Pith. sign in

REVIEW 3 major objections 5 minor 56 references

Boundary element formulation of the Mild-Slope Equation for harmonic water waves propagating over unidirectional variable bathymetries

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A complete boundary-element kernel solves the mild-slope equation over one-directional variable depths.

desk verdict 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. read the letter →

arxiv 2502.00540 v1 pith:SXOMBT6I submitted 2025-02-01 math.NA cs.NAphysics.flu-dyn

Antonio Cerrato , José A. González , Luis Rodríguez-Temblequer This is my paper · ORCID
classification math.NAcs.NAphysics.flu-dyn MSC 76B1565N38
keywords wavepropagationmild-slopeequationHelmholtzboundaryelementmethodfundamentalsolutionGreen'sfunctionvariablebathymetryshoaling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [4.1 (Eq. (55))] 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.
  2. [3.2.1 (Eqs. (23)-(31))] 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.
  3. [4.2 and Section 5.2] 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).
minor comments (5)
  1. [Title page] The affiliation contains a typo: 'Ténica' should be 'Técnica'.
  2. [Section 3.2.1, Eq. (26)] The phrase 'an substituting back' should read 'and substituting back'.
  3. [Figure 2] The labels xi2 = c and xi2 = d on the integration path are introduced but never defined in the text.
  4. [Section 5.1] The reference solution (59) is called 'the analytical solution'; it is actually a WKB/geometric-optics approximation and should be described as such.
  5. [Section 5.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fundamental-solution derivation is independent and externally benchmarked; the one author self-citation is not load-bearing.

full rationale

The derivation is self-contained. Equations (7)-(31) construct the fundamental solution from the Fourier-transformed one-dimensional BVP (9)-(16), solved by the independent FEM system (40)-(44); no parameter is fitted to any target output, and the constants tau, Xi, and M are numerical quadrature choices validated against the constant-depth Hankel kernel in Section 3.3 (Eq. (45)). The BIE (47)-(50) is the standard reciprocity relation, and the matrix coefficients (51)-(52) are defined directly from the kernel. The central claims are benchmarked externally: Section 5.1 against the WKB shoaling solution (59), Section 5.2 against the McCamy-Fuchs analytical cylinder solution, Section 5.3 against Berkhoff's laboratory data and an independent MMSE coupled-mode model, and Section 5.4 is an application study. The only author self-citation is [47], used for the BEM-FEM coupling in Section 5.3; it is an application-level numerical strategy and is not the load-bearing derivation of the kernel or of the BEM equations. The under-specified back-substitution at Eq. (55) is a derivation gap or correctness risk, not a circularity, because the target solution is never used to define the kernel or the system.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central derivation is not fitted to any target solution; the listed free parameters are numerical tuning choices (contour offset, truncation, FFT size, mesh density) chosen by hand. No new physical entity is introduced. The main external input is Belibassakis's Green's function approximation, treated as a domain assumption. The MSE and WKB incident-wave approximations are also domain assumptions inherited from linear wave theory.

