Beats

What happens if the two overlapping waves have frequencies that are not equal, but only slightly different ($f_1 \approx f_2$)? The waves slowly drift from being in phase (constructive interference, large amplitude) to being in antiphase (destructive interference, small amplitude) and back again. The result is a signal whose amplitude pulses slowly in time: these are beats. In sound they are perceived as a periodic “wah-wah”, a regular swelling and fading of volume.

Key formula — Beat frequency

$\ev{f_{\text{beat}} = |f_1 - f_2|}$ The volume pulses as many times per second as the difference between the two frequencies.

Note

Musicians exploit beats to tune their instruments: when two notes are nearly equal, the beat is heard; as you get closer to unison the beat slows down, and when it disappears altogether the two frequencies coincide exactly.

The demonstration is obtained by adding two oscillations of nearby frequency and using the sum-to-product trigonometric identity:

$\cos\alpha + \cos\beta = 2\cos\!\left(\frac{\alpha+\beta}{2}\right)\cos\!\left(\frac{\alpha-\beta}{2}\right)$

The first factor oscillates quickly at the mean frequency $(f_1+f_2)/2$ (the pitch we hear), while the second factor — which modulates the amplitude — oscillates slowly at the frequency $(f_1-f_2)/2$. Since the amplitude reaches a volume maximum twice per cycle of this modulation, the audible beat frequency is precisely $|f_1 - f_2|$.

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Topics: Onde Concepts: Battimenti Skills: Principio di sovrapposizione

Related exercises: Battimenti di due frequenze vicine · Problema — Due altalene accoppiate · Esercizio svolto — Due pendoli accoppiati da un filo