Formation of Beats

When two sound waves of slightly different, but close, frequencies $f_1$ and $f_2$ travel through the same medium and arrive together at a point, their superposition produces a periodic waxing and waning of loudness -- a regular throbbing, loud-soft-loud-soft pattern -- called beats. Taking two waves of equal amplitude $A$, $y_1=A\sin(2\pi f_1 t)$ and $y_2=A\sin(2\pi f_2 t)$, their superposition works out to $y=y_1+y_2=2A\cos\!\left(2\pi\,\frac{f_1-f_2}{2}\,t\right)\sin\!\left(2\pi\,\frac{f_1+f_2}{2}\,t\right)$ This can be read as a wave of the average frequency $\bar f=(f_1+f_2)/2$ (close to either original frequency, and this is the pitch actually heard) whose amplitude is not constant but is itself slowly modulated by the envelope term $2A\cos\!\left(2\pi\,\tfrac{f_1-f_2}{2}\,t\right)$, of the much lower frequency $(f_1-f_2)/2$. Since the loudness heard depends on the size of the amplitude regardless of its sign, the envelope produces a maximum in loudness twice in every one of its own cycles (once when the cosine is $+1$, again half a cycle later when it is $-1$, both giving the same maximum $|{\pm}2A|$) -- so the number of loud beats heard per second, the beat frequency, works out to $f_{\text{beat}} = |f_1-f_2|$ not half that value. Beats are put to direct practical use in tuning musical instruments: a string (or other note source) is compared against a reference of known frequency, and the player adjusts the tension (or other tuning control) until the beats slow and finally vanish, confirming the two frequencies now coincide exactly. Beats are also used to measure an unknown frequency close to a known one: if a known tuning fork and an unknown one together produce $n$ beats per second, the unknown frequency is either $n$ more or $n$ less than the known frequency; to decide which, the unknown fork is slightly loaded with a bit of wax (which always lowers its frequency a l …