Beats
f_beat = |f₁ − f₂| — envelope visible.
Key Notes
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.
Formulas
Beat frequency
Number of beats per second; absolute value.
Resultant amplitude (modulation)
Slowly varying envelope; period = 1/|f₁−f₂|.
Resultant wave
Envelope × carrier.
Important Points
f_beat is ABSOLUTE VALUE — beats happen regardless of which frequency is higher.
Audible beats: 0 < f_beat < ~15 Hz. Higher beats blur into roughness.
Music tuning: adjust until beats vanish ⇒ frequencies match.
Beats are RELATED TO interference but stretched out in time (rather than space).
Heterodyne mixing: beat in EM waves used in radio, laser metrology.
Common pitfall: confusing beat frequency with carrier (average) frequency.
Beats notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.