Superposition

When two or more waves meet in the same region, they can temporarily overlap. While they overlap, their individual displacements combine to produce a resultant displacement.

This behaviour is described by the principle of superposition.

Superposition explains many important wave phenomena, including:

• constructive interference
• destructive interference
• interference patterns
• standing waves
• sound interference
• noise-cancelling technology

An important feature of superposition is that the waves do not normally destroy or permanently change one another. After overlapping, each wave continues travelling with its original shape and direction, provided the medium behaves linearly.

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The Principle of Superposition

The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the sum of the individual displacements of the waves at that point.

In simple form:

resultant displacement = displacement₁ + displacement₂ + ...

For two waves:

y_resultant = y₁ + y₂

Displacement must include direction.

A displacement above the equilibrium position can be treated as positive, while a displacement below equilibrium can be treated as negative.

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Displacement

Displacement describes how far and in which direction a point in a medium has moved from its equilibrium position.

For a transverse wave:

• above equilibrium → positive displacement
• below equilibrium → negative displacement
• on equilibrium → zero displacement

For example:

Wave A displacement = +3 cm

Wave B displacement = +2 cm

Then:

resultant = +3 + (+2)

resultant = +5 cm

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Adding Positive and Negative Displacements

Suppose:

Wave A = +4 cm

Wave B = −3 cm

Then:

resultant = +4 + (−3)

resultant = +1 cm

The point is therefore 1 cm above its equilibrium position.

Superposition is essentially an algebraic addition of displacements.

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Superposition Happens Only During Overlap

Imagine two pulses travelling toward each other along a rope.

Before they meet, each pulse travels independently.

During overlap, their displacements combine.

After they pass through each other, the original pulses continue travelling.

The sequence is:

approach → overlap → separate

The combined shape exists only while the waves overlap.

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Constructive Interference

Constructive interference occurs when overlapping waves have displacements in the same direction.

Their displacements add together.

For example:

Wave A = +3 cm

Wave B = +2 cm

Resultant:

+3 + +2 = +5 cm

The resulting displacement is larger than either individual displacement.

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Complete Constructive Interference

Suppose two identical upward pulses each have an amplitude of:

2 cm

When they overlap perfectly:

resultant displacement = 2 + 2

= 4 cm

For that instant, the resultant pulse has twice the amplitude of either original pulse.

After the overlap, each 2 cm pulse continues travelling.

The waves have not permanently combined into a 4 cm wave.

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Constructive Interference Below Equilibrium

Constructive interference does not necessarily mean the displacement must be upward.

Suppose:

Wave A = −2 cm

Wave B = −3 cm

Then:

resultant = −2 + (−3)

= −5 cm

Both displacements are downward, so they reinforce one another.

The magnitude of the resultant displacement is:

5 cm

This is also constructive interference.

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Destructive Interference

Destructive interference occurs when overlapping waves have displacements in opposite directions.

One displacement is positive while the other is negative.

For example:

Wave A = +5 cm

Wave B = −2 cm

Resultant:

+5 + (−2)

= +3 cm

The waves partially cancel.

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Complete Destructive Interference

Complete destructive interference occurs when two equal but opposite displacements overlap.

Suppose:

Wave A = +4 cm

Wave B = −4 cm

Then:

resultant = +4 + (−4)

= 0 cm

At that instant, the medium may appear completely undisturbed at the overlapping location.

However, the waves have not disappeared permanently.

They continue through one another and reappear as they separate.

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Partial Destructive Interference

Destructive interference does not always produce zero displacement.

Suppose:

Wave A = +6 cm

Wave B = −4 cm

Then:

resultant = +6 − 4

= +2 cm

The waves partially cancel.

Because the upward displacement is larger, the resultant displacement remains upward.

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Constructive vs Destructive Interference

The key is to add the displacements with their signs.

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Interference

Interference is the result of waves overlapping and combining through superposition.

Interference may be:

• constructive
• destructive
• partially constructive
• partially destructive

The exact result depends on the displacement contributed by each wave at each position.

