I am reading about dispersion relations and got stuck with a connection I cannot follow. I have studied the mono-atomic chain which yields a dispersion relation
$w(k) \propto |\sin(\frac{ka}{2})|$.
Now, looking at small $k$, we can approximate that $\sin(ka/2) = ka/2$ so that $w(k)\propto ka$. The textbook states that this limit describes sound waves and this is exactly what I do not quite understand. Which formula is pointing out that we are dealing with sound waves? I have read Why can the dispersion relation for a linear chain of atoms (connected by springs) be written as $\omega(k)=c_s \lvert k\rvert$? and the corresponding comments where it says:
is just the speed of sound, which connects (in your linear dispersion relation) the frequency with the momentum. So, this is just the speed of sound by definition.
and
To elaborate upon what qmd, QuantumMechanics said, the speed of sound is defined either the "phase velocity" ω/k, or the "group velocity", dω/dk. Whenever ω=Ak for some A, one may see that the group velocity and the phase velocity coincide and both are equal to A.
where I understand how the group velocity coincides with the phase velocity but now which of the two corresponds to the speed of sound? or is it necessary for them to coincide in order to have sound waves? I feel like, I just lack a definition of what sound really is.