What you are looking at
The top panel shows two pure waves of slightly different frequency,
f₁
and
f₂. The bottom panel is their
sum — what your ear
actually receives when both play at once. Notice the sum's height swells and shrinks: that slow rise and fall
of loudness is the
beat. Press
🔊 Listen to hear it (the tones are shifted up
to an audible pitch while keeping their difference the same).
Why the loudness pulses
When the two waves are momentarily
in step, their crests line up and add — the sound is
loud. A moment later, because one wave cycles slightly faster, they slip
out of step, crest
meets trough, and they cancel — silence. This alternation repeats over and over. The maths makes it exact:
adding two equal cosines factors into a fast
carrier wave riding inside a slow
envelope:
cos(2πf₁t) + cos(2πf₂t) = 2 cos(2π·Δf/2·t) · cos(2π·f̄·t)
The carrier oscillates at the average frequency f̄ = (f₁+f₂)/2 (the pitch you hear), while the envelope
2cos(2π·Δf/2·t) slowly grows and fades. Each time the envelope passes through zero the sound momentarily
vanishes — those are the gold dots on the plot.
The beat frequency
The envelope reaches full loudness
twice per envelope cycle (once for each hump, positive or
negative), so the loudness pulses arrive at exactly the
difference of the two frequencies:
f_beat = |f₁ − f₂|
Bring the two frequencies together and the beats slow down and stretch out; when they match exactly the beats
vanish entirely. This is precisely how musicians
tune by ear: they adjust a string against a
reference note until the beats slow to a standstill, meaning the frequencies are identical.
Things to try
Start with f₁ = 10 and f₂ = 12 for a clear 2 Hz throb, then slide f₂ toward f₁ and watch the beats get slower
and the envelope stretch out until, at f₂ = f₁, they disappear. Press Listen and do the same by ear — sliding
into tune is unmistakable.