Beats from superposition of close frequencies

What you are seeing: the sum of two cosines of nearby frequencies, $y(t) = \cos(2\pi f_1 t) + \cos(2\pi f_2 t)$. By a standard trig identity this equals $2 \cos(2\pi \bar{f} t) \cos(2\pi f_b t)$, where $\bar{f} = (f_1 + f_2)/2$ is the carrier and $f_b = |f_1 - f_2| / 2$ is the envelope rate. The audible beat rate is $|f_1 - f_2|$, twice the envelope rate, because the ear hears amplitude modulation and a maximum occurs every half-period of the envelope.

Move the $f_2$ slider toward $f_1$ and the envelope stretches out. At $f_1 = f_2$ the beats disappear and the sum collapses to $2\cos(2\pi f t)$.

Figure 1. Beats from two nearby frequencies. Method: closed-form evaluation of $\cos(2\pi f_1 t) + \cos(2\pi f_2 t)$ on a scrolling time window, with the slow envelope drawn over the sum.

$f_1$ (Hz)5.00

$f_2$ (Hz)4.70

speed2

WHAT TO TRY

Move $f_2$ toward $f_1$ and watch the envelope stretch as the beat slows.
Set $f_1 = f_2$ exactly: the beats vanish and the sum is a pure tone.
Pull $f_1$ and $f_2$ far apart and see the spectrum bars separate while the beat speeds up.