$$\begin{align} S_1(t) &= A\sin(k_1x-\omega_1 t)\nonumber\\ S_2(t) &= A\sin(k_2x-\omega_2 t)\nonumber\\ \end{align}\nonumber $$ If we write the angular frequency difference $\delta\omega = \omega_1-\omega_2$ and the average $\bar\omega = (\omega_1+\omega_2)/2$, then it's easy to show that $$\begin{align} \omega_1 &= \bar\omega + \delta\omega/2\nonumber\\ \omega_2 &= \bar\omega - \delta\omega/2\nonumber\\ \end{align}\nonumber $$ Let's look at the interference of these two waves at some point, say x=0. Then the two waves are described by: $$\begin{align} S_1(t) &= A\sin(\omega_1 t)=A\cos(\bar\omega t + \delta\omega t/2)\nonumber\\ S_2(t) &= A\sin(\omega_2 t)=A\cos(\bar\omega t - \delta\omega t/2)\nonumber\\ \end{align}\nonumber $$ If we add these up using the principle of superposition, we get $$S_{tot} = A\sin(\bar\omega t + \delta\omega t/2) + A\sin(\bar\omega t - \delta\omega t/2)\nonumber$$ Next, we expand each of the 2 sines using $$\sin(a\pm b)=\sin(a)\cos(b)\mp \cos(a)\sin(b)\nonumber$$ to get $$\begin{align} S_{tot} &= A\sin(\bar\omega t + \delta\omega t/2) + A\sin(\bar\omega t - \delta\omega t/2)\nonumber\\ &= A\big[\sin(\bar\omega t)\cos(\delta\omega t/2) + \cos(\bar\omega t)\sin(\delta\omega t/2)\big]\nonumber\\ &+ A\big[\sin(\bar\omega t)\cos(\delta\omega t/2) - \cos(\bar\omega t)\sin(\delta\omega t/2)\big]\nonumber\\ &=2A\cos(\delta\omega t/2)\sin(\bar\omega t)\label{beat} \end{align}\nonumber $$ Now, imagine that the 2 frequencies $f_1$ and $f_2$ are such that the difference $\delta f$ is much smaller than the average. For instance, $f_1=400$Hz and $f_2 = 402$Hz. This gives $\bar f=401$Hz and $\delta f = 2$Hz. This looks like a wave with a slow varying amplitude $2A\cos(2\pi\cdot 1Hz)$ and frequency $\bar f=401$Hz. What you will here is a tone that has the average frequency $401$Hz, with an amplitude that "modulates" such that you will hear a peak at 1 cycle per second.

Below, you will see 2 text windows with frequencies for 2 sounds. Push the start button for each wave and you will hear the wave at that frequency. Change the frequency and you will hear the average modulated by the beat frequency $\delta f/2$.