When waves meet, they do not bounce off each other as solid objects do. Their displacements add. That simple idea of superposition explains interference patterns, path difference, and why coherence matters.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
Two pulses on a rope move toward each other. When they overlap, will the rope "choose" one pulse, cancel permanently, or temporarily combine both displacements? Predict what the rope looks like at the instant of overlap.
Type your prediction below. You will revisit it at the end.
Write your prediction in your book. You will revisit it at the end.
Wrong: Momentum is not conserved in collisions with friction.
Right: Momentum is always conserved in isolated systems; friction is an external force, so the system must include the surface.
📚 Core Content
When waves overlap, the medium experiences both disturbances at once, so the displacements add.
If one pulse gives a point on a rope a displacement of +3 cm and another gives it +2 cm at the same instant, the resultant displacement is +5 cm. If one gives +3 cm and the other gives −2 cm, the resultant is +1 cm. We add the displacements algebraically, not by choosing one wave over the other. This principle applies to all wave types — mechanical waves on a string, sound waves in air, and electromagnetic waves in space.
After the waves pass through each other, they continue onward with their original shapes, speeds, and amplitudes completely unchanged. The overlap effect belongs only to the moment of superposition. This is one of the most elegant features of wave behaviour: waves can occupy the same space at the same time without permanently altering each other. Two pulses on a rope will meet, combine according to the superposition principle, and then separate again as if nothing had happened.
If waves meet in phase, they reinforce. If they meet out of phase, they cancel.
Constructive interference happens when the waves line up crest with crest or compression with compression. The resultant amplitude increases, sometimes doubling if the two waves have equal amplitudes. Destructive interference happens when a crest overlaps a trough or a compression overlaps a rarefaction. The resultant amplitude decreases or may become zero at that exact point. It is crucial to remember that destructive interference does not mean the waves have been destroyed — they simply produce a smaller or zero net displacement at that location and time.
Why does this happen? Because each wave continues to oscillate the medium independently. At the point of overlap, the medium is trying to move in two directions at once. If those directions are the same, the motion adds. If they are opposite, the motion subtracts. The energy of the wave is not lost during destructive interference; it is redistributed to regions where constructive interference occurs. This is why an interference pattern shows alternating regions of high and low amplitude.
Path difference compares how much farther one wave travelled than another to reach the same point.
If the difference in path length is one full wavelength, the waves arrive in phase and interfere constructively. If the path difference is half a wavelength, they arrive in antiphase and interfere destructively. This lets us predict bright and dark, loud and quiet, or large and small resultant regions in two-source patterns. The integer $n$ in the equations is called the order of the interference: $n = 0$ gives the central constructive region, $n = 1$ gives the first constructive region on either side, and so on.
Path difference is not about the total distance travelled by each wave — it is about the difference between those distances. Two waves could each travel 10 m and 10.5 m respectively; the path difference of 0.5 m is what matters. If that 0.5 m equals $\lambda/2$ for those waves, destructive interference occurs. This concept is the foundation of all wave interference experiments, from Thomas Young's double-slit experiment with light to acoustic engineering in concert halls.
| Path Difference | Result | Phase Relationship |
|---|---|---|
| $0, \lambda, 2\lambda, 3\lambda, \dots$ | Constructive interference | In phase ($0°, 360°, 720°, \dots$) |
| $\lambda/2, 3\lambda/2, 5\lambda/2, \dots$ | Destructive interference | Antiphase ($180°, 540°, \dots$) |
A stable interference pattern needs sources that stay in step.
Coherent sources have the same frequency and maintain a constant phase relationship. Without coherence, the phase relationship keeps shifting, so the reinforcement and cancellation points wander and the pattern blurs away. You cannot produce a clear interference pattern with two independent light bulbs or two random sound sources because their phases drift relative to each other thousands of times per second.
Coherence comes in two forms. Temporal coherence means the wave maintains a predictable phase over time. Spatial coherence means different points across the wavefront are in phase with each other. For the NSW HSC syllabus, the key requirement is simply that the two sources have the same frequency and a constant phase difference. Laser light is highly coherent, which is why it produces sharp interference patterns. Ordinary white light is incoherent, which is why it does not.
Every wavelength of extra travel corresponds to a full $360°$ phase shift.
A path difference of one full wavelength means the second wave is delayed by exactly one cycle, so its crest still aligns with the first wave's crest — constructive interference. A path difference of half a wavelength means the second wave is delayed by half a cycle, so its crest aligns with the first wave's trough — destructive interference. This direct link between geometry (path difference) and timing (phase) is what makes interference predictable.
