Acoustic Beat

Acoustic beats occur when two sound waves of slightly different frequencies are combined, creating an oscillation in the perceived intensity of the sound. This phenomenon, part of the French Terminale (final-year high school) curriculum, illustrates wave superposition and interference. It has been used for centuries to tune instruments in orchestras.

Discover FizziQ

How to measure it in class

With the FizziQ app, you can create and analyze acoustic beats.

Steps:

Open FizziQ and select the frequency synthesizer
Generate two sounds of close frequencies (e.g. 440 Hz and 442 Hz)
Observe the oscillation of the sound level with the sound level meter
Measure the beat frequency (here 2 Hz) and check that it equals |f₂ - f₁|
Use two slightly detuned tuning forks for an acoustic version

Lab assessment skills involved:

Appropriating: Identify the beat phenomenon as an interference
Analyzing: Predict the beat frequency knowing the source frequencies
Carrying out: Acquire a signal and measure the beat period
Validating: Compare the measured frequency with the theoretical value

Scientific activities on this topic

Possible extensions with FizziQ: tuning two tuning forks by the beat method and measuring the detuning between two instruments.

Learn more

The acoustic beat phenomenon is a direct illustration of wave superposition. It was known and exploited by tuners well before being described mathematically: it was Joseph Sauveur, at the very beginning of the 18th century, who gave its first quantitative analysis and stated that the number of beats per second equals the difference between the frequencies. (Christian Doppler, often wrongly cited here, published in 1842 on a different phenomenon: the Doppler effect, which concerns the frequency shift caused by relative motion between source and observer.)

Acoustic beats occur when two sound waves of slightly different frequencies are combined, creating an oscillation in the perceived intensity of the sound. To explain it, imagine two waves meeting: when they are in phase, they add up and increase the amplitude of the sound (constructive interference), and when they are out of phase, they partially cancel each other (destructive interference), resulting in a cyclic variation of the volume, that is, the beat.

The formula describing this phenomenon is simple: the beat frequency equals the difference in absolute value between the two frequencies involved. If f₁ and f₂ are the frequencies of the two sound waves, then the beat frequency f_b is: f_b = |f₁ − f₂|. The absolute value is not a detail: a frequency is positive, and 440 Hz with 442 Hz gives the same 2 Hz beat as 442 Hz with 440 Hz. This is in fact the limitation of the method for a tuner: the beat alone does not say in which direction the correction must be made.

There are a number of practical applications of the acoustic beat phenomenon. It has been used for centuries to tune instruments precisely in orchestras. When two notes are played together, if they are not at exactly the same frequency, the beat will be heard. Musicians then adjust the frequencies until the beat disappears, signaling that the notes are in unison.

Organ builders also exploit the deliberate detuning of two pipes, but two very different effects must be distinguished. In the so-called undulating stops (voix céleste, unda maris), two pipes separated by 1 to 3 Hz produce a slow beat: the sound is not lower, it undulates. On the other hand, the “acoustic fifth” used to suggest a 16-foot stop with shorter pipes does not rely on beats: it relies on a difference tone, a combination tone created by the nonlinear response of the ear (and of air at high amplitude), which makes a pitch at f₂ − f₁ audible. The beat is a variation in amplitude; the difference tone is a genuinely perceived pitch. Confusing them is a frequent mistake.

Finally, more recently the acoustic beat effect has been popularized by electronic music composers under the name LFO, for Low Frequency Oscillator.

The calculation, and the factor-of-2 trap

Consider two waves of frequencies f₁ and f₂, with the same amplitude and no phase shift: y₁(t) = A·sin(2πf₁t) and y₂(t) = A·sin(2πf₂t). The sum-to-product formulas give:

y(t) = y₁(t) + y₂(t) = 2A · sin(π(f₁+f₂)t) · cos(π(f₁−f₂)t)

(beware: this formula is often copied with (f₁+f₂) in both factors - a classic typo; the slow term is indeed in f₁−f₂.)

The result is interpreted as a fast vibration, the carrier, at the average frequency (f₁+f₂)/2 - this is the pitch you hear - whose amplitude is slowly modulated by the factor cos(π(f₁−f₂)t).

This is where the most frequent confusion lies. The mathematical modulating term cos(π(f₁−f₂)t) does oscillate at the frequency |f₁−f₂|/2. But the ear, like the sound level meter, does not perceive the sign of the amplitude: it perceives its modulus, the envelope |cos(π(f₁−f₂)t)|. Taking the absolute value of a cosine doubles its frequency: the cosine crosses zero twice per period, so the envelope passes through zero twice and through a maximum twice during each period of the modulating term.

The perceived beat frequency is therefore:

f_b = 2 × |f₁−f₂|/2 = |f₁ − f₂|

In other words: the modulating term is at |f₁−f₂|/2, the audible beat is at |f₁−f₂|. Both statements are true, they simply do not refer to the same quantity. With 440 Hz and 442 Hz, the cosine factor oscillates at 1 Hz, but you clearly hear 2 swells of sound per second.

