= Acoustic phonon branch
An acoustic phonon branch has frequency tending to zero as its wavevector tends to zero. A uniform translation of an unconstrained crystal costs no restoring energy, motivating the zero-frequency limit of these displacement modes.
For an ordinary long-wavelength bulk elastic wave, $\omega\simeq c_s|\mathbf k|$, and the group velocity tends to the corresponding sound speed. Longitudinal and transverse branches have different polarizations and can have different speeds. The linear law is a model-dependent extra property: a flexural branch in a two-dimensional sheet can have a different leading dispersion. Thus the definition uses the vanishing frequency, while identifying its slope with a nonzero sound speed requires the appropriate elastic regime.
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