Beats

Consider a wave made up of two components with independent spatial and temporal frequencies ($k_1,k_2$ and $\omega_1,\omega_2$). For simplicity we can assume they both have equal amplitudes, so the wave $f(z,t)$ can be represented as

$f(z,t) = cos(k_1z - \omega_1t) + cos(k_2z - \omega_2t)$
$2f(z,t) = e^{i(k_1z-\omega_1t)} + e^{-i(k_1z-\omega_1t)} + e^{i(k_2z-\omega_2t)} + e^{-i(k_2z-\omega_2t)}$

We can define the average and difference wave properties

$\bar{k} \equiv (k_1+k_2)/2 \qquad \bar{\omega} \equiv (\omega_1+\omega_2)/2$
$\Delta{k} \equiv (k_1-k_2)/2 \qquad \Delta{\omega} \equiv (\omega_1+\omega_2)/2$

so that the above equation becomes

$2f(z,t) = (e^{i(\bar{k}z-\bar{\omega}t)} + e^{-i(\bar{k}z-\bar{\omega}t)})(e^{i(\Delta kz-\Delta \omega t)} + e^{-i(\Delta kz-\Delta \omega t)})$
$f(z,t) = 2cos(\Delta kz - \Delta \omega t)cos(\bar{k} z-\bar{\omega}t)$

The sum of the two waves is equivalent to the product of two different waves. The second wave, the average, will have a higher spatial and temporal frequency than the first, prefactor wave. The low-frequency wave is known as the envelope and the high-frequency wave is known as the carrier.

This simulation demonstrates such a combined waveform. The frequency of the envelope is frequency of the "beat" pattern between the two waves. We can see that the velocity of the envelope, the group velocity, is

$v_g = \Delta\omega/\Delta k$

Minimal Wavepacket

Simulator