arxiv: 2605.09962 · v1 · submitted 2026-05-11 · ❄️ cond-mat.dis-nn

Recognition: 2 theorem links

· Lean Theorem

Attenuation of long-wavelength sound in quenched disordered media

Bingyu Cui , Yuqi Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:10 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords sound attenuationdisordered mediaRayleigh scatteringquenched disorderacoustic wavesBorn approximationdispersion renormalizationelasticity fluctuations

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The pith

Weak spatial fluctuations in elastic moduli reduce the speed of long-wavelength sound while producing Rayleigh attenuation scaling as q to the power d plus one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives how acoustic waves propagate through media with weak quenched disorder in elastic moduli or mass density, focusing on the long-wavelength regime. Analytic calculation of the self-energy in the Born approximation shows that elasticity disorder both slows the waves and attenuates them with the characteristic Rayleigh scaling, while density disorder attenuates but leaves the dispersion relation unchanged to leading order. The results depend only on the variances of the fluctuations and are insensitive to higher details of the disorder distribution. Lattice simulations in one and two dimensions confirm the predicted attenuation rates and velocity shifts. This distinction matters for understanding sound propagation in heterogeneous solids such as glasses or granular materials.

Core claim

For spatially uncorrelated elasticity disorder the sound speed is reduced and the attenuation rate satisfies Γ(q) proportional to q^{d+1}; density disorder produces the same attenuation scaling but does not renormalize the acoustic dispersion to leading order, all within the Born approximation for weak disorder whose results depend solely on the fluctuation variances.

What carries the argument

The disorder-induced self-energy evaluated in the leading Born approximation for the acoustic wave equation with fluctuating moduli and density.

If this is right

Both types of disorder produce Rayleigh-type attenuation Γ(q) proportional to q^{d+1} in d dimensions.
Elasticity disorder lowers the effective sound speed while density disorder does not.
The renormalization and attenuation depend only on the variances of the fluctuations.
Lattice simulations in one and two dimensions match the analytic predictions for both attenuation and dispersion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same framework could be used to predict how sound velocity softening scales with disorder strength in three-dimensional materials.
If spatial correlations in the disorder are introduced, the attenuation exponent or the velocity shift may change.
These results suggest a way to distinguish elasticity versus density heterogeneity by measuring both attenuation and speed in the same sample.

Load-bearing premise

The disorder is weak enough that the Born approximation captures the leading corrections and the fluctuations are spatially uncorrelated.

What would settle it

A measurement in a medium with only density disorder that shows either a shift in sound speed or attenuation scaling other than q to the d plus one in the long-wavelength limit.

Figures

Figures reproduced from arXiv: 2605.09962 by Bingyu Cui, Yuqi Wang.

**Figure 2.** Figure 2: FIG. 2. The propagation of the sound wave in relaxed 1D chains consisting of 20000 lattice points [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The propagation of sound waves in relaxed triangular spring networks consisting of 40000 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The propagation of sound waves in relaxed triangular spring bonds consisting of 40000 [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The propagation of the sound wave in relaxed 1D chains consisting of 20000 lattice points [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The propagation of sound waves in relaxed square spring bonds consisting of 40000 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗

read the original abstract

We derive analytically, and validate numerically, the dispersion renormalization and attenuation of acoustic waves propagating through quenched disordered media in the long-wavelength limit. We consider weak spatial fluctuations in elastic moduli and/or mass density and compute the disorder-induced self-energies within the leading (Born) approximation. For sufficiently weak disorder, the results depend only on the variances of the fluctuations and are therefore insensitive to the detailed form of the underlying random distribution. For spatially uncorrelated elasticity disorder we obtain Rayleigh-type attenuation, $\Gamma(q)\propto q^{d+1}$ , together with a reduction of the sound speed. In contrast, density disorder produces Rayleigh-type attenuation but does not renormalize the acoustic dispersion to leading order. Molecular dynamics simulations and normal-mode analyses of disordered one- and two-dimensional lattices quantitatively confirm the theoretical predictions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper cleanly separates elasticity disorder (which renormalizes sound speed downward) from density disorder (which does not) while both produce Rayleigh attenuation in the weak long-wavelength limit.

