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REVIEW 3 major objections 7 minor 27 references

A conformally mapped numerical wave tank supporting piston and flap wavemakers

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that a double-layered conformal mapping, extended with piston and flap wavemaker maps, gives the first complete conformal-mapping-based numerical representation of a two-dimensional wave flume.

desk verdict A genuinely useful extension of conformal-mapped wave models to piston and flap wavemakers, with solid validation and a few overbroad claims that need tempering. read the letter →

arxiv 2505.13154 v1 pith:KGSXUHSZ submitted 2025-05-19 physics.flu-dyn

Andreas Holm Akselsen This is my paper · ORCID
classification physics.flu-dyn
keywords conformalmappingnumericalwavetankpistonwavemakerflapspuriouswavesreturnflowspectralevolutioncalibration
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

This paper claims that a numerical wave tank for two-dimensional water waves can be built from conformal mapping alone, with both piston and flap wavemakers represented exactly at the mapped boundary. It presents explicit maps for the two wavemaker types and a background potential that enforces the tank walls, and it validates the scheme theoretically and against laboratory measurements in a deep towing tank. If the claim is correct, a simulation can reproduce wavemaker kinematics, spurious waves, return flow, spectral evolution, and wave-height statistics at beyond real-time speed for most wave periods. That would make numerical wave calibration and extreme-event screening practical complements to physical experiments.

What carries the argument

The engine is two nested conformal maps, $z=\bar f(\bar z,t)$ and $\bar z=\bar{\bar f}(\bar{\bar z},t)$, which send the moving wavemaker and bed to fixed straight lines and the free surface to a horizontal line in a rectangle. This leaves only surface dynamics to be integrated explicitly, while the fluid interior and all solid boundaries are handled analytically. The harmonic-extension work is carried by the projection kernels $[\mathcal{C}_h*\mu]$ and $[\mathcal{S}_h*\mu]$ of Eq. (8), which convert boundary data into the complex potential and back out map velocities; wall impermeability is enforced by horizontal mirroring of the domain. Piston motion enters through the dilation map (Eq. 21), flap motion through the iterated kernel map (Eqs. 25–28), and the wall condition through the background potential $\bar W$ of Eq. (29).

What would settle it

Run the same wavemaker signal in a flume at two different widths while keeping depth and paddle geometry fixed; if the phase-resolved surface elevation at 90 meters differs between the two runs, the two-dimensional conformal model cannot be the full description, whereas identical signals would support its phase-resolved claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that a double-layered conformal mapping—first straightening the prescribed wavemaker, bed, and walls into an intermediate plane, then mapping the free surface to a fixed rectangle—can incorporate piston and flap wavemakers exactly. For a piston, the map is a time-dependent dilation about the far corner (Eq. 21); for a flap, the map is built from rotated projection kernels with an iterated displacement (Eqs. 25–28) and repositioned so the waterline meets the paddle face. A precomputed background potential (Eq. 29) enforces impermeability along the paddle and walls. The result, the author argues, is the first conformal-mapping-based numerical representation of a complete wave flume. The paper demonstrates exactly satisfied kinematic conditions at the wavemaker, return flow matching the Stokes-drift/mass-conservation value, and agreement with experiments on phase velocity, spurious-wave amplitudes, spectral evolution, and wave-height distributions at a measurement station 90 meters from the wavemaker.

Load-bearing premise

Everything rests on the real flow being two-dimensional, incompressible, inviscid, and irrotational, so that conformal mapping represents the flume exactly; the paper's own measurements show three-dimensional sloshing at the far station, which would degrade phase-resolved accuracy.

Editorial extensions

If this is right

  • Replaying an experimentally recorded wavemaker signal in the simulation yields phase-resolved surface elevation that tracks the measured harp signals at 90 meters for moderate wave steepness.
  • The model captures the amplitude and arrival of second- and third-order spurious waves, and it reproduces the suppression of those waves when Schäffer's second-order correction is applied to the paddle signal.
  • The simulated wave spectrum at the measurement location, including its evolution along the flume, matches the measured spectrum, while linear wavemaker theory does not capture that evolution.
  • Beyond real-time computation for most tested periods means calibration iterations can be run numerically before physical tests, cutting laboratory time.
  • Wave height statistics at the measurement gauge follow the experimental distributions up to the sample-limited tail, supporting the model's use for statistical wave characterization.

