Two people shout across a room and the sound waves pass straight through each other, unharmed. Ripples from two stones cross and carry on. But where they overlap, something happens: the water climbs higher in some places and lies perfectly flat in others. The rule behind it is almost too simple — you just add the displacements. Everything from noise-cancelling headphones to the double-slit experiment falls out of that one sentence.
📘 What you need to know
Superposition: when two or more waves overlap at a point, the resultant displacement is the sum of the individual displacements
Displacement is a vector, so it can be positive or negative — add them algebraically
Waves in phase reinforce; waves in antiphase cancel
Two equal waves in phase give amplitude 2A; in antiphase they give zero
After overlapping, each wave carries on unchanged — waves do not collide or damage one another
The observable effect of superposition is called interference
The principle of superposition
The principle of superposition
When two or more waves overlap at a point, the displacement at that point is equal to the sum of the displacements of the individual waves
In symbols, at every single point along the wave:
Adding displacementsy = y1 + y2
The crucial word is displacement, not amplitude. Displacement is a vector: a crest counts as positive, a trough as negative. So “adding” two waves can perfectly well mean subtracting, and a big wave plus a big wave can give nothing at all.
Wave 1 y1
add algebraically
Wave 2 y2
gives
Resultant y1 + y2
At the dotted line the red wave is at +0.87A and the blue at +0.50A, so the purple resultant sits at +1.37A. Do that at every point and you have drawn the resultant wave.
Here’s the trick for sketching a resultant. Don’t try to imagine the whole curve at once. Pick a point. Put your finger on wave 1’s height, then on wave 2’s height, add them (watching the signs), and mark a dot. Do that for the crests, the troughs and the zeros first — those are the easy ones — then join the dots smoothly. Examiners give marks for the dots, not the artistry.
The two extremes
Superposition can give any answer between two limits, and both limits are worth knowing cold.
In phase — crest meets crest, trough meets trough. The displacements add, and the resultant is bigger than either wave.
In antiphase — crest meets trough. The displacements have opposite signs and partly (or completely) cancel.
Left: identical waves in phase double the amplitude. Right: waves in antiphase cancel exactly, leaving a flat line. Between these extremes you get everything in between.
Two waves of amplitude A
Phase difference
Resultant amplitude
In phase
0 (or 360°)
2A — maximum
Quarter cycle apart
90°
1.41A
In antiphase
180°
0 — complete cancellation
Two pulses meeting
Superposition is easiest to see with two single pulses travelling towards each other. As they overlap, the medium’s displacement is the algebraic sum of the two. Then — and this is the part people find surprising — the pulses carry straight on, completely unchanged. They pass through each other. They do not bounce, merge or lose anything.
Right column: at the instant of total cancellation the rope is flat everywhere — yet the pulses reappear. The energy is still there, stored in the motion of the rope rather than its displacement.
Where did the energy go? When two pulses cancel completely the displacement is zero, but every particle of the medium is moving fast. The energy is momentarily all kinetic. A moment later the pulses emerge, untouched. Superposition adds displacements, never energies.
✏️ Sketching a resultant wave
Mark the easy points first: where one wave is zero, the resultant equals the other wave.
Where both are crests (or both troughs), add the heights — that’s a maximum of the resultant.
Where a crest meets a trough, subtract. If they’re equal, the resultant is zero there.
Join the dots smoothly. Check the resultant never exceeds A1 + A2.
WE 1
Two waves of the same frequency, with amplitudes 4.0 cm and 3.0 cm, arrive at the same point. Determine the largest and smallest possible amplitude of the resultant wave.
Step 1 — the largest resultant happens when they are in phase
Crest lines up with crest, so the displacements add.
A = 4.0 + 3.0 = 7.0 cmStep 2 — the smallest happens when they are in antiphase
Crest lines up with trough, so one displacement is negative.
A = 4.0 − 3.0 = 1.0 cmBetween 1.0 cm and 7.0 cmNote it never cancels completely — total cancellation needs equal amplitudes as well as antiphase.
WE 2
At an instant, two overlapping waves have the following displacements at three points. Determine the resultant displacement at each. P: y1 = +2.0 cm, y2 = −5.0 cm · Q: y1 = +3.0 cm, y2 = +3.0 cm · R: y1 = −4.0 cm, y2 = +4.0 cm
Step 1 — apply y = y₁ + y₂, keeping the signsStep 2 — point Py = (+2.0) + (−5.0) = −3.0 cmStep 3 — point Qy = (+3.0) + (+3.0) = +6.0 cmStep 4 — point Ry = (−4.0) + (+4.0) = 0P: −3.0 cm | Q: +6.0 cm | R: 0At R the medium is flat, but only for an instant — and only at that point. Zero displacement is not the same as “no wave”.
WE 3
Two waves of equal amplitude 1.0 unit overlap with a phase difference of 90°. At one point each wave has a displacement of 0.71 units. Determine the resultant displacement there, and state whether this is the maximum possible.
Step 1 — add the displacements at that pointy = 0.71 + 0.71 = 1.4 unitsStep 2 — find the resultant amplitude for a 90° phase difference
The two waves are a quarter cycle apart, giving a resultant amplitude of
√(1.0² + 1.0²) = √2 = 1.4 unitsStep 3 — compare
1.4 = 1.4, so this point is the crest of the resultant.
y = 1.4 units, and yes — it is the maximumNote 1.4A, not 2A. Only waves exactly in phase reach 2A.
💡 Top tips
Add displacements, not amplitudes. Signs matter — a trough is negative.
Sketch the resultant by adding at the crests, troughs and zeros first, then joining smoothly.
Total cancellation needs equal amplitudesand antiphase. One alone is not enough.
Waves pass through each other. After overlapping, each is exactly as it was.
The resultant amplitude can never exceed A1 + A2 — a useful check.
⚠ Common mistakes
Thinking waves collide or destroy each other — they superpose, then carry on
Adding amplitudes while ignoring the sign of the displacement
Assuming any two overlapping waves give 2A — that only happens exactly in phase
Believing that at total cancellation the energy vanishes — it is kinetic at that instant
Confusing zero displacement at a point with the wave having stopped
Adding waves of different frequencies as if the pattern were fixed — superposition still applies, but the resultant shape changes with time
Quick recap: The principle of superposition says the resultant displacement where waves overlap is the algebraic sum of their individual displacements. In phase gives a maximum (2A for equal waves); in antiphase gives cancellation (zero for equal waves). Displacement is a vector, so keep the signs. Afterwards the waves separate unchanged.
Superposition is the rule. Its consequence has a name: interference. Next we ask where the reinforcing and cancelling happen — and the answer turns out to depend on how much further one wave has travelled than the other. That’s path difference, and with it we can predict bright and dark fringes exactly. On to Interference of Waves.
Want superposition to feel obvious?
Book a free meeting and we’ll practise sketching resultants and past-paper superposition questions together.
