Physics • Year 11 • Module 3 • Lesson 4
Wave Superposition and Interference
Apply superposition, path-difference conditions, and coherence requirements to real scenarios, data tables, and pulse calculations.
1. Interpret path-difference data — two coherent speakers
Two coherent speakers (S1 and S2) produce sound waves of wavelength 0.80 m. A student measures the distance from each speaker to five positions (P1–P5) along a line. The table shows the measured distances. 8 marks
| Position | Distance from S1 (m) | Distance from S2 (m) | Path difference (m) | Interference type |
|---|---|---|---|---|
| P1 | 2.40 | 2.40 | ||
| P2 | 3.00 | 2.60 | ||
| P3 | 3.20 | 2.40 | ||
| P4 | 4.00 | 2.80 | ||
| P5 | 4.50 | 3.10 |
1.1 Complete the “Path difference” and “Interference type” columns. Show which condition (nλ or (n+½)λ) applies to each. 5 marks (1 per row)
1.2 A student stands at P2 and reports the sound seems unusually quiet. Explain this observation using the superposition principle. 2 marks
1.3 One speaker is turned off. Predict how the interference pattern changes. 1 mark
2. Interpret graph — intensity along an interference pattern
The graph below shows the measured sound intensity at positions along a straight line perpendicular to the axis joining two coherent speakers. Distance is measured from the central axis. 7 marks
Figure 2.1. Sound intensity pattern from two coherent speakers producing 430 Hz waves in air (v = 344 m/s). Illustrative data.
2.1 Describe the pattern shown in the graph. Use the terms “constructive” and “destructive” interference. 2 marks
2.2 The wavelength of the sound is approximately 0.80 m. Identify the spacing between adjacent constructive maxima from the graph and comment on whether this is consistent with the path-difference condition nλ. 2 marks
2.3 A student concludes: “The intensity at the troughs is zero, so all the energy has been lost.” Evaluate this claim using your knowledge of energy conservation in interference. 3 marks
3. Compare constructive and destructive interference across five features
Complete the two-column table. For each feature write a concise description that contrasts the two types of interference. 10 marks (1 per cell)
| Feature | Constructive interference | Destructive interference |
|---|---|---|
| Phase relationship | ||
| Path difference condition | ||
| Resultant amplitude (equal waves) | ||
| Energy at this region | ||
| Real-world example |
4. Predict and justify — noise-cancelling headphones
Noise-cancelling headphones contain a microphone that detects ambient sound, then produces a wave with exactly the same frequency and amplitude as the noise but shifted by exactly half a wavelength (180°) in phase before feeding it to the ear cup.
5 marks
4.1 Predict what the listener hears compared to a person not wearing the headphones. Justify your prediction using the superposition principle and the path-difference condition for destructive interference. 3 marks
4.2 Explain why this noise cancellation would fail if the background noise source suddenly changed frequency and the headphone electronics had a 50 ms delay before updating the cancelling wave. Link your answer to the coherence requirement. 2 marks
Q1.1 — Path-difference table
P1: PD = 0 m = 0λ → Constructive (n = 0). P2: PD = 0.40 m = 0.5λ → Destructive (n = 0). P3: PD = 0.80 m = 1.0λ → Constructive (n = 1). P4: PD = 1.20 m = 1.5λ → Destructive (n = 1). P5: PD = 1.40 m = 1.75λ → neither fully constructive nor fully destructive (intermediate); accept “partial / intermediate interference”.
Q1.2 — Quiet at P2
P2 has a path difference of 0.40 m = 0.5λ, which satisfies the destructive interference condition (n + ½)λ with n = 0. The waves from S1 and S2 arrive at P2 in antiphase — a crest from one source coincides with a trough from the other. By the superposition principle, their displacements cancel algebraically, producing a near-zero resultant amplitude and very low intensity (perceived as unusually quiet) [1 mark for identifying destructive interference at P2 with path difference = 0.5λ; 1 mark for applying the superposition principle to explain cancellation].
Q1.3 — One speaker off
The interference pattern disappears. With only one source, there is no second wave to superpose with. Sound intensity would be roughly uniform (decreasing with distance from the remaining speaker), with no alternating loud and quiet regions.
Q2.1 — Graph description
The graph shows alternating high-intensity peaks (constructive interference) and near-zero troughs (destructive interference) at regular intervals along the lateral axis. The highest peaks occur at 0, ±40 cm, and ±80 cm; the near-zero troughs occur at ±20 cm and ±60 cm. The pattern is symmetric about the central axis.
Q2.2 — Spacing and path-difference condition
Adjacent constructive maxima are separated by approximately 40 cm. Each successive maximum corresponds to an increase of one wavelength in path difference (n = 0, 1, 2, …). The consistent spacing of ~40 cm between constructive maxima is consistent with nλ conditions producing equally spaced regions of reinforcement.
Q2.3 — Energy conservation claim
The student’s claim is incorrect [1]. Destructive interference does not destroy energy; it redistributes it [1]. The energy that is “missing” from the trough positions is transferred to the constructive interference peaks, where the intensity is approximately double that of a single source. Summing the energy across the entire pattern gives the same total as if both speakers were emitting without interference; energy is conserved [1].
Q3 — Compare and contrast table
Phase relationship: Constructive — in phase (0°, 360°, etc.). Destructive — antiphase (180°, 540°, etc.). Path difference condition: Constructive — nλ (n = 0, 1, 2, …). Destructive — (n + ½)λ. Resultant amplitude (equal waves): Constructive — 2A (doubled). Destructive — 0 (zero). Energy at this region: Constructive — above average; energy concentrated here. Destructive — below average / near zero at that point; energy redistributed elsewhere. Real-world example: Constructive — concert hall loud spot / signal boosting in antenna arrays. Destructive — noise-cancelling headphones / quiet spot in Sydney Harbour ferry sound.
Q4.1 — Noise cancellation prediction
The listener hears significantly reduced ambient noise compared to someone not wearing the headphones [1]. By the superposition principle, the anti-noise wave has the same frequency and amplitude but is exactly half a wavelength (180°) out of phase with the incoming noise [1]. This satisfies the condition for complete destructive interference: path difference = (n + ½)λ with n = 0, causing the displacements to cancel algebraically and reduce the resultant amplitude toward zero at the ear [1].
Q4.2 — Failure when frequency changes
For destructive interference to be sustained, the cancelling wave must maintain a constant phase relationship with the noise (coherence requirement) [1]. A 50 ms delay means the electronics takes time to update the anti-noise wave’s frequency. During that delay, the new noise frequency is no longer matched; the anti-noise wave is no longer coherent with (or the correct antiphase of) the noise. The phase difference drifts away from 180°, so the interference is no longer fully destructive and cancellation fails [1].