We can express this mathematically: phase difference $\Delta\phi$ (in degrees) is related to path difference $\Delta x$ by $\Delta\phi = (\Delta x / \lambda) \times 360°$. For example, if $\Delta x = \lambda/4$, then $\Delta\phi = 90°$. The waves are neither fully in phase nor fully out of phase, so the resultant amplitude is intermediate — not doubled, not zero. This is why the intensity of an interference pattern varies smoothly between the bright constructive maxima and the dark destructive minima.
| Path Difference $\Delta x$ | Phase Difference $\Delta\phi$ | Resultant Amplitude (equal waves) |
|---|---|---|
| $0$ | $0°$ | $2A$ (maximum) |
| $\lambda/4$ | $90°$ | $\sqrt{2}A \approx 1.41A$ |
| $\lambda/2$ | $180°$ | $0$ (minimum) |
| $3\lambda/4$ | $270°$ | $\sqrt{2}A \approx 1.41A$ |
| $\lambda$ | $360°$ | $2A$ (maximum) |
Superposition and interference are not classroom abstractions — they shape technology and nature.
Surf lifesavers use interference patterns to locate swimmers in rough water; wave reflections from piers and rocks create complex superposition fields that experienced lifeguards learn to read. Noise-cancelling headphones work by producing a sound wave that is exactly out of phase with ambient noise. The destructive interference inside the ear cup dramatically reduces the perceived sound intensity. Radio antennas are arranged in arrays so that the signals interfere constructively in desired directions and destructively in others, focusing transmission without moving the antenna.
Even in the ocean, swells from distant storms can travel thousands of kilometres and interfere with local wind waves. When the wavelengths and directions match appropriately, the result can be unexpectedly large "rogue" waves — a dramatic example of constructive interference at sea. Understanding superposition helps engineers design safer ships and offshore platforms.
✏️ Worked Examples
Scenario: Two pulses on a rope overlap at one point. Pulse A gives a displacement of +4 cm. Pulse B gives a displacement of −1 cm. Find the resultant displacement.
If Pulse B were −4 cm instead, the resultant would be 0 cm at that instant: complete destructive interference.
Scenario: Two coherent sources send waves of wavelength 0.40 m to a point. The path difference is 0.60 m. Determine whether the interference is constructive or destructive.
If the path difference were 0.80 m instead, that would be $2\lambda$, so the interference would be constructive.
Scenario: Two sources produce waves of wavelength 0.50 m. Source A has frequency 4.0 Hz. Source B has frequency 4.2 Hz. Explain why a stable interference pattern is not observed.
If both sources were driven by the same oscillator at exactly 4.0 Hz, they would be coherent and a stable interference pattern would appear.
Scenario: Two coherent sound sources emit waves with wavelength 0.80 m. A listener is positioned so that the path difference is 0.60 m. Calculate the phase difference between the two waves at the listener's position.
If the path difference were increased to 0.80 m, the phase difference would become 360°, and the interference would be constructive.
Scenario: Three pulses on a string meet at the same point. Pulse A has displacement +5 cm, Pulse B has displacement −2 cm, and Pulse C has displacement −3 cm. Find the resultant displacement and describe the interference.
If Pulse C were +1 cm instead of −3 cm, the resultant would be +4 cm, and the interference would be partially constructive.
Visual Break
🏃 Activities
Find the resultant displacement for each overlap pair:
Classify each as constructive or destructive: $0$, $\lambda/2$, $2\lambda$, $5\lambda/2$, $3\lambda$.
Explain why two random sound sources usually do not produce a stable interference pattern in a room.
A sound engineer places two identical speakers 4 m apart in a Sydney concert venue. A listener stands 6 m from speaker A and 6.5 m from speaker B. The sound frequency is 500 Hz and the speed of sound in air is 340 m/s.
Two coherent sources produce waves with wavelength 0.40 m. A point P is 1.20 m from source 1 and 1.00 m from source 2.
Two circular wave pulses on a still pond are created simultaneously at points A and B, 2.0 m apart. Each pulse has amplitude 3 cm and wavelength 40 cm.
Two coherent sources S1 and S2 are placed 3.0 m apart in a ripple tank. The wavelength is 1.0 m.
Two students are setting up a double-source interference experiment in a ripple tank. They have two independent motors driving the paddles.
Earlier you were asked what the rope looks like when two pulses overlap.
The full answer: the rope temporarily shows the sum of the displacements. That is the principle of superposition. If the displacements reinforce, the overlap is constructive. If they oppose, the overlap is destructive. Afterward, the pulses continue past each other.
Now revisit your prediction. What did you expect waves to do, and what do they actually do?
