Reflection and Refraction of Waves

Waves can change their behaviour when they reach a boundary between two regions or materials.

Two important possibilities are:

• reflection — the wave returns into the original region
• refraction — the wave enters a new region, changes speed, and may change direction

These behaviours occur with many types of waves, including:

• water waves
• sound waves
• seismic waves
• light and other electromagnetic waves

Understanding reflection and refraction allows us to predict how waves behave when their surroundings change.

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Waves at Boundaries

A boundary is a place where the conditions through which a wave travels change.

Examples include:

• the end of a rope
• deep water meeting shallow water
• air meeting glass
• air meeting water
• one rock layer meeting another
• warm air meeting cooler air

When a wave reaches a boundary, its energy may be:

• reflected
• transmitted
• refracted
• absorbed

Often, more than one of these occurs at the same time.

For example, when light reaches a glass window, some light is reflected while some enters the glass.

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Reflection of Waves

Reflection occurs when a wave encounters a boundary and some or all of the wave travels back into the original region.

The incoming wave is called the:

incident wave

The returning wave is called the:

reflected wave

Reflection does not require the wave to change into a different type of wave.

Instead, its direction of propagation changes.

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Reflection of a Wave Pulse

Imagine sending a pulse along a rope toward a boundary.

The pulse travels toward the end of the rope.

When it reaches the boundary, a reflected pulse travels back along the rope.

The exact behaviour of the reflected pulse depends on the type of boundary.

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Reflection at a Fixed Boundary

A fixed boundary is a boundary that cannot move freely.

For example, a rope may be firmly attached to a wall.

When an upward pulse reaches the fixed end, it reflects as a downward pulse.

The reflected pulse is therefore:

inverted

So:

upward incident pulse → downward reflected pulse

The reflected pulse then travels back along the rope.

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Reflection at a Free Boundary

A free boundary can move.

When a pulse reflects from an ideal free end, it is not inverted.

Therefore:

upward incident pulse → upward reflected pulse

The pulse still reverses its direction of travel, but its displacement orientation remains the same.

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Fixed vs Free Boundary

This distinction is particularly useful when studying waves on strings and ropes.

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Reflection of Continuous Waves

Continuous waves can also reflect.

Examples include:

• water waves reflecting from a harbour wall
• sound reflecting from a building
• light reflecting from a mirror
• seismic waves reflecting from boundaries inside Earth

The same basic principle applies:

wave energy returns from a boundary.

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Reflection and the Normal

For waves approaching a flat boundary, angles are measured relative to an imaginary line called the:

normal

The normal is perpendicular to the boundary.

The angle of incidence is measured between the incident wave direction and the normal.

The angle of reflection is measured between the reflected wave direction and the normal.

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The Law of Reflection

For reflection from a suitable flat boundary:

angle of incidence = angle of reflection

or:

θᵢ = θᵣ

For example, if a wave reaches a boundary at:

θᵢ = 35°

then:

θᵣ = 35°

The angles must be measured from the normal.

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Reflection of Water Waves

Water waves can reflect from barriers such as:

• walls
• rocks
• harbour structures
• shorelines

A ripple tank can be used to observe this behaviour.

Straight wavefronts approaching a straight barrier reflect according to the law of reflection.

This makes water waves useful for modelling many general wave behaviours.

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Reflection of Sound

Sound waves can reflect from surfaces.

A reflected sound may be heard as an:

echo

Large, hard surfaces are often effective reflectors of sound.

Examples include:

• cliffs
• walls
• buildings
• tunnel surfaces

Reflection of sound is used in technologies such as:

• sonar
• ultrasound imaging
• echo sounding

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Echoes

Suppose a sound travels toward a wall, reflects, and returns to the observer.

The sound travels:

to the wall + back from the wall

Therefore:

distance to wall = speed × total time / 2

For example, if:

speed of sound = 340 m/s

echo time = 0.60 s

then:

distance = (340 × 0.60)/2

distance = 204/2

distance = 102 m

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Reflection of Light

Light can reflect from surfaces such as:

• mirrors
• glass
• water
• polished metal

Mirrors are designed to produce strong, organized reflection.

The law of reflection applies:

θᵢ = θᵣ

Reflection allows mirrors and many other optical instruments to control the direction of light.

