Beats

When two sound waves of very similar — but not identical — frequencies travel through the same medium, they overlap and interfere with each other. The result is a new wave whose amplitude rises and falls rhythmically, producing a pulsing or throbbing sound known as a beat. You have heard this effect if you have ever stood near two slightly out-of-tune guitars being played together: the sound seems to swell and fade repeatedly. This is not an illusion; it is a direct physical consequence of wave superposition, the principle that displacements from two waves simply add together at every point in space and time. The rate at which the loudness pulses is called the beat frequency. Remarkably, it equals the difference between the two source frequencies, not their average or their sum. If one tuning fork vibrates at 440 Hz and another at 443 Hz, you will hear three beats every second — a slow, detectable pulse that musicians use deliberately to tune their instruments. When the beat frequency drops to zero, the two sources are perfectly in tune. This makes beats one of the most practical tools in acoustics, connecting abstract wave mathematics directly to a musical skill practised for centuries. Beats arise from the mathematics of wave superposition combined with trigonometric addition formulae. The two individual waves alternately reinforce (constructive interference) and cancel (destructive interference) as their phase relationship cycles back and forth. Understanding beats deepens your grasp of wave behaviour generally and introduces ideas — amplitude modulation, resonance, tuning — that reappear across physics, engineering, and telecommunications.

Key takeaways

Superposition of two waves
Applying the sum-to-product identity
Identifying the envelope and beat frequency
Why the absolute value matters
Limiting cases

Superposition of two waves

Consider two sound waves of equal amplitude A but slightly different angular frequencies. At any fixed point, the air displacement due to wave 1 is y₁ = A cos(2πf₁t) and due to wave 2 is y₂ = A cos(2πf₂t). By the principle of superposition the total displacement is y = y₁ + y₂. This is where the mathematics does the heavy lifting.

Applying the sum-to-product identity

Using the trigonometric identity cos P + cos Q = 2 cos((P−Q)/2) cos((P+Q)/2), the combined wave becomes y = 2A cos(2π((f₁−f₂)/2)t) · cos(2π((f₁+f₂)/2)t). The second cosine oscillates at the average frequency — this is the pitch you perceive. The first cosine oscillates much more slowly and acts as a time-varying amplitude (an 'envelope') that modulates the loudness.

Identifying the envelope and beat frequency

The envelope term is 2A cos(2π·(Δf/2)·t) where Δf = f₁ − f₂. Because loudness is proportional to the square of amplitude (not amplitude itself), the ear perceives a maximum whenever the envelope reaches either its positive or negative peak. This happens twice per cycle of the envelope. Therefore the beat frequency — the number of loud pulses per second — is f_beat = |f₁ − f₂|, the absolute difference of the two frequencies.

Why the absolute value matters

It does not matter which frequency is higher. Whether f₁ > f₂ or f₂ > f₁, the number of beats per second is always the positive difference. This means beats cannot tell you which instrument is sharp or flat — only how far apart the frequencies are. A musician must deliberately raise or lower one instrument's pitch and listen for whether the beat rate speeds up or slows down to discover the direction of the error.

Limiting cases

If f₁ = f₂ exactly, Δf = 0 and the beat frequency is zero — the waves are always in phase and produce a steady, constant amplitude sound. If the frequencies differ by more than roughly 15–20 Hz the beats become too rapid for the ear to distinguish individually; instead a roughness or dissonance is perceived. True, countable beats only occur when the frequency difference is small.

The Key Equation

f_{\text{beat}} = |f_1 - f_2|

f_{\text{beat}} — Beat frequency (Hz): The number of amplitude pulses (loud-soft cycles) heard per second.
f_1 — Frequency of wave 1 (Hz): The frequency of the first sound source, e.g. one tuning fork or string.
f_2 — Frequency of wave 2 (Hz): The frequency of the second sound source, which is close but not equal to f₁.
|\cdot| — Absolute value (—): Ensures the result is positive regardless of which frequency is larger.

Worked Example

Problem. A piano tuner strikes a tuning fork of frequency 440 Hz (concert A) and simultaneously plays the A key on a piano. She hears 5 beats per second. After tightening the piano string slightly, she hears 3 beats per second. (a) What were the two possible piano string frequencies before tightening? (b) After tightening, what are the two possible frequencies? (c) How can the tuner determine which possibility is correct?

f_{\text{piano, initial}} = 435\,\text{Hz},\quad f_{\text{piano, after}} = 437\,\text{Hz},\quad f_{\text{beat, target}} = 0\,\text{Hz}

Answer. Before tightening: 435 Hz (the string was flat). After tightening: 437 Hz. The string must be tightened further until beats disappear.

Figure

Diagram

Origins · 1 minHermann von Helmholtz and how this was discovered

Hermann von Helmholtz

1821–1894 · German

Helmholtz provided the rigorous mathematical and physiological foundation for understanding beats and musical consonance in his landmark work 'On the Sensations of Tone' (1863). He showed how the ear analyses complex waveforms, explained why certain frequency combinations produce beats and dissonance, and linked the mathematics of superposition directly to auditory perception. His synthesis of physics, physiology, and music theory defined acoustics as a quantitative science. Helmholtz invented the Helmholtz resonator — a hollow sphere that amplifies a single frequency — to isolate individual overtones by ear, demonstrating that the human ear itself performs a form of Fourier analysis.

Wave Superposition — Beats are a direct application of superposition — the resultant displacement at any point is the algebraic sum of the two individual wave displacements, which creates the amplitude envelope responsible for beats.
Constructive and Destructive Interference — The loud and soft parts of a beat correspond to moments of constructive and destructive interference respectively as the two waves cycle in and out of phase.
Frequency and Period — The beat frequency formula f_beat = |f₁ − f₂| directly uses frequency, and understanding the relationship between frequency and period is essential for calculating beat periods and rates.
Standing Waves and Resonance — Musical instruments rely on resonance to produce stable frequencies; beats reveal when two resonant frequencies are mismatched, making this concept central to instrument tuning.