{"id":"60f9e13f-3e12-439e-b415-2001da17a64c","arxiv_id":"2502.00540","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A boundary element method with a new Green's function kernel solves the mild-slope equation for waves over unidirectional variable bathymetry.","lead":"Scientists built a boundary element method for ocean waves moving over a seabed whose depth changes in one direction. It reproduces shoaling, refraction, diffraction and reflection, and is checked against analytical and experimental wave data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (55) silently drops the ∂√W/∂n term when converting the modified-potential BIE (50) to original variables; for boundaries with an x-normal (cylinder, reflecting walls) the physical no-flux condition is misrepresented.","rationale":"The reader's weakest assumption concerns the accuracy of the approximated inverse Fourier transform of the variable-depth kernel. That is a legitimate numerical concern, but the derivation of the boundary integral equation exposes a more fundamental, and more easily testable, gap: the conversion from the Helmholtz-form BIE (50) to the original-variable system (55) is not a simple substitution unless ccg is constant along the boundary normal. Since the paper never defines H and G, a reader cannot verify that the physical no-flux condition q=0 was actually enforced in the cylinder, wall, and FEM-BEM coupling examples. The constant-depth comparison cannot rule this out because the extra term vanishes identically there. I am not asserting that the authors necessarily made this mistake; if their H and G include the W_n term, the omission is purely a missing derivation. But as written, the central claim is not verifiable from the manuscript, and the proposed test would settle it. This is why the verdict should move to UNVERDICTED rather than ACCEPT or REJECT.","tokens_in":18664,"tokens_out":28207,"duration_ms":296130,"concrete_test":"Recompute the variable-depth circular-cylinder case of Section 5.2 with the same kernel but with the corrected boundary operator: q_hat = sqrt(W) q + (W_n/(2W)) phi_hat on the cylinder, i.e., absorb the term -G_hat W_n/(2W) into the H matrix. Compare the y=0 wake profile in Figure 14 with the version obtained from the literal reading q_hat = sqrt(W) q. If the two profiles differ by more than a few percent, the published H phi = G q system enforces the wrong no-flux condition and the variable-depth scattering results do not support the central claim. A more analytic test is oblique reflection from a straight vertical wall over the Section 5.1 depth profile, where a mode-matching solution gives the reflection coefficient; the omitted term changes it by O((W_n/W) lambda).","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Eq. (55) is not established from the preceding equations. The BIE (47)-(50) is written for the transformed potential phi_hat and its normal derivative q_hat, i.e. the Helmholtz variable. With W=ccg, Eq. (4) gives phi_hat = sqrt(W) phi. Hence q_hat = ∂phi_hat/∂n = sqrt(W) q + (∂sqrt(W)/∂n) phi = sqrt(W) q + (W_n/(2W)) phi_hat. The text says only that 'back-substituting the change of variable introduced in (4)' turns (50) into H phi = G q (55), and it never defines H and G. A simple replacement phi_hat -> sqrt(W) phi, q_hat -> sqrt(W) q is valid only when ∂sqrt(W)/∂n = 0. For a vertical cylinder, or any reflecting boundary whose normal has an x-component, ∂sqrt(W)/∂n ≠ 0 in a unidirectionally variable bathymetry, and the omitted term must be absorbed into H as an extra Robin contribution. If it is not, the rigid-cylinder condition q=0 is imposed instead as q_hat=0, which is a different physical condition, and the reflecting walls and BEM-FEM coupling in Sections 5.3 and 5.4 inherit the same error. The constant-depth validation in Section 3.3 cannot detect this because W is constant there. Thus the central claim rests on a boundary-operator conversion that the submitted text does not derive; if H and G were intended to include the extra term, that definition must be supplied.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a boundary element formulation for the elliptic mild-slope equation in domains where the water depth varies smoothly in a single horizontal direction. The core contribution is a numerical fundamental solution: taking a Fourier transform in the direction parallel to the depth contours reduces the problem to a family of one-dimensional Helmholtz equations with radiation conditions, which are solved by a Galerkin FEM; the inverse transform is evaluated on a deformed contour in the complex Fourier plane using FFTs and asymptotic tail corrections, producing the kernel and its derivatives in Eqs. (24), (27), and (31). The kernel is validated in constant depth against Hankel functions. The BEM is then applied to a shoaling channel, wave scattering by a vertical cylinder in constant and variable depth, an elliptic shoal on a sloping bottom via BEM-FEM coupling, and