Chapter 2 of 8

All 8 chapters
  1. 1Kinds of waves, readunderstood
  2. 2Measuring waves, readunderstood
  3. 3Waves on water, readunderstood
  4. 4Sound and ultrasound, readunderstood
  5. 5Reflection and refraction, readunderstood
  6. 6Lenses, readunderstood
  7. 7The electromagnetic spectrum, readunderstood
  8. 8Waves in communication and health, readunderstood

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

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

  • 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. A graph with the axes displacement (cm) and distance (m). The displacement runs from −6 to 6 cm and the distance from 0 to 2 m in steps of 0.4. Step 1: a smooth wave that rises to 2 cm at 0.4 m, falls to −2 cm at 1.2 m and rises to 2 cm at 2.0 m. Step 2: an arrow from the dashed rest line at 0 up to the first crest, labelled amplitude 2 cm. Step 3: an arrow across from the first crest to the next crest, labelled wavelength 1.6 m, with dashed lines down to the crests.−6−4−2024600.40.81.21.62displacement (cm)distance (m)

    1 Draw the wave. The dashed line through 0 is the rest line.

  2. A graph with the axes displacement (cm) and distance (m). The displacement runs from −6 to 6 cm and the distance from 0 to 2 m in steps of 0.4. Step 1: a smooth wave that rises to 2 cm at 0.4 m, falls to −2 cm at 1.2 m and rises to 2 cm at 2.0 m. Step 2: an arrow from the dashed rest line at 0 up to the first crest, labelled amplitude 2 cm. Step 3: an arrow across from the first crest to the next crest, labelled wavelength 1.6 m, with dashed lines down to the crests.−6−4−2024600.40.81.21.62amplitude 2 cmdisplacement (cm)distance (m)

    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. A graph with the axes displacement (cm) and distance (m). The displacement runs from −6 to 6 cm and the distance from 0 to 2 m in steps of 0.4. Step 1: a smooth wave that rises to 2 cm at 0.4 m, falls to −2 cm at 1.2 m and rises to 2 cm at 2.0 m. Step 2: an arrow from the dashed rest line at 0 up to the first crest, labelled amplitude 2 cm. Step 3: an arrow across from the first crest to the next crest, labelled wavelength 1.6 m, with dashed lines down to the crests.−6−4−2024600.40.81.21.62amplitude 2 cmwavelength 1.6 mdisplacement (cm)distance (m)

    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.

A graph with the axes displacement (cm) and time (s). The time runs from 0 to 2.5 s in steps of 0.5 and the displacement from −4 to 4 cm. A smooth wave rises to 2 cm at 0.5 s, falls to −2 cm at 1.5 s and rises to 2 cm again at 2.5 s. An arrow across from the first crest to the next crest is labelled period, with dashed lines down to the crests.−4−3−2−10123400.511.522.5perioddisplacement (cm)time (s)
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.

A graph with the axes displacement and time, drawn without numbers. Two waves of the same length, one over the other. The solid wave is twice as tall as the dashed wave. The key at the top says loud for the solid wave and quiet for the dashed wave.displacementtimeloudquiet
Same frequency, different amplitude. The taller wave is the louder sound.
A graph with the axes displacement and time, drawn without numbers. Two waves of the same height. The solid wave has twice as many waves in the same time as the dashed wave. The key at the top says high note for the solid wave and low note for the dashed wave.displacementtimehigh notelow note
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

A table with the columns Measure, Symbol and Unit. Row 1: speed, v, m/s. Row 2: frequency, f, Hz. Row 3: wavelength, λ, m. Row 4: period, T, s.MeasureSymbolUnitspeedvm/sfrequencyfHzwavelengthλmperiodTs

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.

A graph with the axes displacement (cm) and distance (m). The distance runs from 0 to 0.75 m in steps of 0.15 and the displacement from −6 to 6 cm. A smooth wave has crests at 0.15 m and 0.75 m. An arrow from crest to crest is labelled wavelength 0.6 m, with dashed lines down to the crests.−6−4−2024600.150.30.450.60.75wavelength 0.6 mdisplacement (cm)distance (m)
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.

