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Wave speed and the wave equation Shake a rope faster: do its waves go faster, or get shorter?

Foundation and Higher · AQA · 45 minutes

Students predict what happens to waves on a rope shaken twice as fast, then try it on a running wave they control. The wave equation, period and units follow, then the required practical from readings to a speed. Slides to present, a vote that shows who's stuck, practice and homework.

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Waves run along a rope at v m/s. Across: how many waves you make each second (Hz); up: how long each wave is (m). Shake twice as fast: what happens to each wave's length? Then move v, a tighter rope: what changes?

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At x = 0, 1, 2 …: Infinity, 4, 2, 1.33, 1, 0.8

The idea in one minute

A wave carries energy from one place to another; the rope, water or air it travels through only moves to and fro and stays where it was. Each wave has an amplitude (the biggest move from the middle), a wavelength λ (from one peak to the next, in metres), a frequency f (how many waves pass a point each second, in hertz) and a period T (the time for one wave, T = 1 ÷ f). Each second f waves pass, each λ long, so the wave moves f × λ metres a second: wave speed v = f λ, in m/s. The speed is set by what the wave travels through, so in the same rope, water or air a higher frequency means a shorter wavelength.

The lesson starts with a prediction before anyone explains: shake a rope faster, and do its waves go faster or get shorter? Students then try it on a running wave (a model, not a substitute for the practical), use the equation with kHz and cm turned into Hz and m, and work from the required practical's readings (a ripple tank, a string, sound) to a speed. It is for Foundation and Higher; the string's loops and the radio question lean towards Higher.

By the end, students can

  • describe transverse and longitudinal waves, and say that a wave carries energy, not the material it travels through
  • name a wave's amplitude, wavelength, frequency and period, and use T = 1 ÷ f
  • use v = f λ and rearrange it, turning kHz into Hz and cm into m first
  • explain why, in the same medium, a higher frequency means a shorter wavelength
  • describe how to measure the speed of water waves, waves on a string and sound in air

Key words

  • transverse
  • longitudinal
  • medium
  • amplitude
  • wavelength (λ)
  • frequency (Hz)
  • period
  • wave speed
  • compression
  • rarefaction

Before this lesson

Students should know speed = distance ÷ time and that 1 km is 1000 m. They should have met waves on a rope or a slinky; transverse and longitudinal are met again here.

The lesson, step by step

30 minutes in total. Each step says what changes for this length.

  1. 1 · Think3 min

    Shake it faster

    Shake a rope's end once a second:

    waves 4 m long run along it.

    Now shake it twice a second.

    A: The waves move twice as fast

    B: Each wave is half as long

    C: Both: faster and shorter

    Vote first. Then say why.

    Write here

    3 min 1 · Hook · students think · the slides say “Think”

    Shake it faster

    Waves on a rope shaken twice as fast: students vote on whether they go faster, get shorter or both, before anyone explains.

    For this length: Vote once, no discussion

    Say

    “Shake one end of a rope once a second and waves 4 m long run along it. Now shake it twice a second. Do the waves move twice as fast, is each wave half as long, or both?”

    Ask
    • What would make the waves go faster along the rope?
    • If you shook it harder but not faster, what would change?
    Watch for

    Saying faster shaking makes faster waves: keep the votes, the next step needs them.

    For the teacher

    Give a minute of silent thinking, then vote. Many say the waves go faster; some say both.

    Don't explain yet. The rope sets the speed, so the waves get shorter: the next step shows why. (Each wave is half as long, 2 m: 4 m/s = 2 Hz × 2 m.)

    Take a vote with Ask the class: faster, shorter or both. Keep the votes on the board for the explain step.

    Shorter: Vote once, no discussion (−2 min).

  2. 2 · Watch8 min

    Waves carry energy, not stuff

    Transverse: the medium moves at right angles

    to the way the wave goes: a rope, ripples on water

    Longitudinal: it moves to and fro along that way:

    sound, in compressions and rarefactions

    A cork on a pond bobs up and down;

    it isn't carried along

    The wave moves on; the medium stays.

    2 · Watch

    Describing a wave

    Amplitude: the middle to a peak

    Wavelength λ: one peak to the next

    Frequency f: waves a second, in Hz

    Period T: seconds for one wave

    T = 1 ÷ f: 4 Hz gives T = 0.25 s

    x: metres along; y: how far it moves

    Move a: what changes, what doesn't?

