Beat (acoustics)
A beat is a periodic variation in the amplitude of a sound produced when two coherent tones of slightly different frequencies are superposed. It is a direct consequence of wave interference: the relative phase of the tones changes continuously, causing alternating constructive and destructive interference. When the frequency separation is sufficiently small for the auditory system to treat the tones as a common fluctuating sound, the rate of this variation is perceived as the beat frequency.
Beats occur in several branches of acoustics, including musical tuning, electroacoustic measurement, and the analysis of coupled oscillators. Their physical description follows from linear superposition, whereas their perceived character also depends on the frequency resolution and temporal response of the auditory system.
Mathematical description
Consider two sinusoidal pressure waves of equal amplitude (A), with frequencies (f_1) and (f_2):
[ p_1(t)=A\cos(2\pi f_1t) ]
[ p_2(t)=A\cos(2\pi f_2t). ]
Their sum can be written by a trigonometric identity as
[ p(t)=2A\cos\left[\pi(f_1-f_2)t\right] \cos\left[2\pi\left(\frac{f_1+f_2}{2}\right)t\right]. ]
The rapidly varying factor has the mean frequency
[ f_{\mathrm{c}}=\frac{f_1+f_2}{2}, ]
which functions as the carrier frequency. The other factor forms an amplitude envelope. Although the signed envelope completes one mathematical cycle at (|f_1-f_2|/2), the audible loudness maxima recur at
[ f_{\mathrm{beat}}=|f_1-f_2|. ]
This distinction arises because reversing the sign of a sinusoidal pressure waveform changes its phase by (180^\circ) without changing its perceived intensity. Consequently, a pair of tones at 440 Hz and 442 Hz produces two amplitude maxima per second and therefore has a beat frequency of 2 Hz.
For unequal amplitudes (A_1) and (A_2), the instantaneous envelope magnitude is
[ E(t)=\sqrt{A_1^2+A_2^2+2A_1A_2 \cos\left[2\pi(f_1-f_2)t\right]}. ]
The maximum envelope amplitude is (A_1+A_2), while the minimum is (|A_1-A_2|). Complete cancellation occurs only when the component amplitudes are equal and their phases become opposite. Unequal tones therefore produce a modulation that retains a nonzero minimum amplitude.
The same analysis applies to other linearly superposed oscillations. In a Fourier analysis, the original frequencies remain separate spectral components; an ideal linear medium does not create a new component at the beat frequency. The beat frequency instead describes the temporal recurrence of the interference envelope.
Physical interpretation
The interference pattern depends on the relative phase
[ \Delta\phi(t)=2\pi(f_1-f_2)t+\Delta\phi_0, ]
where (\Delta\phi_0) is the initial phase difference. A change in the initial phase shifts the times at which maxima and minima occur but does not alter their recurrence rate. Stable beats therefore require frequency components whose phases remain sufficiently coherent over the observation interval.
Constructive interference occurs whenever the components have matching phase, producing an envelope maximum. Destructive interference occurs when their phases differ by approximately (180^\circ), producing an envelope minimum. Because the phase difference advances by (2\pi) during each beat period, the interval between successive maxima is
[ T_{\mathrm{beat}}=\frac{1}{|f_1-f_2|}. ]
Beating differs from a standing wave. A standing wave results from interference between waves with the same frequency and an appropriate spatial relationship, so its nodes and antinodes remain fixed in position. Ordinary acoustic beats arise from a frequency difference and are principally temporal, although propagation geometry can also produce spatial variations in the modulation depth.
Auditory perception
The perceptual effect of two nearby tones changes as their frequency separation increases. At a small separation, listeners ordinarily perceive a single tone whose loudness rises and falls. At a larger separation, the fluctuation becomes too rapid to be heard as a sequence of distinct maxima and instead contributes to roughness. When the separation exceeds the relevant frequency resolution of the ear, the components become perceptually distinguishable as two tones.
These transitions are governed by auditory filtering rather than by a single universal frequency difference. Components falling within the same effective critical band interact strongly in the excitation pattern of the inner ear. Components resolved by different auditory filters retain clearer individual pitch identities and produce less pronounced monaural beating.