read the letter

The main point is that weak, uncorrelated fluctuations in elastic moduli produce both attenuation scaling as q to the d+1 and a reduction in sound speed, whereas density fluctuations give the same attenuation but no leading dispersion shift. This follows from the disorder-averaged self-energy in the Born approximation, with results depending only on the variances of the fluctuations rather than their detailed distribution. The authors back the analytics with molecular dynamics and normal-mode checks on one- and two-dimensional lattices that match the predictions quantitatively. That separation and the explicit forms are the concrete advance over prior scattering treatments. The calculation is standard perturbation theory applied to acoustic waves, but the distinction between the two disorder types is stated clearly and the numerics supply an independent check. The obvious limits are the restriction to long wavelengths and weak disorder where the Born term suffices, plus the assumption of delta-correlated fluctuations and the fact that the lattice tests stop at two dimensions. No higher-order terms or three-dimensional validation are given. Readers working on elastic or acoustic wave propagation in disordered condensed-matter systems will find the expressions useful for the Rayleigh regime. The work is narrow but the analytics plus supporting simulations are solid enough that it deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The manuscript derives the long-wavelength dispersion renormalization and attenuation of acoustic waves in quenched disordered media with weak spatial fluctuations in elastic moduli and/or mass density. Using the leading Born approximation for the disorder-averaged self-energy, it obtains Rayleigh-type attenuation Γ(q) ∝ q^{d+1} for both disorder types, together with a downward renormalization of the sound speed only for elasticity disorder (vanishing to leading order for density disorder). The analytic results depend only on the variances of the fluctuations and are confirmed quantitatively by molecular dynamics simulations and normal-mode analyses on one- and two-dimensional disordered lattices.

Significance. If the central claims hold, the work supplies a transparent, parameter-free analytic framework (depending solely on disorder variances) that cleanly distinguishes the effects of elasticity versus density disorder on acoustic propagation. The combination of a standard perturbative derivation with direct numerical validation on lattices provides an independent check and could serve as a useful reference for phonon attenuation in heterogeneous or amorphous materials.

minor comments (2)

The abstract and introduction would benefit from a brief statement of the precise definition of 'quenched' disorder and the continuum versus lattice formulation used in the analytic sections, to clarify the scope for readers unfamiliar with the subfield.
In the numerical sections, the reported disorder strengths (variances) should be explicitly compared to the analytic weak-disorder assumption to allow readers to assess the quantitative agreement range.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of the analytic framework and numerical validation, and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation computes the disorder-averaged self-energy to leading Born order for weak delta-correlated fluctuations in elastic moduli or density. The imaginary part of the self-energy produces the Rayleigh attenuation Γ(q)∝q^{d+1} via the on-shell phase-space integral in d dimensions, while the real part yields sound-speed renormalization only for elasticity disorder (vanishing for density disorder after principal-value integration). These expressions depend solely on the input variances, as required by the perturbative order, and are confirmed by independent molecular-dynamics and normal-mode simulations on 1D/2D lattices. No load-bearing step reduces to a self-citation, fitted parameter renamed as prediction, or self-definitional relation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the Born approximation applied to weak spatial fluctuations whose variances are treated as given inputs; no new entities are postulated and no parameters are fitted beyond the disorder variances themselves.

axioms (1)

domain assumption Leading-order Born approximation for disorder-induced self-energies is valid for sufficiently weak fluctuations
Invoked to compute renormalization and attenuation; stated to hold when disorder is weak enough that results depend only on variances.

pith-pipeline@v0.9.0 · 5426 in / 1236 out tokens · 28220 ms · 2026-05-12T04:10:03.276082+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For spatial dimension d≥2... Ω²_{L,T}(q) = [1 - Σ C_{ab} σ_{ab}²] ω₀²(q) ... Γ_{L,T}(q)∝q^{d+1}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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