Physics › 🌊 Waves & Sound

Superposition & Interference

Drop two stones in a still pond and watch the ripples pass clean through each other — yet for one shimmering instant, they add up into something new.

High schoolAP Physics 1 level
💡
The big idea: When two or more waves occupy the same space, they don't collide like particles — they simply add. This "superposition" means the same two waves can pile up into a bigger wave, cancel to nothing, or land anywhere in between, depending only on how far out of step they are when they meet. That single idea explains dead spots at a concert, the colors in a soap bubble, and the double-slit pattern that first hinted light is a wave.
🎯 By the end, you'll be able to
  • State the superposition principle and use it to find the resultant displacement where two waves overlap.
  • Calculate the path difference between two sources and use it to predict constructive or destructive interference at a point.
  • Explain why coherence — a fixed phase relationship between sources — is required to see a stable interference pattern.
  • Use the interactive simulation to connect changes in wavelength and source spacing to the spacing of interference fringes.
📎 You should already know
  • Wave basics: amplitude, wavelength, frequency, and period
  • The wave speed equation v = fλ
  • Reading a sine-wave graph (displacement vs. position or time)

Two Ripples Meet

Toss two stones into a still pond a few feet apart. Each one sends out a circle of ripples, and where those circles cross, something interesting happens: for an instant the water rises higher than either ripple alone would make it, or barely moves at all — and then both ripples keep spreading outward, completely unbothered, as if the crossing never happened.

That's the single strangest and most useful fact about waves: unlike two marbles, two waves can occupy the same point in space at the same time. What you see there is neither wave alone — it's both, added together.

🔑 The Superposition Principle

When two or more waves overlap at a point, the resulting displacement is the algebraic sum of the displacements each wave would have produced on its own. Crests (positive) and troughs (negative) simply add with their signs. Once the waves pass each other, each continues on exactly as it would have if it had traveled alone — the overlap changes nothing permanent.

\[ y_{\text{total}}(x,t) = y_1(x,t) + y_2(x,t) \]
The total displacement at any point is the sum of each individual wave's displacement there — this is the superposition principle in one line.
\[ \Delta = n\lambda, \quad n = 0, 1, 2, \dots \]
Constructive interference (in-phase sources): the path difference Δ between the two waves is a whole number of wavelengths, so crest lines up with crest.
\[ \Delta = \left(n + \tfrac{1}{2}\right)\lambda, \quad n = 0, 1, 2, \dots \]
Destructive interference (in-phase sources): the path difference is a half-integer number of wavelengths, so crest lines up with trough.
🎮 Interactive: Two-Source Interference Pattern LIVE
Drag the two sources and adjust wavelength and spacing. Watch the bright/loud lines (constructive) and dark/quiet lines (destructive) shift as the path difference to each point changes.

Why Path Difference Is the Whole Story

Call the distance from source A to some point d₁, and the distance from source B to that same point d₂. The path difference is \( \Delta = |d_1 - d_2| \) — literally, how much extra distance one wave had to travel to reach that point.

That extra distance is what determines whether the two waves arrive in step or out of step. If Δ happens to be a whole number of wavelengths, both waves have completed a whole number of extra cycles, so they arrive crest-to-crest: constructive interference. If Δ is a whole number of wavelengths plus half, one wave arrives exactly upside-down relative to the other: destructive interference.

