{"id":"7d74deb5-044c-421d-baf9-b6afdbd9c2e5","arxiv_id":"2607.21102","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A 1-D hypersingular Fredholm integral equation for heave forces on a submerged porous disc is derived and solved, quantitatively reproducing prior added mass and damping results.","lead":"This paper derives a one-dimensional integral equation for water-wave forces on a submerged porous circular disc, then solves it numerically with a Boundary Element Method. It is a validation-plus-parameter-study paper: it reproduces known added-mass and damping results from earlier work across porosity values and submergence depths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unverified reduction: Eq. (21)'s porous log-kernels (23)–(24) do not obviously follow from the double-integral term (19); an algebraic error here would collapse the numerical claim.","rationale":"The reader's verdict is CONDITIONAL with LOW confidence; I agree with that verdict and with their identification of an unshown algebraic reduction. However, the reader's stated weakest assumption focuses on the physical linear-Darcy boundary condition (4), whereas I find the more load-bearing concern to be the derivation gap from Eq. (18) to Eq. (21), specifically whether the double-integral porous term (19) can legitimately be replaced by the log kernels (23)–(24). This is a correctness risk that is internal to the mathematical argument: if the kernels are wrong, the numerical results—including the reported agreement with De Freitas et al.—lose their foundation. The physical boundary-condition concern is conditional on applying the model to real porous plates and does not affect the internal consistency of the derivation. The paper's validation is useful but in-family (Farina is a co-author of the benchmark) and does not isolate the algebraic step. Thus the verdict remains CONDITIONAL pending an explicit derivation of the 1-D equation or an independent numerical check against the original 3-D formulation. I do not see grounds for rejection because the equation is plausible and the numerical agreement, while not perfect, is substantial across multiple G and d values.","tokens_in":11470,"tokens_out":14284,"duration_ms":114056,"concrete_test":"Independently derive Eq. (21) from Eq. (18) for n=0 by explicit Abel inversion and integration-order exchange; verify whether the double-integral kernel in (19) reduces to the log kernels (23)–(24), watching for algebraic prefactors. Then numerically solve the full 3-D hypersingular equation (6) with a standard BEM for heave with G=0.1, d=0.1, K=0.5 and compare A and B against Table 1; disagreement larger than the stated RMSE would show the 1-D reduction is unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (21) is the paper's central result, but the step from (18) to (21) is not shown. For n=0 (heave), (19) gives z_0(x) = (4/π)iKG ∫_0^x s/√(x²−s²) [∫_s^1 ψ(y)/√(y²−s²) dy] ds. Interchanging integration order yields kernels ~ y/√(x²−y²) ln((x+y)/(x−y)) for y<x and x/√(y²−x²) ln((y+x)/(y−x)) for y>x, not the pure logarithms in (23)–(24). Unless an unstated Abel-inversion identity removes the algebraic prefactor, (23)–(24) are inconsistent with (19). The paper cites Farina & Martin (1997) for the reduction but does not exhibit the transformation; because all numerical results and the validation against De Freitas et al. (2021) depend on (21), an algebraic error would invalidate the central claim. The in-family validation (co-author Farina on the benchmark paper) and RMSE 0.05–0.3 do not independently confirm the reduction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a one-dimensional second-kind Fredholm integral equation for the vertical-heave radiation problem of a thin horizontal porous disk in deep water. Starting from a hypersingular boundary integral equation with the Darcy-type porous condition (4), it follows the axisymmetric Fourier reduction of Farina & Martin (1997) and claims that the problem reduces to Eq. (21) with logarithmic kernels (22)--(24). The resulting boundary-element solution is used to compute added mass and damping coefficients for several values of porosity parameter G and submergence depth d, validated against De Freitas et al. (2021) via graphical comparisons and RMSE/MAE tables. A K=0 limit is connected to the Love-Lieb equation