When do we stop hearing a beat?

The beat is audible only if the frequency gap remains small. In practice:

below about 15 Hz of difference, you perceive a single sound whose intensity fluctuates: this is the beat;
between about 15 and 30 Hz, the fluctuation becomes too fast to follow and gives way to an unpleasant sensation of roughness;
beyond that, the ear separates the two sounds and hears two distinct pitches. There is no longer a perceived beat, even though the mathematical superposition remains exactly the same.

This point is important: the beat is not produced by the ear, it is really present in the pressure signal. But its perception depends on the ear’s temporal and frequency resolution.

Beats and interference: the same phenomenon, in two different dimensions

Beats are interference in time: at a fixed point, the amplitude varies over the seconds. Classic interference patterns (two loudspeakers in phase, Young’s slits) are interference in space: at a given instant, the amplitude varies with location. In both cases, the cause is the same - the superposition of two waves and the evolving phase difference.

Six science experiments with the tone generator (frequency generator) (FizziQ blog) describes how to produce the two pure tones needed.

School level and curricula

Terminale (final year of French high school) - Physics and Chemistry specialty

Theme: Waves and signals
Expected skill: Analyze recordings of sound signals using a microphone and an oscilloscope or an acquisition interface

Formula

Superposition of two pure tones of equal amplitude:

y(t) = 2A · sin(2π · ((f₁+f₂)/2) · t) · cos(2π · ((f₁−f₂)/2) · t)

Carrier frequency, that is, the perceived pitch:

f_carrier = (f₁ + f₂) / 2

Perceived beat frequency (the one counted by ear or on the sound level meter):

f_b = |f₁ − f₂|

Beat period:

T_b = 1 / |f₁ − f₂|

where:

f₁, f₂: frequencies of the two superposed pure tones (Hz)
f_b: beat frequency, number of sound swells per second (Hz)
T_b: time between two successive intensity maxima (s)
A: amplitude of each of the two waves (pressure unit, Pa)
y(t): resulting acoustic overpressure (Pa)

Application examples

Tuning an instrument. A 440 Hz tuning fork and a piano string at 437 Hz give f_b = 3 Hz, one swell every 0.33 s, very easy to count. The tuner tightens the string until the beats slow down and then disappear. The method is remarkably sensitive: counting one beat every 10 seconds means setting the string to within 0.1 Hz, that is 0.02% - far better than what the ear can do by comparing two isolated pitches.
Two recorders in class. Two students playing the same A are never exactly in tune: the typical gap of 2 to 5 Hz produces an audible beat, which FizziQ’s sound level meter records as a regular undulation of the level.
Undulating organ stop. Two pipes detuned by 1 to 2 Hz produce the characteristic slow vibrato of the voix céleste.
Synchronizing aircraft engines. On a twin-engine propeller aircraft, a slight difference in engine speed between the two engines creates a beat of a few hertz clearly audible in the cabin; pilots synchronize the propellers precisely to remove it.
Frequency measurement by heterodyning. In electronics, the unknown signal is mixed with a known reference signal and the beat frequency is measured, which is much lower and therefore easier to count. This is the principle of the superheterodyne radio receiver, transposed from acoustics.
Checking string tension. A guitar string tuned to 110 Hz that is loosened until it gives 1 beat per second against a reference is detuned by 1 Hz, about 16 hundredths of a semitone.

FAQ

Q: Is the beat frequency |f₁ − f₂| or |f₁ − f₂|/2? A: Both formulas circulate because they describe two different things. The modulating term in the calculation, cos(π(f₁−f₂)t), oscillates at |f₁−f₂|/2. But the ear perceives the envelope, that is, the absolute value of this term, which passes through a maximum twice per period. The audible beat is therefore at |f₁ − f₂|. This is the value measured in class.

Q: What note do we hear when two sounds beat? A: A single pitch, that of the average frequency (f₁+f₂)/2, whose volume fluctuates. With 440 and 442 Hz, you hear an A of 441 Hz that swells and fades twice per second - not two notes, and certainly not a 2 Hz note, which would in any case be inaudible.

Q: Does the beat exist if no one is listening, or is it an illusion of the ear? A: It really exists in the signal: a microphone or a pressure sensor records the amplitude modulation. It is not an auditory illusion, unlike the difference tone which, for its part, arises from the nonlinearity of the ear.

Q: Why can we no longer hear a beat when the frequencies are too far apart? A: Beyond about twenty hertz of difference, the modulation becomes too fast to be followed by the ear, which ends up separating the two sounds into two distinct pitches. The modulation nonetheless remains present in the signal, and spectral analysis recovers it.

Q: Do we need two sounds of equal amplitude to observe a beat? A: No, but that is the clearest case. If the amplitudes differ, the envelope no longer goes down to zero: the sound never cancels out completely and the fluctuation is less pronounced. Its frequency, however, remains |f₁ − f₂|.

Interference - Sound Wave Addition - Pure Tone - Sound synthesizer - Fundamental Frequency - Musical Note Frequencies - Doppler Effect - Active Noise Reduction