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Refraction of Waves

Refraction occurs when a wave enters a region where its speed changes and, if it enters at an angle, its direction changes.

Refraction therefore involves:

change in wave speed

and often:

change in wave direction

Examples include:

• water waves moving from deep to shallow water
• light moving from air into glass
• sound moving through air at different temperatures
• seismic waves moving between layers inside Earth

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Why Waves Refract

Consider a straight wavefront approaching a boundary at an angle.

Suppose one side of the wavefront enters the new region first.

If the wave travels more slowly there, that side slows down while the rest of the wavefront continues at its original speed.

The wavefront rotates.

As a result, the direction of wave travel changes.

This is the basic mechanism of refraction.

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Refraction Requires a Change in Wave Speed

A wave changes direction during refraction because its speed changes across the boundary.

If the speed does not change, there is no refraction due to the boundary.

Therefore, the key idea is:

change in wave speed → possible change in direction

The amount of bending depends on the difference in wave speeds and the angle at which the wave reaches the boundary.

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Entering a Slower Region

When a wave enters a region where it travels more slowly at an angle, it bends:

toward the normal

Therefore:

faster → slower = toward the normal

For example, water waves moving from deeper water into sufficiently shallow water slow down.

If they enter the shallow region at an angle, their direction bends toward the normal.

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Entering a Faster Region

When a wave enters a region where it travels faster at an angle, it bends:

away from the normal

Therefore:

slower → faster = away from the normal

For example, shallow-water waves moving into deeper water generally speed up and bend away from the normal.

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A Useful Refraction Rule

Remember:

slower → toward the normal

faster → away from the normal

This applies when the wave crosses the boundary at an angle.

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What Happens at 0°?

Suppose a wave travels directly along the normal.

It reaches the boundary at:

0° to the normal

Its speed may change.

Its wavelength may also change.

However, it does not bend sideways because the entire wavefront reaches the boundary simultaneously.

Therefore:

refraction can occur without a change in direction.

The wave can still experience a change in speed and wavelength.

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Frequency During Refraction

When a wave crosses into a different region, its frequency remains constant.

The source determines the frequency.

Crossing the boundary does not suddenly change how frequently the source produces waves.

Therefore:

f₁ = f₂

The wave's speed and wavelength may change, but its frequency does not.

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Wave Speed and Wavelength

The wave equation is:

v = fλ

where:

v = wave speed

f = frequency

λ = wavelength

During refraction:

frequency stays constant

Therefore, if speed changes:

wavelength must change

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Entering a Slower Region

Suppose a wave enters a slower region.

Since:

v = fλ

and f remains constant:

v decreases → λ decreases

Therefore:

slower wave → shorter wavelength

The wavefronts become closer together.

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Entering a Faster Region

If a wave enters a faster region:

v increases

while:

f remains constant

Therefore:

λ increases

So:

faster wave → longer wavelength

The wavefronts become farther apart.

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Predicting Wavelength Changes

A useful summary is:

This relationship is one of the most important ideas in wave refraction.

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Worked Example: Slower Region

A water wave has:

f = 4 Hz

v = 2.0 m/s

Calculate its wavelength.

Use:

v = fλ

Rearrange:

λ = v/f

λ = 2.0/4

λ = 0.50 m

The wave enters shallower water where its speed decreases to:

1.2 m/s

The frequency remains:

4 Hz

New wavelength:

λ = 1.2/4

λ = 0.30 m

Therefore:

• speed decreased
• frequency stayed constant
• wavelength decreased

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Worked Example: Faster Region

A wave has:

f = 5 Hz

and travels at:

v = 3 m/s

Its wavelength is:

λ = v/f

λ = 3/5

λ = 0.60 m

The wave enters a region where its speed becomes:

5 m/s

Frequency remains:

5 Hz

New wavelength:

λ = 5/5

λ = 1.0 m

The wavelength increased because the wave speed increased.

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Reading Wavefront Diagrams

A wavefront joins points on a wave that are at the same stage of their cycle.

For water waves, each line in a wavefront diagram may represent a crest.

The spacing between adjacent wavefronts represents:

wavelength

Therefore:

closer wavefronts → shorter wavelength

farther-apart wavefronts → longer wavelength

This makes wavefront diagrams particularly useful for studying refraction.

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Direction of Travel and Wavefronts

The direction in which a wave travels is perpendicular to its wavefronts.

