  We employ numerical simulations to understand the evolution of elastic
standing waves in disordered frictional disk systems, where the dispersion
relations of rotational sound modes are analyzed in detail. As in the case of
frictional particles on a lattice, the rotational modes exhibit an
"optical-like" dispersion relation in the high frequency regime, representing a
shoulder of the vibrational density of states and fast oscillations of the
autocorrelations of rotational velocities. A lattice-based model describes the
dispersion relations of the rotational modes for small wave numbers. The
rotational modes are perfectly explained by the model if tangential elastic
forces between the disks in contact are large enough. If the tangential forces
are comparable with or smaller than normal forces, the model fails for short
wave lengths. However, the dispersion relation of the rotational modes then
follows the model prediction for transverse modes, implying that the fast
oscillations of disks' rotations switch to acoustic sound behavior. We evidence
such a transition of the rotational modes by analyzing the eigen vectors of
disordered frictional disks and identify upper and lower limits of the
frequency-bands. We find that those are not reversed over the whole range of
tangential stiffness as a remarkable difference from the rotational sound in
frictional particles on a lattice.
