Beats from Superposition of Close Frequencies

What you are seeing: two waves of nearly equal frequency added together, $y = \cos(k_1 x - \omega_1 t) + \cos(k_2 x - \omega_2 t)$. By a standard trig identity this is a fast carrier $\cos(\bar{k} x - \bar{\omega} t)$ riding inside a slow group envelope. The top panel shows the pattern travelling through space (deep-water ripples, $\omega = \sqrt{g k}$): the balls sit on the carrier crests and stream forward at the phase velocity $v_p = \bar{\omega}/\bar{k}$, while the whole group creeps along at the group velocity $v_g = d\omega/dk$. For deep water $v_g = v_p / 2$ exactly, so the crests visibly overtake the group, appearing at its rear and dying at its front.

Listen at one fixed point instead and you hear the temporal beat: the middle and lower panels show $\cos(2\pi f_1 t) + \cos(2\pi f_2 t)$ swell and fade at the audible beat rate $|f_1 - f_2|$. Move $f_2$ toward $f_1$ and the group stretches and slows; at $f_1 = f_2$ the beat vanishes and phase and group move together.

Figure 1. Beats and wave groups from two nearby frequencies. Top: the superposition travelling in space through a dispersive medium, with carrier crests (phase velocity) outrunning the group envelope (group velocity); for deep-water dispersion $v_g = v_p/2$. Middle and bottom: the same two tones at a fixed point in time, the audible beat at rate $|f_1 - f_2|$, with the frequency spectrum and the envelope period. Method: closed-form superposition; $\omega = \sqrt{g k}$.

$f_1$ (Hz)5.00

$f_2$ (Hz)4.70

speed3

WHAT TO TRY

Watch the balls on the crests race forward through the group, twice as fast as the group itself drifts: phase velocity against group velocity.
Move $f_2$ toward $f_1$ and the group stretches out and slows; the audible beat in the lower panels slows with it.
Set $f_1 = f_2$ exactly: the group fills the whole window, the beat vanishes, and phase and group move together.
Pull $f_1$ and $f_2$ far apart and see the spectrum bars separate while the beat speeds up.