= Wavepacket derivation of group velocity
Superpose two nearby harmonic waves. The identity
$$
\cos(k_1x-\omega_1t)+\cos(k_2x-\omega_2t)
=2\cos\!\left(\frac{\Delta k\,x-\Delta\omega\,t}{2}\right)
\cos(\bar kx-\bar\omega t)
$$
separates a rapid carrier from a slow beat envelope. A constant point of the envelope moves at $\Delta\omega/\Delta k$, which tends to
$$
\boxed{v_g=\frac{d\omega}{dk}}
$$
as the components approach one another. Equivalently, expand a narrow <wave packet> spectrum around $k_0$: $\omega(k)=\omega_0+v_g(k-k_0)+\cdots$. Factoring the carrier out of its <Fourier transform> representation leaves an envelope depending on $x-v_gt$. This interpretation requires a narrow spectrum and little attenuation or distortion; higher derivatives of the <dispersion relation> can change the envelope shape. The carrier <phase velocity> is $v_p=\omega/k$ and need not equal $v_g$.
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