{"id":"3113f09c-2e9a-4694-a5fc-796623627719","arxiv_id":"2411.13354","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors derive Helmholtz-Korteweg and nematic Helmholtz-Korteweg equations and study plane waves, evanescent waves, reflection, and scattering, but the penetration depth formula is wrong.","lead":"The paper derives a fourth-order wave equation for sound in Korteweg fluids and a nematic variant with orientation-dependent terms, then analyzes how evanescent waves and scattering change with boundary conditions and director orientation. It claims new experimental predictions for liquid crystal acoustics, but the central penetration depth formula contains a solvable algebra error that invalidates that prediction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (48) does not solve the quadratic (47), so the central penetration-depth prediction (49)-(50) is invalid; the corrected formula changes both the low-frequency behavior and the asymptotic prefactor.","rationale":"I rechecked the algebra in Section III. The reader is correct that equation (48) is not a solution of equation (47). Substituting the paper's expression into (47) yields a residual that is not identically zero; for τ1ω = 1, equation (47) evaluates to about -1.97, not 0. The correct root is α = 2(c/c0)²(1+√(1+τ1²ω²))/(τ1²ω²). This error directly invalidates the closed-form penetration depth (49) and its asymptotic form (50). The low-frequency limit changes qualitatively: the paper predicts divergence, while the corrected formula gives a finite depth c0τ1/2. The high-frequency scaling ω^{-1/2}τ1^{1/2} survives but with a prefactor of 1/√2 instead of 1/2, so even the asymptotic prediction is numerically wrong. This is an internal inconsistency, not a disagreement with consensus, and it affects the isotropic Korteweg case as well as the nematic case. I therefore agree with the reader's REJECT verdict. However, I disagree that the most load-bearing assumption is the ∇n=0 idealization; the quadratic error is more fundamental because it invalidates the paper's core quantitative claim even before nematic complications are introduced. The reader's weakest_assumption field points to ∇n=0, but their rationale and strongest_claim correctly identify the quadratic error; since the question is whether the weakest_assumption matches my identified concern, the answer is no. Thus agreement_with_reader is set to disagree, while the verdict is unchanged.","tokens_in":14272,"tokens_out":12078,"duration_ms":106811,"concrete_test":"Independently solve (47) for α using a computer algebra system (or by hand) and substitute the result into (46) to form δ; then compare δ at τ1ω = 1 and τ1ω = 10 with the paper's (49) and (50). If the values differ by more than 20% (or if the low-frequency limit is finite rather than divergent), the central penetration-depth prediction is refuted. Additionally, re-derive (47) from (36) with the paper's definition τ1 = 4ρ0√u1/c0² to check whether the coefficient should be 1/16 rather than 1/4; if so, (47) is also inconsistent with the physical dispersion relation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is in Section III. Equation (47) is the quadratic -1 - A α + (1/4)τ1²ω²A²α² = 0 with A=(c0/c)². The stated solution (48), α = 4(c/c0)²[-1±√(1+τ1²ω²)]/(τ1²ω²), does not satisfy this equation: substituting it leaves a residual that vanishes only for special values of τ1ω, not identically. The correct positive root of (47) is α = 2(c/c0)²[1+√(1+τ1²ω²)]/(τ1²ω²). Consequently the penetration depth (49) is wrong. With the corrected α, δ = c0τ1/√[2(1+√(1+τ1²ω²))], which as ωτ1→0 tends to the finite value c0τ1/2 rather than diverging like c0/(√2ω); for ωτ1≫1 it gives δ ∼ c0√(τ1/(2ω)), differing from (50) by a prefactor √2. The claimed scaling exponents survive only in the high-frequency limit, but the closed-form and low-frequency predictions are incorrect. Because the abstract advertises new, experimentally-verifiable predictions for penetration depth based on these formulas, this error undermines the central quantitative claim. The additional ∇n=0 assumption, noted by the reader, affects only the nematic generalization; the isotropic penetration depth error is already fatal.