Science

What are waves in physics, and what is the wave equation?

Quick answer

A wave carries energy from place to place without carrying matter. Waves can be transverse or longitudinal, and wave speed = frequency × wavelength (v = fλ).

Ages 11–16Secondary

In one sentence

A wave is a vibration that carries energy from one place to another, without carrying the material itself along with it.

Explained

Drop a stone in a pond and ripples spread out to the edge. A leaf floating on the water bobs up and down as each ripple passes, but it does not travel to the edge with them. That is the key idea: a wave moves energy, not matter. The water only vibrates about where it is.

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Water ripples, a vibrating guitar string and the sound reaching your ear are all waves.

There are two ways a wave can vibrate:

Transverse waves: the vibrations are at right angles (perpendicular) to the direction the wave travels. Ripples on water, a wave sent along a rope, and light are transverse. They have high points called crests (peaks) and low points called troughs.
Longitudinal waves: the vibrations go backwards and forwards in the same direction the wave travels. Sound is longitudinal. The particles bunch up in compressions and spread out in rarefactions. Push and pull one end of a stretched slinky spring and you can see it.

Waves are also grouped by what they travel through. Mechanical waves, such as sound, water waves and earthquake waves, need a material (a medium). Electromagnetic waves, such as light and radio, can cross empty space. That is why there is no sound in space, but sunlight still reaches us. See the electromagnetic spectrum.

Every wave can be described with a few measurements:

Amplitude: the largest distance a point moves from its rest position. It is measured from the middle to a crest, not from a trough to a crest. A bigger amplitude carries more energy: a louder sound or a taller water wave.
Wavelength (λ): the length of one complete wave, for example from one crest to the next, or one compression to the next. Measured in metres.
Frequency (f): how many waves pass a point each second, measured in hertz (Hz). 50 Hz means 50 waves every second.
Period (T): the time for one complete wave, in seconds. T = 1 ÷ f, so a 4 Hz wave has a period of 0.25 s.

These are joined together by the wave equation:

wave speed = frequency × wavelength, v = f × λ

If 3 waves pass you every second and each is 2 m long, then 6 m of wave passes you each second: the speed is 6 m/s. For one kind of wave in one material the speed stays about the same, so a higher frequency means a shorter wavelength.

Sound waves and hearing. When something vibrates, it pushes the air into compressions and rarefactions that spread out at about 340 m/s. In your ear, the sound makes your eardrum vibrate. Three tiny bones pass the vibrations to the cochlea, a coiled tube of fluid where hair cells turn them into nerve signals for the brain. The frequency of a sound decides its pitch (high or low) and the amplitude decides its loudness. Young people can usually hear from about 20 Hz to 20,000 Hz; the top limit often drops with age. Sound above 20,000 Hz is called ultrasound: bats and dolphins use it, and doctors use it for scans.

Examples

Wave	Type	Typical speed	Notes
Sound in air	longitudinal, mechanical	about 340 m/s	faster in water (about 1,500 m/s) and in solids
Ripples on water	transverse, mechanical	a few cm/s to m/s	shows crests and troughs clearly
Wave on a guitar string	transverse, mechanical	depends on tension	makes the air vibrate, giving sound
Light	transverse, electromagnetic	3.0 × 10⁸ m/s in a vacuum	needs no material
Earthquake P-waves and S-waves	longitudinal and transverse	several km/s	P-waves arrive first

Worked example: the note A that orchestras tune to has a frequency of 440 Hz. In air at 340 m/s, its wavelength is λ = v ÷ f = 340 ÷ 440 ≈ 0.77 m.

Common mistakes

Measuring amplitude from trough to crest. That gives twice the amplitude. Measure from the middle (rest) line to a crest.
Thinking the material travels with the wave. The leaf on the pond only bobs up and down. Energy moves; the water stays roughly where it is.
Mixing up frequency and period. Frequency is waves per second; period is seconds per wave. They are opposites: T = 1 ÷ f.
Forgetting units. Use metres and hertz in v = f × λ: 30 cm is 0.3 m, and 2 kHz is 2,000 Hz.

Step by step

Write the wave equationV = f × λ. Rearrange if needed: f = v ÷ λ or λ = v ÷ f.
Find the numbers you knowCount waves per second for frequency, or measure crest to crest for wavelength.
Convert unitsLengths in metres, frequency in hertz, time in seconds.
CalculatePut the numbers in and work it out, with the unit: m/s, Hz or m.
Check it makes senseSound in air should come out near 340 m/s; a higher frequency should give a shorter wavelength.

To measure the speed of sound, stand a measured distance from a big wall, clap, and time the echo: the sound goes there and back, so speed = (2 × distance) ÷ time. In a ripple tank you can measure the wavelength with a ruler across several ripples and find the frequency by counting ripples for 10 seconds.

