Proximity to Jamming Governs Acoustic Attenuation in Damped Packings

Fig. 1 — Experimental and field measurements of acoustic attenuation coefficient vs. frequency from marine sediment studies.

Fig. 1. Measured spatial attenuation (how fast a sound wave weakens) versus frequency, compiled from field and lab experiments on water- and oil-saturated sediments. Over roughly a million-fold span of frequency (tens of Hz to ~1 MHz) the data follow a clean quadratic law ($\alpha \propto f^2$) at low frequency but flatten into a nearly linear law ($\alpha \propto f$) at higher frequency. No grain-scale model had previously explained that transition.

The problem, in one sentence

When you send a sound wave through water-saturated sand, gravel, or mud, the wave grows fainter with distance. That fading — its attenuation — has long been observed to grow faster with frequency at low frequencies (quadratically) and slower at high frequencies (roughly linearly). Nobody had a grain-scale explanation for that crossover.

Why it is hard

Fluid-saturated granular solids are jamming systems: a pile of solid grains (much stiffer than the liquid between them) held in a disordered, just-barely-stable contact network, like sand packed a little too tightly. The way such a network is jammed is known to control its vibrations, yet existing acoustic models (Biot–Stoll, Grain Shear) treat the solid skeleton as a smooth, uniform material and ignore that messy contact network. As a result they cannot, from first principles, produce the flat, high-frequency linear attenuation that the experiments show.

What this research does

Using particle-based (“discrete element”) numerical simulations, the authors modeled a fluid-saturated packing grain by grain, including the liquid and realistic energy loss at each grain–grain contact. They then followed sound waves — and the underlying vibrational normal modes — of that packing as they swept three knobs: the frequency, the confining pressure, and the grain-contact damping.

The central finding: a pressure-controlled crossover

Both the vibrational modes and the propagating waves undergo a sharp, coordinated transition at one and the same critical frequency that is set entirely by the system’s proximity to jamming — physically, the confining pressure:

\[\hat{\omega}_c = \hat{P}^{1/2}\]

Below this frequency, grains move together in coherent, wave-like fashion. Attenuation is ordinary and viscous: it scales quadratically with frequency and linearly with grain-contact damping ($\hat{\alpha} \propto \hat{\gamma}\,\hat{\omega}^2$). This matches classical, fluid-based theory.
Above this frequency, the grains move incoherently, scattering off one another. Attenuation instead scales linearly with frequency and only weakly (sub-linearly) with contact damping — the mysterious regime that experiments had seen but no one could explain.

The fact that the same pressure-set frequency governs both the modes and the waves, in both shear and compression waves, is the signature that the mechanism is geometric (the structure of the contact network near jamming) rather than a property of the fluid.

Why this matters

This picture is named Jammed-Network Scattering (JNS), and it makes a striking, parameter-free prediction. Plugging in realistic numbers for water-saturated silica sand, the predicted crossover comes out to

\[f_c \approx \frac{c}{2\pi d}\left[\tfrac{3}{4}(1-\nu^2)\frac{\Delta\rho\, g\, h}{E}\right]^{1/3} \;\approx\; 2.03\ \text{kHz}\]

which lands almost exactly where the experimental data switch from the quadratic to the linear regime. In other words, the long-standing $\alpha \propto f$ behavior of sediments is not an unexplained fluid mystery — it is the natural result of sound travelling near the jamming point, scattering off a disordered contact network. That reframes the acoustics of the ocean floor, soil, and granular materials, and points the way toward grain-scale, physics-based (rather than empirically fit) acoustic models.

The figures

Every figure from the manuscript is shown below, grouped by the story each tells.

The experiments this explains

Fig. 1 — Attenuation data. Experimental and field measurements of the spatial attenuation coefficient $\alpha$ versus frequency from marine sediment studies. At low frequency the data follow $\alpha \propto f^2$ (viscous); at higher frequency they bend over to a nearly linear $\alpha \propto f$.

Fig. 1 — Attenuation vs. frequency from marine sediment experiments.

Fig. 2 — The simulation setup. Oscillation simulations that generate the wave-propagation data. Particles within one diameter of the wall at $x=0$ are driven sinusoidally. Panel (a) is a compression wave at large pressure; panel (b) is a shear wave at small pressure.

Fig. 2(a) — Compression wave at high pressure.

The modes: the origin of two regimes

Fig. 3 — Modes and the crossover. The damped vibrational modes plotted in the attenuation–frequency plane ($\hat{\beta}_i/\hat{\gamma}$ vs. $\hat{\omega}_i$) across many pressures ($\hat{P}$, symbol color) and damping levels. The top row is raw frequency; the bottom row is scaled by $\hat{\omega}_c$, which collapses the curves. The data show a clean quadratic rise at low frequency ($\hat{\beta}_i \propto \hat{\omega}_i^2$) that bends to a flatter scaling above the crossover $\hat{\omega}_c = \hat{P}^{1/2}$.

Fig. 3a — Decay rate vs. frequency at two pressures.