Reading between the lines

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

  • Editorial inference: the same mapping construction could be adapted to other wavemaker geometries—multi-hinged paddles, directional wavemakers, or absorbing wavemakers—turning the method into a general testbed for wavemaker theory rather than a two-case construction.
  • Editorial inference: the unexplained small phase shifts at 90 meters could be tested by varying the flume width; if the shifts scale with width, they are three-dimensional in origin, and if not, a missing two-dimensional mechanism such as a wall boundary-layer current is implicated.
  • Editorial inference: because the model runs beyond real time and preserves full nonlinearity, it could generate the many synthetic realizations needed to estimate extreme wave-height statistics for a specific flume, complementing limited physical ensembles.
  • Editorial inference: the Schwarz-Christoffel flap variant of Appendix A, which avoids the Gibbs noise of the kernel map at the hinge, could extend reliable flap angles beyond the roughly 35-degree convergence limit noted for the projection-kernel iteration.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper extends a double-layered conformal mapping method for two-dimensional potential-flow water waves so that it can represent piston- and flap-type wavemakers as moving boundaries in a numerical wave tank. It gives explicit conformal maps for piston motion (Eq. 21) and flap motion (Eqs. 25-28), a background potential satisfying the wall conditions (Eq. 29), and validates the resulting model against exact steady wave solutions, second-order wavemaker theory, return-flow predictions, and laboratory experiments in a towing tank. The paper claims that this provides, for the first time based on conformal mapping, a complete numerical representation of wave flumes, that the model reproduces spurious waves and spectral evolution, and that it runs faster than real time except for the shortest tested periods.

Significance. If the technical content is sound, this is a useful contribution to numerical wave tank methodology. The conformal mapping construction avoids discretizing the interior and solid boundaries, the free surface is treated nonlinearly without order truncation, and the wavemaker kinematics are satisfied by construction. The kinematic condition checks in Fig. 6, the comparison with SSGW phase velocities in Fig. 7, and the spectral and statistical comparisons in Figs. 15-18 are valuable demonstrations, and the inclusion of code listings is a practical strength. However, the central claim of a 'complete numerical representation of wave flumes' is broader than what the two-dimensional model and the presented validation can support, and one validation formula appears to be in error. These issues are local and fixable, so the paper merits revision rather than rejection.

major comments (3)
  1. [§5, Eq. (34)] The return-flow formula appears to under-predict the depth-averaged Stokes drift by a factor of two. For a monochromatic wave the depth-integrated Stokes transport is (1/2)|a|^2 ω coth(kh), so mass conservation in a closed tank gives U0 = −(1/(2h))|a|^2 ω coth(kh) = −g k |a|^2/(2hω). Equation (34) has g/(4h) instead. Unless a_n is defined with a non-standard amplitude convention, which is not stated, the comparison in Fig. 8 uses a target that is a factor of two too small and cannot serve as a quantitative validation of return flow. Please correct Eq. (34) or define the amplitude convention precisely and recompute the comparison.
  2. [Abstract; §§6.2-6.3; §7] The abstract's claim of a 'complete numerical representation of wave flumes' and §8's statement that phase-resolved signals agree 'even at considerable distances' are stronger than the evidence supports. Section 6.2 attributes observed phase shifts and energy discrepancies to 'three-dimensional effects and the excitation of transverse sloshing modes', §6.3 invokes three-dimensional effects, paddle gaps, and measurement inaccuracies to explain missing energy, and §7 concedes that the model 'lacks three-dimensional effects' and questions whether 'exact phase-resolved predictions' are attainable. The bottom panels of Figs. 12 and 13 show transverse-gauge deviations of order 0.05-0.1 m against wave heights of 0.15-0.3 m. The demonstrated capabilities are spectral and statistical fidelity in a two-dimensional setting; the 'complete' and phase-resolved wording should be qualified accordingly.
  3. [§6.1, Fig. 9] A substantial part of the validation is not independently checkable. The text states that third-order spurious-wave predictions are included 'using the author's own, as yet unpublished, extensive wavemaker theory', and the return-flow comparison in Fig. 8 relies on Akselsen (2025b). Since the paper's central claims include accurate reproduction of wavemaker characteristics and spurious waves, the third-order comparison in Fig. 9 and the return-flow target in Fig. 8 function as validation against the author's own constructions. Please include a derivation or a citable reference for the third-order theory, or restrict the relevant validation claims to the parts that do not depend on it.
minor comments (7)
  1. [§2.2, Eq. (15)] The sentence preceding Eq. (15), 'which is to eb evaluated', contains a typo and should read 'which is to be evaluated'.
  2. [Figure 7 caption] The caption contains 'Sokes' second definition' and should read 'Stokes' second definition'.
  3. [Affiliation] The affiliation line lists 'Trønderlag'; the correct Norwegian county name is 'Trøndelag'.
  4. [§6.1, paragraph after Fig. 9] The phrase 'three-multidimensional effects' is likely a typo for 'three-dimensional effects'.
  5. [§6.3, paragraph after Fig. 14] The sentence 'Simulated wavemaker motions are identical to those applied during the Identical wavemaker motions are applied...' is grammatically broken and appears to be a duplicated partial sentence.
  6. [Appendix B, Listing 2] In the MATLAB code, the line 'nx = size(nu,1);' uses the variable 'nu', whereas the function argument is named 'mu'; this would cause an error when the function is called.
  7. [Figure 5 caption] The caption lists the third panel angle as 30 degrees twice; please confirm whether the third panel is meant to be -30 degrees.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor by-construction kinematic check and auxiliary self-citations, but the central wavemaker model is validated against external experiments and SSGW.