a harbor resonance example. The paper claims that the formulation reproduces shoaling, refraction, diffraction, and reflection for bathymetric slopes up to 1:3.","tokens_in":18934,"tokens_out":10280,"duration_ms":104318,"significance":"If the central claims are correct, the work offers a practical BEM alternative to domain discretizations for coastal wave propagation over straight, parallel contours, with an extension to locally two-dimensional bathymetries through BEM-FEM coupling. The paper is commendably explicit: the kernel derivation is written out step by step, the constant-depth kernel is checked against the Hankel fundamental solution, the cylinder scattering is checked against the McCamy-Fuchs solution, and the elliptic-shoal results are compared with laboratory data and an independent MMSE solution. No parameters are fitted to the target results; the numerical parameters tau, Xi, and M are fixed a priori. The principal weakness is that the conversion of the boundary integral equation back to original variables is not derived, and several truncations in the inverse-transform evaluation are used without error estimates or a variable-depth convergence study.","major_comments":[{"comment":"The passage from the modified-potential system (50) to H phi = G q is not established. With W = ccg, Eq. (4) gives phi_hat = sqrt(W) phi and hence q_hat = partial phi_hat / partial n = sqrt(W) q + (partial sqrt(W) / partial n) phi. The extra term is nonzero on any boundary segment whose normal has an x-component, which includes the cylinder boundary in Section 5.2, the BEM-FEM interface in Section 5.3, and the harbor quay walls in Section 5.4. If the matrices H and G in (55) are meant to absorb this term, their definitions must be supplied; if they are not, the rigid-wall condition q = 0 is replaced by q_hat = 0, which is not the same physical condition when partial sqrt(W) / partial n is nonzero. The constant-depth comparison in Section 3.3 cannot detect this because W is constant there. This point is load-bearing for all variable-depth examples.","section":"4.1 (Eq. (55))"},{"comment":"The inverse-transform derivation relies on unquantified approximations: the integrals along C1+ and the hyperbolic-sine terms in (23), (25), and (28) are dropped for small tau, and the semi-infinite tail is replaced by the asymptotic form (12) with Xi = 6 k_hat* and M = 4096 set by experience. No error estimate or convergence study is provided for a case with genuinely variable kappa(x); the constant-depth validation in Section 3.3 exercises the FFT and contour treatment but not the variable-coefficient one-dimensional solver, since the constant-depth problem has the exact solution (11). Because the kernel appears in every BEM matrix entry, the claimed accuracy for slopes up to 1:3 depends on these truncations being benign. Please add a systematic study of the kernel error as a function of tau, Xi, and M on a nontrivial variable-depth profile, or an a posteriori check against a highly resolved reference solution for one of the Section 5 examples.","section":"3.2.1 (Eqs. (23)-(31))"},{"comment":"The incident field (59) used in the scattering formulation is a WKB/geometric-optics approximation to the MSE solution, not an exact solution of the modified Helmholtz equation (5) for phi_hat. The BIE (57) is exact only when phi_hat_in satisfies the Helmholtz equation with the same variable coefficient; otherwise the formulation solves a scattering problem with an inconsistent free term. The discrepancy may be small for mild slopes but should be stated and estimated, especially since the variable-depth cylinder results in Section 5.2 have no independent reference and the 'analytical solution' used in Section 5.1 is the same WKB expression (59).","section":"4.2 and Section 5.2"}],"minor_comments":[{"comment":"The affiliation contains a typo: 'Ténica' should be 'Técnica'.","section":"Title page"},{"comment":"The phrase 'an substituting back' should read 'and substituting back'.","section":"Section 3.2.1, Eq. (26)"},{"comment":"The labels xi2 = c and xi2 = d on the integration path are introduced but never defined in the text.","section":"Figure 2"},{"comment":"The reference solution (59) is called 'the analytical solution'; it is actually a WKB/geometric-optics approximation and should be described as such.","section":"Section 5.1"},{"comment":"The harbor example is an application without an independent validation or error measure; the abstract's claim of 'excellent agreement' should be restricted to the validated examples.","section":"Section 5.4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be a reprint of the authors' 2016 EABE paper, as indicated by the footnote; the arXiv posting should be transparent about this. The concern about Eq. (55) is substantive: if the implementation actually uses the correct transformation, the text can be repaired by defining H and G, but if not, the variable-depth results may need to be recomputed. The inverse-transform truncation issue is also real and should be addressed with a numerical convergence study."