3. Wave Speed

Wave Speed

Waves transfer energy from one place to another. They can travel through materials, such as sound travelling through air or waves travelling along a rope, and some waves, such as electromagnetic waves, can also travel through a vacuum.

One of the most important quantities used to describe a wave is its wave speed. Wave speed tells us how quickly the disturbance or wave pattern travels from one location to another.

Learning Targets

• I can define wave speed.
• I can use the wave equation v = λf.
• I can calculate wave speed, frequency, or wavelength.
• I can explain how wave speed depends on the medium.
• I can solve problems involving the wave equation.

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What Is Wave Speed?

Wave speed is the distance travelled by a wave per unit time.

Like other speeds, it can be described using:

Speed = Distance / Time

The SI unit for wave speed is:

metres per second (m/s)

For example, if a wave travels 20 m in 4 s:

Wave speed = 20 / 4

Wave speed = 5 m/s

This means the wave pattern travels 5 metres every second.

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Waves Transfer Energy

A wave transfers energy, but it does not normally transport matter from the source all the way to the destination.

Imagine producing waves along a rope.

The wave travels along the rope, but the individual sections of rope mainly move around their equilibrium positions.

Similarly, when a water wave moves across the surface of a pond, the water itself does not simply travel across the pond with the wave.

Wave speed describes how quickly this disturbance and its energy travel.

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The Wave Equation

Wave speed is connected to frequency and wavelength.

The relationship is:

v = λf

where:

• v = wave speed in metres per second (m/s)
• λ = wavelength in metres (m)
• f = frequency in hertz (Hz)

This is one of the most important equations in wave physics.

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Understanding the Wave Equation

The wave equation makes sense if we think about what frequency and wavelength mean.

Suppose:

Wavelength = 3 m

and:

Frequency = 2 Hz

A frequency of 2 Hz means that two complete waves pass a point every second.

Each wave is 3 m long.

Therefore, in one second:

Distance travelled = 2 × 3 m

Distance travelled = 6 m

So:

Wave speed = 6 m/s

Using the equation:

v = λf

v = 3 × 2

v = 6 m/s

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Wavelength

Wavelength is the distance between two equivalent points on consecutive waves.

The symbol for wavelength is the Greek letter:

λ

pronounced lambda.

The SI unit is:

metre (m)

For a transverse wave, wavelength can be measured:

• Crest to crest
• Trough to trough
• Between any two equivalent points on consecutive waves

For a longitudinal wave, wavelength can be measured:

• Compression to compression
• Rarefaction to rarefaction

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Frequency

Frequency is the number of complete waves or oscillations passing a point each second.

The symbol is:

f

The unit is:

hertz (Hz)

For example:

5 Hz = 5 waves per second

100 Hz = 100 waves per second

1000 Hz = 1000 waves per second

A higher frequency means that more complete waves pass a point each second.

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Comparing Frequency and Wavelength

When waves travel through the same medium at the same wave speed, frequency and wavelength are inversely related.

If frequency increases:

Wavelength decreases

If frequency decreases:

Wavelength increases

This follows from:

v = λf

If v remains constant, increasing one of the other quantities requires the other to decrease.

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Rearranging the Wave Equation

We can use the wave equation to calculate three different quantities.

To calculate wave speed:

v = λf

To calculate wavelength:

λ = v / f

To calculate frequency:

f = v / λ

Before solving a problem, identify which quantity you are trying to find and choose the correct form of the equation.

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Worked Example — Calculating Wave Speed

A wave has:

Wavelength = 4 m

Frequency = 3 Hz

Calculate its speed.

Use:

v = λf

Substitute:

v = 4 × 3

v = 12 m/s

Therefore:

Wave speed = 12 m/s

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Worked Example — Calculating Frequency

A wave travels at:

20 m/s

and has a wavelength of:

5 m

Calculate its frequency.

Use:

f = v / λ

Substitute:

f = 20 / 5

f = 4 Hz

Therefore:

Frequency = 4 Hz

This means four complete waves pass a point every second.

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Worked Example — Calculating Wavelength

A sound wave travels at:

340 m/s

and has a frequency of:

170 Hz

Calculate its wavelength.

Use:

λ = v / f

Substitute:

λ = 340 / 170

λ = 2 m

Therefore:

Wavelength = 2 m

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Unit Conversions

Before using the wave equation, quantities should normally be converted into appropriate SI units.

Wavelength should usually be in:

metres (m)

Frequency should be in:

hertz (Hz)

Wave speed will then be in:

metres per second (m/s)

Useful conversions include:

1 km = 1000 m

1 cm = 0.01 m

1 mm = 0.001 m

1 kHz = 1000 Hz

1 MHz = 1,000,000 Hz

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Worked Example — Converting Units

A wave has a wavelength of:

50 cm

and a frequency of:

8 Hz

First convert the wavelength:

50 cm = 0.50 m

Now use:

v = λf

v = 0.50 × 8

v = 4 m/s

Therefore:

Wave speed = 4 m/s

Converting the units first prevents a common calculation error.

