Beats occur when two sound waves with nearby frequencies are added together. Instead of hearing two completely separate tones, the listener often hears a single tone whose loudness rises and falls in time. This periodic change in loudness is called beat modulation.
If the two tones are
\[
y_1(t)=\sin(2\pi f_1 t),\qquad
y_2(t)=\sin(2\pi f_2 t),
\]
then the combined signal is
\[
y(t)=y_1(t)+y_2(t).
\]
Using a trigonometric identity, this sum can be rewritten as
\[
y(t)=2\cos\!\big(\pi(f_1-f_2)t\big)\sin\!\Big(2\pi\frac{f_1+f_2}{2}t\Big).
\]
This formula is very important because it separates the motion into two parts: a fast oscillation at the average frequency
\[
f_{avg}=\frac{f_1+f_2}{2},
\]
and a slow amplitude modulation controlled by the cosine factor.
The beat frequency is the rate at which the loudness maxima repeat:
\[
f_{beat}=|f_1-f_2|.
\]
The corresponding beat period is
\[
T_{beat}=\frac{1}{f_{beat}}.
\]
So if two tones are 440 Hz and 442 Hz, then
\[
f_{beat}=|440-442|=2\ \text{Hz},
\]
which means the sound grows louder and softer 2 times each second. The beat period is
\[
T_{beat}=\frac{1}{2}=0.5\ \text{s}.
\]
Beats are easiest to notice when the two frequencies are close. If the difference is too large, the ear tends to interpret the result as two distinct tones rather than a slowly varying loudness pattern. If the frequencies are exactly equal, then
\[
f_{beat}=0,
\]
so there is no beat modulation at all. The two signals simply combine into a single steady tone of fixed amplitude.
In music, beats are extremely useful for tuning instruments. When a piano string or guitar string is slightly out of tune with a reference tone, the player can listen for beats. As the instrument is adjusted, the beat frequency decreases. Perfect tuning corresponds to the disappearance of the beats.
This calculator assumes equal-amplitude pure tones and uses the standard classroom beat formula. At university level, the topic extends to unequal amplitudes, multiple tones, Fourier analysis, spectral envelopes, and interference in more general wave systems. Even so, the simple formula
\[
f_{beat}=|f_1-f_2|
\]
remains the core idea and explains the basic modulation pattern very well.
The interactive graph in this tool shows the summed waveform together with its envelope, and the animation highlights the current state of the beat cycle. The optional audio preview is included to make the result easier to hear as well as see.
