carry it across land and under the sea. Sound does not travel all the way.

Keep it

Copy this into your exercise book.

A message changes a carrier wave. Digital: pulses of 1 and 0. Calls: radio and microwaves, light in fibres. UV, X-rays and gamma rays ionise and can damage DNA. Dose matters.

Draw: A phone, a mast, a cable under the sea and a second phone, with the kind of wave written on each arrow.

A bump runs past a ribbon

Send a bump down a rope

A pushed row of buttons

A rope wave on a distance graph

The period of a swinging stone

Find the speed of ripples in a tray

Ripples over a plate

Measure the speed of sound with an echo

Test the law of reflection

The silver surface of water

Make a picture with a lens

Feel infrared from a warm mug

See a remote control with a phone camera

Send a message with a torch

Key words, A to Z