The figure also visualizes individual modes: a plane-wave-type mode (a), a mode near the transition (c), a localized “scattering” mode (d), and a plane-wave mode in a large packing (e).

Fig. 3b — A typical plane-wave-type mode.

Fig. 4 — Modes across the whole parameter space. The same attenuation–frequency plot for a wide range of damping values, with colors spanning the full pressure range so the universal crossover at $\hat{\omega}_c = \hat{P}^{1/2}$ is obvious.

Fig. 4 — Modes across all pressures and damping levels.

Wave speed: a self-consistency check

Fig. 5 — Wavespeed. Panel (a): dimensionless wavespeed $\hat{c}$ versus grain-contact damping $\hat{\gamma}$, showing the waves are nearly insensitive to damping in the low-damping regime. Panels (b) and (c): fitted shear- and compression-wave speeds versus pressure, each recovering the well-established $P^{1/2}$ scaling of jammed packings — confirming the simulation reproduces known elastic physics before damping and scattering enter the picture.

Fig. 5(b) — Shear wavespeed vs. pressure.

Fig. 5(c) — Compression wavespeed vs. pressure.

Where dissipation comes from

Fig. 6 — Attenuation versus grain-contact damping. Attenuation $\hat{\alpha}$ versus $\hat{\gamma}$ at moderate pressure, for (a) compression and (b) shear waves. Below $\hat{\omega}_c$ the curves are evenly spaced and grow linearly with $\hat{\gamma}$ (viscous); above $\hat{\omega}_c$ the damping dependence weakens markedly, signaling scattering-dominated attenuation.

Fig. 6(a) — Compression attenuation vs. damping.

Fig. 6(b) — Shear attenuation vs. damping.

Fig. 7 — Compression waves in the two regimes. Panel (a,b): instantaneous particle displacement along the channel (longitudinal in blue, transverse in orange) for $\hat{\omega}<\hat{\omega}_c$ (a) and $\hat{\omega}>\hat{\omega}_c$ (b). Panel (c,d): oscillation amplitude vs. position. Panel (e,f): the wrapped phase. In the viscous regime (left column) the longitudinal motion is smooth and coherent with $A_y \ll A_x$; in the scattering regime (right column) the motion becomes disordered and $A_y \sim A_x$ — with a near-complete loss of phase coherence.

Fig. 7(a) — Compression displacement, viscous regime.

Fig. 7(b) — Compression displacement, scattering regime.

Fig. 7(c) — Compression amplitude, viscous regime.

Fig. 7(d) — Compression amplitude, scattering regime.

Fig. 7(e) — Compression phase, viscous regime.

Fig. 7(f) — Compression phase, scattering regime.

Fig. 8 — The same diagnostics for shear waves, at the same parameters. In the viscous regime $A_x \ll A_y$ and the transverse phase shifts coherently (with notably more attenuation than the compression case); in the scattering regime $A_x \sim A_y$ and both phase components become disordered.

Fig. 8(a) — Shear displacement, viscous regime.

Fig. 8(b) — Shear displacement, scattering regime.

Fig. 8(c) — Shear amplitude, viscous regime.

Fig. 8(d) — Shear amplitude, scattering regime.

Fig. 8(e) — Shear phase, viscous regime.

Fig. 8(f) — Shear phase, scattering regime.

Attenuation versus frequency

Fig. 9 — Shear-wave attenuation vs. frequency. Attenuation normalized by $\hat{\gamma}$ (top) and by a sublinear coefficient $\hat{\gamma}^{a}$ (bottom), plotted against $\hat{\omega}/\hat{\omega}_c$ for a range of $\hat{P}$ and $\hat{\gamma}$. Below $\hat{\omega}_c$ the data collapse onto a single slope-2 curve ($\hat{\alpha}\propto\hat{\gamma}\hat{\omega}^2$); above it they spread, indicating the breakdown of linear damping. Bracketed reference lines mark slopes 1 and 1/3.

Fig. 9 — Shear attenuation vs. frequency.

Fig. 10 — Compression-wave attenuation vs. frequency. Attenuation normalized by $\hat{\gamma}$ (top) and by $\hat{\gamma}^{b}\hat{P}^{1/4}$ (bottom) versus $\hat{\omega}/\hat{\omega}_c$. Below $\hat{\omega}_c$ the same $\hat{\alpha}\propto\hat{\gamma}\hat{\omega}^2$ behavior appears, but — unlike shear — a residual pressure dependence remains. This $P^{1/4}$ scaling is the same pressure dependence seen in the compression-wave speed above.

Fig. 10 — Compression attenuation vs. frequency.

The simulation framework

The figures above were generated with GranMA (Granular Media Acoustics), a particle-based simulation and post-processing framework that pairs a MATLAB molecular-dynamics engine with a Julia analysis backend. It handles packing generation, wave-propagation simulation, and data analysis, and is where all of these results were produced.

Manuscript and figures: jammed-network-scattering
Simulation framework: GranMA

Pre-publication (arXiv): arXiv:2608.28157
Related post on this site: An arXiv preview of this paper →