Beat frequency

In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies.

When tuning instruments that can produce sustained tones, beats can readily be recognized. Tuning two tones to a unison will present a peculiar effect: when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a tremolo as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible.

This phenomenon is best known in acoustics or music, though it can be found in any linear system: "According to the law of superposition, two tones sounding simultaneously are superimposed in a very simple way: one adds their amplitudes". 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 one wave 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 envelope of the maxima and minima form a wave whose frequency is half the difference between the frequencies of the two original waves. Consider two sine waves of unit amplitude:

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