Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 2 / b / d / Solution

There are three acoustic branches, not two. For propagation parallel to the tetragonal $c$ axis, one branch is a longitudinal wave, polarized parallel to $c$. The other branches are transverse waves whose polarizations lie along the equivalent basal directions. The fourfold symmetry about $c$ makes those transverse modes degenerate along that axis.

Consequently at a generic nonzero wave vector along $c$ there are only two distinct acoustic frequencies: one longitudinal frequency and one doubly degenerate transverse frequency. At the zone center all acoustic frequencies go to zero as required by uniform translation invariance. The degeneracy would generally lift if propagation moved away from the symmetry axis or the basal tetragonal symmetry were broken.