Therefore, if wavefronts rotate at a boundary, the direction of travel also changes.

When interpreting a diagram:

1. identify the wavefronts
2. identify their spacing
3. draw or imagine a line perpendicular to them
4. use that line to determine the direction of wave travel

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Water Waves in Deep and Shallow Water

Water-wave speed depends on water depth.

In sufficiently shallow water, waves travel more slowly than they do in deeper water.

Therefore, when water waves travel:

deep → shallow

they generally:

• slow down
• keep the same frequency
• decrease in wavelength
• bend toward the normal if entering at an angle

When they travel:

shallow → deep

they generally:

• speed up
• keep the same frequency
• increase in wavelength
• bend away from the normal if entering at an angle

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Why Ocean Waves Turn Toward the Shore

Ocean waves often approach a coastline at an angle.

The part of the wavefront that reaches shallow water first slows down first.

The deeper-water part continues moving faster.

This rotates the wavefront.

As a result, waves tend to become more nearly parallel to the shoreline as they approach.

This is an important real-world example of:

wave refraction

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Refraction of Light

Light also refracts when its speed changes.

For example, light travels more slowly in glass than in air.

When light enters glass from air at an angle:

• speed decreases
• wavelength decreases
• frequency remains constant
• ray bends toward the normal

When it returns from glass to air:

• speed increases
• wavelength increases
• frequency remains constant
• ray bends away from the normal

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Refractive Index

For light, the speed change can be described using refractive index:

n = c/v

where:

n = refractive index

c = speed of light in vacuum

v = speed of light in the material

A higher refractive index corresponds to a lower speed of light in that material.

Therefore, refractive index helps predict how light will refract at a boundary.

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Refraction of Sound

Sound can also refract.

The speed of sound depends on properties of the medium, including temperature.

If different layers of air have different temperatures, sound may travel at different speeds through them.

This can cause sound waves to bend.

This helps explain why sound can sometimes be heard unusually well over long distances under certain atmospheric conditions.

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Refraction of Seismic Waves

Seismic waves travel through Earth's interior.

Their speeds change as they encounter regions with different physical properties.

As a result, seismic waves:

• refract
• reflect
• change direction

Scientists analyze these changes to investigate Earth's internal structure.

This provides evidence about:

• the mantle
• outer core
• inner core
• boundaries between Earth's layers

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Reflection and Refraction Can Occur Together

A wave reaching a boundary does not always either reflect or refract.

Both can occur simultaneously.

For example, when light strikes a glass window:

• some light reflects from the surface
• some enters the glass and refracts

Similarly, seismic waves reaching an internal Earth boundary may be partly reflected and partly transmitted and refracted.

Wave energy can therefore be divided at a boundary.

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Comparing Reflection and Refraction

The main distinction is:

reflection → wave returns

refraction → wave enters a new region and changes speed

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Reflection vs Refraction in a Window

Imagine looking through a glass window at night.

You may see:

• objects outside through the glass
• a faint reflection of yourself

The light reaching the window has been divided.

Some light:

reflects

Some light:

enters the glass and refracts

This everyday example shows that reflection and refraction can occur at the same boundary.

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Reflection vs Refraction of Water Waves

Suppose water waves reach a region of shallow water next to a barrier.

At the barrier:

reflection may send waves back.

At the deep-to-shallow boundary:

refraction may change their speed, wavelength, and direction.

The two phenomena involve different physical processes even though both can change the direction of a wave.

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Reflection vs Refraction of Sound

Sound reflection can produce:

echoes

Sound refraction can produce:

bending through regions where sound speed differs

For example:

• a cliff can reflect sound
• atmospheric temperature differences can refract sound

This demonstrates that reflection and refraction are general wave behaviours rather than properties unique to light.

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Reflection vs Refraction of Seismic Waves

When seismic waves encounter a boundary inside Earth:

some energy may:

reflect back

while some may:

enter the next layer and refract

By detecting these waves at different locations, scientists can infer properties of Earth's interior.

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Sketching Reflection

To sketch reflection:

1. Draw the boundary.
2. Draw the normal perpendicular to the boundary.
3. Draw the incident direction.
4. Measure or estimate the angle of incidence from the normal.
5. Draw the reflected direction on the other side of the normal.
6. Make the angle of reflection equal to the angle of incidence.
7. Add arrows showing direction of travel.