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Superposition Point by Point

When entire wave shapes overlap, you must add their displacements point by point.

Suppose two waves overlap.

At one position:

Wave A = +2 cm
Wave B = +3 cm

Resultant:

+5 cm

At another position:

Wave A = +1 cm
Wave B = −1 cm

Resultant:

0 cm

At another position:

Wave A = −2 cm
Wave B = −1 cm

Resultant:

−3 cm

The final waveform is constructed by repeating this process across the entire overlapping region.

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A Superposition Table

A table can help when constructing a resultant wave.

For every position:

resultant = Wave A + Wave B

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Sketching a Superposition Diagram

To sketch the resultant of two overlapping waves:

1. Draw the equilibrium line.
2. Identify corresponding positions on both waves.
3. Determine the displacement of Wave A at each position.
4. Determine the displacement of Wave B at the same position.
5. Add the two displacements algebraically.
6. Plot the resultant displacement.
7. Repeat across the entire overlap.
8. Connect the points smoothly if the waves are continuous.

Do not simply add the amplitudes unless the corresponding peaks or troughs are actually overlapping.

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Example: Two Upward Pulses

Suppose two upward pulses approach each other.

Pulse A has amplitude:

+3 cm

Pulse B has amplitude:

+2 cm

When their peaks overlap perfectly:

resultant = +3 + +2

= +5 cm

At that instant, the resultant pulse reaches 5 cm above equilibrium.

This is:

constructive interference

Afterward, the pulses separate and continue travelling.

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Example: Upward and Downward Pulses

Suppose:

Pulse A = +4 cm

Pulse B = −3 cm

When their largest displacements overlap:

resultant = +4 + (−3)

= +1 cm

The resultant is 1 cm upward.

This is:

partial destructive interference

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Example: Complete Cancellation

Two pulses have equal amplitudes but opposite directions:

Pulse A = +5 cm

Pulse B = −5 cm

During complete overlap:

resultant = +5 + (−5)

= 0 cm

The rope may temporarily appear flat.

Moments later, the two pulses emerge and continue travelling.

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Why Waves Pass Through One Another

In many ordinary wave systems, each wave causes its own displacement independently.

During overlap, these displacements add.

Once the waves no longer overlap, each continues to propagate according to the properties of the medium.

Therefore, superposition is temporary.

The waves do not normally:

• bounce off one another
• permanently merge
• permanently cancel
• exchange identities

Instead, they pass through one another.

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Waves Are Not Solid Objects

It can be tempting to imagine two pulses as objects colliding.

But waves are disturbances, not solid objects.

When two balls collide, they can exert forces on each other and change their motion.

When two ordinary small-amplitude waves overlap in a linear medium, the medium simply responds to the combined displacement.

Afterward, each disturbance continues.

This is why two sound waves, for example, can travel through the same region simultaneously.

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Superposition and Energy

Complete destructive interference does not mean that the wave energy has simply vanished.

During interference, energy can be redistributed within the wave system.

In sustained interference patterns, some regions may have greater wave intensity while others have less.

Therefore:

zero displacement at one point does not necessarily mean zero energy everywhere.

The original waves can continue after the interference occurs.

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Superposition of Continuous Waves

Superposition applies not only to individual pulses but also to continuous waves.

When continuous waves overlap, their displacements continually add.

Depending on their relative positions or phase, they may produce:

• constructive interference
• destructive interference
• alternating combinations

This can create stable interference patterns under suitable conditions.

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Waves in Phase

Two waves are in phase when corresponding points in their cycles line up.

For example:

• crest aligns with crest
• trough aligns with trough

If two waves of the same frequency are in phase, they produce strong constructive interference.

For identical waves:

crest + crest → larger crest

trough + trough → deeper trough

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Waves Out of Phase

Two waves can be out of phase.

If a crest from one wave aligns with a trough of another, destructive interference occurs.

For identical waves exactly half a cycle apart:

crest + equal trough → zero displacement

This corresponds to complete destructive interference.

