Noise-cancelling headphones work by adding a second sound wave that is the perfect mirror image of the noise, so the two cancel to silence. The same principle makes two overlapping ripples build into a bigger crest in one place and flatten to nothing in another. That single rule, just add the displacements, is the principle of superposition, and it explains all interference.
Two pulses travel towards each other along a rope. One has a crest of $+3\ \text{cm}$, the other a trough of $-3\ \text{cm}$, and they meet at the same point.
Before reading: at the instant they exactly overlap, what is the displacement of the rope at that point? Write your prediction and one sentence explaining why.
Where two waves overlap, just add their displacements
When two waves meet, you find the result at any point simply by adding the displacement of one wave to the displacement of the other, no new physics, just addition.
The principle of superposition states that when two or more waves overlap at a point, the resultant displacement is the algebraic sum of the individual displacements:
$$y_{\text{total}} = y_1 + y_2$$
The waves only add while they overlap; afterwards each carries on unchanged. This single rule describes every interference effect.
Principle of superposition: where two or more waves overlap, the resultant displacement at each point is the algebraic sum of the individual displacements, $y_{\text{total}} = y_1 + y_2$. After overlapping, each wave continues unchanged.
Pause, copy the highlighted principle into your book before moving on.
Did you get this? True or false: after two waves overlap and superpose, they merge permanently into one new wave.
Why the same rule gives reinforcement in one place and cancellation in another
We just saw that overlapping waves add. That raises a question: why does adding sometimes give a bigger wave and sometimes nothing at all? This card answers it → it depends on whether the waves are in phase or out of phase.
Whether two waves reinforce or cancel comes down to their alignment: crest-on-crest builds up, crest-on-trough cancels.
At one point, wave 1 has a displacement of $+4\ \text{cm}$ and wave 2 has a displacement of $-1\ \text{cm}$. At a second point, both waves have a displacement of $+4\ \text{cm}$. Find the resultant displacement at each point and state the type of interference.
Constructive interference occurs when waves are in phase (crest on crest) and displacements add to a larger amplitude; destructive interference occurs when waves are out of phase by half a wavelength (crest on trough) and displacements cancel. The resultant is always $y_1 + y_2$.
Add the highlighted conditions to your notes before the check below.
Quick check: Two waves meet so that a crest of one lines up with a trough of the other, and both have amplitude $5\ \text{cm}$. The resultant displacement is:
Pattern A, Calculate then sketch
Two pulses travelling along a rope overlap. At point P, pulse 1 = $+2\ \text{cm}$ and pulse 2 = $+5\ \text{cm}$. At point Q, pulse 1 = $+6\ \text{cm}$ and pulse 2 = $-6\ \text{cm}$. In your book: (1) use superposition to find the resultant at P and at Q; (2) state the type of interference at each point; (3) sketch the resultant shape at Q in one sentence of description.
Two truths, one lie: Two truths and a lie about the principle of superposition. Which statement is the lie?
Dot point: Model the principle of superposition
Seeing superposition directly with springs, ropes and ripple tanks
We just learned the conditions for constructive and destructive interference. That raises a question: how can you model superposition so the addition of displacements is visible? This card answers it → send pulses towards each other and watch them combine.
You can model superposition by sending two wave pulses towards each other along a spring or rope and observing the brief moment when they add.
Ways to model the principle of superposition:
Each model confirms the same rule: where the waves overlap, displacements add; afterwards the waves separate and travel on as if nothing happened. A flat resultant at an instant describes displacement, not the full energy state: motion and deformation of the medium also matter.
Superposition can be modelled with pulses on a spring or rope (taller pulse when same-side pulses meet, momentary flat when opposite-side pulses meet) or two in-phase sources in a ripple tank (lines of large waves and lines of calm). Each shows how displacements add and that the individual waves continue unchanged after overlap; displacement alone does not determine the energy density.
Pause, copy the highlighted modelling summary into your book before the check below.
Last check: Decide if the statement is true or false.
At a point of total destructive interference, the energy of the two waves is permanently destroyed.
Fill the gaps: Complete the account of interference, then press Check.
When two waves arrive phase, their displacements add and the resultant amplitude is large: this is interference. When they arrive exactly out of phase, the displacements subtract and the amplitude is small or zero: this is interference. In both cases the resultant is found by applying the principle of .
Dot point: Use the principle of superposition to explain the constructive interference and destructive interference of waves
Superposition adds displacements. Displacement is not energy.
