Year 11 PhysicsModule 3⏱ ~40 min5 MC · 3 Short AnswerLesson 4 of 18
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.
Today's hook: In 1989, Amar Bose patented noise-cancelling headphones using destructive interference, a microphone captures the 340 Hz aircraft cabin drone and the chip generates an exact anti-phase copy; when the two waves superpose the amplitude cancels by ~25 dB. That 25 dB reduction relies entirely on one principle: at every point in the ear canal, the resultant displacement is the algebraic sum of the two waves. When they are 180° out of phase and equal in amplitude, the sum is zero. Superposition is the rule; ANC headphones are the proof.
0/5TASKS
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Warm up first
Warm up first
Three quick questions from earlier lessons. Pulling old material back to mind before you learn something new makes the new material stick better, so this is not busywork.
Worksheets
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:
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Learning Intentions
Learning Intentions
goals
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
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Key Terms
Key Terms
vocab
Superposition principleThe resultant displacement at any point is the algebraic sum of the individual wave displacements.
Constructive interferenceTwo waves overlap in phase; resultant amplitude is the sum of individual amplitudes.
Destructive interferenceTwo waves overlap anti-phase; resultant amplitude is the difference (zero if equal).
Path differenceThe difference in distances each wave travels from its source to a point of interference.
Cross-lesson links: Superposition from this lesson is the mechanism behind standing waves (L07, L08), beats (L10), and standing waves in pipes (L11). The Bose QuietComfort 1989 (Amar Bose, 340 Hz drone, 25 dB cancellation by destructive interference) is the most direct commercial application. Constructive interference also underpins diffraction patterns (L06).
Misconceptions to fix
✗ Wrong: Waves destroy each other when they cancel out.
✓ Right: After passing through each other, waves are unchanged. Superposition is temporary.
✓ Right: Energy is redistributed in space, more here, less elsewhere, not created.
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The Superposition Principle
Core Content
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The Superposition Principle
+5 XP
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:
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Drive It: Where the Energy Goes When Two Waves Cancel
+XP for exploring
The table above gives three rows. This gives the continuum between them, and the one thing a table cannot show: what happens during the overlap.
Interactive · Wave superposition lab
Try this, in order. 1. Press Freeze at overlap and read all three energy figures before you press Play — the total, the kinetic share and the potential share. 2. Press Play and watch what comes out the other side. 3. Switch to Two waves, set both phases equal, then drag wave 2's phase to 180 degrees and watch the thin curves while the gold one flattens. 4. Move the two frequencies 1 Hz apart and count the beats. 5. In Standing wave, hunt for the four driving frequencies below 8 Hz that build a pattern, then drag the probe onto a node and then onto an antinode and compare both readings.
Predict then reveal+8 XP
1 · Predict
2 · Reveal
3 · Compare
A pulse of +4 cm and a pulse of −4 cm travel toward each other on the string. At the exact instant they overlap, the string is flat everywhere. Predict: what has happened to the energy that was in those two pulses?
50%
Actual answer
Nothing has happened to it. The simulator's total energy readout holds at 17.112 mJ through the whole run, and at the flat instant it reads 100.0 per cent kinetic and 0.0 per cent potential.
A stretched string stores potential energy in its slope, as one half times tension times the slope squared, and kinetic energy in its transverse speed, as one half times mass per metre times the speed squared. A flat string has zero slope everywhere, so its potential energy really is zero — but flat is exactly the moment every point is moving fastest, because one pulse is pulling each point up while the other pulls it down and their velocity contributions add instead of cancelling. So the energy is all kinetic, the total is untouched, and a fraction of a second later both pulses reappear at 4.00 cm each, unchanged. Reverse the sign of one pulse and the simulator gives you the mirror image: an 8.0 cm crest, 0.0 per cent kinetic, 100.0 per cent potential, and the same total. The sign changes how the energy is split, never how much there is.
How close was your prediction?
Good. Hold on to the rule underneath it: superposition adds displacements, and displacement is not energy.
This is the usual way it goes wrong. "Cancelled" gets read as "gone", but the only thing that cancelled was the displacement — and a flat string moving at speed is not an empty one.
In the simulator, press Freeze at overlap with the default +4.0 cm and −4.0 cm pulses. Read the total energy, the kinetic share and the potential share. Which set of three did you get?
After the two pulses cancel at the overlap instant, the waves are gone and the string stays flat.
In the simulator's standing-wave mode, the average energy flowing past the probe is zero at an antinode as well as at a node.
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Path Difference and Interference Conditions
3
Path Difference and Interference Conditions
+5 XP
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.
Interference conditions (same frequency, same phase source)
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.
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Activity 4, Superposition Sketches
Activities
Activity 4, Superposition Sketches
ApplyBand 3
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
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Activity 5, ANC Explanation
Activity 5, ANC Explanation
ExplainBand 4
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:
✓
Show what you have learned
Multiple Choice, superposition and interference
+5 XP
Short Answer, 10 marks
+5 XP
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.
✓
Retrieve, reflect and finish
Check what actually stuck
How did your thinking change?
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.