Superposition & Interference
Waves pass through each other untouched, adding as they overlap.
Lesson at a glanceShow the key points
- Two pulses moving in opposite directions pass through each other and emerge unchanged.
- Principle of superposition: the disturbance at a point is the (vector) sum of the disturbances each wave would produce alone. Valid for small (linear) waves.
- Equal and opposite pulses momentarily cancel: the string is straight, but its particles are moving.
- Two sine waves of the same frequency travelling the same way with phase difference δ give a sine wave of amplitude A = √(A₁² + A₂² + 2A₁A₂ cos δ) (15.18), tan ε = A₂ sin δ/(A₁ + A₂ cos δ) (15.19).
- Constructive interference: δ = 2nπ, A = A₁ + A₂. Destructive: δ = (2n + 1)π, A = |A₁ − A₂| (15.20).
Two billiard balls collide and bounce. Two waves meet, overlap, and then carry on as if they had never met. That ghostly ability is what makes interference possible.
Pulses that pass through each other
Two people holding opposite ends of a string each flick a pulse toward the other. The pulses meet, overlap and separate, and each emerges with exactly its original shape, travelling exactly as before. While they overlap, the displacement at each point is simply the sum of what each pulse alone would produce: 0.2 cm up from one and 0.4 cm up from the other gives 0.6 cm up.
If one pulse is an upside-down copy of the other, there is an instant when they cancel everywhere and the string is perfectly straight. But the particles are still moving, and a moment later the two pulses re-emerge. If one person shakes up and down and the other side to side, the displacements are at right angles and add as vectors.
This holds for small disturbances (linear waves). Stretch a string too far and displacements no longer simply add; such nonlinear waves are beyond this course. When waves overlap we say they interfere, and superposition makes interference remarkably simple.
Two sine waves going the same way
Two sources send waves of the same frequency (so the same k) along +x, one started a little later or placed elsewhere, so they differ in phase by δ:
Exactly as when we combined two SHMs, the sum is another sine wave with
Two waves y₁ = (3.0 mm) sin k(x − vt) and y₂ = (2.0 mm) sin k(x − vt + x₀) travel together, with k = (π/2) cm⁻¹. Find the resultant amplitude for x₀ = 1.0 cm and for x₀ = 2.0 cm.
- 1Waves pass through each other unchanged; where they overlap, displacements add (vectorially).
- 2Superposition holds for small (linear) waves.
- 3Same-frequency waves in the same direction: .
- 4Constructive at δ = 2nπ (A₁ + A₂); destructive at δ = (2n + 1)π (|A₁ − A₂|).
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.