free parameters (4)
  • tau (contour offset) = tau = Delta xi = Xi/N
    Complex contour offset in the Fourier plane; the small-tau assumption is used to drop several integrals in the inverse transform. Section 3.2.2.
  • Xi (Fourier truncation) = 6*khat_max
    Truncation of the Fourier integral, chosen large enough that the asymptotic tail applies. Section 3.2.2.
  • M (FFT sample count) = 4096
    Number of FFT points over the interval [-Xi, Xi], selected as a power of two. Section 3.2.2.
  • elements per wavelength = at least 20
    Mesh resolution rule for the 1D FEM and the BEM meshes; influences accuracy. Sections 3.2.3 and 5.
assumptions (5)
  • domain assumption The mild-slope equation accurately describes linear water waves over slowly varying bathymetry, with slopes up to about 1:3 for general incidence.
    Core model in Eq. (2); validity limits from Berkhoff and Booij are cited in Section 1.
  • standard math Bergmann's change of variable (4) recasts the MSE exactly as a Helmholtz equation with modified wavenumber (6), for sufficiently smooth depth profiles.
    Section 2, Eqs. (4)-(6).
  • domain assumption The complex-plane inverse Fourier transform approximations, including the neglect of integrals along C1+ and portions of C2+ and the asymptotic tail, are accurate for the chosen parameters tau and Xi.
    Section 3.2.1, Eqs. (23)-(31); directly validated only for constant depth in Section 3.3.
  • domain assumption Outside the interval [a,b] the depth is constant, so the Sommerfeld conditions (14) and analytic continuations (15)-(16) apply.
    Section 3, Eqs. (13)-(16).
  • domain assumption The WKB-type incident wave (59) with shoaling and refraction coefficients is a valid representation of the incoming field for very slowly varying bathymetry.
    Section 5, Eq. (59); used in the channel and scattering examples.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Boundary element formulation of the Mild-Slope Equation for harmonic water waves propagating over unidirectional variable bathymetries." pith.science (2026). https://pith.science/paper/SXOMBT6I

@misc{pith2026250200540,
  author       = {Pith},
  title        = {Pith review of: Boundary element formulation of the Mild-Slope Equation for harmonic water waves propagating over unidirectional variable bathymetries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXOMBT6I}},
  note         = {Machine review of arXiv:2502.00540}
}
read the original abstract

This paper presents a boundary element formulation for the solution of the Mild-Slope equation in wave propagation problems with variable water depth in one direction. Based on the Green's function approximation proposed by Belibassakis \cite{Belibassakis2000}, a complete fundamental-solution kernel is developed and combined with a boundary element scheme for the solution of water wave propagation problems in closed and open domains where the bathymetry changes arbitrarily and smoothly in a preferential direction. The ability of the proposed formulation to accurately represent wave phenomena like refraction, reflection, diffraction and shoaling, is demonstrated with the solution of some example problems, in which arbitrary geometries and variable seabed profiles with slopes up to 1:3 are considered. The obtained results are also compared with theoretical solutions, showing an excellent agreement that demonstrates its potential.

Figures

Figures reproduced from arXiv: 2502.00540 by the authors.

Figure 1
Figure 1. Wave number variation in the x-direction for a fixed wave frequency due to a monotonically decreasing water [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Antisymmetric integration path C in the complex plane avoiding the ±ˆk1 and ±ˆk3 roots lying on the real axis 3.1 Inverse transform in the complex plane To compute the inverse Fourier transforms (IFT) given by equations (17-19), Belibassakis et al. [3, 4] propose a truncation of the infinite interval of integration and to use a discrete Fast Fourier Transform (FFT) algorithm that approximates the integral on a finit… view at source ↗
Figure 3
Figure 3. Approximation and decomposition into linear paths of the integration path in the complex [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Comparison of analytical and numerical fundamental solution [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Comparison of analytical and numerical fundamental solution derivatives [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Behavior of the Green’s function ψ(r, ro; ˆk) (thin solid line), the kernel ℘( ˆkor) (dashed line) and the regularized integrand (thick solid line). The source point is located at xo = 60m and the bathymetry is defined in Section 5.1 with a wave period T = 5s an equati…
Figure 7
Figure 7. Figure 7: Water depth h(x) is supposed to decrease monotonically along the channel from 14m to 0.5m between x = 0m and x = 70m. This means that, for the considered initial wave-length of 39 m, waves travel from intermediate water-depths to shallow waters. In the transition zone,…
Figure 7
Figure 7. Figure 7: Channel with variable water depth h(x) in the longitudinal direction. Model of the closed rectangular domain (a). Boundary conditions and associated wave-number k(x) in the x-direction (b) 15 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Shoaling effect in a channel with monotonically decreasing water depth. Real and imaginary parts of the wave [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 10
Figure 10. Figure 10: As expected, we obtain a diffraction pattern that decays inversely with the square root of the radial distance [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 9
Figure 9. Figure 9: Rigid circular cylinder mounted on waters with variable depth [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Interference patterns obtained using BEM for a cylinder in constant water depth [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Wave scattering by a cylinder in constant water depth. Comparison with the analytical solution of the [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: BEM solution for a decreasing water depth in the [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: BEM solution for a decreasing water depth in the [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Wave scattering by a cylinder in variable water depth. Normalized magnitude of the velocity potential [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: Scattering produced by an elliptic bank on a sloping bottom. Absolute value of the normalized wave height [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Normalized wave height along a longitudinal section A-A’ (top) and a transversal section B-B’ (bottom), see [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: Nearshore bathymetric mapping and boundary conditions used for the harbor simulation. Water depth [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: WAF diagram calculated using BEM for the harbor of Chipiona example under inclined incident waves and [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    (eds.) Abramowitz M., Handbook of mathematical functions, 10 ed., NBS, 1972

