Rotational sound in disordered granular materials

We numerically study the evolution of elastic standing waves in disordered disk systems with a focus on the dispersion relations of rotational sound. As on a lattice, the rotational mode exhibits an optical-like dispersion relation in the high frequency regime, representing a shoulder in the vibrational density of states and fast oscillations of the autocorrelations of rotational velocities. If tangential stiffness between the disks is large enough, a lattice-based model perfectly describes the dispersion relation of the rotational mode. If it is comparable to or smaller than the normal stiffness, the model fails for short wavelengths. However, the dispersion relation then follows the model prediction for the transverse mode, implying that the fast oscillations of disks' rotations switch to acousticlike behavior. We evidence such a transition from rotational to transverse modes by analyzing their respective participation of different degrees of freedom to the eigenvectors.