  We derive the phonon dynamics of magnetic metals in the presence of strong
spin-orbit coupling. We show that both a dissipationless viscosity and a
dissipative viscosity arise in the dynamics. While the dissipationless
viscosity splits the dispersion of left-handed and right-handed circularly
polarized phonons, the dissipative viscosity damps them differently, inducing
circular phonon dichroism. The effect offers a new degree of manipulation of
phonons, i.e., the control of the phonon polarization. We investigate the
effect in Weyl semimetals. We find that there exists strong circular phonon
dichroism in Weyl semimetals breaking both the time-reversal and the inversion
symmetry, making them potential materials for realizing the acoustic circular
polarizer.
