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.
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.
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)