This might seem like a trivial question but it is not for me. So, I was reading on group and phase velocities from A.P. French where he calculates the phase and group velocities for a superposition of sine waves of different speed and wavelength. I will write down a brief analysis:
$$y(x,t)=A\sin(k_1x-\omega_1t) + A\sin(k_2x-\omega_2t) $$ which simplifies to
Now, what I usually find in literature is that for a general wave,the velocity of a wave is defined as $v=\omega/k$ and here we see that by simplifying the superposition, we get a slow moving and a fast moving term and
(1) For the slow moving wave which represents the group envelope, we call the velocity as group velocity $v_g=\Delta \omega/\Delta k=\partial\omega/\partial k$ (for waves with small differences in $\omega$ and $k$).
(2) For the fast moving wave which represents the ripples, we call the velocity as phase velocity $v_g=\bar \omega/\bar k=\omega/ k$ (if $\omega$ is given as a function of $k$).
What I don't understand in this analysis is
Why this definition of velocity? Why did we just divide the factors of x and t and call that as velocity? For a single sine wave, I understand how we can find the displacement of a maxima or minima and see how much it moves in some time t and define that as velocity (just like mentioned here). But is there any similar treatment possible for this?
How did we identify which one was the group and which one was the phase velocity? Also, it is not very intuitive at first site for a person who didn't know this before that there are actually 2 velocities embodied in such a solution?
I would be very grateful if someone could possibly have an answer to these questions.