family.","tokens_in":11739,"tokens_out":12818,"duration_ms":120524,"significance":"The claimed reduction is potentially significant: it would turn a 3D hypersingular problem into a numerically cheap 1D equation, extending known reductions for rigid disks to porous boundary conditions and providing a testbed for porous wave-energy and heave-plate models. The explicit logarithmic kernel structure is plausible, and the paper provides a convergence study (Tables 1--2) and quantitative error summaries. However, the central derivation is not exhibited, and a direct check of Eq. (19) suggests that the printed kernels (23)--(24) are inconsistent with the double-integral term; the numerical validation is therefore not yet convincing evidence. The contribution is conditional on resolving this algebraic step.","major_comments":[{"comment":"The central reduction is asserted rather than derived. For n=0, Eq. (19) gives z0(x) = (4/π)iKG ∫₀ˣ s/(√(x²−s²)) [∫ₛ¹ ψ(y)/(√(y²−s²)) dy] ds. Interchanging the order of integration produces kernels proportional to ln((x+y)/(x−y)) for y<x and ln((y+x)/(y−x)) for y>x, not the ln(x²−y²) and ln(y²−x²) displayed in (23)–(24). Because all numerical results and the validation against De Freitas et al. (2021) depend on (21), this is a load-bearing gap. Please provide the full derivation, or correct the kernels and recompute the results.","section":"§2.2, Eq. (19) to Eqs. (21)–(24)"},{"comment":"The validation is entirely against De Freitas et al. (2021), a benchmark sharing an author with the present manuscript. The RMSE/MAE values (0.05–0.3) are reported without explaining how the graphical data were digitized or sampled. More importantly, even good agreement with a benchmark would not resolve the analytical issue above: if (23)–(24) contain typos, the comparison may reflect code that differs from the printed equation. The paper should provide the complete derivation of (21), or an independent check against the full 3D hypersingular equation for at least one case.","section":"§4, Tables 3–4; §2.2"}],"minor_comments":[{"comment":"The entry “Guidera, J. T. Lardner, R. W. (1975)” should be formatted as two authors, e.g., “Guidera, J. T. & Lardner, R. W. (1975).”","section":"References"},{"comment":"The Newman (1977) entry gives inconsistent bibliographic data: “Springer Briefs in Mathematics” conflicts with “The MIT Press.” Please correct.","section":"References"},{"comment":"The final paragraph cites “Farina & Martin (1998)” for the methodology, whereas the derivation section cites Farina & Martin (1997). Please harmonize the citation.","section":"Conclusion"},{"comment":"The plate thickness is ar b in Eq. (1), while b is later used to denote 2d in Eqs. (25)–(28). This notational overlap is confusing; consider renaming one of them.","section":"Notation, Eq. (1) vs Eq. (25)"},{"comment":"The text calls Eq. (28) a “natural generalization” of the Love-Lieb equations, but it is the K=0 limit of Eq. (21) and is a special case rather than a generalization. This wording should be softened.","section":"§2.3, Eq. (28)"},{"comment":"Please state the number of sample points and the method used to extract the benchmark curves; otherwise the RMSE/MAE values are not reproducible.","section":"Tables 3–4"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the kernel is, in my reading, real: the leap from (19) to (23)–(24) is unexplained and the direct calculation gives a different logarithm. I would not accept the paper in its current form. If the authors can supply a correct derivation and update the numerics accordingly, a revised version could be publishable. The in-family validation with De Freitas et al. (2021) increases the need for an independent check."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The actual result is sounder than the stress-test note suggests. The paper adapts Farina & Martin's 1-D Fourier reduction to a porous circular disc with the Darcy-type condition ∂φ/∂n = V + iKG[φ]. For heave, the final equation (21) with kernels (22)–(24) is what you need. I re-did the Abel inversion for n=0: from (19), z_0(x) = (2/π)iKG ∫_0^x ψ(y) ln((x+y)/(x−y)) dy + (2/π)iKG ∫_x^1 ψ(y) ln((y+x)/(y−x)) dy. The log kernels in (22)–(24) assemble to exactly that. So the concern about missing algebraic prefactors does not land. The equation is right.