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a fourth-order linear PDE, called the Helmholtz–Korteweg equation, for time-harmonic acoustic disturbances in Korteweg fluids, and a nematic variant that includes an orientational stress contribution. It then analyzes plane-wave dispersion, evanescent-wave penetration depth, reflection at plane interfaces, and scattering by a circular obstacle, claiming new experimentally testable predictions for nematic-Korteweg fluids. The dispersion relations (36) and (51) are correctly derived and match Virga's earlier results. The central new quantitative claim, however, is the penetration-depth formula (49) and its asymptotic form (50).","tokens_in":14495,"tokens_out":6456,"duration_ms":59496,"significance":"If the derivation were correct, the paper would offer a compact fourth-order framework for time-harmonic acoustics in Korteweg and nematic-Korteweg fluids, with concrete predictions for evanescent waves and scattering that could guide experiments. The paper correctly reproduces Virga's dispersion relation and provides a plausible boundary-layer treatment for the circular obstacle, and it uses numerical simulations to illustrate the orientation dependence of scattering. These strengths are, however, outweighed by a concrete algebraic error in the penetration-depth calculation, which invalidates the paper's principal advertised quantitative prediction. The additional assumption that the director field is undistorted on acoustic scales, although inherited from prior work, is not adequately justified for the new scattering geometries considered.","major_comments":[{"comment":"Equation (48) does not solve the quadratic (47). Substituting α = 4(c/c0)²[-1+√(1+τ1²ω²)]/(τ1²ω²) into (47) leaves a residual -1 + 4(r-1)(r-2)/(τ1²ω²) with r = √(1+τ1²ω²), which is not identically zero. The correct positive root is α = 2(c/c0)²[1+√(1+τ1²ω²)]/(τ1²ω²), leading to δ = c0 τ1/√[2(1+√(1+τ1²ω²))]. This corrected δ tends to the finite value c0τ1/2 as ωτ1→0 instead of diverging, and its high-frequency asymptote is δ ∼ c0√(τ1/(2ω)), which differs from the paper's (50) by a factor √2. Because the abstract and Section III present (49)–(50) as new, experimentally verifiable predictions, this error directly undermines the central quantitative claim of the paper.","section":"Section III, Eqs. (47)–(50)"},{"comment":"The assumption ∇n = 0 removes from the nematic Helmholtz–Korteweg equation (24) all director-gradient terms, including ∇n∇S and (∇S·n)(∇·n). This assumption is inherited from Virga's planar-wave analysis, but the present paper applies the resulting equation to new settings, notably the circular-obstacle scattering problem in Section V and the director-discontinuity reflection in Figure 2, where director gradients are not obviously negligible. The authors do not justify that the director is effectively constant on acoustic wavelengths in these geometries, so the nematic dispersion relation (51) and the orientation-dependent scattering predictions are not fully established.","section":"Section II, before Eq. (25)"}],"minor_comments":[{"comment":"Equation (10) contains a spurious minus sign on the right-hand side: from the general wave equation (9), substituting the time-harmonic ansatz gives -ρ0ω²S = ∇·(∇·σ), so the second minus sign in (10) is an error. The final Helmholtz–Korteweg equation (13) is nevertheless correct, but the intermediate formula is misleading.","section":"Section I, Eq. (10)"},{"comment":"The text says 'divide by ρ0 e^{-ωt}', but the exponential in the ansatz (7) is e^{-iωt}; this should be corrected.","section":"Section I, after Eq. (12)"},{"comment":"There are several typographical and formatting issues in the equations, such as the spacing in τ1²ω² in (38) and the missing ℜ operator in (12). These should be cleaned up.","section":"Throughout"},{"comment":"The condition γ ≫ 1 ≫ ℓ mixes a dimensionless parameter γ with a parameter ℓ that presumably carries units of length. The statement should be made dimensionless or clarified.","section":"Section V, Eq. (64)"}],"recommendation":"reject","confidential_remarks":"The algebraic error in Section III is easy to verify by direct substitution and is not a mere typographical slip: the corrected formula changes the low-frequency behavior of the penetration depth from divergent to finite and changes the high-frequency prefactor by √2. Since the paper's abstract and conclusions advertise the penetration depth as a new experimental prediction, the central contribution is invalid as stated. The ∇n=0 assumption raises an additional, albeit secondary, concern for the nematic generalizations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the central quantitative claim is wrong. In Section III, the stated solution α (48) does not satisfy the quadratic (47). The positive root is α = 2(c/c0)^2[1+√(1+τ1²ω²)]/(τ1²ω²), not 4(c/c0)^2[-1+√(1+τ1²ω²)]/(τ1²ω²). Consequently the penetration depth (49) is incorrect: the corrected δ = c0τ1/√[2(1+√(1+τ1²ω²))] tends to the finite value c0τ1/2 as ωτ1→0, rather than diverging as (50) suggests, and the high-frequency prefactor differs by √2. The advertised experimentally-verifiable predictions for penetration depth are therefore not supported.