Wave Interference

How the model works

When two waves overlap, the medium's displacement is the sum of the individual displacements (the principle of superposition):

y₁ = A₁ sin(k₁x − ω₁t)
y₂ = A₂ sin(k₂x − ω₂t + φ)
y = y₁ + y₂

Both waves travel at the same speed c (set to 1 m/s), so the wavenumber is k = 2πf/c.

• Same frequency, φ = 0: peaks line up, giving constructive interference with amplitude A₁ + A₂.
• Same frequency, φ = 180°: peaks meet troughs, giving destructive interference with amplitude |A₁ − A₂|.
• Slightly different frequencies: the sum swells and fades. These are beats, at the beat frequency |f₁ − f₂|.

For equal frequencies, the resultant amplitude is √(A₁² + A₂² + 2A₁A₂ cos φ).

Ideal linear, non-dispersive medium with no damping. Real media can be dispersive or non-linear at large amplitude.

Guided experiments

1. Perfect cancellation. Equal amplitudes, equal frequencies, φ = 180°. Prediction: the sum is flat everywhere, all the time. The waves still exist; their displacements add to zero. Set φ = 180°
2. Beats. f₁ = 1.0 Hz and f₂ = 1.2 Hz. Prediction: an envelope pattern repeats every 5 s, since 1/0.2 Hz = 5 s. Watch the probe trace on the lower graph. Make beats
3. Unequal amplitudes. A₁ = 1, A₂ = 0.5, φ = 180°. Prediction: no full cancellation; the resultant amplitude is 0.5. Partial cancellation

Challenge

Two equal waves (A = 1) have a phase difference of 120°. What is the resultant amplitude? What phase difference gives an amplitude of exactly 1?

References

• OpenStax, University Physics Volume 1, §16.5 “Interference of Waves” and §17.7 “Beats”.