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good news first: this paper gives coastal engineers something they did not have before—a boundary-only BEM for the mild-slope equation with unidirectional variable bathymetry. It builds on Belibassakis's Green's function but contributes the missing x- and y-derivative kernels, recasts them as symmetric FFT integrals, and validates the scheme on several standard problems. The constant-depth kernel check against the Hankel solution is clean, and the shoaling-channel, cylinder, elliptic-shoal, and harbor examples are the right kind of evidence. For a methods paper, the derivations in Sections 3.2.1–3.2.3 are unusually explicit; I could follow the path from (23) to (31) without much guesswork.\n\nThe biggest hole is the step from (50) to (55). The BIE is derived for the transformed potential φ̂ and its normal derivative q̂. After the change φ̂ = √W ϕ, the relation between q̂ and q is q̂ = √W q + (∂√W/∂n) ϕ, not just √W q. The paper simply says \"back-substituting\" and writes Hϕ = Gq, without ever defining the new H and G. If H and G are just the old matrices with nodal values rescaled, the ∂√W/∂n term is silently dropped, and the zero-flux condition on reflecting walls and the cylinder is misrepresented—exactly the situations where the boundary normal has an x-component. The constant-depth test cannot catch this because W is constant. I suspect the authors' actual code includes the term (the validations look too good otherwise), but the paper needs to show the algebra. This is not a fatal flaw, but it is a load-bearing derivation that must be supplied.\n\nSecond, the inverse-transform approximation rests on unquantified truncations: dropping the C1+ and part of the C2+ integrals for small τ, and replacing the tail with asymptotic formulas. The parameters τ = Δξ, Ξ = 6k̂*, M = 4096 are stated but not justified by convergence or sensitivity studies, and the kernel itself is never benchmarked against an independent variable-depth Green's function. These are addressable in revision.\n\nThird, the paper ships no code or data, and comparisons are mainly visual; there are no error tables or convergence rates. For a methods paper that is not disqualifying, but it makes independent verification harder. The citation pattern is fine; the only self-reference is a prior BEM-FEM paper that is not load-bearing.\n\nWho should read this: researchers and engineers working on BEM for wave problems, especially harbor and coastal applications. The central idea is sound and the examples are encouraging. I would send it to review, with the requirement that the authors derive Eq. (55) properly and provide a sensitivity analysis for the kernel parameters.","headline":"Useful boundary-only MSE solver with an explicit kernel derivation and decent validation, but the back-substitution in Eq. (55) is under-derived and needs attention.","tokens_in":19505,"tokens_out":7033,"would_cite":false,"duration_ms":67823,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76B15","65N38"],"pacs":[],"model":"deepseek-v4-flash","headline":"A complete boundary-element kernel solves the mild-slope equation over one-directional variable depths.","keywords":["wave propagation","mild-slope equation","Helmholtz equation","boundary element method","fundamental solution","Green's function","variable bathymetry","shoaling"],"falsifier":"Evaluate the kernel from equations (24), (27) and (31) for an exactly solvable non-constant depth profile whose one-dimensional Fourier-domain problem has a closed-form solution, and compare against the exact Green's function over a range of $\\xi$ truncations; if the error does not shrink as $N$, $\\Xi$, and $M$ increase, or if the kernel error exceeds the reported BEM discrepancy on the shoaling channel, the central claim would be refuted. A simpler observational check would be to run the full BEM on a 1:3 sloping channel with much finer boundary meshes and confirm that the wave amplification factor converges to the analytical solution rather than deviating systematically.","tokens_in":18408,"feed_emoji":"🌊","tokens_out":7837,"duration_ms":70525,"temperature":0.7,"pith_summary":"The paper sets out to show that the Mild-Slope Equation for water waves over a seabed whose depth varies smoothly in one preferred direction can be solved by a boundary element method built on a complete fundamental-solution kernel. It constructs the kernel by Fourier-transforming in the alongshore direction, solving a family of one-dimensional wave problems over the variable-depth interval, and returning to physical space with FFT-based integrals plus analytic far-field tails. This kernel is then fed into a standard boundary element scheme, and tested on a shoaling channel, wave scattering by a cylinder, an elliptic shoal on a sloping bottom, and a harbor. If the claim is right, the