    2 · Watch

    The wave equation: v = f λ

    wave speed (m/s) = frequency (Hz) × wavelength (m)

    Each second f waves pass, each λ long:

    the wave goes f × λ metres a second

    2 Hz, 3 m: v = 2 × 3 = 6 m/s

    Rearranged: f = v ÷ λ and λ = v ÷ f

    Sound, 340 m/s at 680 Hz: λ = 340 ÷ 680 = 0.5 m

    Frequency in Hz, wavelength in metres

    2 · Watch

    Try it: shake it faster

    Now 1 Hz at 4 m/s: λ = 4 m

    Predict λ at 2 Hz, then move f

    Which f makes λ = 1 m?

    Switch to Longitudinal:

    what changes, and what doesn't?

    Predict first: 4 = 2 × ?

    Wave speedWavelength 4 m (v = f λ: 4 = 1 × 4) · period 1 s

    2 · Watch

    Same medium, same speed

    The speed is set by what the wave goes through:

    how tight the rope is, how deep the water

    Shake faster: more waves, each one shorter

    The hook: 2 Hz on the 4 m/s rope, λ = 2 m

    Sound in air: about 340 m/s

    A higher note: a shorter wavelength

    Same medium: frequency up, wavelength down

    2 · Watch

    Units first, then the equation

    kHz × 1000 → Hz; MHz × 1 000 000 → Hz

    cm ÷ 100 → m; ms ÷ 1000 → s

    4 kHz, 8.5 cm: 4000 × 0.085 = 340 m/s

    f = v ÷ λ λ = v ÷ f T = 1 ÷ f

    An echo goes there and back:

    distance = speed × time, twice the distance to the wall

    Hz and metres before you multiply

    8 min 2 · Explain · you explain · the slides say “Watch”

    Describing a wave, and v = f λ

    A wave carries energy, not the medium; amplitude, wavelength, frequency and period; v = f λ; the same medium keeps the speed; units first.

    For this length: Skip the longitudinal part of Try it; say the echo in a line, not its slide lines

    Say

    “A wave moves energy along. The rope, the water or the air stays where it is. Each second f waves go past, each one λ long: so the wave goes f × λ metres a second.”

    Ask
    • A cork on a pond as ripples pass: where does it go?
    • Each second 2 waves pass, each 3 m long. How far has the wave gone?
    • On the same rope, you double the frequency. What has to happen to the wavelength?
    Watch for

    Measuring amplitude from peak to trough, and wavelength from a peak to a trough: show both on the graph.

    The hook, worked

    1 Hz, λ = 4 m: v = 1 × 4 = 4 m/s

    Same rope, same 4 m/s; now 2 Hz

    4 = 2 × λ, so λ = 4 ÷ 2 = 2 m: each wave half as long

    Sound in air

    Sound at 340 m/s, a note of 680 Hz

    λ = v ÷ f = 340 ÷ 680 = 0.5 m

    For the teacher

    Transverse: the medium moves at right angles to the way the wave goes (a rope, ripples on water). Longitudinal: it moves to and fro along it (sound, a slinky pushed and pulled), in compressions and rarefactions. Either way the wave carries energy and the medium stays where it was: a cork bobs, it isn't carried away.

    On the wave's graph: amplitude from the middle to a peak (not peak to trough); wavelength from one peak to the next, or between any two points at the same place in the wave; frequency, waves a second, in hertz; period, seconds for one wave, T = 1 ÷ f.

    Wave speed: each second f waves pass, each λ long, so the wave goes f × λ metres. v = f λ, speed in m/s, frequency in Hz, wavelength in m. Rearranged: f = v ÷ λ, λ = v ÷ f.

    Try it: the running wave starts at 1 Hz and 4 m/s, a wavelength of 4 m, as in the hook. Ask for a prediction at 2 Hz before anyone moves f; it shows 2 m. Then which frequency gives 1 m (4 Hz), and what switching to longitudinal changes (how the medium moves, not v = f λ).

    The hook answered: the speed is set by what the wave travels through (how tight the rope is, how deep the water), not by how fast you shake it. Same medium, same speed: double the frequency, half the wavelength. Sound in air goes at about 340 m/s, so a higher note has a shorter wavelength.

    Units first: kHz × 1000 to Hz, MHz × 1 000 000; cm ÷ 100 to m; ms ÷ 1000 to s. Then multiply or divide.

    Before the Try it slide, ask the class to predict the wavelength at 2 Hz; then move f on the running wave together.

    Shorter: Skip the longitudinal part of Try it; say the echo in a line, not its slide lines (−4 min).

  3. 3 · Your turn8 min

    Required practical: ripple tank

    A motor makes ripples; a lamp throws them onto paper

    f: count waves passing a mark in 10 s, then ÷ 10

    λ: a strobe makes them look still; measure across

    several waves with a ruler, then ÷ how many

    v = f λ, with λ in metres

    30 waves in 10 s; 5 waves cover 40 cm: v = ?