  1. Write the equation: v = fλ.
  2. Put in the numbers: v = 2.5 Hz × 0.6 m.
  3. 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.

Two rows of upright lines, which are layers of air. In the top row the layers bunch together in two places and spread out in others. The bunched places are labelled compression and the spread places are labelled rarefaction. An arrow from one compression to the next is labelled wavelength. In the lower row, labelled a moment later, the bunches have moved to the right. An arrow at the foot is labelled wave travels.wavelengthcompressionrarefactiona moment laterwave travels
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?

  1. Change the frequency to hertz: 0.8 kHz = 0.8 × 1 000 = 800 Hz.
  2. Rearrange v = fλ to λ = v ÷ f.
  3. 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.

  1. 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.
  2. Find the frequency: f = 1 ÷ T = 1 ÷ 2 = 0.5 Hz.
  3. The wavelength is λ = 1.6 m.
  4. 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.

A graph with the axes displacement (cm) and distance (m). A smooth wave with a crest of 2 cm at 0.4 m and a trough of −2 cm at 1.2 m. One arrow from the dashed rest line at 0 up to the crest is labelled amplitude 2 cm. A second arrow at 1.2 m from the trough up to 2 cm is labelled 4 cm.−6−4−2024600.40.81.21.62amplitude 2 cm4 cmdisplacement (cm)distance (m)
The arrow from trough to crest is 4 cm, twice the amplitude. Measure from the rest line.
A graph with the axes displacement (cm) and distance (m). A smooth wave with crests of 2 cm at 0.4 m and 2.0 m and a trough of −2 cm at 1.2 m. A long arrow from the first crest to the next crest is labelled wavelength 1.6 m. A shorter arrow below the wave, from the first crest to the trough, is labelled 0.8 m, with dashed lines up to the crest and the trough.−6−4−2024600.40.81.21.62wavelength 1.6 m0.8 mdisplacement (cm)distance (m)
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.

  1. Tie the stone firmly to the string. Tie the other end of the string up high so the stone hangs in the air.
  2. 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.
  3. Start the clock when the stone passes its lowest point. Count 10 full swings and stop the clock.
  4. Work out the period: T = time for 10 swings ÷ 10.
  5. 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 table with the columns Measure and 1 m string. Row 1: 10 swings, 20 s. Row 2: T = 20 ÷ 10, 2 s. Row 3: f = 1 ÷ T, 0.5 Hz.Measure1 m string10 swings20 sT = 20 ÷ 102 sf = 1 ÷ T0.5 Hz

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.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The graph shows a wave on a rope. What is its amplitude?

    A graph with the axes displacement (cm) and distance (m). The displacement runs from −4 to 4 cm and the distance from 0 to 1.5 m in steps of 0.3. A smooth wave rises to 3 cm at 0.3 m, falls to −3 cm at 0.9 m and rises again to 3 cm at 1.5 m.−4−3−2−10123400.30.60.91.21.5displacement (cm)distance (m)
    Tap your answer.

    1. Not this one. That is the height from the trough to the crest, which is twice the amplitude. Measure from the rest line to the crest. Try another.

    2. Not this one. That is a length along the other axis. It is the wavelength, not a height. Try another.

    3. Not this one. The crest is higher than 2 cm on the displacement axis. Read the crest against the numbers on that axis. Try another.

    4. Yes, that is right. The crest is 3 cm above the rest line at 0, so the amplitude is 3 cm. The trough is 3 cm below it.

    See the answer

    3 cm

    The crest is 3 cm above the rest line at 0, so the amplitude is 3 cm. The trough is 3 cm below it.

