Two sine waves at slightly different frequencies (2.0 and 2.3 Hz) producing visible beats. The composite swells and fades as the components drift in and out of phase.
The Core Idea
When two waves share the same medium, they don't collide — they add. The resulting displacement at every point is simply the sum of what each wave would produce alone. This principle of superposition is one of the most powerful ideas in physics: it underlies interference patterns, beat frequencies, standing waves, and the entire framework of linear systems.
The Mathematics
For two sinusoidal waves travelling in the same direction:
y₁ = A sin(k₁x − ω₁t)
y₂ = A sin(k₂x − ω₂t)
Superposition gives:
y = y₁ + y₂ = 2A cos(Δk·x/2 − Δω·t/2) · sin(k̄x − ω̄t)
where Δω = ω₁ − ω₂ and k̄, ω̄ are the averages. The result is a fast oscillation at the average frequency, modulated by a slow envelope at the beat frequency Δf = |f₁ − f₂|. This is why tuning a guitar by ear works: you listen for the beats slow to zero as the strings approach unison.
Higher frequencies with a 0.5 Hz difference. The beat envelope is clearly visible as periodic amplitude modulation.
Beats in action
Adjust the frequency sliders and watch how the beat rate changes. The closer the two frequencies, the slower the beats — when they match exactly, the beats vanish and you hear a pure tone.
Standing waves
When two waves of identical frequency travel in opposite directions, superposition produces a standing wave — nodes (perpetual zero displacement) and antinodes (maximum oscillation) locked in space:
y = 2A sin(kx) cos(ωt)
The spatial pattern sin(kx) never moves; only the amplitude oscillates in time. This is the physics of guitar strings, organ pipes, and microwave cavities.
Constructive and destructive interference
The key insight is all about phase:
- Constructive: waves in phase → amplitudes add → louder, brighter, bigger.
- Destructive: waves out of phase → amplitudes cancel → quieter, darker, smaller.
- Complete cancellation: equal amplitude, 180° phase difference → total silence at that point.
Same frequency — pure constructive interference with double amplitude.
Try setting both frequencies to the same value above — the two waves are always in phase, producing pure constructive interference with double amplitude.
Beyond sine waves
Superposition only holds for linear systems. In nonlinear media (shock waves, certain optical crystals, general relativity), waves interact and produce new frequencies — harmonics, sum-and-difference tones, solitons. But for the vast majority of engineering and physics (acoustics, optics, quantum mechanics, signal processing), superposition is the rule.
Further reading
- Wave interference (Wikipedia)
- Beat (acoustics)
- Crawford, Waves (Berkeley Physics Course, Vol. 3)