 Solution

At long wavelength, the acoustic phonon branch moves the two atoms in a cell nearly in phase: $u_M\simeq u_m$. In the $q\to 0$ limit this is rigid translation and $\omega \to 0$. The optical phonon branch moves them oppositely, with stationary cell centre of mass: $M{u}_M+m{u}_m=0$. Its $q=0$ frequency is finite because adjacent unlike atoms change their separation. For unequal masses the lighter atom has the larger optical displacement. Three-dimensional crystals have three acoustic branches and, with two atoms per primitive cell, three optical branches; longitudinal and transverse directions describe displacement relative to the wavevector. The name optical does not mean that every optical phonon branch is infrared-active: coupling also depends on effective charges and symmetry.