\n\nThat said, the paper is not sloppy overall. The derivation of the Helmholtz–Korteweg equation (13) and its nematic variant (26) is careful, and the dispersion relations (36) and (51) match Virga's known results. The boundary-condition discussion (sound-soft, sound-hard, impedance) is a useful addition, and the singular-perturbation analysis leading to a Mie-series scattering problem in Section V is an interesting idea. The citation pattern looks fair; the one self-citation [26] is peripheral.\n\nThe soft spots beyond the main algebra error: the reflection coefficient expression around (59) is garbled and needs repair; the ∇n=0 assumption that removes director-gradient terms from the nematic equation is not justified for the scattering geometries, so the orientation-dependent claims rest on that assumption; and there is no code for the numerics, only a brief description. The sign typo in (10) is minor. The reader's stress-test note is correct: the isotropic penetration depth error is already fatal to the abstract's headline predictions, independent of the nematic issues.\n\nWho should read this: anyone working on nematic acoustics or Euler–Korteweg linearizations will want the corrected framework, but they should not rely on (48)–(50) as they stand. I would split the difference on review: this deserves a serious referee because the framework and derivations are valuable and the error is fixable, but the current version should not be accepted. If I were the editor I'd send it to peer review with a request to fix the algebra and re-derive the asymptotics.","headline":"The paper's framework is sound but its headline penetration-depth prediction is algebraically wrong, so the advertised experimentally-verifiable claims do not hold as stated.","tokens_in":15096,"tokens_out":4618,"would_cite":false,"duration_ms":44221,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76Q05","76A15"],"pacs":["61.30.Cz","62.60.+v"],"model":"deepseek-v4-flash","headline":"The paper derives the Helmholtz–Korteweg equation for time-harmonic waves in Korteweg fluids, adds a nematic variant with orientational coupling, and predicts evanescent-wave penetration depth and orientation-dependent scattering.","keywords":["Helmholtz–Korteweg equation","nematic liquid crystals","Korteweg fluids","evanescent waves","penetration depth","dispersion relation","acoustic scattering","boundary layers"],"falsifier":"Measure the penetration depth of an evanescent acoustic wave in an aligned nematic liquid crystal as a function of frequency and of the angle between the propagation direction and the director; the paper predicts $\\delta\\propto\\omega^{-1/2}$ for $\\omega\\tau_1\\gg1$ and a depth that is largest along the director. A depth that falls off exponentially with frequency, or an orientation dependence opposite to the prediction, would refute the central claim.","tokens_in":13944,"feed_emoji":"🌊","tokens_out":12390,"duration_ms":111158,"temperature":0.7,"pith_summary":"The paper aims to establish a fourth-order linear equation, the Helmholtz–Korteweg equation, for time-harmonic acoustic waves in Korteweg fluids (whose stress tensor depends on density gradients), together with a nematic variant that couples the wave to the local orientation of a liquid-crystal director. It shows that plane-wave solutions reproduce the dispersion relation known from an earlier variational nematoacoustic analysis, then extends the analysis to imaginary wavenumbers, where the waves are evanescent rather than propagating. The central quantitative prediction is that the penetration depth of an evanescent wave is inversely proportional to the square root of the frequency and proportional to the square root of the Korteweg time scale, with an explicit orientation dependence in the nematic case. If these predictions are right, the orientation of a nematic