    Open the Practical quiz: readings to a speed.

    3 · Your turn

    A string, and sound in air

    A vibration generator shakes a string at f Hz

    Change f until a still pattern of loops forms

    Each loop is half a wave: λ = 2 × one loop

    Sound: two microphones d apart, a timer: v = d ÷ t

    Echo: clap at a wall; the sound goes there and back

    Speed = 2 × distance to the wall ÷ time

    Measure across many; divide by how many.

    3 · Your turn

    Worked, then yours

    I do: 0.5 Hz, 3 m: v = 0.5 × 3 = 1.5 m/s

    You do: 4 Hz, 0.75 m: its speed?

    I do: 340 m/s, 850 Hz: λ = 340 ÷ 850 = 0.4 m

    You do: 1500 m/s in water, 3000 Hz: λ?

    I do: 2.5 kHz, 13.6 cm: 2500 × 0.136 = 340 m/s

    You do: 50 kHz, 3.0 cm, in water: its speed?

    Hz and metres first; then v = f λ.

    Write here

    Its answers' slide comes with your copy.

    8 min 3 · Your turn · students write

    The practical, then yours

    The required practical from readings to a speed (a ripple tank, a string, sound), each answered at once, then twins of worked examples.

    For this length: The ripple tank questions and the first two pairs only

    Say

    “Each question is a reading from the practical. Turn the readings into hertz and metres before you use v = f λ.”

    Ask
    • Why count the waves for 10 seconds, not 1?
    • Why measure across five waves and divide, not one?
    Watch for

    Dividing the count the wrong way round (10 ÷ 30), or leaving the wavelength in centimetres.

    Wave speed

    Pond ripples: 0.5 Hz, λ = 3 m

    v = f λ = 0.5 × 3 = 1.5 m/s

    Then theirs

    A rope: 4 Hz, wavelength 0.75 m. Its speed?

    Rearrange for λ

    Sound in air: 340 m/s, 850 Hz

    λ = v ÷ f = 340 ÷ 850 = 0.4 m

    Then theirs

    Sound in water: 1500 m/s, 3000 Hz. Its wavelength?

    Prefixes first

    A note: 2.5 kHz, wavelength 13.6 cm

    2.5 kHz = 2500 Hz; 13.6 cm = 0.136 m

    v = 2500 × 0.136 = 340 m/s

    Then theirs

    A dolphin's click in water: 50 kHz, wavelength 3.0 cm. Its speed?

    Each question is a reading from the required practical on waves, or from measuring the speed of sound. Turn readings into hertz, metres and seconds first, and show your working on paper.

    1. In a ripple tank, Amira counts 30 waves passing a mark in 10 seconds. What is their frequency, in Hz?
    2. With the strobe on, the waves look still. Five whole waves cover 40 cm on her ruler. What is the wavelength, in metres?
    3. Use her frequency (3 Hz) and wavelength (0.08 m). What is the speed of the waves, in m/s?
    4. Why does Amira measure across five waves and divide by 5, not measure one wave?
      1. One wave is short, so a small slip with the ruler is a big part of it; across five, the slip is shared out
      2. The waves further from the bar are longer, so she needs an average
      3. Measuring across five waves gives the frequency as well
    5. A vibration generator shakes a string at 40 Hz. The still pattern has 4 loops along 1.2 m of string, and each loop is half a wavelength. What is the speed of the waves on the string, in m/s?
    6. Kai stands 85 m from a tall wall and claps. The echo comes back 0.50 s later. What is the speed of sound, in m/s?
    7. Two microphones are 1.36 m apart, in line with a loudspeaker. A timer shows the sound reaches the second one 4.0 ms after the first. What is the speed of sound, in m/s?
    For the teacher

    Ripple tank: a motor dips a bar in the water; a lamp above throws the ripples' shadows onto paper below. Frequency: count the waves passing a mark in 10 s and divide by 10. Wavelength: a strobe at the right rate makes the waves look still (or take a photo with a ruler in it); measure across several waves and divide by how many. Then v = f λ.

    Waves on a string: a vibration generator shakes a stretched string at a frequency set on the signal generator. Change the frequency until a still pattern of loops forms; each loop is half a wavelength, so measure across all the loops, divide by how many, and double it.

    Sound in air: two microphones a measured distance apart and a timer that starts and stops as the sound reaches each, v = d ÷ t; or an echo from a wall, where the sound goes there and back (distance = speed × time, with twice the distance to the wall).

    Work the examples aloud while the slower ones finish the quiz; the twins check they multiply, rearrange, and turn kHz and cm into Hz and m first.

