Some confusion on group and phase velocity of a general wave group

Question

While discussing a wave group, it has always been turned up as taking only two waves of different $k$ values, the resultant being a composition of a wave of larger $k$ modulated by an envelope of smaller $k$. The phase velocity is explained from the movement of phase of ripples across the envelope whereas group velocity, by the motion of the envelope itself.

For a general wave group having an arbitrarily large number of component waves with $\omega=\omega(k)$, the group velocity is $v_g=\frac{d\omega}{dk}$. Now, depending on the nature of $\omega(k)$, $v_g$ may also have a functional dependence on $k$.

Naturally the question arises that at which value of $k$, the group velocity is to be calculated. That is my first question.

The second question is whether we can assign a phase velocity $v_p$ for the wave group. What provokes me to ask this is that still we can explain the movement of the phase of ripples across the group envelope. If the answer is "YES", then the first question is applicable to $v_p$ also.

Personally, I have an intuitive guess which may be completely wrong. Nevertheless, I want to share it. In the $k$-spectrum, a $k$-value near the shorter end decides the shape of the envelope and responsible for it's motion and hence has role in calculating $v_g$. On the other hand, some $k$-value at the larger end decides $v_p$.

Looking for help.

Answer 1

The assumption behind the definition of a wave packet is that it is quasi-monochromatic. Namely, a central dominant wavenumber (equivalently, a wavelength) can be clearly identified. Then, the group velocity is just the first term of the Taylor series of $\omega(k)$ calculated around the central wavenumber $k_0$.

$$ v_g = \left.\frac{d\omega}{dk}\right|_{k_0} $$

If can define also the zero-order term of the Fourier series of $\omega(k)$ as the phase velocity of the group. Namely, $v_p = \omega(k_0)/k_0$.

Alternatively, you can define the phase velocity for the monochromatic components of the wavepacket. In other words, you can expand the wavepacket in the Fourier series. Each element of the Fourier series is a monochromatic wave with a clearly defined phase velocity.