Sound propagation in a lattice of elastic beads: Time of flight, dispersion relation and time-frequency analysis

We study sound propagation in a model granular medium, which is a triangular array of nominally identical spherical beads under isotropic stress. Because of the point-like nature of the contacts between the beads, the slightest polydispersity makes the lattice of effective contacts random. This randomness evolves with the overall stress applied on the boundaries, and we use detection of longitudinal burst waves, with gaussian envelope, as a probe for the medium. At low and moderate stress, the velocity dependency on the applied stress exhibits clear discrepancies with Hertzian behavior, which shows that the contact lattice is indeed random.Time-frequency analysis gives full access to the dispersion relation of the lattice, both for long and short waves. For long waves, the time-of-flight is shown to be identical to the group delay, as expected. This method also allows measurements for short waves, which probe small-scale heterogeneities in the contact lattice: At high stress, almost all possible contacts are effective, and time-of-flight measurements indicate almost perfect Hertzian behavior. Group delay measurements for short waves, on the contrary, reveal persistent small-scale disorder. We discuss in some details the algorithms used for time-frequency analysis (Wigner-Ville distributions, pseudo Wigner–Ville distributions, reassignment method).