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.