    Students open the Practical quiz on their own devices; each answer is checked at once. Open the answers slide once most have done the twins.

    Shorter: The ripple tank questions and the first two pairs only (−4 min).

    Longer: Pairs plan the echo method for the school field: what to measure, and how to make it more accurate (+3 min).

  4. 4 · Vote and practise7 min

    Vote first

    A sound has a frequency of 2 kHz

    and a wavelength of 17 cm. What is its speed?

    A: 34 m/s

    B: 340 m/s

    C: 34 000 m/s

    D: 0.34 m/s

    Choose on your own, then say why.

    4 · Vote and practise

    Practise: show your working

    1. 5 Hz, wavelength 2 m: the speed?

    2. 50 Hz: the period?

    3. Ripples, a guitar string or sound: longitudinal?

    4. 0.25 kHz, 60 cm: the speed in m/s?

    5. 330 m/s, wavelength 1.5 m: the frequency?

    Stuck on 4? Hz and metres first.

    Write here

    4 · Vote and practise

    Reason it out

    6. A higher note in the same room: what changes?

    7. A cork as ripples pass: where does it go?

    8. Shake harder, not faster: what changes?

    9. Sonar: echo after 0.80 s at 1500 m/s: how deep?

    10. 100 MHz radio at 300 000 000 m/s: λ?

    Stretch: period 0.04 s, λ = 0.5 m: f and v?

    Same medium, same speed.

    7 min 4 · Who's stuck? · students vote · the slides say “Vote and practise”

    Vote, then practise

    One question shows who uses kHz and cm as they are; then practice in levels, and period, frequency and speed together as the stretch.

    For this length: After the vote, questions 1–5 only

    Say

    “Choose an answer on your own. Nobody talks until everyone has chosen.”

    Ask
    • What is 2 kHz in hertz? 17 cm in metres?
    Watch for

    Multiplying the numbers as they stand: ask for hertz and metres first.

    Show your working on paper. Turn kHz into Hz and cm into m first. Give speeds in m/s, wavelengths in m, frequencies in Hz and periods in s. The stretch is optional.

    The question that shows who's stuck

    1. A sound has a frequency of 2 kHz and a wavelength of 17 cm. What is its speed?
      1. 34 m/s
      2. 340 m/s
      3. 34 000 m/s
      4. 0.34 m/s
    In your Classroom, as they answer (an example)
    • A7
    • B12
    • C6
    • D4

    Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.

    1. A wave has a frequency of 5 Hz and a wavelength of 2 m. What is its speed, in m/s?
    2. A wave has a frequency of 50 Hz. What is its period, in seconds?
    3. Which of these is a longitudinal wave?
      1. Ripples on a pond
      2. A wave on a guitar string
      3. Sound travelling through air
    4. A wave has a frequency of 0.25 kHz and a wavelength of 60 cm. In hertz, f = ___ Hz; in metres, λ = ___ m; so its speed is v = ___ m/s.
    5. Sound travels at 330 m/s in cold air. A note has a wavelength of 1.5 m. What is its frequency, in Hz?
    6. A singer goes from a low note to a higher note in the same room. Compared with the low note, the higher note's sound waves…
      1. go faster, with the same wavelength
      2. go at the same speed, with a shorter wavelength
      3. go at the same speed, with a longer wavelength
      4. go faster, with a longer wavelength
    7. A cork floats on a pond. Ripples pass it for a minute. What does the cork do?
      1. It is carried along with the ripples to the edge of the pond
      2. It bobs up and down, staying about where it was
      3. It stays completely still while the ripples pass under it
    8. Ben shakes a rope harder, but just as often each second. What changes?
      1. The amplitude: each wave is taller and carries more energy
      2. The wavelength: each wave gets longer
      3. The speed: the waves move along faster
    9. A ship's sonar sends a pulse of sound down to the seabed. The echo comes back 0.80 s later. Sound travels at 1500 m/s in seawater. How deep is the sea there, in metres?
    10. A radio station broadcasts at 100 MHz. Radio waves travel at 300 000 000 m/s. What is the wavelength, in metres?
    11. Stretch: a wave's period is 0.04 s and its wavelength is 0.5 m. Its frequency is ___ Hz, so its speed is ___ m/s.

    Support · Write the three quantities in a line with their units: v = ? m/s, f = ? Hz, λ = ? m. Change each into those units before anything else, then put them into v = f λ.

    For the teacher

    Everyone chooses before anyone explains. Then ask in turn: what is the frequency in hertz? The wavelength in metres? What is one times the other?