Use the simulation. Press Freeze at overlap with the default $+4.0\ \text{cm}$ and $-4.0\ \text{cm}$ pulses. Read the total energy, the kinetic share and the potential share. Which set did you get?
Use the simulation. True or false: after the two pulses cancel at the overlap instant, the waves are gone and the string stays flat.
A fresh set drawn from this lesson's question bank, feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence, that tells the system what to drill next.
UnderstandBand 4(4 marks) 1. State the principle of superposition and use it to explain how two waves can produce both constructive interference and destructive interference. Include the phase condition for each.
1 mark: states superposition (resultant = sum of displacements) · 1 mark: constructive = in phase (crest on crest), amplitudes add · 1 mark: destructive = out of phase by half a wavelength (crest on trough), cancel · 1 mark: links both to adding displacements
ApplyBand 4(3 marks) 2. Two pulses travel towards each other along the same string. Pulse A has an amplitude of $+4\ \text{cm}$ and pulse B has an amplitude of $-3\ \text{cm}$. State the resultant displacement at the instant their peaks coincide, and describe what the two pulses do immediately afterwards.
1 mark: resultant displacement $= +1\ \text{cm}$ · 1 mark: obtained by adding the displacements algebraically, including sign · 1 mark: both pulses continue past each other unchanged in shape, amplitude and speed
Dot point: Model the principle of superposition
AnalyseBand 5(3 marks) 3. Two equal waves travel in opposite directions along a string and form a standing wave. At a node the resultant displacement is always zero. Does this prove that no energy is present at the node? Explain your answer using the string's motion or deformation, and distinguish energy density from energy transfer.
1 mark: zero resultant displacement alone does not determine the energy density · 1 mark: at times when the standing-wave node has nonzero slope, the string stores elastic potential energy (it is greatest at maximum deformation) · 1 mark: distinguish energy stored locally (density) from energy transferred past a point (flux; equal counter-propagating waves have zero net average flux)
Dot point: Use the principle of superposition to explain the constructive interference and destructive interference of waves
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set from the lesson bank.
Q1 (4 marks): The principle of superposition states that when two or more waves overlap at a point, the resultant displacement is the algebraic sum of the individual displacements ($y_{\text{total}} = y_1 + y_2$). Constructive interference occurs when the two waves arrive in phase, a crest of one lines up with a crest of the other, so their displacements add to give a larger amplitude (e.g. $+3 + 3 = +6$). Destructive interference occurs when the waves are out of phase by half a wavelength, a crest of one lines up with a trough of the other, so their displacements cancel, partly or completely (e.g. $+3 + (-3) = 0$). In both cases the result follows directly from adding the displacements at that point.
Q2 (3 marks): The principle of superposition states that where two waves overlap, the resultant displacement at any point is the algebraic sum of the displacements each wave would produce alone, taking direction into account through the sign. Here the displacements are $+4\ \text{cm}$ and $-3\ \text{cm}$, so at the instant the peaks coincide the resultant displacement is $(+4) + (-3) = \mathbf{+1\ \text{cm}}$. Immediately afterwards the two pulses pass through each other and continue unchanged, each retaining its original shape, amplitude, speed and direction of travel. Superposition affects the medium only while the pulses overlap; it does not alter the pulses themselves.
Q3 (3 marks): No. A node has zero resultant displacement, but displacement alone does not tell us the local energy density. In a standing wave on a string, the node remains at rest, so its transverse kinetic energy density is zero. At instants when the standing-wave shape is not flat, neighboring string segments have opposite displacements, so the slope at the node is nonzero and the string stores elastic potential energy there. At maximum deformation the node's slope and elastic potential energy are greatest; at the flat-string instant the slope and this potential-energy term are zero. Therefore zero displacement does not establish zero energy at all times. Energy density describes energy stored per length at a location, while energy flux describes transfer of energy along the string. For two equal counter-propagating waves forming a standing wave, the average net energy flux is zero even though the standing-wave pattern stores energy. Cancellation respects energy conservation.
You can now apply the principle of superposition, distinguish constructive from destructive interference, and model both. Next lesson uses superposition to explain the full interference pattern, the alternating bands of maxima and minima two sources create.
↩ Back to focus-area hubAt the start you predicted the displacement when a $+3\ \text{cm}$ crest meets a $-3\ \text{cm}$ trough. By superposition the result is zero: $(+3) + (-3) = 0$, total destructive interference at that instant. If you predicted $+6$ or $-6$, revisit Card 1, you add the displacements with their signs.
A short spaced-retrieval set, mixing this lesson with earlier ones. It gets harder to guess and easier to remember.
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