📝 Worked example: Two wave pulses travel toward each other on a rope. At the instant they meet, pulse 1 has displacement +4 cm and pulse 2 has displacement +3 cm. (a) What is the total displacement at that instant? (b) If pulse 2 had instead been inverted (displacement −3 cm) at that same instant, what would the total be?
  1. Apply the superposition principle: add the individual displacements algebraically.
  2. (a) Both pulses are positive (same side): \( y_{\text{total}} = +4\text{ cm} + 3\text{ cm} = +7\text{ cm} \). This is constructive — the pulses reinforce.
  3. (b) One pulse is now negative: \( y_{\text{total}} = +4\text{ cm} + (-3\text{ cm}) = +1\text{ cm} \). The pulses partially cancel, but because their amplitudes aren't equal, they don't fully cancel to zero.
✓ (a) +7 cm (fully constructive). (b) +1 cm (partial destructive interference — complete cancellation needs equal amplitudes).
📝 Worked example: Two small loudspeakers, A and B, emit sound in phase at a frequency of 850 Hz. The speed of sound in the room is 340 m/s. A listener stands 8.00 m from speaker A and 6.80 m from speaker B. Does the listener hear a loud sound or a quiet spot?
  1. Find the wavelength: \( \lambda = \dfrac{v}{f} = \dfrac{340\text{ m/s}}{850\text{ Hz}} = 0.400\text{ m} \).
  2. Find the path difference: \( \Delta = |d_1 - d_2| = |8.00\text{ m} - 6.80\text{ m}| = 1.20\text{ m} \).
  3. Express Δ in wavelengths: \( \dfrac{\Delta}{\lambda} = \dfrac{1.20}{0.400} = 3.00 \), a whole number.
  4. Since \( \Delta = 3\lambda \) (n = 3, an integer), the condition for constructive interference is satisfied.
✓ The listener stands at a loud spot: constructive interference (Δ = 3λ exactly).
⚠️ Two Traps Worth Avoiding

Destructive interference doesn't destroy energy. Where two waves cancel, the energy doesn't vanish — it shows up as extra amplitude somewhere else in the pattern. Superposition redistributes energy; it never deletes it.

The Δ = nλ rule assumes the sources start in phase. If the two sources are already out of step with each other, you have to fold that starting offset into the condition. And none of this produces a stable pattern unless the sources are coherent — same frequency, with a phase relationship that stays fixed over time. Two ordinary light bulbs flicker in and out of step so fast that any interference pattern they make is smeared into invisibility; a laser split into two beams (or two speakers driven by the same signal) stays coherent, which is why the pattern holds still long enough to see or hear.

Check your understanding

1. According to the superposition principle, when two waves overlap at a point, the resulting displacement is:
Superposition says overlapping waves simply add, with sign — crests add to crests, troughs subtract from crests, and so on. There's no collision; each wave keeps traveling afterward unaffected.
2. Two coherent, in-phase sources produce waves with wavelength λ = 2.0 m. At a certain point, the path difference is 5.0 m. What happens at that point?
Divide the path difference by the wavelength: 5.0 m / 2.0 m = 2.5, a half-integer. That matches Δ = (n + 1/2)λ with n = 2, so the waves arrive out of step and destructively interfere.
3. Why do two independent light bulbs never produce a visible interference pattern, even though light is a wave?
Two separate ordinary sources emit light with rapidly and randomly shifting phase relative to each other. The interference pattern still exists instant to instant, but it reshuffles far faster than any eye or detector can track, so it averages out to a smooth glow. Coherent sources (like a beam split by a double slit) keep a fixed phase relationship, so their pattern holds still.
4. For destructive interference between two coherent sources of equal amplitude that start in phase, the path difference Δ must equal:
A path difference of a whole number of wavelengths lines crest up with crest (constructive). Adding an extra half-wavelength shifts one wave exactly out of step with the other, so (n + 1/2)λ gives destructive interference.
✅ Key takeaways
  • Superposition: where two waves overlap, the total displacement is just the algebraic sum of each wave's displacement — and each wave carries on afterward as if the other were never there.
  • Constructive interference (louder sound, brighter light) occurs when the path difference between two in-phase sources is a whole number of wavelengths: Δ = nλ.
  • Destructive interference (quiet, dark) occurs when the path difference is a half-integer number of wavelengths: Δ = (n + 1/2)λ.
  • A steady, observable interference pattern needs coherent sources — same frequency and a phase relationship that doesn't drift over time.