Correct definition of an 'acoustic mode'? I am reading 'The Oxford Solid State Basics' by S.H.Simon in which on page 92 defines an acoustic mode as:

... any mode that has linear dispersion as $k\rightarrow 0$.

Whilst on page 94 he defines it as:

... one mode will be acoustic (goes to zero energy at $k=0$).

Unless all modes that tend to zero do so linearly and vice versa then these two definitions don't overlap. Thus my question is as follows: does one of these conditions imply the other and if not what is the correct definition for an acoustic mode?
 A: No, one does not imply the other, and I disagree with the first definition.
For example, the dispersion relation of the ZA mode in graphene goes to zero like $x^2$, so energy goes to zero as $k \to 0$ but does not do so linearly.
The 'A' in 'ZA' stands for acoustic, so that's an example of a nonlinear acoustic mode.
(That said, the first definition has some merit. The slope of a linear dispersion relation as $k \to 0$ is the speed of sound, which is a constant -- at least in isotropic materials. "Acoustic" modes get their name because they behave like sound at long wavelengths, and non-linear dispersion relations don't have a speed of sound. So there is logic in saying that non-linear dispersion relations are not acoustic. However, I don't think that's the common definition.)
A: The vibrational modes that have linear dispersion close to $k=0$ are acoustic modes where the slope of the dispersion curve is the speed of sound in the material (different for different directions of $k$).
The frequencies of optical modes do not go to zero at $k=0$. I would guess the dispersion to be quadratic.  