    • If you chose A: That is the height from the trough to the crest, which is twice the amplitude. Measure from the rest line to the crest.
    • If you chose B: That is a length along the other axis. It is the wavelength, not a height.
    • If you chose C: The crest is higher than 2 cm on the displacement axis. Read the crest against the numbers on that axis.

  2. The graph shows how a ribbon on a rope moves as time goes by. What is the frequency of the wave?

    A graph with the axes displacement (cm) and time (s). The time runs from 0 to 1 s in steps of 0.2 and the displacement from −3 to 3 cm. A smooth wave has a crest of 2 cm at 0.2 s and the next crest at 1 s.−3−2−1012300.20.40.60.81displacement (cm)time (s)
    Tap your answer.

    1. Yes, that is right. The crests are at 0.2 s and 1.0 s, so the period is T = 0.8 s. The frequency is f = 1 ÷ 0.8 = 1.25 Hz.

    2. Not this one. You used 0.2 s, the time of the first crest. The period is the time between two crests. Try another.

    3. Not this one. 0.8 s is the period. The frequency is 1 divided by the period, and its unit is Hz. Try another.

    See the answer

    1.25 Hz

    The crests are at 0.2 s and 1.0 s, so the period is T = 0.8 s. The frequency is f = 1 ÷ 0.8 = 1.25 Hz.

    • If you chose B: You used 0.2 s, the time of the first crest. The period is the time between two crests.
    • If you chose C: 0.8 s is the period. The frequency is 1 divided by the period, and its unit is Hz.

  3. The graph is a photograph of a wave on a rope. What is its wavelength?

    A graph with the axes displacement (cm) and distance (m). The distance runs from 0 to 3 m in steps of 0.6 and the displacement from −3 to 3 cm. A smooth wave has a crest of 2 cm at 0.6 m, a trough of −2 cm at 1.8 m and the next crest at 3 m.−3−2−1012300.61.21.82.43displacement (cm)distance (m)
    Tap your answer.

    1. Not this one. That is the whole length of the picture, which holds more than one wave. Measure from crest to the next crest. Try another.

    2. Yes, that is right. The crests are at 0.6 m and 3.0 m. The distance from one crest to the next is 3.0 − 0.6 = 2.4 m.

    3. Not this one. That is the distance from crest to trough, which is only half a wavelength. Try another.

    4. Not this one. 2 cm is a height, the amplitude. A wavelength is a length along the rope. Try another.

    See the answer

    2.4 m

    The crests are at 0.6 m and 3.0 m. The distance from one crest to the next is 3.0 − 0.6 = 2.4 m.

    • If you chose A: That is the whole length of the picture, which holds more than one wave. Measure from crest to the next crest.
    • If you chose C: That is the distance from crest to trough, which is only half a wavelength.
    • If you chose D: 2 cm is a height, the amplitude. A wavelength is a length along the rope.

  4. A wave has a frequency of 20 Hz and a wavelength of 0.5 m. What is its speed?

    Tap your answer.

    1. Yes, that is right. v = fλ = 20 Hz × 0.5 m = 10 m/s.

    2. Not this one. You divided the frequency by the wavelength. The wave equation is a product: v = fλ. Try another.

    3. Not this one. You added 20 and 0.5. The speed is the frequency times the wavelength. Try another.

    4. Not this one. You divided the wavelength by the frequency. Multiply the two instead. Try another.

    See the answer

    10 m/s

    v = fλ = 20 Hz × 0.5 m = 10 m/s.

    • If you chose B: You divided the frequency by the wavelength. The wave equation is a product: v = fλ.
    • If you chose C: You added 20 and 0.5. The speed is the frequency times the wavelength.
    • If you chose D: You divided the wavelength by the frequency. Multiply the two instead.

  5. Ravi shakes a rope faster, so its waves have a higher frequency. The rope and its tension stay the same. What happens?

    Tap your answer.

    1. Not this one. The speed is set by the rope and its tension, not by how fast you shake it. A faster shake only makes more waves each second. Try another.

