= Solution
{c}
There are two <chemical atoms> in the <primitive unit cell>, hence six displacement coordinates and six <phonon branches> at each <wavevector>. Three rigid <translations> cost no restoring <potential energy> in the long-<wavelength> limit. Consequently
$$
\boxed{3\text{ acoustic phonon branches}.}
$$
For propagation along $[100]$, one <acoustic phonon> has <longitudinal polarization> along $x$ and two have <transverse polarization> along $y,z$. The cubic <crystal symmetry> makes those two transverse directions equivalent. In terms of the <elastic stiffness tensor> and the mass <density>,
$$
v_L=\sqrt{\frac{C_{11}}{\rho_m}},\qquad
v_{T,y}=v_{T,z}=\sqrt{\frac{C_{44}}{\rho_m}}.
$$
Thus there are \b[two distinct acoustic velocities] in general, one longitudinal and a doubly degenerate transverse value. This <degeneracy of normal modes> concerns the two <transverse polarizations>; it does not imply equality with the longitudinal <speed of sound>.
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