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Phase Difference

Phase difference describes how far apart two waves are in their cycles.

For two identical sinusoidal waves:

0° phase difference → maximum constructive interference

180° phase difference → maximum destructive interference

Intermediate phase differences produce partial interference.

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Path Difference

Interference can also depend on the difference in the distances travelled by two waves.

This is called the:

path difference

For waves of wavelength λ, constructive interference commonly occurs when the path difference is:

0, λ, 2λ, 3λ, ...

Destructive interference commonly occurs when the path difference is:

λ/2, 3λ/2, 5λ/2, ...

These relationships are particularly useful when studying interference patterns.

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Sound-Wave Interference

Sound waves obey the principle of superposition.

Suppose two sound waves reach the same location.

The pressure variations produced by the waves combine.

If they reinforce:

constructive interference

can produce greater sound intensity.

If they oppose:

destructive interference

can reduce the sound intensity.

This principle has important technological applications.

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Noise-Cancelling Headphones

Active noise-cancelling headphones use destructive interference.

A simplified process is:

1. A microphone detects surrounding sound.
2. Electronics analyze the sound.
3. The headphones produce another sound wave.
4. This wave is designed to oppose the unwanted sound.
5. The two pressure disturbances superpose.
6. Destructive interference reduces the unwanted sound at the listener's ear.

This works especially well for relatively steady, low-frequency background noise such as:

• aircraft engine noise
• vehicle noise
• ventilation systems

Real noise cancellation is more complex than simply producing one perfectly inverted wave everywhere.

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Interference of Water Waves

Water waves can also superpose.

If two sets of water waves overlap:

• crest + crest → constructive interference
• trough + trough → constructive interference
• crest + trough → destructive interference

A ripple tank can be used to observe these patterns.

Regions of strong and weak wave motion can appear across the water surface.

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Interference of Light

Light waves also obey the principle of superposition.

When coherent light waves overlap, they can produce:

• bright regions from constructive interference
• dark regions from destructive interference

Interference experiments provide strong evidence for the wave behaviour of light.

Applications include:

• interferometers
• precision measurement
• optical coatings
• spectroscopy
• holography

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Thin-Film Colours

The colours seen in:

• soap bubbles
• thin oil films
• some coated lenses

can involve interference.

Light reflects from different surfaces of the thin layer.

The reflected waves travel slightly different distances.

When they recombine, some wavelengths interfere constructively while others interfere destructively.

This can produce changing colours.

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Standing Waves

Superposition is also responsible for standing waves.

A standing wave can form when two waves of the same frequency and similar amplitude travel in opposite directions.

Their continuous superposition produces locations called:

• nodes
• antinodes

At a node, displacement remains zero.

At an antinode, the oscillation has maximum amplitude.

Standing waves are important in:

• strings
• musical instruments
• air columns
• resonant systems

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Superposition of More Than Two Waves

The principle is not limited to two waves.

Suppose three waves overlap at one point:

Wave A = +2 cm

Wave B = −4 cm

Wave C = +3 cm

Then:

resultant = +2 − 4 + 3

resultant = +1 cm

The same principle works for any number of overlapping waves:

add all individual displacements.

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Worked Example: Three Waves

At one instant, three waves produce these displacements at the same point:

Wave A = +5 mm

Wave B = −2 mm

Wave C = −1 mm

Resultant:

+5 + (−2) + (−1)

= 5 − 2 − 1

= +2 mm

The resultant displacement is:

2 mm above equilibrium

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Worked Example: Predicting Interference

Two identical sinusoidal waves have amplitudes of:

4 cm

Their crests arrive at the same point at the same time.

At that instant:

resultant = +4 + +4

= +8 cm

This is maximum constructive interference.

Later, suppose a crest from one wave overlaps an equal trough from the other:

resultant = +4 + (−4)

= 0 cm

This is complete destructive interference.

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Reading Superposition Diagrams

When interpreting a superposition diagram, do not focus only on the peaks.

Look at each horizontal position.

Ask:

What is Wave A's displacement here?