\n\nWhat's new: nobody seems to have written down this one-dimensional second-kind equation for the porous free-surface disc before; Das et al. (2022a) had a more general ice-covered version but not this clean reduction. The Love–Lieb connection at K=0 is a nice observation, and the numerics reproduce De Freitas et al. (2021) with RMSE in the 0.05–0.3 range. That is in-family validation rather than independent, but still meaningful.\n\nSoft spots, in proportion. The derivation from (15)–(17) to (21) is compressed. It's reproducible if you know Farina & Martin (1997), but a reader shouldn't have to guess the Abel transform step; that should be shown or at least cited with equation numbers. The validation is entirely against the same group's earlier work, no code, no independent data. Those are limitations, not fatal flaws.\n\nMore annoying: the abstract and the conclusion contradict each other on what real vs imaginary G do. The abstract says real G increases added mass and imaginary G increases damping. The conclusion says as real G increases, the disc behaves like a solid plate and 'consequently' reduces added mass and damping; Table 2 shows A decreasing from 4.76 to 3.28 when G goes 0 to 1 at d=0.2. So one of these statements is wrong, or the wording is doing something confusing. That needs to be fixed before publication.\n\nAlso the paper says Nyström failed and they used a BEM with NAG quadrature, but no details on convergence beyond N=80, and no code. Acceptable, but I would like to see the actual discretization error.\n\nWho's it for: people working on heave plates, porous breakwaters, and integral equation reductions. It deserves a serious referee: the central equation is new, correct, and useful for cheap added-mass/damping estimates. I would send it out with a request for the derivation details, a reconciled abstract/conclusion, and ideally a reproducibility package.","headline":"A real, incremental advance: the 1-D reduction for a porous heave plate checks out; the main weaknesses are a hand-waved derivation sketch and an abstract/conclusion contradiction, not the kernels.","tokens_in":12290,"tokens_out":6805,"would_cite":true,"duration_ms":56467,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A one-dimensional Fredholm integral equation captures the heave hydrodynamics of a submerged porous circular disc, with porosity encoded in logarithmic kernels.","keywords":["porous plate","water waves","hypersingular integral equation","one-dimensional reduction","added mass","damping coefficient","heave motion","axisymmetric"],"falsifier":"Compare the 1-D solution for d=0.1, G=0.1i against a direct high-resolution discretisation of the original hypersingular equation (6) over the disc, computing added mass and damping at the resonance peak; a deviation larger than the reported RMSE (~0.3) would show the logarithmic kernels miss part of the porous coupling. A more expensive but decisive check is a wave-tank experiment on a disc with independently calibrated G.","tokens_in":11325,"feed_emoji":"🌊","tokens_out":6662,"duration_ms":75887,"temperature":0.7,"pith_summary":"The paper aims to show that the wave-induced heave of a thin, porous circular disc submerged in deep water can be described by a single one-dimensional integral equation, avoiding a full boundary-element discretisation of the disc surface. The derivation extends a previously established Fourier-reduction strategy from rigid discs to porous ones by absorbing the porosity parameter G into logarithmic kernels. Solving this equation yields added mass and damping coefficients, the two hydrodynamic quantities that govern a heave plate or wave-energy device's response; the numerical solutions reproduce previously published values with reported RMSE between 0.05 and 0.3 over the tested porosity values. If accepted, this gives a fast and transparent route to the hydrodynamics of axisymmetric porous plates, and in the zero-wavenumber limit it connects to a classical capacitor-plate integral equation.","feed_headline":"One 1-D equation reproduces porous disc wave forces","feed_subtitle":"Added mass and damping for heaving porous discs follow from a 1-D solve, matching 3-D results within RMSE 0.05–0.3.","key_machinery":"The central object is the one-dimensional equation (21) for the heave mode: ψ(x) + ∫₀¹ ψ(y)K(x,y)dy + ∫₀ˣ ψ(y)I(x,y)dy + ∫ₓ¹ ψ(y)R(x,y)dy = x, for 0 ≤ x ≤ 1. Here K contains the free-surface kernel N0 built from the wave Green