Anomalous acoustic absorption and structural control of the transparency window in dense granular suspensions

Ultrasonic transmission through a random close packing of glass beads in water exhibits, below $kd\approx\pi$ ($k$ the wavenumber in water, $d$ the bead diameter), a transparency band in which disorder correlations — encoded in the static structure factor $S(q)$, the Fourier signature of bead-position correlations — suppress scattering so strongly that a low-frequency ballistic pulse propagates through the packing (Tourin et al., Phys. Rev. E 113, 025411 (2026)). Inside this band the residual attenuation is set not by scattering but by absorption, and the measured absorption coefficient follows a power law $\alpha_a\sim f^\gamma$ with $\gamma\approx1.4$ — nearly triple the viscous boundary-layer prediction $\gamma=0.5$, outside every previously reported experimental range ($0.5<\gamma<1$), and accompanied by more than an order-of-magnitude discrepancy in absolute absorption length ($\ell_a\approx20$ mm measured versus 400 mm predicted at 0.3 MHz for $d=1.5$ mm beads at volume fraction 64%). The open problem is to turn this anomaly into a mechanism-level, predictive account of acoustic attenuation in dense granular suspensions: identify the dominant dissipative process(es) inside the transparency band, map how volume fraction, short-range order as quantified by $S(q)$, bead material, size, and polydispersity control the window (including the near-close-packing regime where the analytic Percus–Yevick hard-sphere structure factor — the standard closed-form model of pair correlations in an equilibrium hard-sphere fluid — fails), and incorporate absorption into the structure-factor-corrected and GCPA (Generalized Coherent Potential Approximation) theoretical frameworks so that extinction lengths are predicted jointly from correlation-modified scattering and frequency-dependent absorption. As of August 2026, no published work addresses any of these three parts.

Background

An ultrasonic pulse sent through a random packing of millimetric solid beads in water is strongly scattered: each bead scatters sound, and after many scattering events the acoustic energy travels diffusively as a long incoherent tail (the coda), while the coherent ballistic pulse launched by the source decays exponentially with propagation distance. The characteristic lengths are the scattering mean free path $\ell_s$ (mean distance between scattering events) and the absorption mean free path $\ell_a$ (distance over which acoustic energy is irreversibly converted to heat); the coherent intensity decays over the extinction length, defined by $1/\ell_{\mathrm{ext}}=1/\ell_a+1/\ell_s$, and the absorption coefficient is $\alpha_a=1/(2\ell_a)$. In a dilute suspension the scattering attenuation is well predicted by the Independent Scattering Approximation (ISA), which adds up the scattering cross sections of individual beads as if their positions were uncorrelated. In a dense packing the bead positions are strongly correlated, and the standard correction multiplies the single-particle form factor $F(q)$ by the static structure factor — the Fourier signature of bead-position correlations, $S(q)=(1/N)|\sum_j e^{i\mathbf{q}\cdot\mathbf{r}_j}|^2$ — inside the scattering cross-section integral, an approach introduced for correlated disordered media in optics by Saulnier et al. and by Fraden and Maret and reviewed by Vynck et al. Where $S(q)<1$ at small wave numbers, destructive interference between waves scattered by different beads suppresses the effective cross section; in the extreme case of hyperuniform point patterns (disordered configurations whose large-scale density fluctuations vanish), this suppression can render an otherwise opaque medium nearly transparent, as demonstrated for light by Leseur, Pierrat and Carminati. For acoustic excitations in dense colloidal suspensions, the reference effective-medium theory is the Generalized Coherent Potential Approximation (GCPA) of Jing, Sheng and Zhou — a self-consistent multiple-scattering scheme valid up to high volume fractions including close packing, although it does not explicitly display disorder correlations and, in its standard form, accounts for attenuation by scattering only. Two experimental baselines exist for ultrasound in water-saturated glass-bead packings: Salin and Schön and Weaver and Sachse measured absorption exponents in the range $0.5<\gamma<1$ (with $\alpha_a\sim f^\gamma$), consistent with viscous dissipation localized in thin boundary layers at the bead surfaces, for which theory gives $\gamma=0.5$; and Page, Schriemer, Bailey and Weitz tested the diffusion approximation for multiply scattered sound in similar samples but reported no extinction-length measurements below $kd\approx4$, leaving the low-frequency regime unexplored.