What is Wave B's displacement here?

Then calculate:

A + B

Repeat across the diagram.

This produces the complete resultant waveform.

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Predicting the Sign of the Resultant

A useful shortcut is to compare the displacements.

If both are positive:

resultant is positive and larger in magnitude

If both are negative:

resultant is negative and larger in magnitude

If one is positive and one negative:

subtract their magnitudes

The larger displacement determines the direction of the resultant.

If they are equal and opposite:

resultant = 0

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After the Waves Overlap

Consider two pulses:

Pulse A travels right.

Pulse B travels left.

During overlap, a resultant waveform appears.

After overlap:

• Pulse A continues right
• Pulse B continues left
• each returns to its original form in an ideal linear medium

The waves have passed through one another.

This is one of the most important ideas in superposition.

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Temporary Cancellation Does Not Destroy the Waves

Suppose two identical opposite pulses overlap perfectly.

For an instant:

resultant displacement = 0

It may look as though both waves have disappeared.

They have not.

A moment later, they separate and become visible again.

The cancellation applied to the resultant displacement during overlap, not to the permanent existence of the two waves.

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Limits of the Simple Superposition Model

The simple principle of superposition works extremely well when the system behaves linearly.

This is a good approximation for many:

• small waves on strings
• sound waves
• water-wave experiments
• electromagnetic waves

For very large disturbances or strongly nonlinear materials, waves can interact in more complicated ways.

At the level of most introductory wave problems, however, we assume:

waves pass through one another unchanged after superposition.

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A Complete Pulse Example

Two pulses move toward each other.

Pulse A has a maximum displacement of:

+6 cm

Pulse B has a maximum displacement of:

−4 cm

Before overlap, each pulse has its own shape.

At complete overlap:

resultant maximum displacement = +6 + (−4)

= +2 cm

This is partial destructive interference.

After overlap:

• the +6 cm pulse continues in its original direction
• the −4 cm pulse continues in its original direction
• each has its original shape and amplitude

The +2 cm pulse exists only as the temporary resultant during overlap.

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A Reliable Strategy for Superposition Problems

When two or more waves overlap:

1. Draw or identify the equilibrium line.
2. Choose a position on the waves.
3. Determine each wave's displacement at that position.
4. Assign positive and negative signs correctly.
5. Add the displacements algebraically.
6. Plot the resultant displacement.
7. Repeat at several positions.
8. Connect the resultant points smoothly when appropriate.
9. Identify whether the interference is constructive or destructive.
10. Remember that the resultant exists only while the waves overlap.
11. After overlap, draw the original waves continuing in their original directions.
12. Check that no wave has mysteriously disappeared or changed permanently.

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Common Mistakes

Adding amplitudes instead of displacements

Superposition requires the displacement at each position to be added, including its direction.

Ignoring negative displacement

A trough or downward pulse should normally be represented with a negative displacement.

Thinking constructive interference always points upward

Two downward displacements can also reinforce each other.

Thinking destructive interference always gives zero

Only equal and opposite displacements cancel completely.

Thinking zero displacement means the waves have disappeared

The waves can still be present and continue after overlap.

Thinking waves bounce off each other

In the simple linear model, waves pass through one another.

Thinking the larger resultant pulse is permanent

It exists only while the appropriate waves overlap.

Adding only the peaks

The entire resultant waveform must be found by adding displacements point by point.

Confusing interference with reflection

Interference occurs when waves overlap.

Reflection occurs when a wave encounters a boundary and returns.

Thinking destructive interference destroys energy

Interference redistributes wave effects; cancellation of displacement at a point does not mean all wave energy has vanished.

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Key Terms

Superposition — principle stating that overlapping wave displacements add to produce a resultant displacement.

Displacement — distance and direction of a point from its equilibrium position.

Equilibrium position — undisturbed or central position of a vibrating medium.

Resultant displacement — combined displacement produced by all waves at a particular point.

Interference — effect produced when waves overlap and superpose.

Constructive interference — interference in which wave displacements reinforce one another.

