# Beats

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## Beats

vibrations with a periodically changing amplitude resulting from the superposition of two harmonic oscillations with slightly different, but close, frequencies. Beats arise as a consequence of the fact that the phase difference between two vibrations with different frequencies is constantly varying such that both vibrations are in phase at one moment, after some period of time are out of phase, are then again in phase, and so on. If A1 and A2 are the amplitudes of two superposed vibrations, then, given identical vibration phases, the amplitude of the resulting vibration reaches its greatest magnitude, A1 + A 2; when the vibration phases are opposite, the resulting vibration amplitude falls to its lowest magnitude, A1 - A2. In the simplest case, when the amplitudes of both vibrations are equal, their sum reaches the magnitude of 2A, given identical vibration phases, and falls to zero when they are in opposing phases. The result of superposing the vibrations can be written as

where ω1 and ω2 are the respective angular frequencies of two superposed harmonic vibrations. (The initial phases of both vibrations are postulated to be equal to zero, as they do not play a role in forming the beats; a role is played only by the difference in phase between both vibrations, and this difference always varies between 0 and 2π.)

If ω1 and ω2 differ slightly, then in expression (1) the value

can be considered as the slowly changing amplitude of the vibration

The angular frequency Ω = ω1 – ω2 is called the angular frequency of the beats. That is, as long as the frequency ω1 + ω2 is much greater than the frequency of the beats, the variable magnitude (2) can rightly be considered as the vibration amplitude (3), since magnitude (2), although not constant (as an amplitude should be), changes only slowly. As frequencies ω1 and ω2 become closer, the vibration frequency decreases, disappearing when ω1→ω2 (“zero” beats); this principle is used in tuning musical instruments. In radio technology a heterodyne receiver is called a beat receiver. The essence of such a receiver is that if two harmonic vibrations are fed into a nonlinear element (detector), then a harmonic vibration is obtained with a difference frequency Ω. Since the difference frequency is much lower than the frequency of the received vibrations, it can be perceived as sound at several frequency correlations.

Determining the frequency of a beat tone between a measured and standard vibration is one of the more precise methods for comparing the measured magnitude with the standard and is widely used in practice. With the aid of beats it is possible to discover the most minute differences in frequency; for this reason, the “beat method” is used in various devices for measuring frequencies, capacity, inductivity, and the like.

### REFERENCE

Gorelik, G. S. Kolebaniia i volny, 2nd ed. Moscow, 1959.

S. E. KHAIKIN

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