I'm trying to define different velocities given a wave, but in this case I don't know how to identify the energy at the different terms that we have. I'm studying the propagation of waves of similar frequencies, i.e. beatings, that we can express,
$$ E(x,t)=\underbrace{2E_0}_{\text{Amplitude}}\underbrace{\cos(\bar{\omega}t-\bar{k}x)}_{\text{Modulated wave}}\underbrace{\cos \left(\frac{\Delta \omega}{2}t-\frac{\Delta k}{2}x\right)}_{\text{Modulator wave}} $$
Where we defined $\bar{\omega}=\frac{\omega_1+\omega_1}{2}$, $\Delta \omega=\omega_2-\omega_1$, and the same for the wave number $k_1$ and $k_2$.
For the velocities, the group velocity and phase velocity, we can identify these as the velocities of the modulated and modulator wave, but I don't know how to treat the energy in this type of wave.
$\bullet$ Phase velocity $\mathbf{ v_p}$ velocity of modulated wave i.e. $v_p=\frac{\bar{\omega}}{\bar{k}}$
$\bullet$ Group velocity $\mathbf{ v_g}$ velocity of modulator wave i.e. $v_g=\frac{\Delta \omega}{\Delta k}$
$\bullet$ Energy velocity (?) I want to know how the energy propagates in these cases.