In this context, Tourin, Abraham, Palla, Le Ber, Pierrat, Benech, Negreira and Jia reported the observation of an acoustic transparency band in two and three dimensions: in 2D, random packings of copper cylinders at 72% surface coverage, and in 3D, random close packing of slightly polydisperse glass beads ($d=1.5\pm0.1$ mm, volume fraction $\Phi_V=64\%$) immersed in water. Within a low-frequency band extending up to $kd\sim\pi$ (about 0.5 MHz), correlation-induced suppression of scattering — captured equally by the structure-factor-corrected cross section and by GCPA, which agree with each other — lets a ballistic pulse propagate with little scattering; the ballistic speed ($\approx1700$ m/s, close to the speed of sound in water) shows that the wave travels through the interstitial fluid rather than the grain-contact skeleton. The analytic Percus–Yevick (PY) hard-sphere structure factor — the standard closed-form approximate solution for the pair-correlation structure of an equilibrium hard-sphere fluid — reproduces this prediction at moderate volume fractions but overestimates the first diffraction peak near close packing, where structure factors from simulations of jammed packings (mechanically stable disordered sphere packings; Donev et al.) must be used instead; the PY breakdown near random close packing (the densest disordered arrangement of spheres) is documented by Katzav et al. Inside the transparency band the predicted scattering mean free path exceeds one hundred bead diameters, so the measured extinction length (10–30 mm across the band) must be set by absorption, giving $\ell_a\approx20$ mm at 0.3 MHz — more than an order of magnitude shorter than the $\ell_a=400$ mm implied by the thin viscous boundary-layer formula $\alpha_a=\frac{\Phi_V}{1-\Phi_V}d^{-1}c^{-1}\sqrt{4\pi\mu f/\rho}$ (with $\mu$ and $\rho$ the viscosity and density of water and $c$ the sound speed). Strikingly, the measured absorption follows $\alpha_a\sim f^\gamma$ with $\gamma\approx1.4$: inconsistent with the boundary-layer prediction $\gamma=0.5$, with the prior experimental range $0.5<\gamma<1$, and with any conventional model of acoustic dissipation in dense granular media. The authors' own precursor experiment (Abraham et al., using time reversal — recording the scattered wavefield and re-emitting it reversed in time to refocus energy onto its source — to probe ultrasound diffusion in the same suspensions) had targeted the diffusive regime, not this ballistic band.

Problem Statement

Establish a mechanism-level, predictive account of acoustic attenuation in dense granular suspensions — one coherent objective whose three linked parts jointly support it, posed by the open issues of Tourin et al. (2026):

(1) Mechanism of the anomalous absorption. Identify the dominant dissipative process or processes that produce the measured absorption law $\alpha_a\sim f^\gamma$ with $\gamma\approx1.4$ and the absorption length $\ell_a\approx20$ mm at 0.3 MHz inside the transparency band of water-saturated random-close-packed glass beads ($d=1.5$ mm, $\Phi_V=64\%$), where the thin viscous boundary-layer model predicts $\gamma=0.5$ and $\ell_a=400$ mm and all prior experiments gave $0.5<\gamma<1$.

(2) Structural control of the transparency window. Determine systematically how the position, width, and absorption behavior of the transparency band depend on the packing's volume fraction, its degree of short-range order as quantified by the structure factor $S(q)$ (including the near-close-packing regime where the analytic Percus–Yevick $S(q)$ breaks down), and on bead material, size, and polydispersity. Map quantitatively how variations of $S(q)$ control the window; the dependence on the volume fraction, bead material, size, and polydispersity is to be characterized qualitatively, in terms of the sign and principal trends of each effect.

(3) Predictive theory including absorption. Incorporate the identified absorption mechanism into the theoretical frameworks for correlated dense suspensions — the structure-factor-corrected scattering integral and the GCPA effective-medium theory — so as to predict extinction lengths that jointly treat correlation-modified scattering and frequency-dependent absorption, both inside the transparency band and in the scattering-dominated regime above it.

The three parts are linked: the overall aim is an account in which measured extinction lengths across the transparency band follow from packing structure and material parameters. (A further direction raised by the source's framing — whether engineered short-range order or hyperuniform-like packings could deliberately tune the window and minimize absorption — goes beyond the explicitly stated open issues and is not required for completion.)

References

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