Beat (acoustics) - Mathematics and Physics of Beat Tones

Mathematics and Physics of Beat Tones

This phenomenon manifests acoustically. If a graph is drawn to show the function corresponding to the total sound of two strings, it can be seen that maxima and minima are no longer constant as when a pure note is played, but change over time: when the two waves are nearly 180 degrees out of phase the maxima of each cancel the minima of the other, whereas when they are nearly in phase their maxima sum up, raising the perceived volume.

It can be proven (see List of trigonometric identities) that the successive values of maxima and minima form a wave whose frequency equals the difference between the frequencies of the two starting waves. Let's demonstrate the simplest case, between two sine waves of unit amplitude:

If the two starting frequencies are quite close (usually differences of the order of few hertz), the frequency of the cosine of the right side of the expression above, that is (f₁−f₂)/2, is often too slow to be perceived as a pitch. Instead, it is perceived as a periodic variation of the sine in the expression above (it can be said, the cosine factor is an envelope for the sine wave), whose frequency is (f₁ + f₂)/2, that is, the average of the two frequencies. However, because the sine part of the right side function alternates between negative and positive values many times during one period of the cosine part, only the absolute value of the envelope is relevant. Therefore the frequency of the envelope is twice the frequency of the cosine, which means the beat frequency is:

This can be seen on the diagram on the right.

A physical interpretation is that when equals one, the two waves are in phase and they interfere constructively. When it is zero, they are out of phase and interfere destructively. Beats occur also in more complex sounds, or in sounds of different volumes, though calculating them mathematically is not so easy.

Beating can also be heard between notes that are near to, but not exactly, a harmonic interval, due to some harmonic of the first note beating with a harmonic of the second note. For example, in the case of perfect fifth, the third harmonic (i.e. second overtone) of the bass note beats with the second harmonic (first overtone) of the other note. As well as with out-of tune notes, this can also happen with some correctly tuned equal temperament intervals, because of the differences between them and the corresponding just intonation intervals: see Harmonic series (music)#Harmonics and tuning.