Group velocity of acoustic waves in strongly scattering media: Dependence on the volume fraction of scatterers

Abstract

We study, both experimentally and theoretically, the ballistic propagation of ultrasonic wave pulses through a random strongly scattering medium as a function of the volume fraction of the scatterers. The scattering medium consists of a liquid suspension of monodisperse glass beads, whose concentration is varied by controlling the upward flow of the liquid in a fluidized bed. At intermediate frequencies, where the acoustic wavelength \ensuremath{\lambda} is comparable to the size of the glass bead scatterers, very strong scattering is observed, with the scattering mean free path reaching values as low as \ensuremath{\lambda}/2. At high volume fractions of scatterers, the scattering results in pronounced dispersion, as demonstrated experimentally by the strong frequency dependence found in both the phase and group velocities. However, as the volume fraction is lowered, the dispersion is substantially reduced, in marked contrast to recent predictions for electromagnetic waves. Our experimental results are explained quantitatively by a theoretical model, based on a spectral function approach, that accounts for the renormalization of the scattering within the medium, an effect that is greatest when the concentration of scatterers is largest. The mechanisms underlying the frequency dependence of the velocities and their dependence on volume fraction are further elucidated by examining the ultrasonic energy density, both inside the scatterers and in the surrounding fluid. This allows us to show that the velocities are substantially slowed down both by (i) resonant scattering from the glass spheres, where energy is trapped within the solid scatterers, and by (ii) tortuosity effects, where the wave energy is largely confined to the tortuous fluid pathways. These results demonstrate convincingly why the phase and group velocities of acoustic waves vary strongly with frequency at high volume fractions of scatterers, but only show weak dispersive effects at low volume fractions. Furthermore, our microscopic picture of the dispersion gives a simple physical explanation of why its volume fraction dependence is opposite to that expected for light and other electromagnetic waves, where the velocity inside the scatterers is normally less than in the surrounding medium.