Acoustic measurement of a granular density of modes

In glasses and other disordered materials, measurements of the vibrational density of states reveal that an excess number of long-wavelength (low-frequency) modes, as compared to the Debye scaling seen in crystalline materials, is associated with a loss of mechanical rigidity. In this paper, we present a novel technique for measuring the density of modes (DOM) in a real granular material, in which we mimic thermal excitations using white noise acoustic waves. The resulting vibrations are detected with piezoelectric sensors embedded inside a subset of the particles, from which we are able to compute the DOM via the spectrum of the velocity autocorrelation function, a technique previously applied in thermal systems. The velocity distribution for individual particles is observed to be Gaussian, but the ensemble distribution is non-Gaussian due to varying widths of the individual distributions. In spite of this deviation from a true thermal system, we find that the DOM exhibits several thermal-like features, including Debye scaling in a compressed hexagonally ordered packing, and an increase in low-frequency modes as the confining pressure is decreased. In disordered packings, we find that a characteristic frequency $f_c$ increases with pressure, but more weakly than has been observed in simulations of frictionless packings.