Two close sine tones produce amplitude beats at their frequency difference

Evidence

When F1 is equal to F2, a single note is heard. As soon as F2 is moved higher or lower than F1, a sound with clearly undulating amplitude variations known as “beats” is heard. The frequency of the beats is equal to (F2 − F1), or (F1 − F2) if F1 is greater than F2, and the amplitude varies between (A1 + A2) and (A1 − A2), or (A1 + A2) and (A2 − A1) if A2 is greater than A1. Note that when the amplitudes are equal (A1 = A2), the amplitude of the beats varies between (2 × A1) and 0. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 574

When two tones close to each other in frequency are added, the resulting broadband signal contains amplitude fluctuation at frequencies corresponding to the difference in the frequencies. This fluctuation is called beating. The resulting perception of the auditory event is also interesting, as it depends relatively systematically on the difference in frequency. This is shown conceptually in Figure 11.6. — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1235

Consider track 1 on the accompanying CD. Suppose two pure tones, or sine waves, with amplitudes A1 and A2 and frequencies F1 and F2, respectively, are sounded together. If F1 is fixed and F2 is changed slowly from being equal to or in unison with F1 either upward or downward in frequency, the following is generally heard (see Figure 2.7). When F1 is equal to F2, a single note is heard. As soon as F2 is moved higher or lower than F1, a sound with clearly undulating amplitude variations known as “beats” is heard. The frequency of the beats is equal to (F2 − F1), or (F1 − F2) if F1 is greater than F2, and the amplitude varies between (A1 + A2) and (A1 − A2), or (A1 + A2) and (A2 − A1) if A2 is greater than A1. Note that when the amplitudes are equal (A1 = A2), the amplitude of the beats varies between (2 × A1) and 0. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 574

Categorization

• XREF: Beats phenomenon connects to tuning practice for musicians and to how detuned instruments create chorus/flange effects in audio engineering.
• DEFINE: Defines ‘beating’ as the amplitude fluctuation arising from two nearby-frequency tones.
• corroborated: 3 independent passages support this claim — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 574; Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1235