    Questions 1–3 build fluency; 4–5 change the look (prefixes, rearranging); 6–8 ask for reasons; 9–10 are exam-style. The stretch goes from a period to a frequency to a speed.

    Students show their working on paper or the board; the quiz keeps their final answers.

    If many chose 34 or 34 000 m/s, write kHz → Hz and cm → m on the board and do the vote question together before they practise.

    Shorter: After the vote, questions 1–5 only (−5 min).

    Longer: The stretch, then each student writes a question with kHz and cm for a partner (+8 min).

  5. 5 · Show what you know4 min

    The rope again

    A wave on a rope: 2.5 Hz, wavelength 1.6 m

    What is its speed?

    The same rope is shaken at 5 Hz.

    What is the wavelength now?

    Same rope, same speed.

    Write here

    4 min 5 · Exit · students show · the slides say “Show what you know”

    The rope again

    A wave's speed, then the same rope shaken faster, shows who keeps the speed and shortens the wavelength.

    Say

    “Same rope, same speed.”

    Watch for

    Doubling the wavelength with the frequency.

    Write v = f λ with each quantity and its unit first.

    1. A wave on a rope has a frequency of 2.5 Hz and a wavelength of 1.6 m. What is its speed, in m/s?
    2. The same rope is now shaken at 5 Hz. What is the wavelength now, in metres?
    For the teacher

    A wave on a rope has frequency 2.5 Hz and wavelength 1.6 m: its speed?

    The same rope is shaken at 5 Hz: the wavelength now?

    Those who doubled the wavelength need the running wave again next lesson; set the homework.

    Longer: Why do a ripple tank's waves get shorter when its motor runs faster? Answer with v = f λ. (+4 min).

One to one: let the student drive the running wave while you only ask what they predict before each move; then ask them to explain the ripple tank method aloud, as if to a classmate, before any numbers.

Homework: The wave equation

Write v = f λ (or T = 1 ÷ f) first, turn every reading into Hz, m and s, then work it out. Give units with every answer.

  1. A water wave has a frequency of 0.4 Hz and a wavelength of 5 m. Find its speed.
  2. A wave has a frequency of 20 Hz. Find its period.
  3. A note of 425 Hz travels through air at 340 m/s. Find its wavelength.
  4. A sound has a frequency of 1.25 kHz and a wavelength of 27.2 cm. Find its speed in m/s.
  5. Mia stands 102 m from a cliff and claps. The echo comes back 0.60 s later. Find the speed of sound.
  6. Explain, using v = f λ, why a higher note in the same room has a shorter wavelength.
  7. Describe how you would measure the frequency, the wavelength and then the speed of water waves in a ripple tank.

Print the worksheet with working space and homework. Teacher answers come with your Library copy.

Teaching wave speed and the wave equation: common questions

Is the wave equation on Foundation and Higher, and in combined science?

Yes. In AQA GCSE Physics (8463) and AQA Combined Science: Trilogy (8464), wave speed, frequency, wavelength, period and the required practical on waves are for both tiers. The whole lesson suits both; the loops on a string and the radio question lean towards Higher.

Do students have to remember v = f λ?

Yes: wave speed = frequency × wavelength is one of the equations AQA expects students to recall and apply, and to rearrange. Learning T = 1 ÷ f alongside it costs little and comes up in the same questions.

Why doesn't the wave go faster when the frequency goes up?

The speed depends on what the wave travels through: how tight and heavy a rope is, how deep the water, the air's temperature. Shaking faster makes more waves each second, but the rope carries each one along at the same speed, so they are closer together: shorter. That is the hook's answer, and the running wave shows it.

What speed of sound should students use?

The one the question gives. In air it is about 330 to 340 m/s, a little faster on a warm day; the lesson uses 340 m/s. Sound is faster in water (about 1500 m/s) and faster still in solids.

Is the running wave a substitute for the required practical?

No. It is a model to predict with: it shows the wavelength change as the frequency does, and a particle that only moves up and down or to and fro. Students still need to do the ripple tank and the string, or at least describe them, and the practical quiz works from readings like theirs.

Can I use the lesson without creating an account?

Yes. Read the lesson, print the worksheet and open its slides without signing in; nothing you draw there is kept. A free teacher account keeps the whole lesson in your Library to change and teach, with the quizzes, the answers and the answer slides.

Curriculum reference: AQA GCSE Physics 8463: 4.6.1.1 (transverse and longitudinal waves; a wave transfers energy, not the medium) and 4.6.1.2 (properties of waves: amplitude, wavelength, frequency, period T = 1 ÷ f, wave speed v = f λ, measuring the speed of sound in air and of ripples on water, and the required practical on waves in a ripple tank and on a stretched string).

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