    2. Not this one. With v fixed, v = fλ means a bigger f needs a smaller λ. So the wavelength shortens. Try another.

    3. Yes, that is right. The speed stays the same because the rope is the same. Since v = fλ, a higher f must go with a shorter λ.

    See the answer

    The waves travel at the same speed and the wavelength gets shorter.

    The speed stays the same because the rope is the same. Since v = fλ, a higher f must go with a shorter λ.

    • If you chose A: The speed is set by the rope and its tension, not by how fast you shake it. A faster shake only makes more waves each second.
    • If you chose B: With v fixed, v = fλ means a bigger f needs a smaller λ. So the wavelength shortens.

  6. A guitar string vibrates 250 times each second. What is its period?

    Tap your answer.

    1. Not this one. 250 is the frequency in hertz. The period is 1 divided by it. Try another.

    2. Not this one. A period of 0.25 s would be a frequency of 4 Hz. Work out 1 ÷ 250 again. Try another.

    3. Not this one. You divided 1 000 by 250. The period is 1 ÷ 250, much shorter than a second. Try another.

    4. Yes, that is right. T = 1 ÷ f = 1 ÷ 250 = 0.004 s. A vibration this fast takes a very short time.

    See the answer

    0.004 s

    T = 1 ÷ f = 1 ÷ 250 = 0.004 s. A vibration this fast takes a very short time.

    • If you chose A: 250 is the frequency in hertz. The period is 1 divided by it.
    • If you chose B: A period of 0.25 s would be a frequency of 4 Hz. Work out 1 ÷ 250 again.
    • If you chose C: You divided 1 000 by 250. The period is 1 ÷ 250, much shorter than a second.

  7. A drum is hit harder, so the note is louder, but it stays the same note. What changes in the wave?

    Tap your answer.

    1. Not this one. A bigger frequency is a higher note, not a louder one. Pitch depends on frequency. Go to S3.9.8. Try another.

    2. Yes, that is right. Loudness depends on the amplitude. The note, which is the frequency, does not change, and the wavelength is v ÷ f, so it does not change either.

    3. Not this one. The note is the same, so the frequency is the same. The speed of sound in air is fixed, so the wavelength is the same too. Try another.

    See the answer

    The amplitude gets bigger. The frequency and wavelength stay the same.

    Loudness depends on the amplitude. The note, which is the frequency, does not change, and the wavelength is v ÷ f, so it does not change either.

    • If you chose A: A bigger frequency is a higher note, not a louder one. Pitch depends on frequency. Go to S3.9.8.
    • If you chose C: The note is the same, so the frequency is the same. The speed of sound in air is fixed, so the wavelength is the same too.

  8. From an earlier chapter

    In which kind of wave does the vibration run along the same line as the way the wave travels?

    Tap your answer.

    1. Not this one. In a transverse wave the vibration is at right angles to the way the wave travels, as in a rope. Go to S4.10.1. Try another.

    2. Not this one. Waves carry energy, not matter. The material only shakes about its place. Go to S4.10.1. Try another.

    3. Yes, that is right. In a longitudinal wave, such as sound or a push along a spring, the vibration is along the direction of travel.

    See the answer

    A longitudinal wave

    In a longitudinal wave, such as sound or a push along a spring, the vibration is along the direction of travel.

    • If you chose A: In a transverse wave the vibration is at right angles to the way the wave travels, as in a rope. Go to S4.10.1.
    • If you chose B: Waves carry energy, not matter. The material only shakes about its place. Go to S4.10.1.

  9. A wave has a frequency of 4 Hz and a wavelength of 0.75 m. Find its speed.

    See the answer

    3 m/s

    v = fλ = 4 Hz × 0.75 m = 3 m/s.

  10. A wave has a frequency of 40 Hz. What is its period?

    See the answer

    0.025 s

    T = 1 ÷ f = 1 ÷ 40 = 0.025 s.

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.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (951 KB). It opens on any phone.