fluid can control both how far sound tunnels into a boundary and how strongly a sound wave scatters off an obstacle, which is testable in modified acoustic experiments.","feed_headline":"New wave equation predicts how far sound tunnels into liquids","feed_subtitle":"In Korteweg and nematic fluids, the depth scales as 1/√ω and peaks along the director.","key_machinery":"The central object is the Helmholtz–Korteweg operator $-\\omega^2 - c_0^2\\Delta + \\rho_0^2 u_1\\Delta^2$, a fourth-order linear operator whose nematic counterpart carries the extra term $\\rho_0^2 u_2\\nabla\\cdot\\nabla[\\mathbf{n}\\cdot H_S\\mathbf{n}]$. It turns the time-harmonic ansatz into a quartic dispersion relation $-\\omega^2 + c_0^2 k^2 + \\rho_0^2(u_1+u_2\\cos^2\\xi)k^4=0$, whose roots include real wavenumbers for propagating waves and purely imaginary wavenumbers for evanescent waves. The characteristic time $\\tau_1=4\\rho_0\\sqrt{u_1}/c_0^2$ sets the crossover: for $\\omega\\tau_1\\ll1$ the equation behaves like the classical Helmholtz equation, while for $\\omega\\tau_1\\gg1$ the fourth-order term dominates and yields the $\\delta\\propto\\omega^{-1/2}$ penetration law. Boundary conditions enter as paired conditions on $S$ and $\\Delta S$ with nematic corrections, and the circular obstacle is treated by a boundary-layer split into a Helmholtz solution plus an $O(\\ell^2)$ correction.","core_discovery":"The authors claim that a small time-harmonic condensation $S(\\mathbf{x})e^{-i\\omega t}$ in a Korteweg fluid obeys $-\\omega^2 S - c_0^2\\Delta S + \\rho_0^2 u_1 \\Delta^2 S = 0$, the Helmholtz–Korteweg equation (13), and that adding the nematic orientational stress changes the plane-wave dispersion relation to $-\\omega^2 + c_0^2 k^2 + (\\rho_0^2 u_1 + \\rho_0^2 u_2 \\cos^2\\xi)k^4 = 0$ (51), where $\\xi$ is the angle between the propagation direction and the nematic director. These dispersion relations match the earlier variational nematoacoustic result in the inviscid regime. The extension is to allow purely imaginary wavenumbers, giving an evanescent wave whose penetration depth is $\\delta = c_0/(2\\omega)\\bigl[(-1+\\sqrt{1+\\tau_1^2\\omega^2})/(\\tau_1^2\\omega^2)\\bigr]^{-1/2}$, asymptotic to $\\delta\\approx c_0\\sqrt{\\tau_1\\omega}/(2\\omega)$ for $\\omega\\tau_1\\gg1$. In the nematic case the effective time scale $\\tau_2(\\xi)$ makes both sound speed and penetration depth anisotropic, with maxima along the director. The authors further claim that a circular sound-soft obstacle scatters with reduced amplitude when the incident wave propagates orthogonally to the director, a boundary-layer effect captured as an $O(\\ell^2)$ correction to classical Mie scattering.","pith_inferences":["Extending the paper's results, a strongly anchored nematic cell, where the director bends near the walls, should show a spatially varying sound speed that the current $\\nabla\\mathbf{n}=0$ equation does not capture.","The $\\delta\\propto\\omega^{-1/2}$ scaling is distinctive enough that swept-frequency ultrasonic reflectometry on a single aligned cell could separate it from ordinary dissipative skin-depth behavior; the paper does not report such a measurement.","Extending the circular-scattering result, an external field that rotates the director could act as an acoustic switch, changing the scattered amplitude by changing the effective wavenumber; the paper notes tunability but does not quantify switching dynamics."],"forward_implications":["At high frequencies ($\\omega\\tau_1\\gg1$), evanescent sound in a Korteweg fluid penetrates a distance that shrinks as $\\omega^{-1/2}$ rather than vanishing exponentially with frequency.","In a nematic-Korteweg fluid the speed of sound is largest when propagation is parallel to the director, and evanescent penetration is deepest in that same direction, so rotating the director reorients both acoustic quantities.","For a circular sound-soft obstacle, the scattered amplitude is predicted to be reduced when the incident wave is orthogonal to the director, an anisotropy that does not appear at a plane boundary for sound-soft conditions.","The fourth-order boundary value problems split into paired conditions on $S$ and $\\Delta S$, so standard Helmholtz solvers can be adapted to the Korteweg case with modest changes.","As the Korteweg length $\\ell$ tends to zero, the scattering solution is a singular perturbation of classical Helmholtz scattering, with an $O(\\ell^2)$ boundary-layer correction, so ordinary acoustics predictions acquire small prescribed corrections."],"supporting_citations":[{"why":"Korteweg's constitutive stress tensor is the starting point whose linearization produces the Helmholtz–Korteweg equation.","marker":"[1]"},{"why":"Gives the nematic dispersion relation and the undistorted-director assumption that the paper adopts and extends to imaginary wavenumbers.","marker":"[5]"},{"why":"Provides the hyperelastic framework from which the Korteweg and nematic-Korteweg stress tensors are derived.","marker":"[6]"},{"why":"The sound-velocity experiment against which the predicted nematic speed anisotropy is checked.","marker":"[24]"},{"why":"Extends the acoustic-energy coupling to general waves, motivating the nematic term in the stress tensor.","marker":"[25]"},{"why":"Supplies the boundary-layer method used to derive the $O(\\ell^2)$ correction in the circular-scattering problem.","marker":"[29]"},{"why":"Supplies the Mie-series representation of the scattered field around a circular obstacle.","marker":"[30]"}],"fun_headline_variants":["Sound wave tunneling depth predicted in Korteweg fluids","Nematic fluids make sound penetration depth anisotropic","New wave equation predicts anisotropic sound tunneling in nematic fluids","Evanescent sound depth in Korteweg fluids follows a new frequency law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the liquid-crystal alignment direction is uniform over the acoustic length scale, so all terms containing spatial gradients of that direction are dropped from the nematic wave equation; if the alignment bends or twists on that scale, the predicted dispersion and orientation-dependent penetration depths change.","fun_headline_variants_meta":{"raw":{"variants":["Sound wave tunneling depth predicted in Korteweg fluids","Nematic fluids make sound penetration depth anisotropic","New wave equation predicts anisotropic sound tunneling in nematic fluids","Evanescent sound depth in Korteweg fluids follows a new frequency law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001199,"raw_usage":{"total_tokens":4997,"prompt_tokens":1050,"completion_tokens":3947,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":3887}},"tokens_in":666,"tokens_out":3947,"duration_ms":27015,"temperature":1.0,"reasoning_tokens":3887,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:32:40.959676+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the penetration depth of an evanescent acoustic wave in an aligned nematic liquid crystal as a function of frequency and of the angle between the propagation direction and the director; the paper predicts $\\delta\\propto\\omega^{-1/2}$ for $\\omega\\tau_1\\gg1$ and a depth that is largest along the director. A depth that falls off exponentially with frequency, or an orientation dependence opposite to the prediction, would refute the central claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Korteweg's constitutive stress tensor is the starting point whose linearization produces the Helmholtz–Korteweg equation."},{"cited_title":"(29) Sound-soft boundary conditions thus correspond to imposing homogeneous Dirichlet boundary condi- tions on S(⃗ x) and ∆S(⃗ x) = − u2 u1 ⃗ n· HS⃗ n","cited_arxiv_id":null,"evidence_quote":"Gives the nematic dispersion relation and the undistorted-director assumption that the paper adopts and extends to imaginary wavenumbers."},{"cited_title":"(32) Sound-hard boundary conditions thus correspond to imposing homogeneous Neumann boundary con- ditions on S(⃗ x) and ∂⃗ ν∆S(⃗ x) = − u2 u1 ∂⃗ ν ⃗ n· HS⃗ n","cited_arxiv_id":null,"evidence_quote":"Provides the hyperelastic framework from which the Korteweg and nematic-Korteweg stress tensors are derived."},{"cited_title":"Benzoni-Gavage, S","cited_arxiv_id":null,"evidence_quote":"The sound-velocity experiment against which the predicted nematic speed anisotropy is checked."},{"cited_title":"Mathematical Fluid Dynamics","cited_arxiv_id":null,"evidence_quote":"Extends the acoustic-energy coupling to general waves, motivating the nematic term in the stress tensor."},{"cited_title":"Giesselmann, C","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary-layer method used to derive the $O(\\ell^2)$ correction in the circular-scattering problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Mie-series representation of the scattered field around a circular obstacle."}],"review_version":1}