Remember:

θᵢ = θᵣ

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Sketching Refraction

To sketch refraction:

1. Draw the boundary.
2. Draw the normal.
3. Draw the incident wave direction.
4. Determine whether the new region is faster or slower.
5. If slower, bend the wave toward the normal.
6. If faster, bend the wave away from the normal.
7. Keep frequency unchanged.
8. Change the wavelength appropriately.
9. Add arrows showing direction of travel.

For wavefront diagrams, also change the spacing between wavefronts.

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A Complete Refraction Example

Water waves travel from deep water into shallow water.

Before the boundary:

v = 2.4 m/s

f = 3 Hz

Calculate wavelength:

λ = v/f

λ = 2.4/3

λ = 0.80 m

In shallow water:

v = 1.5 m/s

Frequency remains:

3 Hz

New wavelength:

λ = 1.5/3

λ = 0.50 m

Therefore:

• speed decreases from 2.4 m/s to 1.5 m/s
• frequency remains 3 Hz
• wavelength decreases from 0.80 m to 0.50 m
• if the wave enters at an angle, it bends toward the normal

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A Complete Boundary Example

A wave approaches a boundary.

Part of the wave returns to the original region.

Another part enters the new region and slows down.

The returning part demonstrates:

reflection

The transmitted part demonstrates:

refraction

Because the transmitted wave slows down:

its wavelength decreases

because frequency remains constant.

If it crosses at an angle:

it bends toward the normal

Several wave behaviours can therefore be observed at the same boundary.

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Applications of Wave Reflection

Reflection is useful in many technologies.

Examples include:

Sonar

Reflected sound determines distance to underwater objects.

Medical ultrasound

Reflected ultrasound provides information about structures inside the body.

Mirrors

Reflected light forms images.

Radar

Reflected electromagnetic waves can locate objects and determine motion.

Seismology

Reflected seismic waves reveal underground structures.

Echo sounding

Reflected sound measures water depth.

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Applications of Wave Refraction

Refraction also has many applications.

Examples include:

Lenses

Refraction focuses or spreads light.

Eyeglasses

Lenses redirect light to improve image formation in the eye.

Cameras

Lens systems refract light onto sensors.

Microscopes

Lenses use refraction to form magnified images.

Telescopes

Refracting telescopes use lenses to collect and focus light.

Seismology

Refraction helps scientists investigate Earth's internal structure.

Coastal engineering

Water-wave refraction helps predict how wave energy reaches shorelines.

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Wave Behaviour Around Coastlines

Refraction can redistribute wave energy along a coastline.

As waves approach irregular shores:

• wavefronts bend
• energy may become concentrated around headlands
• energy may become more spread out in bays

This can influence:

• erosion
• sediment transport
• beach formation
• harbour design

Wave refraction is therefore important in coastal science and engineering.

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A Reliable Strategy for Boundary Problems

When a wave reaches a boundary:

1. Identify the incident wave.
2. Identify the boundary.
3. Determine whether the wave reflects, enters the new region, or both.
4. If it reflects, apply the reflection rule.
5. If it enters another region, compare wave speeds.
6. Determine whether the wave speeds up or slows down.
7. Remember that frequency stays constant.
8. Use v = fλ to predict the wavelength change.
9. If the wave slows, wavelength decreases.
10. If the wave speeds up, wavelength increases.
11. If it enters at an angle, predict the direction of bending.
12. Slower means toward the normal.
13. Faster means away from the normal.
14. Check that the final diagram is consistent with all of these predictions.

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Common Mistakes

Thinking reflection and refraction are the same

Reflection sends a wave back. Refraction involves transmission into a region where wave speed changes.

Thinking a reflected wave always becomes inverted

Inversion depends on the type of boundary. A rope pulse at a fixed end inverts; at a free end it does not.

Measuring angles from the boundary

Angles are measured from the normal.

Thinking frequency changes during refraction

Frequency remains constant when the wave crosses the boundary.

Thinking wavelength remains constant

If speed changes while frequency remains constant, wavelength must change.

Thinking slower waves have longer wavelengths during refraction

With constant frequency:

slower → shorter wavelength

Thinking faster waves bend toward the normal

For ordinary refraction:

faster → away from normal

slower → toward normal

Thinking a wave must change direction to undergo refraction

A wave entering along the normal can change speed and wavelength without changing direction.