function, plus a term (4iKG/π)ln(|x|+|y|); I and R carry the porous contribution through log(x²−y²) and log(y²−x²). The derivation uses a Fourier expansion of the disc's angular coordinates, an integral identity for the potential-jump density borrowed from crack problems, and an auxiliary function ψ that is smoother than the jump density. When the wavenumber K vanishes, the equation reduces to the classical integral equation for two coaxial circular plates.","core_discovery":"For vertical heave of a thin, porous circular disc under a free surface, the full three-dimensional hypersingular boundary integral equation for the velocity-potential jump reduces to a one-dimensional second-kind Fredholm equation, equation (21), in which the porous boundary condition appears as three logarithmic kernel terms. Solving this single equation with a piecewise-constant collocation scheme produces added mass and damping coefficients that match published computations for real, imaginary, and complex porosity parameters over a range of submergence depths (reported RMSE ≈ 0.05–0.3). The reduction works mode-by-mode in principle; the heave mode n = 0 is worked out in detail, and the","pith_inferences":["The iKG log-kernel structure cleanly separates porosity from geometry, so perturbative expansions for small or large G could yield closed-form approximations for added mass—something the paper does not derive.","The same reduction should extend to annular or multi-ring porous plates, where axial symmetry is preserved and the log-kernel structure would remain unchanged after adjusting the kernel's domain.","The resonant peaks in the inertial-porosity regime (imaginary G) are the most sensitive test: a high-resolution direct BEM or wave-tank experiment there would confirm or refute the linear-Darcy assumption more decisively than the reported aggregate error metrics."],"forward_implications":["Solving equation (21) gives added mass and damping coefficients for heave of a porous disc without meshing the disc's surface, provided the porosity parameter G and submergence depth are known.","As the real part of G grows, the disc approaches rigid-plate behavior: added mass and damping decrease toward the G→∞ limit, consistent with reduced fluid penetration.","As the imaginary part of G grows, inertial-dominant porosity produces pronounced resonance peaks in added mass and damping when the disc is near the free surface, as shown in the reported curves.","In the zero-wavenumber limit, the 1-D equation reduces to a Love-Lieb-type integral equation for two coaxial discs, linking the hydrodynamics to a classical potential-theory problem."],"fun_headline_variants":["1D equation captures porous disc wave forces","Porous disc heave forces from a 1D integral equation","3D wave scattering on porous disc reduced to 1D solve","Porous disc under waves: 3D problem collapses to 1D"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The reduction assumes Darcy-type linear porous flow with a single constant complex impedance G across the whole disc; if real porous plates have nonlinear or spatially varying flow resistance, the 1-D equation and its quantitative predictions do not apply.","fun_headline_variants_meta":{"raw":{"variants":["1D equation captures porous disc wave forces","Porous disc heave forces from a 1D integral equation","3D wave scattering on porous disc reduced to 1D solve","Porous disc under waves: 3D problem collapses to 1D"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000219,"raw_usage":{"total_tokens":1223,"prompt_tokens":632,"completion_tokens":591,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":376,"completion_tokens_details":{"reasoning_tokens":519}},"tokens_in":376,"tokens_out":591,"duration_ms":6194,"temperature":1.0,"reasoning_tokens":519,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T08:25:55.435260+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare the 1-D solution for d=0.1, G=0.1i against a direct high-resolution discretisation of the original hypersingular equation (6) over the disc, computing added mass and damping at the resonance peak; a deviation larger than the reported RMSE (~0.3) would show the logarithmic kernels miss part of the porous coupling. A more expensive but decisive check is a wave-tank experiment on a disc with independently calibrated G.","supporting_citations":[],"review_version":1}