    Stegun I.A. (eds.) Abramowitz M., Handbook of mathematical functions, 10 ed., NBS, 1972

    work page 1972

  2. [2]

    Noureddine Atalla and Franck Sgard, Finite element and boundary methods in structural acoustics and vibration, CRC Press, 2015

    work page 2015

  3. [3]

    Belibassakis, The Green’s function of the mild-slope equation: The case of a monotonic bed profile, Wave Motion 32 (2000), 339–361

    K.A. Belibassakis, The Green’s function of the mild-slope equation: The case of a monotonic bed profile, Wave Motion 32 (2000), 339–361

    work page 2000

  4. [4]

    Belibassakis and G.A

    K.A. Belibassakis and G.A. Athanassoulis, Three-dimensional Green’s function for harmonic water waves over a bottom topography with different depths at infinity, Journal of Fluid Mechanics 510 (2004), 267 – 302

    work page 2004

  5. [5]

    Belibassakis, G.A

    K.A. Belibassakis, G.A. Athanassoulis, and Th.P. Gerostathis,A coupled-mode model for the refraction-diffraction of linear waves over steep three-dimensional bathymetry, Applied Ocean Research 23 (2001), no. 6, 319 – 336

    work page 2001

  6. [6]

    Beltrami, G

    G. Beltrami, G. Bellotti, P. Girolamo, and P. Sammarco,Treatment of wave breaking and total absorption in a mild-slope equation fem model, Journal of Waterway, Port, Coastal, and Ocean Engineering127 (2001), no. 5, 263–271

    work page 2001

  7. [7]

    Bergmann, The equation in a medium with a variable index of refraction, The Journal of the Acoustical Society of America 17 (1946), no

    Peter G. Bergmann, The equation in a medium with a variable index of refraction, The Journal of the Acoustical Society of America 17 (1946), no. 4, 329 – 333

    work page 1946

  8. [8]

    Berkhoff, N

    J.C.W. Berkhoff, N. Booy, and A.C. Radder, Verification of numerical wave propagation models for simple harmonic linear water waves, Coastal Engineering 6 (1982), no. 3, 255 – 279

    work page 1982

Show all 56 references
  1. [9]

    Berkhoff, Computation of combined refraction - diffraction, Proceedings of 13th International Conference on Coastal Engineering, ASCE, 1972

    W. Berkhoff, Computation of combined refraction - diffraction, Proceedings of 13th International Conference on Coastal Engineering, ASCE, 1972

    1972

  2. [10]

    wave diffraction and refraction, Ph.D

    Willem Berkhoff, Mathematical models for simple harmonic linear water waves. wave diffraction and refraction, Ph.D. thesis, Delft Hydraulics Laboratory, 1976

    1976

  3. [11]

    Bettess and O

    P. Bettess and O. C. Zienkiewicz,Diffraction and refraction of surface waves using finite and infinite elements, International Journal for Numerical Methods in Engineering 11 (1977), no. 8, 1271–1290

    1977

  4. [12]

