Beat Frequency Calculator — Acoustic Beat Period & Envelope

Formula

How this is calculated

When two sound waves of slightly different frequencies f₁ and f₂ are played simultaneously, they periodically reinforce and cancel each other — a phenomenon called wave interference. The mathematical identity sin(2πf₁t) + sin(2πf₂t) = 2 cos(π(f₁−f₂)t) × sin(2π·f_avg·t) shows that the composite wave oscillates at the average frequency f_avg but with an amplitude that fluctuates at half the frequency difference. The ear hears this amplitude variation as a pulsing or beating sound, with f_beat = |f₁ − f₂| peaks of loudness per second.

Beats are used extensively in music: a tuner hears beats between a reference pitch and the instrument and tightens or loosens the string until the beating disappears (f_beat → 0 Hz). At 1–3 Hz the beats are easily audible pulsations; at 5–8 Hz they produce roughness or dissonance; above about 15 Hz the human auditory system can no longer track the individual pulses and the sensation blurs into a continuous tone.

The chart plots the amplitude envelope |2 cos(π·f_beat·t)| against time in milliseconds, showing three full beat cycles. This is the "outer shape" of the waveform — the physical sound oscillates at f_avg beneath the envelope, but at typical audio frequencies (hundreds of Hz) those individual cycles are too dense to resolve at this time scale.