Wave Interference Simulator

When two waves meet, they don't bounce off each other — they pass through, adding point by point. This is the principle of superposition. In this simulator, two identical waves travel on the same rope with an adjustable phase shift between them. Change the phase and watch the combined wave grow, shrink, or vanish entirely.

Frequency 1 Hz

Phase shift (wave 1) 0 rad

Resultant amplitude at probe 0.0Destructive interference

Controls

Frequency: The frequency of both waves in Hz. Higher frequency means more wave crests per second.
Phase shift: How far wave 2 is shifted relative to wave 1, in degrees. 0° = perfectly in step; 180° = perfectly out of step.

The physics

y = y_1 + y_2

Superposition: the total displacement is the sum of the two waves

y = 2A\cos\left(\frac{\phi}{2}\right)\sin\left(kx - \omega t + \frac{\phi}{2}\right)

Combined wave of two equal waves with phase shift φ

A_{total} = 2A\cos\left(\frac{\phi}{2}\right)

Resulting amplitude depends only on the phase shift

Assumptions

Both waves have the same amplitude and frequency
The medium is uniform — no damping or reflections
One-dimensional waves on the same line
Linear superposition holds (small amplitudes)

Guided activity

Set the phase shift to 0°. What is the combined amplitude compared to a single wave? This is fully constructive interference.
Now set the phase shift to 180°. What happens to the combined wave? This is fully destructive interference.
Try 90°. Predict the amplitude first using A_total = 2A·cos(φ/2), then check the simulator.
Slowly drag the phase shift from 0° to 360° and describe how the amplitude changes. Where are the zeros?

Challenge questions

At which phase shifts (between 0° and 360°) does the combined wave completely vanish? Prove it with the amplitude formula.
Real noise-cancelling headphones use destructive interference. Why must the anti-noise wave match the incoming sound's frequency exactly?
What would change if the two waves had different frequencies? (Hint: you would get beats — a slow pulsing of loud and quiet.)