Wave Superposition and Interference
In 1989, Amar Bose filed the patent for the first commercial noise-cancelling headphone — the Bose QuietComfort. The device uses a microphone to sample ambient noise, then generates an anti-noise signal that is exactly 180° out of phase. When the two waves superpose, destructive interference reduces the dominant 340 Hz aircraft cabin drone by approximately 25 dB — the equivalent of cutting perceived loudness by a factor of roughly 18. Superposition is not just a physics concept; it is the operating principle of every pair of noise-cancelling headphones sold today.
Practise this lesson
Three printable worksheets covering superposition and interference.
When two waves meet and their crests coincide, what do you think happens to the resultant displacement? What if a crest meets a trough? Predict both outcomes.
Warm-up — the superposition principle states that at any point, the resultant displacement is:
Know
- Superposition principle: resultant = algebraic sum
- Constructive interference: in phase → max amplitude
- Destructive interference: anti-phase → zero amplitude
Understand
- Why waves can add or cancel without destroying each other
- How path difference determines the type of interference
- How ANC headphones use destructive interference
Can Do
- Draw resultant waveforms using superposition
- Identify constructive and destructive interference from diagrams
- Explain interference effects in real applications
Core Content
When two waves overlap, displacements add algebraically
Put on a pair of noise-cancelling headphones in a noisy environment. Before you switch them on, you hear the full roar of cabin noise or traffic. The moment you switch them on, the sound drops dramatically — not because the headphones block sound like earmuffs, but because a second sound wave is being played into your ear canal that is exactly opposite to the first. The two waves are simultaneously in the same space. At every point in the canal, the displacements of both waves add together — and when they are equal in magnitude but opposite in sign, they sum to zero. You have just witnessed superposition.
The superposition principle applies to all linear waves. At every point and at every instant, add the individual displacements with their signs to find the resultant. This is the only rule needed to work out all interference patterns.
| Situation | Phase relation | Result |
|---|---|---|
| Crest meets crest | In phase (0°) | Constructive — doubled amplitude |
| Crest meets trough | Anti-phase (180°) | Destructive — zero amplitude (if equal) |
| Partial overlap | Between 0° and 180° | Partial — between minimum and maximum |
The superposition principle states that the resultant displacement at any point equals the algebraic sum of individual wave displacements. Constructive interference (in phase, $\Delta\phi = 0$) produces maximum amplitude; destructive interference (anti-phase, $\Delta\phi = 180°$) produces zero amplitude if equal.
Pause — copy the highlighted principle into your book before moving on.
Two identical waves, each of amplitude 3 cm, meet in anti-phase. The resultant amplitude is:
Path difference determines which type of interference occurs
We just saw that superposition produces constructive or destructive interference depending on phase. That raises a question: in a two-source experiment, what determines the phase at each point? This card answers it → path difference $\Delta d$ sets the phase; whole-number wavelengths give constructive, half-integer give destructive.
For two sources of identical waves, the type of interference at a point depends on the path difference $\Delta d$ — the difference in distance each wave travels from its source.
Constructive: $\Delta d = n\lambda$ ($n = 0, 1, 2, \ldots$)
Destructive: $\Delta d = (n + \tfrac{1}{2})\lambda$ ($n = 0, 1, 2, \ldots$)
These conditions are why a double-slit or two-source experiment creates regular bands of constructive (bright / loud) and destructive (dark / quiet) regions.
Constructive interference occurs when path difference $\Delta d = n\lambda$ ($n = 0, 1, 2, \ldots$); destructive interference occurs when $\Delta d = (n + \tfrac{1}{2})\lambda$. Path difference is the key quantity for predicting interference at any point.
Add the highlighted interference conditions to your notes before the check below.
A path difference of $2\lambda$ causes constructive interference.
After destructive interference, both waves cease to exist.
Activities
Two waves of equal amplitude meet. Sketch the resultant for each case:
- Crest coincides with crest (in phase)
- Crest coincides with trough (anti-phase)
- Crest of wave A coincides with the midpoint of wave B
Explain how active noise-cancelling headphones use destructive interference to reduce unwanted sound. Your answer should reference: superposition, anti-phase, path difference / phase shift.
Which of the following is the odd one out in relation to the superposition principle?
Complete: Constructive interference occurs when path difference = $n\_\_$ where $n$ is a whole number. (Enter the symbol.)
Path difference $= 2.5\lambda$ produces:
UnderstandBand 3(3 marks) 1. State the superposition principle and distinguish between constructive and destructive interference.
ApplyBand 4(3 marks) 2. Two sources emit waves of wavelength 0.4 m. A point is 0.2 m closer to source A than source B. State the type of interference at this point and justify your answer.
AnalyseBand 6(4 marks) 3. Explain why energy is not created or destroyed during constructive interference, even though amplitude doubles.
Show all answers
Short Answer — Model Answers
Q1 (3 marks): Superposition: resultant displacement = algebraic sum of individual displacements. Constructive: waves in phase; amplitudes add (resultant maximum). Destructive: waves anti-phase; amplitudes cancel (resultant zero if equal).
Q2 (3 marks): $\Delta d = 0.2$ m $= 0.5\lambda$. This is a half-wavelength path difference, so the condition for destructive interference is met ($\Delta d = (n+\tfrac{1}{2})\lambda$ with $n=0$). Destructive interference.
Q3 (4 marks): When two waves meet constructively at one region, they must also meet destructively at adjacent regions. The total energy redistributes spatially — concentrated regions of double amplitude are balanced by regions of zero amplitude. The total energy integrated across all space is unchanged.
Amar Bose's 1989 QuietComfort patent made the physics you just learned into a commercial product. The dominant 340 Hz aircraft cabin noise is reduced by ~25 dB through destructive interference: the anti-noise signal is generated at exactly 180° out of phase, so crest meets trough and the algebraic sum is near zero. Your Think First prediction was about crests and troughs meeting — and that is exactly the mechanism. Constructive interference: crest + crest → doubled amplitude. Destructive interference: crest + trough → zero amplitude (if equal). Both are temporary — the waves re-emerge unchanged after passing through each other.