Destructive interference — interference in which wave displacements oppose one another.

Complete constructive interference — maximum reinforcement when appropriate matching wave displacements overlap.

Complete destructive interference — cancellation to zero resultant displacement when equal and opposite displacements overlap.

Partial destructive interference — incomplete cancellation caused by opposite displacements of unequal magnitude.

Phase — position of a wave within its cycle.

In phase — waves whose corresponding points in their cycles align.

Out of phase — waves whose corresponding points do not align.

Phase difference — difference between the positions of two waves within their cycles.

Path difference — difference between the distances travelled by two waves before reaching a location.

Pulse — single disturbance travelling through a medium.

Resultant wave — temporary combined waveform produced by superposition.

Node — location in a standing wave where displacement remains zero.

Antinode — location in a standing wave where oscillation amplitude is greatest.

Standing wave — wave pattern produced by superposition of suitable waves travelling in opposite directions.

Linear medium — medium in which individual wave effects can be added using the principle of superposition.

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Key Takeaways

• Superposition occurs when two or more waves overlap.
• The resultant displacement is the algebraic sum of the individual displacements.
• Positive and negative directions must be included when adding displacements.
• Superposition must be applied point by point across overlapping waves.
• Constructive interference occurs when displacements reinforce each other.
• Constructive interference can occur above or below equilibrium.
• Destructive interference occurs when displacements oppose each other.
• Complete destructive interference requires equal and opposite displacements.
• Partial destructive interference occurs when opposite displacements have different magnitudes.
• A zero resultant displacement does not mean that the original waves have permanently disappeared.
• In an ideal linear medium, waves continue unchanged after passing through one another.
• The resultant waveform exists only while the original waves overlap.
• Waves should not be treated like solid objects colliding with one another.
• Continuous waves can also undergo constructive and destructive interference.
• Waves that are in phase tend to reinforce each other.
• Waves 180° out of phase can produce complete destructive interference when their amplitudes are equal.
• Path difference can determine whether interference is constructive or destructive.
• Sound, water waves, and light can all demonstrate interference.
• Active noise cancellation uses destructive interference.
• Thin-film colours can result from interference of light.
• Standing waves are produced through superposition of waves travelling in opposite directions.
• Superposition can involve more than two waves.
• The principle of superposition is fundamental to understanding how multiple waves can occupy the same region at the same time.

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Check Your Understanding

1. State the principle of superposition in your own words.
2. Two waves produce displacements of +3 cm and +5 cm at the same point. Calculate the resultant displacement and identify the type of interference.
3. Two waves produce displacements of −2 cm and −4 cm. Calculate the resultant displacement. Is this constructive or destructive interference?
4. At one point, Wave A has a displacement of +7 cm and Wave B has a displacement of −3 cm. Calculate the resultant displacement.
5. Two waves each have amplitudes of 6 cm. One produces a displacement of +6 cm while the other produces −6 cm at the same point. What is the resultant displacement, and what type of interference occurs?
6. Explain the difference between constructive and destructive interference.
7. Explain why destructive interference does not always result in zero displacement.
8. Describe how you would construct a resultant wave from two overlapping wave diagrams.
9. Two pulses travelling in opposite directions overlap completely and temporarily produce a flat rope. Explain what happens to the pulses immediately afterward.
10. Explain why waves can pass through one another rather than behaving like two solid objects in a collision.
11. Three waves produce displacements of +4 mm, −7 mm, and +5 mm at the same point. Calculate the resultant displacement.
12. Explain what is meant by waves being in phase and describe the interference expected when two identical waves are perfectly in phase.
13. Explain how active noise-cancelling headphones use the principle of superposition.
14. A student says, "When two waves undergo complete destructive interference, both waves are destroyed and their energy disappears." Explain why this statement is incorrect.
15. Two identical pulses travel toward each other on a rope. One pulse points upward and the other downward. Sketch or describe the wave pattern before overlap, during complete overlap, and after overlap, and explain how the sequence demonstrates the principle of superposition.