Beats

Beats: when two waves of slightly different frequencies superpose, the resultant amplitude OSCILLATES at a low 'beat' frequency.

Beat frequency f_beat = |f₁ − f₂|. Always non-negative.

Used in musical tuning — listen for beats between two strings; tune until beats disappear (f₁ = f₂).

Mathematically: y₁ + y₂ = 2A·cos(2π·(f₁−f₂)/2·t)·cos(2π·(f₁+f₂)/2·t). Slow envelope × fast carrier.

Envelope frequency = (f₁−f₂)/2; we hear beats at TWICE this rate because |envelope| has period halved.

Beat phenomenon is a TIME-DOMAIN manifestation of constructive/destructive interference.

Audible beat range: f_beat ~ 0-15 Hz. Higher beats are perceived as roughness or two distinct tones.

Lasers: heterodyne detection uses beat frequencies between two slightly detuned lasers.