Tuning Fork Beats — Oscilloscope (2D)

Tuning Fork 1

Frequency f1 440.0 Hz

Tuning Fork 2

Frequency f2 445.0 Hz

Sound

Volume 30%

Adjust the sliders — beats appear the moment f1 ≠ f2.

Live Readout

Beat frequency |f1−f2|

5.00 Hz

Beat period

0.200 s

How it works

Two tuning forks ringing at close but unequal frequencies f1 and f2 each produce a pure sine tone. When the two tones reach your ear together, their pressure waves add up: x(t) = sin(2πf1·t) + sin(2πf2·t).

Using the trig sum-to-product identity, that sum equals 2·cos(2π·(f1−f2)/2·t)·sin(2π·(f1+f2)/2·t) — a tone at the average frequency (f1+f2)/2 whose loudness is modulated by a slow envelope oscillating at half the difference frequency. Because loudness depends on the envelope's absolute value, the ear perceives the volume swelling and fading at |f1−f2| Hz — the beat frequency.

Equal frequencies (f1 = f2): the envelope is constant — no beats, just a steady tone.
Small mismatch (a few Hz): slow, clearly audible "wa-wa-wa" throbbing — exactly how a piano tuner brings two strings into tune, listening for the beats to slow down and vanish.
Large mismatch: the beats become too fast to hear as pulsing and the ear starts perceiving two separate pitches instead.

The oscilloscope below plots fork 1 and fork 2 individually (top) and their summed waveform with its beat envelope (bottom) in real time.