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Wave Speed and the Medium

The speed of a wave often depends on the medium through which it travels.

A medium is the material through which a wave travels.

Examples include:

• Air
• Water
• Steel
• Glass
• A rope
• A spring

The properties of the medium determine how quickly a disturbance can be transferred from one part of the material to another.

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Sound Waves and the Medium

Sound is a mechanical wave, meaning it requires a medium.

Sound travels by causing particles in the medium to vibrate.

The vibrations are passed from particle to particle.

Sound generally travels faster through solids than through liquids and faster through liquids than through gases, although the exact speed depends on the properties and conditions of the material.

Typical values are approximately:

These values can vary with temperature, pressure, composition, and other conditions.

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Why Does Sound Travel at Different Speeds?

Wave speed depends on how the particles of the medium interact.

In solids, particles are strongly connected and disturbances can often be transmitted rapidly.

In gases, particles are much farther apart and the mechanical properties are very different.

However, simply saying that particles are "closer together" is not a complete explanation.

The speed depends on properties such as the medium's:

• Elasticity
• Density
• Temperature

Different types of waves depend on different properties of their medium.

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Waves on a String

The speed of a wave travelling along a string depends on properties of the string.

Two particularly important factors are:

• Tension
• Mass per unit length

Increasing the tension generally increases the wave speed.

A heavier string per unit length generally has a lower wave speed for the same tension.

This can be observed by producing waves on ropes or strings with different tensions.

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Electromagnetic Waves

Electromagnetic waves are different from mechanical waves because they do not require a material medium.

They can travel through a vacuum.

Examples include:

• Radio waves
• Microwaves
• Infrared
• Visible light
• Ultraviolet
• X-rays
• Gamma rays

In a vacuum, all electromagnetic waves travel at approximately:

3.00 × 10⁸ m/s

This is the speed of light in a vacuum.

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Light Travelling Through Materials

Although electromagnetic waves travel fastest in a vacuum, they travel more slowly through materials such as:

• Water
• Glass
• Plastic

When light passes from one material into another, its speed can change.

This change in speed is responsible for refraction.

For example, light entering glass from air usually slows down and changes direction unless it enters perpendicular to the boundary.

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What Happens When a Wave Enters a New Medium?

When a wave crosses into another medium, its frequency normally remains unchanged because the source still produces oscillations at the same rate.

However, the wave speed may change.

Because:

v = λf

if:

frequency stays constant

and:

wave speed changes

then:

wavelength must change

This is an important application of the wave equation.

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Example — Wave Entering a Slower Medium

Suppose a wave has:

Frequency = 5 Hz

In Medium A:

Wave speed = 20 m/s

Therefore:

λ = v/f

λ = 20/5

λ = 4 m

The wave enters Medium B where its speed becomes:

10 m/s

The frequency remains:

5 Hz

Therefore:

λ = 10/5

λ = 2 m

So when the wave slows down:

Wavelength decreases from 4 m to 2 m

The frequency remains unchanged.

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Reading Wave Diagrams

Wave diagrams often allow wavelength to be measured directly.

For a transverse wave, identify two neighbouring:

• Crests
• Troughs
• Equivalent zero-crossing points

The horizontal distance between them is one wavelength.

Be careful not to confuse:

Amplitude with wavelength.

Amplitude is measured from the equilibrium position vertically to a crest or trough.

Wavelength is measured between corresponding points along the direction of wave travel.

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Worked Example — Reading a Wave

Suppose a diagram shows four complete wavelengths occupying a distance of:

12 m

One wavelength is:

λ = 12 / 4

λ = 3 m

If the frequency is:

6 Hz

then:

v = λf

v = 3 × 6

v = 18 m/s

Therefore:

Wave speed = 18 m/s

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Frequency, Wavelength, and Wave Speed

It is important to understand which quantities determine which.

For a wave travelling in a particular medium, the properties of the medium largely determine the wave speed.

The source determines the frequency.

The wavelength then adjusts according to:

v = λf

For example, if the source frequency increases while the wave remains in the same medium:

Frequency increases

Wave speed remains approximately constant

Wavelength decreases

This distinction is important when analyzing wave behaviour.

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Worked Example — Changing Frequency

A wave travels along a rope at:

24 m/s

Initially, its frequency is:

4 Hz

Its wavelength is:

λ = 24/4

λ = 6 m

The source frequency increases to:

8 Hz

If the rope conditions remain unchanged, the wave speed remains:

24 m/s

The new wavelength is:

λ = 24/8

λ = 3 m

Therefore:

Frequency doubled

Wavelength halved

Wave speed remained unchanged

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Wave Speed in Everyday Life

Wave speed is important in many technologies and natural phenomena.