    R. P. Bonet, Refraction and diffraction of water waves using finite elements with a DNL boundary condition , Ocean Engineering 63 (2013), 77–89

    2013

  5. [13]

    Booij, A note on the accuracy of the mild-slope equation, Coastal Engineering 7 (1983), no

    N. Booij, A note on the accuracy of the mild-slope equation, Coastal Engineering 7 (1983), no. 3, 191 – 203

    1983

  6. [14]

    Chamberlain and D

    P.G. Chamberlain and D. Porter,The modified mild-slope equation, Journal Fluid Mechanics 291 (1995), 393 – 407

    1995

  7. [15]

    Chandrasekera and Kwok Fai Cheung, Extended linear refraction-diffraction model , Journal of Waterway, Port, Coastal, and Ocean Engineering123 (1997), no

    Carmela N. Chandrasekera and Kwok Fai Cheung, Extended linear refraction-diffraction model , Journal of Waterway, Port, Coastal, and Ocean Engineering123 (1997), no. 5, 280 – 286

    1997

  8. [16]

    Chen, Effects of bottom friction and boundary absorption on water wave scattering, Applied Ocean Research 8 (1986), no

    H.S. Chen, Effects of bottom friction and boundary absorption on water wave scattering, Applied Ocean Research 8 (1986), no. 2, 99 – 104. 25

    1986

  9. [17]

    Chen and C.C

    H.S. Chen and C.C. Mei, Oscillations and wave forces in a man-made harbor in the open sea, Symposium on Naval Hydrodynamics 10th, Cambridge, June 1974

    1974

  10. [18]

    thesis, Department of Civil Engineering, University of Maine, August 2002

    Wei Chen,Finite element modeling of wave transformation in harbors and coastal regions with complex bathymetry and ambient currents, Ph.D. thesis, Department of Civil Engineering, University of Maine, August 2002

    2002

  11. [19]

    mild-slope

    Graham J.M. Copeland, A practical alternative to the "mild-slope" wave equation, Coastal Engineering 9 (1985), no. 2, 125 – 149

    1985

  12. [20]

    José Domínguez, Boundary elements in dynamics, har/dsk ed., International series on computational engineering, Computational Mechanics Publications; Elsevier Applied Science, 1993

    1993

  13. [21]

    1, 1 – 29

    Dan Givoli, Non-reflecting boundary conditions, Journal of Computational Physics 94 (1991), no. 1, 1 – 29

    1991

  14. [22]

    Keller, Non-reflecting boundary conditions for elastic waves, Wave Motion 12 (1990), no

    Dan Givoli and Joseph B. Keller, Non-reflecting boundary conditions for elastic waves, Wave Motion 12 (1990), no. 3, 261 – 279

    1990

  15. [23]

    Hauguel, A combined FE-BIE method for water waves, 16th International Conference on Coastal Engineering, 1978, pp

    A. Hauguel, A combined FE-BIE method for water waves, 16th International Conference on Coastal Engineering, 1978, pp. 715–721

    1978

  16. [24]

    11, 1246 – 1257

    Sung-Shan Hsiao, Chun-Ming Chang, and Chih-Chung Wen, Solution for wave propagation through a circular cylinder mounted on different topography ripple-bed profile shoals using DRBEM, Engineering Analysis with Boundary Elements 33 (2009), no. 11, 1246 – 1257

    2009

  17. [25]

    Tai-Wen Hsu and Chi-Chung Wen,A study of using parabolic model to describe wave breaking and wide-angle wave incidence, J. Chin. Inst. Eng. 23 (2000), no. 4, 515–527

    2000

  18. [26]

    11, 1479 – 1498

    Tai-Wen Hsu and Chih-Chung Wen,A parabolic equation extended to account for rapidly varying topography, Ocean Engineering 28 (2001), no. 11, 1479 – 1498

    2001

  19. [27]