Annotate your prediction in your book with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
✅ Check Your Understanding
1. The principle of superposition states that when waves overlap:
2. Two equal waves meet exactly in antiphase. The result is:
3. A path difference of $2\lambda$ gives:
4. Coherent sources must have:
5. A point receives waves with path difference $3\lambda/2$. The interference is:
6. A student says, "Destructive interference means the waves disappear forever." The best correction is:
7. Explain the difference between constructive and destructive interference using phase relationship. 3 MARKS
8. Two pulses overlap at a point. One has displacement +6 cm and the other −4 cm. Find the resultant displacement and identify the type of interference. 3 MARKS
9. Explain why two-source interference patterns require coherent sources. 4 MARKS
1. B — superposition means displacements add algebraically.
2. D — equal waves in antiphase cancel completely.
3. A — $2\lambda$ matches the constructive condition.
4. C — coherence requires same frequency and constant phase relationship.
5. B — $3\lambda/2$ is a destructive condition.
6. C — cancellation is during overlap only.
Q7 (3 marks): Constructive interference occurs when two waves meet in phase, meaning their crests align with crests and their troughs align with troughs. In this situation, the displacements add algebraically, producing a resultant amplitude that is larger than either individual amplitude. Destructive interference occurs when two waves meet out of phase by half a cycle — a crest aligns with a trough. Here, the displacements oppose each other, producing a smaller resultant amplitude or, if the waves have equal amplitudes, complete cancellation at that point. It is important to note that this cancellation is temporary and local; the waves continue past each other unchanged.
Q8 (3 marks): Using the principle of superposition, the resultant displacement at the point of overlap is the algebraic sum of the individual displacements: Resultant = +6 cm + (−4 cm) = +2 cm. Because the two pulses have opposite signs, they partially cancel each other. However, since the magnitudes are not equal, the cancellation is incomplete. This type of overlap is called partial destructive interference. If the second pulse had been −6 cm, the resultant would have been 0 cm, representing complete destructive interference.
Q9 (4 marks): A stable interference pattern requires coherent sources because the locations of constructive and destructive interference are determined by the phase relationship between the two waves. Coherent sources maintain the same frequency and a constant phase difference, which means that points of reinforcement and cancellation stay in fixed positions in space. If the sources were not coherent, their phase relationship would drift continuously. The constructive regions would move around, and the pattern would blur out into a uniform average intensity, making it impossible to observe distinct bright and dark (or loud and quiet) regions. This is why laser light, which is highly coherent, produces sharp interference patterns, while two independent light bulbs do not.
Activity 4: Wavelength $\lambda = v/f = 340/500 = 0.68\ \text{m}$. Path difference = 6.5 − 6.0 = 0.5 m. Number of wavelengths = 0.5 / 0.68 ≈ 0.74$\lambda$. This is not an integer or half-integer multiple, so the interference is intermediate — neither fully constructive nor fully destructive. It will be closer to constructive than destructive because 0.74 is closer to 1.0 than to 0.5.
Activity 5: Path difference = 1.20 − 1.00 = 0.20 m. As a multiple of wavelength: $0.20 / 0.40 = 0.5\lambda$. This matches the destructive interference condition $(n + 1/2)\lambda$ with $n = 0$. Therefore point P experiences destructive interference.
Activity 6: (1) At the midpoint, the waves from A and B travel equal distances, so the path difference is zero. The pulses meet in phase and interfere constructively, producing a temporary amplitude of 6 cm. (2) After passing through each other, each pulse continues with its original amplitude of 3 cm, speed, and wavelength unchanged. (3) Path difference = 1.2 − 1.0 = 0.2 m = 0.2 / 0.40 = 0.5$\lambda$. This is destructive interference.
Activity 7: (1) Path difference = 5.0 − 4.0 = 1.0 m. (2) Since $\lambda = 1.0$ m, the path difference equals exactly one wavelength. This satisfies the constructive interference condition $n\lambda$ with $n = 1$. Therefore point P experiences constructive interference. (3) On the diagram, constructive interference points should be located where the path difference is 0, ±1.0 m, ±2.0 m, etc. Destructive interference points should be located where the path difference is ±0.5 m, ±1.5 m, etc.
Activity 8: (1) Two independent motors will have slightly different frequencies and their phase relationship will drift continuously. This means the positions of constructive and destructive interference will keep moving, so no stable pattern can be observed. (2) They could drive both paddles from the same motor or use a single mechanical linkage so that both paddles move in phase at exactly the same frequency. (3) If the frequencies are even slightly different, the phase difference between the sources changes over time. A constant phase difference is essential for stable interference; similar frequencies are not sufficient because the phase drift destroys the pattern.
Tick when you have finished the activities and checked the answers.