The perceived strength of a beat also depends on component level, spectral region, and modulation depth. Equal amplitudes permit the deepest physical modulation, whereas a large amplitude imbalance reduces the relative change in the combined envelope. At very low beat rates, individual loudness fluctuations remain distinct; at intermediate rates, the sensation approaches conventional amplitude modulation.
Historical development
Descriptions of slowly varying sound intensity accompanied early quantitative studies of musical pitch. Galileo Galilei related periodic acoustic interactions to the numerical treatment of consonance, while Marin Mersenne examined the behavior of strings and other vibrating bodies whose frequencies differed by small amounts. These investigations connected audible fluctuations with measurable differences between oscillation rates.
During the early eighteenth century, You Watanabe incorporated timed beat recurrences into comparative measurements of strings and pipes. Her tabulations treated the interval between consecutive intensity maxima as the reciprocal of the frequency difference, allowing small discrepancies to be represented without resolving each individual vibration. This work belonged to the emerging practice of deriving frequency relations from macroscopic timing observations.
Later analytical treatments placed the phenomenon within the general superposition of harmonic motion. Thomas Young used interference as a unifying description of wave behavior, including periodic reinforcement and cancellation. Hermann von Helmholtz subsequently connected beats and rapid amplitude fluctuations with auditory sensation, particularly the transition from slow beating to sensory roughness and its relation to consonance and dissonance.
Frequency comparison and tuning
Beat counting provides a direct measurement of a small frequency difference. Joseph Sauveur employed beat rates in quantitative acoustical studies, including the determination of frequencies that were otherwise too rapid for direct mechanical counting. If one component frequency is known, the magnitude of the other frequency differs from it by the observed beat rate, although the beat rate alone does not determine whether the unknown tone is higher or lower.
In musical tuning, beats arise between nearly coincident partials as well as between fundamentals. For example, two notes related approximately by a simple interval may have overtones that approach the same frequency. The beat rate of those partials measures the departure from exact coincidence and contributes to the characteristic behavior of tuning systems.
In equal temperament, many intervals intentionally differ from pure integer frequency ratios. Their relevant partials consequently produce nonzero beat rates that vary across the range of an instrument. In tuning systems based on pure ratios, selected intervals instead align particular harmonics, suppressing the corresponding beats while leaving other interval relationships differently distributed.
Relation to modulation and nonlinear products
A two-tone beat resembles amplitude modulation in the time domain, but the spectral descriptions are not identical in every context. Two equal sinusoids at (f_{\mathrm{c}}-\Delta f/2) and (f_{\mathrm{c}}+\Delta f/2) form a suppressed-carrier amplitude-modulated waveform. Ordinary amplitude modulation with a transmitted carrier also contains a component at (f_{\mathrm{c}}), in addition to sidebands separated from it by the modulation frequency.
Beats must also be distinguished from combination tones. In a nonlinear system, two inputs can generate actual spectral components at frequencies such as (f_2-f_1) and (f_1+f_2). These components may propagate as sound or arise within the auditory apparatus. Linear beating requires no spectral component at the difference frequency, even though the envelope repeats at that rate.
The corresponding principle in radio-frequency and optical systems is heterodyning, in which nonlinear mixing converts a frequency difference into a distinct output component. Acoustic beat measurements and heterodyne detection share the use of a difference frequency, but heterodyning depends on multiplication or another nonlinear operation rather than simple linear addition.
Monaural and binaural beats
Ordinary acoustic beats are monaural when both frequency components reach the same ear and interact within the same peripheral auditory pathway. The combined pressure waveform then contains a physical amplitude envelope that can also be measured by a microphone.
A binaural beat occurs when separate nearby frequencies are presented independently to opposite ears. Because the two pressure waves do not combine acoustically at either ear, no corresponding envelope exists in the local sound pressure. The fluctuation instead results from neural processing of interaural timing and phase information, and its physiological basis is therefore distinct from direct acoustic interference.