    3–4, 281 – 298

    Ken ichiro Hamanaka, Open, partial reflection and incident-absorbing boundary conditions in wave analysis with a boundary integral method, Coastal Engineering 30 (1997), no. 3–4, 281 – 298

    1997

  20. [28]

    3, 193 – 214

    Michael Isaacson and Shiqin Qu, Waves in a harbour with partially reflecting boundaries, Coastal Engineering 14 (1990), no. 3, 193 – 214

    1990

  21. [29]

    Keller and Dan Givoli, Exact non-reflecting boundary conditions, Journal of Computational Physics 82 (1989), no

    Joseph B. Keller and Dan Givoli, Exact non-reflecting boundary conditions, Journal of Computational Physics 82 (1989), no. 1, 172 – 192

    1989

  22. [30]

    Kirby,A general wave equation for waves over rippled beds, Journal of Fluid Mechanics 162 (1986), 171 – 186

    James T. Kirby,A general wave equation for waves over rippled beds, Journal of Fluid Mechanics 162 (1986), 171 – 186

    1986

  23. [31]

    Lau and Z

    S.L. Lau and Z. Ji, Efficient 3-d infinite element for water wave diffraction problems, International Journal for Numerical Methods in Engineering 28 (1989), no. 6, 1371–1387, cited By (since 1996)6

    1989

  24. [32]

    Changhoon Lee, Woo Sun Park, Yong-Sik Cho, and Kyung Doug Suh,Hyperbolic mild-slope equations extended to account for rapidly varying topography, Coastal Engineering 34 (1998), 243 – 257

    1998

  25. [33]

    Wang, and Sieun Eom, Boundary element modeling of multidirectional random waves in a harbor with a rectangular navigation channel, Ocean Engineering 36 (2009), no

    Hong Sik Lee, Sung Duk Kim, K.-H. Wang, and Sieun Eom, Boundary element modeling of multidirectional random waves in a harbor with a rectangular navigation channel, Ocean Engineering 36 (2009), no. 17-18, 1287 – 1294

    2009

  26. [34]

    1, 39 – 58

    Hong Sik Lee and A.Neil Williams, Boundary element modeling of multidirectional random waves in a harbor with partially reflecting boundaries, Ocean Engineering 29 (2002), no. 1, 39 – 58

    2002

  27. [35]

    Li and K

    B. Li and K. Anastasiou, Efficient elliptic solvers for the mild-slope equation using the multigrid technique , Coastal Engineering 16 (1992), no. 3, 245 – 266

    1992

  28. [36]

    3, 227 – 242

    Bin Li, An evolution equation for water waves, Coastal Engineering 23 (1994), no. 3, 227 – 242

    1994

  29. [37]

    Huanwen Liu, A modified GDRBEM model for wave scattering, International Conference on Estuaries and Coasts, 11 2003, pp. 749–755

    2003

  30. [38]

    12, 2775 – 2791

    Shu-Xue Liu, Bing Sun, Zhong-Bin Sun, and Jin-Xuan Li, Self-adaptive FEM numerical modeling of the mild-slope equation, Applied Mathematical Modelling 32 (2008), no. 12, 2775 – 2791

    2008

  31. [39]

    Maa, T.-W

    J.P.-Y . Maa, T.-W. Hsu, and D.-Y . Lee,The RIDE model: an enhanced computer program for wave transformation, Ocean Engineering 29 (2002), no. 11, 1441 – 1458

    2002

  32. [40]

    Massel, Extended refraction-diffraction equation for surface waves, Coastal Engineering 19 (1993), 97 – 126

    Stanislaw R. Massel, Extended refraction-diffraction equation for surface waves, Coastal Engineering 19 (1993), 97 – 126

    1993

  33. [41]

    McCamy and R.A

    R.C. McCamy and R.A. Fuchs, Wave forces on piles; a diffraction theory, Beach Erosion Board, Techn. Memo- randum, no. 69, 1954. 26

    1954

  34. [42]