Practice questions

Waves and v = f × λ

Try each one, then check your answer.

A wave has a frequency of 5 Hz and a wavelength of 2 m. Find its speed.
Show answer
v = 5 × 2 = 10 m/s
A wave has f = 50 Hz and λ = 0.3 m. Find its speed.
Show answer
v = 50 × 0.3 = 15 m/s
Sound travels at 340 m/s. Find the wavelength of a 170 Hz sound.
Show answer
λ = 340 ÷ 170 = 2 m
A sound wave in air (340 m/s) has a wavelength of 0.5 m. Find its frequency.
Show answer
f = 340 ÷ 0.5 = 680 Hz
What is the period of a wave with a frequency of 4 Hz?
Show answer
T = 1 ÷ 4 = 0.25 s
A wave has a period of 0.02 s. What is its frequency?
Show answer
f = 1 ÷ 0.02 = 50 Hz
12 waves pass a buoy in 3 seconds. What is the frequency?
Show answer
12 ÷ 3 = 4 Hz
Exactly 3 complete waves fit along 1.5 m of rope. What is the wavelength?
Show answer
1.5 ÷ 3 = 0.5 m
Is sound in air transverse or longitudinal?
Show answer
Longitudinal: the air vibrates along the direction the sound travels.
Are ripples on a pond transverse or longitudinal?
Show answer
Transverse: the water moves up and down while the wave moves across.
A crest is 4 cm above the rest line and a trough is 4 cm below it. What is the amplitude?
Show answer
4 cm (not 8 cm: measure from the rest line)
The frequency of a wave doubles but its speed stays the same. What happens to the wavelength?
Show answer
λ = v ÷ f, so the wavelength halves.
Can people hear a sound of 25,000 Hz?
Show answer
No: it is above about 20,000 Hz, so it is ultrasound.
A water wave has a period of 2 s and a wavelength of 8 m. Find its frequency and speed.
Show answer
f = 1 ÷ 2 = 0.5 Hz; v = 0.5 × 8 = 4 m/s
A guitar string makes a 330 Hz note in air at 340 m/s. Find the wavelength of the sound.
Show answer
λ = 340 ÷ 330 ≈ 1.03 m

Word problems

Waves in real life

Try each one, then check your answer.

You shout towards a cliff and hear the echo 2 s later. Sound travels at 340 m/s. How far away is the cliff?
Show answer
The sound goes there and back: 340 × 2 = 680 m in total, so the cliff is 340 m away.
A ship's sonar pulse returns from the seabed after 0.8 s. Sound travels at 1,500 m/s in seawater. How deep is the sea?
Show answer
1,500 × 0.8 = 1,200 m there and back, so the depth is 600 m.
You hear thunder 6 s after you see the lightning. About how far away is the storm?
Show answer
The light arrives almost at once; sound covers 340 × 6 = 2,040 m, about 2 km.
A surfer counts 6 waves reaching the shore in 1 minute. The crests are 15 m apart. Find the frequency and speed of the waves.
Show answer
f = 6 ÷ 60 = 0.1 Hz; v = 0.1 × 15 = 1.5 m/s
In a ripple tank, 10 ripples span 30 cm and the frequency is 8 Hz. Find the wave speed.
Show answer
λ = 30 ÷ 10 = 3 cm = 0.03 m; v = 8 × 0.03 = 0.24 m/s

Questions people ask

What is the difference between transverse and longitudinal waves?

In a transverse wave the vibrations are at right angles to the direction of travel, like ripples or light. In a longitudinal wave they are along the direction of travel, like sound, with compressions and rarefactions.

What is the wave equation?

Wave speed = frequency × wavelength, v = f × λ, with speed in m/s, frequency in Hz and wavelength in m. It works for every kind of wave, from sound to light.

What is wavelength?

The length of one complete wave: the distance from one crest to the next crest (or one compression to the next). Its symbol is the Greek letter λ, lambda.

What is frequency?

The number of complete waves that pass a point every second, measured in hertz. A higher-frequency sound has a higher pitch.

How do we hear sound waves?

Sound waves make the eardrum vibrate. Three small bones pass the vibrations to the cochlea, where tiny hair cells turn them into nerve signals that the brain hears as sound.

What is the range of human hearing?

About 20 Hz to 20,000 Hz for young people. The highest sounds we can hear usually get lower as we get older, and very loud sounds can damage the hair cells for good.

Can sound travel through space?

No. Sound is a mechanical wave and needs particles to pass the vibrations along. Space is almost empty, so it is silent. Light and radio waves can cross it.

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