method offers a natural way to handle open and partially reflecting boundaries for depth-varying coastal regions, reproducing shoaling, diffraction, refraction and reflection on slopes up to 1:3.","feed_headline":"Mild-slope waves solved by a boundary-element kernel","feed_subtitle":"It reproduces shoaling, diffraction, refraction and reflection on seabed slopes up to 1:3.","key_machinery":"The central object is the fundamental solution (Green's function) of the reduced Helmholtz equation $\\nabla^2\\hat{\\phi} + \\hat{k}(x)^2\\hat{\\phi} = 0$, where $\\hat{k}$ is the local modified wave number derived from water depth through the dispersion relation. The construction transforms the equation in the variable $y$ parallel to the depth contours, obtaining a one-dimensional radiating problem in $x$ for each Fourier parameter $\\xi$; this problem is solved with a Galerkin finite element scheme on the interval where depth varies and matched to constant-depth analytic solutions outside. The inverse Fourier transform is evaluated along an antisymmetric contour in the complex $\\xi$-plane that is offset by a small negative imaginary shift, split into a short segment, a finite FFT interval, and a semi-infinite tail. The decisive simplifications are to drop the short-segment integrals for small $\\tau$ and to evaluate the tail with asymptotic formulas, so the kernel's accuracy rests on how well these represent the true transform.","core_discovery":"The central claim is that equations (24), (27) and (31) form a complete fundamental-solution kernel for the Helmholtz-type form of the Mild-Slope Equation with a modified wave number depending on x alone. The completeness matters because the boundary element method needs both the kernel and its normal derivative evaluated on the boundary, and previous suggestions supplied only an approximation for the potential itself. Once the kernel and the two derivatives are available, the direct boundary integral equation and its scattering version follow in the usual way. The paper reports agreement with the analytical shoaling solution in a channel, with the analytic cylinder diffraction pattern at constant depth, and reasonably good agreement with laboratory data for the elliptic-shoal problem, showing combined shoaling, refraction and diffraction behind a cylinder on a sloping seabed.","pith_inferences":["A direct kernel-level check against an exactly solvable variable-depth profile (for example a profile whose one-dimensional Fourier problem has a closed form) would separate kernel approximation error from BEM discretization error; the paper validates the kernel only for constant depth.","The fixed parameter choice $\\tau = \\Delta\\xi$, $\\Xi = 6\\hat{k}^*$, and $M = 4096$ is empirical; mapping the sensitivity of the kernel to these parameters across a range of profiles would show when the dropped short-segment integrals and the asymptotic tails cease to be negligible.","The accuracy ceiling of 1:3 slopes is inherited from the Mild-Slope Equation itself, so beyond that slope any error is likely dominated by the physical model rather than by the numerical kernel.","Because the transform method uses only the one-dimensional profile between the two constant-depth half-spaces, it should also work for piecewise-smooth profiles if the one-dimensional problem in the middle is solved with enough resolution; the paper demonstrates monotone smooth profiles only."],"forward_implications":["A boundary element method equipped with this kernel reproduces shoaling, diffraction, refraction and reflection for smooth one-directional depth profiles with slopes up to 1:3.","Open and partially reflecting boundaries can be represented without the absorbing-boundary machinery that finite element and finite difference schemes need, because the fundamental solution already obeys the radiation condition.","The method supplies a variable-depth incident field and free-term vector for scattering problems, so obstacle and harbor response can be computed directly in the open-domain formulation.","The same Helmholtz reduction is compatible with modified mild-slope models, so the kernel construction extends beyond the classical Mild-Slope Equation by changing the expression of the modified wave number.","The kernel enables BEM-FEM coupling for bathymetric irregularities in two directions, as demonstrated by the elliptic-shoal verification."],"supporting_citations":[{"why":"supplies the Green's function approximation and FFT evaluation procedure that this kernel construction extends.","marker":"[3]"},{"why":"introduces the Mild-Slope Equation that is the target model.","marker":"[9]"},{"why":"provides the change of variable that rewrites the Mild-Slope Equation as a Helmholtz equation.","marker":"[7]"},{"why":"provides the complex-plane inverse transform technique used to reduce aliasing.","marker":"[4]"},{"why":"establishes the 1:3 slope accuracy range that the paper adopts as its target