Energy Velocity of Diffusing Waves in Strongly Scattering Media

Abstract

Measurements of the diffusive transport of multiply scattered ultrasonic waves show that the energy velocity is very similar in magnitude and frequency dependence to the group velocity. Our data are accurately described using a theoretical model that accounts for the renormalization of scattering by the coupling between neighboring scatterers, quantitatively predicting the scattering delay that causes the strong frequency dependence of these velocities seen in our experiments. This gives a unified physical picture of the velocities of energy transport by both diffusive and ballistic waves.[ S0031-9007(97) PACS numbers: 43.35. + d, 43.20. + g, 62.30. + d In recent years, there has been a tremendous revival of interest in the propagation of classical waves through inhomogeneous media containing random scatterers [1,2]. When the scattering is strong, the propagation is typically very well described using the diffusion approximation, and this has successfully facilitated the interpretation of a wide range of fascinating wave phenomena. Even though the transport is diffusive, it is, nevertheless, still essential to define several propagation velocities. These include the group ͑y g ͒ and phase ͑y p ͒ velocities which characterize the ballistic, or unscattered, component of the incident waves. For diffusive waves, the relevant velocity is the energy velocity y e which is defined as the ratio of the energy flux to the energy density [3], and which corresponds to the average local velocity of energy transport in the diffusion process, since y e is related to the wave diffusion coefficient D by D y e l ‫ء‬ ͞3. Here l ‫ء‬ is the transport mean free path, or the distance the waves must propagate until their direction is randomized. Within this scenario, the energy velocity represents a distinctly different type of velocity, as wave coherence is no longer relevant. The first calculations for diffusive waves by the Amsterdam group [4], carried out for low volume fractions of scatterers f, correctly accounted for the extremely low value of y e found in light scattering experiments; they also predicted a strong frequency dependence for y e in the vicinity of Mie resonances as a result of the temporary storage of wave energy inside the scatterer [4][5][6][7]. However, these calculations fail as f is increased. Following an idea proposed by Sheng [2], recent Coherent Potential Approximation (CPA) calculations [8,9] have attempted to rectify this deficiency for higher volume fractions, and have suggested that the large variations of y e with frequency are washed out with increasing f; these results are purported to be in excellent agreement [8,9] with the limited experimental data that currently exist for y e [10,11]. However, in the intermediate frequency regime where resonant scattering occurs, such CPA theories become suspect [12], raising questions about their reliability. In addition to these questions about the behavior of y e , there have been conflicting ideas about the possible connection between the energy velocity ...

Comment on “Energy Velocity of Diffusing Waves in Strongly Scattering Media”

“…In Fig. 1, we present the results of y E as a function of frequency for the acoustic case for three concentrations, confirming that the behavior observed by Schriemer et al [1] is qualitatively reproduced. In addition, the new effective medium theory gives the correct long-wavelength limit for the effective dielectric constant e e for both the scalar and EM cases.…”

“…For larger f s the dips are smeared out, as observed in several experiments. Schriemer et al [1] concluded that our new effective medium approach gives the wrong results for acoustic waves, since their measurements indicate considerable structure in y E at high f s . However, their setup corresponds to low dielectric scatterers in a high dielectric background.…”

“…In a recent Letter, Schriemer et al [1] report on the measurements of the mean free paths (transport and scattering), and the energy and group velocities of a suspension of glass beads immersed in water. By using an effective medium model [2] based on a spectral function approach, they can explain all of their experimental results.…”

“…Sheng and collaborators [1,2] calculated the effective medium by finding the maximum of the imaginary part of the effective Green's function, the so-called spectral function. However, finding this maximum, although physically very reasonable, is only an "ansatz" like the new effective medium theory [3,4] and has not rigorously been derived from a microscopic theory.…”

Sound Propagation in Suspensions of Colloidal Spheres with Viscous Coupling

“…Anderson localization of light has recently indeed been observed [2]. By contrast, to date only a few experiments on sound propagation in the strongly scattering regime have been performed [3][4][5][6]. However, these experiments yielded astonishing results.…”