For a 2D square lattice with lattice constant a, the acoustic branch satisfies this dispersion. At small q (long wavelength), ω ≈ v_s·|q| (linear, sound-like). At the Brillouin zone boundary q=π/a, ω reaches its maximum 2√(k/m).
Phonons are quantized lattice vibrations. Acoustic branches arise from in-phase motion of all atoms in a unit cell. The group velocity v_g = dω/dq gives the speed of sound, which vanishes at the zone boundary — a standing wave.