regime.","marker":"[13]"},{"why":"supplies the analytical incident-wave solution used as the reference and free-term input in the shoaling and scattering tests.","marker":"[44]"},{"why":"gives the analytic cylinder diffraction solution used to validate the constant-depth case.","marker":"[41]"},{"why":"provides the laboratory data used for the elliptic-shoal verification.","marker":"[8]"},{"why":"provides the coupled-mode numerical results used as a second comparison for the elliptic-shoal case.","marker":"[5]"},{"why":"gives the special quadrature used to evaluate the weakly singular boundary integrals in the BEM.","marker":"[20]"}],"fun_headline_variants":["Boundary element kernel tames mild-slope waves over variable seabeds","New BEM kernel for mild-slope waves on sloping bathymetry","BEM formulation solves mild-slope equation for slopes up to 1:3","Boundary element method captures all wave phenomena on 1:3 seabed","Complete fundamental kernel enables BEM for variable-depth waves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the approximations used to assemble the kernel—dropping the short contour integrals for small $\\tau$ and replacing the semi-infinite tail by asymptotic formulas—stay accurate for every smooth depth profile in the claimed slope range, even though the kernel itself is only directly checked against a constant-depth solution.","fun_headline_variants_meta":{"raw":{"variants":["Boundary element kernel tames mild-slope waves over variable seabeds","New BEM kernel for mild-slope waves on sloping bathymetry","BEM formulation solves mild-slope equation for slopes up to 1:3","Boundary element method captures all wave phenomena on 1:3 seabed","Complete fundamental kernel enables BEM for variable-depth waves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000775,"raw_usage":{"total_tokens":3379,"prompt_tokens":845,"completion_tokens":2534,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":461,"completion_tokens_details":{"reasoning_tokens":2439}},"tokens_in":461,"tokens_out":2534,"duration_ms":16531,"temperature":1.0,"reasoning_tokens":2439,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T18:36:32.375027+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the kernel from equations (24), (27) and (31) for an exactly solvable non-constant depth profile whose one-dimensional Fourier-domain problem has a closed-form solution, and compare against the exact Green's function over a range of $\\xi$ truncations; if the error does not shrink as $N$, $\\Xi$, and $M$ increase, or if the kernel error exceeds the reported BEM discrepancy on the shoaling channel, the central claim would be refuted. A simpler observational check would be to run the full BEM on a 1:3 sloping channel with much finer boundary meshes and confirm that the wave amplification factor converges to the analytical solution rather than deviating systematically.","supporting_citations":[{"cited_title":"Belibassakis, The Green’s function of the mild-slope equation: The case of a monotonic bed profile, Wave Motion 32 (2000), 339–361","cited_arxiv_id":null,"evidence_quote":"supplies the Green's function approximation and FFT evaluation procedure that this kernel construction extends."},{"cited_title":"Berkhoff, Computation of combined refraction - diffraction, Proceedings of 13th International Conference on Coastal Engineering, ASCE, 1972","cited_arxiv_id":null,"evidence_quote":"introduces the Mild-Slope Equation that is the target model."},{"cited_title":"Bergmann, The equation in a medium with a variable index of refraction, The Journal of the Acoustical Society of America 17 (1946), no","cited_arxiv_id":null,"evidence_quote":"provides the change of variable that rewrites the Mild-Slope Equation as a Helmholtz equation."},{"cited_title":"Belibassakis and G.A","cited_arxiv_id":null,"evidence_quote":"provides the complex-plane inverse transform technique used to reduce aliasing."},{"cited_title":"Booij, A note on the accuracy of the mild-slope equation, Coastal Engineering 7 (1983), no","cited_arxiv_id":null,"evidence_quote":"establishes the 1:3 slope accuracy range that the paper adopts as its target regime."},{"cited_title":"Panchang, B","cited_arxiv_id":null,"evidence_quote":"supplies the analytical incident-wave solution used as the reference and free-term input in the shoaling and scattering tests."},{"cited_title":"McCamy and R.A","cited_arxiv_id":null,"evidence_quote":"gives the analytic cylinder diffraction solution used to validate the constant-depth case."},{"cited_title":"Berkhoff, N","cited_arxiv_id":null,"evidence_quote":"provides the laboratory data used for the elliptic-shoal verification."},{"cited_title":"Belibassakis, G.A","cited_arxiv_id":null,"evidence_quote":"provides the coupled-mode numerical results used as a second comparison for the elliptic-shoal case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the special quadrature used to evaluate the weakly singular boundary integrals in the BEM."}],"review_version":1}