Beat Frequency — Physica

Equivalent forms

f_b = |f_1 - f_2|

T_{beat} = \frac{1}{|f_1 - f_2|}

The audible pulse rate is simply the arithmetic difference of two pitches — interference made hearable.

Symbol	Name	SI unit	Dimension	Range
$f_1$	First frequency Frequency of the first wave or source	Hz	T^-1	1 – 1000
$f_2$	Second frequency Frequency of the second wave or source	Hz	T^-1	1 – 1000
$f_beat$	Beat frequencyoutput Number of amplitude pulses heard per second	Hz	T^-1	0 – 100

Unit systems

Symbol	SI	CGS	Imperial
$f_1$	Hz	Hz	Hz
$f_2$	Hz	Hz	Hz
$f_beat$	Hz	Hz	Hz

Where it holds

Valid for the linear superposition of two sinusoids of comparable amplitude and nearly equal frequency. Large amplitudes or nonlinear media introduce sum/difference tones beyond the simple beat.

Discovery

Joseph Sauveur · 1701

Sauveur, who coined the word 'acoustics', systematically studied the beating of organ pipes and showed that the beat rate equals the difference of the two pipe frequencies, using it as a precise way to tune instruments.

Try this

Why do two slightly out-of-tune guitar strings produce a throbbing 'wah-wah' you can hear?

Two tuning forks vibrate at 256 Hz and 260 Hz. How many beats per second does a listener hear?

Research status: stable

Real-world applications

Tuning musical instruments by reducing beats to zero against a reference pitch.
Heterodyne radio receivers mix the signal with a local oscillator to a fixed intermediate frequency.
Laser interferometry and LIGO-style readouts use optical beat notes to measure tiny frequency shifts.
Doppler radar extracts target speed from the beat between transmitted and reflected waves.

Common misconceptions

The beat frequency is the average of the two frequencies — it is the difference.
Beats only occur in sound — any two waves (light, microwaves, water) beat; optical beats power heterodyne detection.
The envelope frequency equals the beat frequency — the envelope is at half the beat rate, but loudness peaks twice per envelope cycle.

Experimental verification

Strike two tuning forks differing by a few Hz and the throbbing loudness is timed with a stopwatch; the pulse count per second matches |f_1 - f_2| to within measurement error. Oscilloscopes display the classic amplitude-modulated envelope.

Derivation

Add two equal-amplitude cosines:

\cos (2*pi*f_1*t) + \cos (2*pi*f_2*t) = 2*\cos (2*pi*((f_1-f_2)/2)*t)*\cos (2*pi*((f_1+f_2)/2)*t)

.

The fast carrier oscillates at the average frequency; the slow envelope oscillates at (f_1-f_2)/2.

Since intensity (loudness) tracks the squared envelope, the amplitude maxima recur twice per envelope cycle, giving a beat rate of |f_1 - f_2|.

Limiting cases

f_1 = f_2

⟶ f_beat

= 0

Identical frequencies stay in phase forever — no beating, just a steady tone.

|f_1 - f_2| > 20 Hz⟶ Beats blur into roughnessAbove

\sim 20\,\mathrm{Hz}

the ear stops resolving individual pulses and perceives a rough or dissonant tone.

What if…

What if you tighten one string until the beats vanish?

Zero beats means $f_1 = f_2$ — the strings are perfectly in tune. This is exactly how musicians tune by ear.

What if the two frequencies differ by 200 Hz?

The 'beat' is now itself an audible 200 Hz difference tone rather than a slow throb; the ear no longer resolves separate pulses.

1

Two tuning forks

Given ·

f 1:: 256
f 2:: 260

Find · f_beat

Steps

Identify the two source frequencies: $f_1 = 256\,\mathrm{Hz}$ , $f_2 = 260\,\mathrm{Hz}$ .
Apply f_beat $= |f_1 - f_2$ |.
f_beat $= |256 - 260| = 4\,\mathrm{Hz}$ .

Result · f_beat

= |256 - 260| = 4\,\mathrm{Hz}

, so the listener hears 4 loudness pulses every second.

Wave Equation→Standing Wave Frequencies→Doppler Effect→