Communication

Radio signals carry information over large distances.

Medical imaging

Ultrasound uses sound waves travelling through body tissues.

Earthquakes

Seismic waves travel at different speeds through Earth's interior.

Music

Sound waves travel through air from instruments to listeners.

Sonar

Sound waves travel through water and reflect from objects.

Fibre-optic communication

Light waves transmit information through optical fibres.

Understanding wave speed allows scientists and engineers to calculate distances, locate objects, study materials, and transmit information.

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Measuring Distance Using Waves

Wave speed can be used to determine distance.

Suppose a sound pulse travels toward an object and reflects back.

If we know:

• Wave speed
• Total travel time

we can calculate the total distance travelled.

However, the wave travels:

To the object + back again

Therefore:

Distance to object = Total wave distance / 2

This principle is used in:

• Sonar
• Ultrasound
• Echo sounding
• Some distance sensors

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Worked Example — Sonar

A sonar pulse travels through water at:

1500 m/s

The echo returns after:

0.40 s

First calculate the total distance travelled:

Distance = speed × time

Distance = 1500 × 0.40

Distance = 600 m

But this includes the journey to the object and back.

Therefore:

Distance to object = 600 / 2

Distance to object = 300 m

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Common Mistakes

Mistake: Confusing wave speed with frequency

Wave speed is measured in:

m/s

Frequency is measured in:

Hz

They describe different properties.

Mistake: Confusing wavelength with amplitude

Wavelength is measured along the direction of wave travel.

Amplitude measures maximum displacement from equilibrium.

Mistake: Using centimetres directly in the wave equation

Convert centimetres to metres when SI units are required.

For example:

25 cm = 0.25 m

Mistake: Assuming higher frequency always means higher wave speed

In the same medium, changing frequency does not necessarily change wave speed.

Instead, wavelength often changes.

Mistake: Thinking frequency changes when a wave crosses a boundary

The frequency normally remains unchanged because it is determined by the source.

Wave speed and wavelength can change.

Mistake: Assuming all waves travel at the same speed

Wave speed depends on the type of wave and the medium through which it travels.

Mistake: Using the wrong form of the equation

Remember:

v = λf

λ = v/f

f = v/λ

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Key Terms

Wave speed – the distance travelled by a wave per unit time.

Wavelength – the distance between equivalent points on consecutive waves.

Frequency – the number of complete oscillations or waves passing a point each second.

Hertz (Hz) – the unit of frequency; one hertz means one cycle per second.

Medium – a material through which a mechanical wave travels.

Mechanical wave – a wave that requires a material medium.

Electromagnetic wave – a wave that can travel through a vacuum.

Amplitude – the maximum displacement from the equilibrium position.

Refraction – a change in direction caused by a change in wave speed when a wave enters another medium at an angle.

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Check Your Understanding

1. Define wave speed.

2. State the wave equation and identify each quantity.

3. What is the SI unit of wave speed?

4. A wave has a wavelength of 5 m and a frequency of 4 Hz. Calculate its speed.

5. A wave travels at 30 m/s and has a wavelength of 6 m. Calculate its frequency.

6. A wave travels at 200 m/s and has a frequency of 50 Hz. Calculate its wavelength.

7. Convert 75 cm into metres.

8. A wave has a wavelength of 75 cm and a frequency of 12 Hz. Calculate its wave speed.

9. Explain why the speed of a mechanical wave can change when it enters a different medium.

10. A wave enters a new medium and slows down. Its frequency remains constant. What happens to its wavelength?

11. Explain why increasing the frequency of a wave on the same rope does not necessarily increase its wave speed.

12. A sonar pulse travels through water at 1500 m/s and returns after 0.20 s. Calculate the distance to the reflecting object.

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Key Takeaways

• Wave speed describes how quickly a wave disturbance travels.
• Wave speed is measured in metres per second (m/s).
• The wave equation connects wave speed, wavelength, and frequency: v = λf.
• Wavelength is measured in metres.
• Frequency is measured in hertz.
• The wave equation can be rearranged to calculate wave speed, wavelength, or frequency.
• Unit conversions should be completed before substituting values into calculations.
• Mechanical wave speed depends on the properties of the medium.
• Sound travels at different speeds through gases, liquids, and solids.
• Electromagnetic waves travel at approximately 3.00 × 10⁸ m/s in a vacuum.
• When a wave enters a different medium, its speed and wavelength can change while its frequency remains constant.
• In a particular medium with constant wave speed, increasing frequency results in a shorter wavelength.
• Wave-speed calculations have practical applications in sound, communications, sonar, ultrasound, seismology, and optics.

Did You Know?

Scientists can use the speeds of seismic waves to investigate Earth's interior. Because different seismic waves travel at different speeds and behave differently as they pass through Earth's layers, their arrival times provide evidence about structures deep beneath the surface that humans cannot observe directly.