Thinking reflection and refraction cannot happen together

A boundary can reflect some wave energy while transmitting and refracting the rest.

Thinking wavefront spacing represents frequency

Wavefront spacing represents wavelength.

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Key Terms

Boundary — location where the conditions through which a wave travels change.

Incident wave — wave approaching a boundary.

Reflected wave — wave travelling back from a boundary.

Transmitted wave — portion of a wave that passes into another region.

Reflection — return of a wave from a boundary.

Refraction — change in wave behaviour caused by a change in wave speed when entering another region, often producing a change in direction.

Normal — imaginary line perpendicular to a boundary.

Angle of incidence — angle between the incident direction and the normal.

Angle of reflection — angle between the reflected direction and the normal.

Fixed boundary — boundary that cannot move freely and can invert a reflected rope pulse.

Free boundary — boundary able to move, producing a non-inverted reflected pulse in the ideal rope model.

Wave speed (v) — speed at which a wave disturbance propagates.

Frequency (f) — number of complete wave cycles passing a point each second.

Wavelength (λ) — distance between corresponding points on consecutive wave cycles.

Wavefront — line joining points on a wave that are at the same stage of oscillation.

Refractive index — quantity describing the reduction of light speed in a material relative to vacuum.

Echo — reflected sound that reaches an observer after the original sound.

Transmission — passage of wave energy through a boundary into another region.

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Key Takeaways

• Waves can reflect and refract when they encounter boundaries.
• Reflection occurs when wave energy returns into the original region.
• Refraction occurs when a wave enters a region where its speed changes.
• A reflected rope pulse inverts at an ideal fixed end.
• A reflected rope pulse does not invert at an ideal free end.
• For regular reflection, the angle of incidence equals the angle of reflection.
• Angles are measured from the normal.
• Refraction occurs because wave speed changes.
• A wave entering a slower region at an angle bends toward the normal.
• A wave entering a faster region at an angle bends away from the normal.
• A wave entering along the normal can change speed without changing direction.
• Frequency remains constant when a wave crosses a boundary.
• The wave equation is v = fλ.
• If speed decreases while frequency remains constant, wavelength decreases.
• If speed increases while frequency remains constant, wavelength increases.
• Wavefront spacing represents wavelength.
• Water waves slow and shorten in wavelength when moving into sufficiently shallow water.
• Light refracts when its speed changes between optical media.
• Sound can refract when sound speed varies through the atmosphere or another medium.
• Seismic waves reflect and refract at boundaries inside Earth.
• Reflection and refraction can occur simultaneously at the same boundary.
• Echoes, mirrors, sonar, and ultrasound demonstrate reflection.
• Lenses, water-wave bending, atmospheric sound bending, and seismic-wave paths demonstrate refraction.
• Comparing speed, wavelength, and direction provides a reliable way to analyze wave behaviour at boundaries.

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Check Your Understanding

1. Define reflection of a wave.
2. Explain what happens when a wave pulse reflects from an ideal fixed boundary and compare this with reflection from a free boundary.
3. State the law of reflection and explain how the angles are measured.
4. Define refraction and explain why a wave can change direction when it enters a new region.
5. Water waves travel from deep water into shallower water at an angle. Predict what happens to their speed, frequency, wavelength, and direction.
6. Why does the frequency of a wave remain constant when it crosses a boundary?
7. A wave has a frequency of 5 Hz and travels at 4 m/s. Calculate its wavelength.
8. The wave in Question 7 enters another region where its speed decreases to 2.5 m/s. Calculate its new wavelength.
9. Explain why wavefronts become closer together when a wave enters a slower region.
10. A wave enters a faster region at an angle. Predict how its direction and wavelength change.
11. Explain why a wave travelling directly along the normal can undergo refraction without changing direction.
12. Compare reflection and refraction, identifying at least two similarities and two differences.
13. Explain how both reflection and refraction can occur when light reaches a glass window.
14. Describe one example of reflection involving sound, light, water waves, or seismic waves, and explain the physical process involved.
15. Ocean waves approach a beach at an angle. One end of each wavefront reaches shallow water before the other. Explain how this causes the wavefront to rotate, predict the changes in speed and wavelength, and explain why the waves tend to become more nearly parallel to the shoreline.