When two sinusoidal waves of slightly different frequencies f₁ and f₂ are superimposed, the result is a wave whose instantaneous amplitude oscillates slowly at the difference frequency f_beat = |f₁ − f₂|. Mathematically, the sum y = A·sin(2πf₁t) + A·sin(2πf₂t) factors as y = 2A·cos(π·Δf·t)·sin(2π·f_avg·t), showing a carrier wave at the average frequency f_avg = (f₁+f₂)/2 modulated by an envelope oscillating at f_beat/2 (though the audible pulsing rate is f_beat, because the ear is sensitive to amplitude regardless of sign). Beat frequencies are heard as a rhythmic throbbing and are used by musicians to tune instruments by ear — when two strings or pipes are in perfect unison, the beats disappear entirely.
Set f₁ and f₂ individually (range 400–500 Hz), adjust amplitude and time window, and watch the individual waves (blue and red), their superposition (white), and the slow beat envelope (gold). The stat panel reports f_beat and the beat period T_beat = 1/f_beat in milliseconds. Click "Play Beats" to hear the audible pulsing effect through your device's speakers.
What is the beat frequency formula?
The beat frequency is simply f_beat = |f₁ − f₂|. For example, two tuning forks at 440 Hz and 443 Hz produce 3 beats per second — you hear one complete volume swell every 1/3 ≈ 333 ms. This follows from the sum-to-product identity: sin(2πf₁t) + sin(2πf₂t) = 2·cos(π(f₁−f₂)t)·sin(π(f₁+f₂)t), where the cosine factor is the slowly varying envelope.
How do musicians use beats to tune instruments?
A musician plays two notes simultaneously — for instance, two strings on a violin — and listens for the pulsing beat. Adjusting the tuning peg speeds up or slows down the beats; when the beat frequency reaches zero, the notes are in unison and the strings are in tune. This method can detect mistunings as small as 0.5 Hz (one beat every 2 seconds), far finer than most people can distinguish pitch directly. Piano tuners use the same technique, listening for beats between partials of adjacent notes to set equal temperament.
Why does the amplitude of the combined wave vary between 0 and 2A?
When f₁ and f₂ are in phase (peaks aligned), they add constructively to give amplitude 2A. Half a beat period later, they are exactly out of phase (one peak aligns with the other's trough) and cancel to give amplitude 0. The envelope 2A|cos(π·Δf·t)| traces this modulation. For equal amplitudes, the signal completely disappears at each null — a clear demonstration of wave interference and superposition.
Beats are a temporal interference phenomenon — two coherent sources at slightly different frequencies produce constructive and destructive interference that alternates in time at a single point in space. Spatial interference (as in a double-slit experiment) produces alternating bright and dark fringes fixed in space for two sources at the same frequency but different positions. Both arise from the same superposition principle, but beats unfold in time rather than space.
Yes. Beats occur whenever two oscillating quantities at slightly different frequencies are added together. In optics, two laser beams of slightly different frequencies interfere to produce optical beats at the difference frequency — the basis of heterodyne detection, used in telecommunications and laser Doppler velocimetry. In radio engineering, the superheterodyne receiver beats the incoming RF signal against a local oscillator to produce a fixed intermediate frequency (IF) regardless of the station frequency.
Combination tones (Tartini tones) are additional pitches heard when two loud tones are played simultaneously. The most prominent is the difference tone at f₁ − f₂, which is related to beats but distinct: combination tones arise from nonlinear distortion in the ear and auditory system at any frequency difference, whilst beats are a linear superposition effect perceptible only when f_beat is low enough to be heard as a rhythm (roughly below 15–20 Hz). Above ~15 Hz, beats transform perceptually into a roughness or dissonance rather than distinct pulses.
Beats provide an extremely sensitive way to compare two oscillation frequencies. In the Pound-Rebka experiment (1959), which verified gravitational redshift predicted by general relativity, the 14.4 keV gamma-ray frequency from a source at the top of a 22.5 m tower was compared to a detector at the bottom by Doppler-shifting the source and observing when the Mössbauer absorption resonance (effectively a beat null) occurred. The fractional frequency shift of 2.46×10⁻¹⁵ was confirmed — one of the first precision tests of general relativity.
The human auditory system can track amplitude modulation as separate beats up to about 15–20 Hz; above this rate, the pulsing fuses perceptually into a continuous sensation of roughness or dissonance. This transition frequency, around 15 Hz, corresponds approximately to the frequency resolution of the auditory nerve fibres' phase locking. In music theory, intervals producing beat rates above ~15 Hz (such as a minor second at A = 440 and 466 Hz giving 26 beats/s) are generally perceived as dissonant.
AM radio works by multiplying a carrier wave at a fixed radio frequency (e.g., 900 kHz) by an audio signal (0–15 kHz), producing sidebands at f_carrier ± f_audio. The receiver demodulates (extracts the envelope), recovering the audio. This is mathematically identical to beats: the beat formula y = 2A·cos(π·Δf·t)·sin(2π·f_avg·t) is exactly an AM signal where the carrier is at f_avg and the modulating frequency is Δf/2. The difference is intentionality — AM broadcasting uses controlled modulation; musical beats are an unintended consequence of two near-identical sources.
Noise-cancelling headphones use a microphone to sample ambient noise, then play an inverted (180° phase-shifted) copy of that noise through the speaker — destructive interference reduces the unwanted sound. This is interference at a single point (the ear canal) rather than beats, because the goal is complete cancellation (Δf = 0, f_beat = 0). However, the underlying principle is identical: superposition of waves that are out of phase produces a smaller (ideally zero) resultant amplitude.