🔊 Sound waves
Sound travels through air at about 340 m/s. Every sound in air travels at this speed, whether it is a high note or a low note.
Properties of Waves » The Wave Equation
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 3.5, 3.7
- The wave equation: wave speed = frequency × wavelength
- How to rearrange it to find frequency or wavelength
- How to deal with kHz, MHz, centimetres and standard form
- How to use it for water waves, sound waves, radio waves and light
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Unlock This CourseOne equation for every wave
In Describing Waves you met frequency and wavelength. Multiply them together and you get the speed of the wave. This works for every kind of wave, transverse or longitudinal.
The wave equation
wave speed = frequency × wavelength
v = f × λ
v is the wave speed in metres per second (m/s), f is the frequency in hertz (Hz) and λ (the Greek letter lambda) is the wavelength in metres (m).
This equation is not given in the exam. You must learn it.
Here is why it works. The frequency tells you how many complete waves leave the source each second. The wavelength tells you how long each wave is. Suppose a source sends out 3 waves each second and each wave is 2 m long. In one second, the front of the first wave has moved 3 × 2 = 6 m. So the wave speed is 6 m/s.
Key terms:
- Wave speed (v): the distance a wave travels each second.
- Kilohertz (kHz): 1 kHz = 1000 Hz.
- Megahertz (MHz): 1 MHz = 1 000 000 Hz.
Rearranging the equation
Exam questions can ask for any one of the three quantities. You can rearrange the equation or use a formula triangle with v at the top and f and λ side by side at the bottom.
| To find | Use | In words |
|---|---|---|
| Wave speed | v = f × λ | frequency × wavelength |
| Frequency | f = v ÷ λ | wave speed ÷ wavelength |
| Wavelength | λ = v ÷ f | wave speed ÷ frequency |
Set out every calculation in three lines: the equation, the substitution (putting the numbers in) and the answer with its unit.
Get the units right first
The equation only gives a speed in m/s if the frequency is in hertz and the wavelength is in metres. Convert before you substitute.
| Given in | Change to | How | Example |
|---|---|---|---|
| kilohertz (kHz) | Hz | multiply by 1000 | 1.7 kHz = 1700 Hz |
| megahertz (MHz) | Hz | multiply by 1 000 000 | 100 MHz = 100 000 000 Hz |
| centimetres (cm) | m | divide by 100 | 1.5 cm = 0.015 m |
Using the equation in different contexts
The exam can set the wave equation in any context. Two speeds come up again and again, so it helps to know them. Questions usually give you the speed, but check that your answers look sensible against these values.
💡 Electromagnetic waves
Light and radio waves are both electromagnetic waves. They travel through air at 3 × 108 m/s. That is 300 000 000 m/s, written in standard form.
You will meet the whole family of electromagnetic waves in The Electromagnetic Spectrum.
Worked example 1: water waves (finding speed)
Question: Ripples on a pond have a frequency of 10 Hz and a wavelength of 1.5 cm. Calculate the speed of the ripples.
Convert: 1.5 cm = 0.015 m
v = f × λ
v = 10 × 0.015
v = 0.15 m/s
Worked example 2: water waves (finding frequency)
Question: Waves on the sea travel at 6 m/s. Their wavelength is 12 m. Calculate the frequency of the waves.
f = v ÷ λ
f = 6 ÷ 12
f = 0.5 Hz
This means half a wave passes each second.
Worked example 3: sound waves (finding wavelength)
Question: A whistle makes a sound of frequency 1.7 kHz. The speed of sound in air is 340 m/s. Calculate the wavelength of the sound.
Convert: 1.7 kHz = 1700 Hz
λ = v ÷ f
λ = 340 ÷ 1700
λ = 0.2 m
Worked example 4: radio waves (standard form)
Question: A radio station broadcasts at a frequency of 100 MHz. Radio waves travel at 3 × 108 m/s. Calculate the wavelength of the radio waves.
Convert: 100 MHz = 100 000 000 Hz = 1 × 108 Hz
λ = v ÷ f
λ = (3 × 108) ÷ (1 × 108)
λ = 3 m
Worked example 5: light (standard form)
Question: Orange light has a frequency of 5 × 1014 Hz. The speed of light is 3 × 108 m/s. Calculate the wavelength of the light.
λ = v ÷ f
λ = (3 × 108) ÷ (5 × 1014)
λ = 6 × 10−7 m
This is 0.000 000 6 m, far too small to see. On a calculator, use the ×10x (or EXP) button to enter each number, and put brackets round the number you are dividing by.
Same speed, different frequency
All sounds in air travel at the same speed. So if the frequency goes up, the wavelength must go down, because f × λ has to give the same answer.
| Sound in air | Frequency | Wavelength (340 ÷ f) | Speed |
|---|---|---|---|
| Low note | 170 Hz | 2 m | 340 m/s |
| Middle note | 340 Hz | 1 m | 340 m/s |
| High note | 680 Hz | 0.5 m | 340 m/s |
Each time the frequency doubles, the wavelength halves. The same is true for electromagnetic waves: they all travel at 3 × 108 m/s in air, so a higher frequency always means a shorter wavelength.
Common mistakes
- Leaving the wavelength in centimetres. Divide by 100 first, or your speed will be 100 times too big.
- Forgetting kHz and MHz. 100 MHz is not 100 Hz. Multiply by 1 000 000.
- Dividing the wrong way round. The wave speed always goes on top: f = v ÷ λ and λ = v ÷ f.
- Using 340 m/s for light or radio waves. 340 m/s is for sound in air. Light and radio waves travel at 3 × 108 m/s.
- Missing the unit. Speed in m/s, frequency in Hz, wavelength in m.
Exam-style question
(a) State the equation linking wave speed, frequency and wavelength. [1 mark]
(b) A dolphin makes a sound of frequency 75 kHz. The speed of sound in sea water is 1500 m/s. Calculate the wavelength of the sound in sea water. [3 marks]
(c) A radio wave has a wavelength of 1500 m. The speed of radio waves is 3 × 108 m/s. Calculate the frequency of the radio wave. Give your answer in kHz. [3 marks]
(d) The sound from the dolphin passes from the sea water into the air. The speed of the sound decreases but its frequency does not change. Explain what happens to the wavelength of the sound. [2 marks]
Model answer
(a) wave speed = frequency × wavelength, or v = f × λ (1).
(b) 75 kHz = 75 000 Hz (1). λ = v ÷ f = 1500 ÷ 75 000 (1). λ = 0.02 m (1).
(c) f = v ÷ λ = (3 × 108) ÷ 1500 (1). f = 200 000 Hz (1). f = 200 kHz (1).
(d) The wavelength decreases (1), because wavelength = wave speed ÷ frequency, and the speed is smaller while the frequency is the same (1).
Exam tip
In parts (b) and (c) the unit prefix is where marks are lost. Circle 'kHz' in the question before you start: in (b) you convert it at the beginning, and in (c) you convert your answer at the end by dividing by 1000.