Waves/Beats

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[edit] Beats

Suppose two sound waves of different frequency impinge on your ear at the same time. The displacement perceived by your ear is the superposition of these two waves, with time dependence A(t) = sin(ω₁t) + sin(ω₂t) = 2sin(ω₀t)cos(Δωt), (2.19)

where we now have ω₀ = (ω₁ + ω₂) / 2 and Δω = (ω₂ - ω₁) / 2. What you actually hear is a tone with angular frequency ω₀ which fades in and out with period T_beat = π / Δω = 2π / (ω₂ - ω₁) = 1 / (f₂ - f₁). (2.20)

The beat frequency is simply f_beat = 1 / T_beat = f₂ - f₁. (2.21)

Note how beats are the time analog of wave packets -- the mathematics are the same except that frequency replaces wavenumber and time replaces space.

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