    Mei, The applied dynamics of ocean surface waves, John Wiley, Chichester, 1983

    C.C. Mei, The applied dynamics of ocean surface waves, John Wiley, Chichester, 1983

    1983

  35. [43]

    Naserizadeh, H

    R. Naserizadeh, H. B. Bingham, and A. Noorzad, A coupled boundary element-finite difference solution of the elliptic modified mild slope equation, Engineering Analysis with Boundary Elements 35 (2011), no. 1, 25 – 33

    2011

  36. [44]

    Panchang, B

    V . Panchang, B. Xu W. Chen, K. Schelenker, Z. Demirbilek, and M. Okihiro,Exterior bathymetry effects in elliptic harbor wave models, Journal of Waterway, Port, Coastal, and Ocean Engineering126 (2000), 71–78

    2000

  37. [45]

    Panchang, Bryan R

    Vijay G. Panchang, Bryan R. Pearce, Ge Wei, and Benoit Cushman-Roisin, Solution of the mild-slope wave problem by iteration, Applied Ocean Research 13 (1991), no. 4, 187 – 199

    1991

  38. [46]

    Porter and D.J

    D. Porter and D.J. Staziker, Extensions of the mild-slope equation , Journal of Fluid Mechanics 300 (1995), 367–382

    1995

  39. [47]

    Rodríguez-Tembleque, José A

    L. Rodríguez-Tembleque, José A. González, and A. Cerrato,Partitioned solution strategies for coupled BEM–FEM acoustic fluid–structure interaction problems, Comput. Struct. 152 (2015), 45 – 58

    2015

  40. [48]

    Shaw and W

    R.P. Shaw and W. Falby,FEBIE - A combined finite element-boundary integral equation method, Computers & Fluids 6 (1978), no. 3, 153 – 160

    1978

  41. [49]

    Steward and V

    D. Steward and V . Panchang,Improved coastal boundary condition for surface water waves, Ocean Engineering 28 (2001), no. 1, 139 – 157

    2001

  42. [50]

    Kyung Doug Suh, Changhoon Lee, and Woo Sun Park,Time-dependent equations for wave propagation on rapidly varying topography, Coastal Engineering 32 (1997), 91 – 117

    1997

  43. [51]

    T.-K. Tsay, W. Zhu, and P.L.-F. Liu, A finite element model for wave refraction, diffraction, reflection and dissipation, Applied Ocean Research 11 (1989), no. 1, 33 – 38

    1989

  44. [52]

    Liu,A finite element model for wave refraction and diffraction, Applied Ocean Research 5 (1983), no

    Ting-Kuei Tsay and Philip L-F. Liu,A finite element model for wave refraction and diffraction, Applied Ocean Research 5 (1983), no. 1, 30 – 37

    1983

  45. [53]

    Wu,Boundary element acoustics, fundamentals and computer codes, 5 ed., vol

    T.W. Wu,Boundary element acoustics, fundamentals and computer codes, 5 ed., vol. 3, WIT Press, Boston, 2000

    2000

  46. [54]

    Zhu, A new DRBEM model for wave refraction and diffraction, Engineering Analysis with Boundary Elements 12 (1993), no

    S. Zhu, A new DRBEM model for wave refraction and diffraction, Engineering Analysis with Boundary Elements 12 (1993), no. 4, 261–274

    1993

  47. [55]

    5, 377 – 390

    Song-Ping Zhu, Huan-Wen Liu, and Ke Chen, A general DRBEM model for wave refraction and diffraction , Engineering Analysis with Boundary Elements 24 (2000), no. 5, 377 – 390

    2000

  48. [56]

    Marchant, A perturbation DRBEM model for weakly nonlinear wave run-ups around islands, Engineering Analysis with Boundary Elements 33 (2009), no

    Song-Ping Zhu, Huan-Wen Liu, and Timothy R. Marchant, A perturbation DRBEM model for weakly nonlinear wave run-ups around islands, Engineering Analysis with Boundary Elements 33 (2009), no. 1, 63 – 76. 27

    2009

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.