  1. self definitional [Section 5, first paragraph, Figure 6]
    "We begin by validating the kinematic wavemaker boundary condition, which, by design, should be exactly satisfied. Figure 6 confirms this, showing examples from both the piston mapping in section 3, and the flap mapping in section 4."

    The kinematic condition is not an independent prediction: Eq. (6) is imposed by construction through the mapping and the background potential, and the maps (21)-(29) are designed to satisfy it. The figure therefore checks that the implementation is consistent with its own defining equations (a code verification), rather than testing the model against new physics. The text explicitly concedes the outcome is guaranteed ('by design'). This is a tautological validation step, though it is not used as evidence for the central experimental claims.

full rationale

The central derivation of the wavemaker mappings is self-contained: Eqs. (21)-(29) are constructed from the model in Section 2 and validated against external experiments (SINTEF tank) and independent SSGW exact solutions. The return-flow comparison (Eq. 34) cites the author's own Akselsen (2025b), but that formula is parameter-free, analytically derived from mass conservation and Schäffer theory, and is not fitted to the simulation; the self-citation is therefore auxiliary, not load-bearing. The third-order wavemaker theory used in Figure 9 is unpublished and self-referential, but the same figure contains experimental amplitudes as external corroboration, so the central spurious-wave claim does not reduce to the self-citation. The only genuine by-construction item is the kinematic boundary-condition check (Figure 6), which the paper itself describes as guaranteed by design; it is an implementation consistency test, not an independent prediction. No parameter is fitted and no predicted quantity is defined in terms of the target output, so the overall circularity is low.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The main burden is numerical rather than physical: damping and beach parameters are chosen by hand, and the potential-flow assumptions are standard for this branch of wave modeling. The background potential split is an ad hoc construction that the paper assumes to be result-independent. No new physical entities are introduced.

free parameters (2)
  • Modal damping coefficient r and cutoff wavenumber kd = r=0.01 with kd=0.5 kmax for gentle cases; r=0.25 with kd=0.25 kmax for the steep case 80084
    These stabilization parameters are hand-chosen in Sections 6.1 and 6.2. The steeper case requires broader damping, and the paper admits it modifies the wave train front, so the parameters affect the validation results.
  • Numerical beach absorption intensity nu_0 and length L_b = not specified
    Section 2.4 states these should be chosen to suppress reflections, but no values or sensitivity study are given, leaving an unspecified numerical choice in the method.
assumptions (6)
  • domain assumption The fluid is incompressible, inviscid, and irrotational, so a velocity potential exists and Laplace's equation holds.
    Invoked implicitly at the start of Section 2.1 through the complex potential w = phi + i psi and Bernoulli's equation in Section 2.2.
  • standard math Conformal mappings preserve harmonicity of the velocity potential.
    Used in Eq. 3 to define transformed potentials in the intermediate and final planes.
  • domain assumption The kinematic boundary condition can be written as particle impermeability: a particle on the boundary remains on it (Eq. 1).
    This is the starting point for the whole mapping formulation, stated in Section 2.1.
  • domain assumption The free surface pressure is zero, expressed through the Bernoulli equation (Eq. 15).
    Imposed in Section 2.2 to evolve the surface potential.
  • ad hoc to paper The split of the total potential into w plus W is arbitrary and does not affect the physical result.
    Section 2.3 says both f and W are prescribed only along wall and bed boundaries and may be assigned according to preference. The model assumes the final wave field is independent of this choice.
  • standard math The projection kernels (8) with mirroring (16) correctly enforce wall conditions on a discrete Fourier grid.
    The properties (9) are used to construct solutions (10) and (12); these are standard spectral identities for periodic band-limited functions.

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Cite this review

Pith. "Pith review of A conformally mapped numerical wave tank supporting piston and flap wavemakers." pith.science (2026). https://pith.science/paper/KGSXUHSZ

@misc{pith2026250513154,
  author       = {Pith},
  title        = {Pith review of: A conformally mapped numerical wave tank supporting piston and flap wavemakers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KGSXUHSZ}},
  note         = {Machine review of arXiv:2505.13154}
}
read the original abstract

This paper advances the development of the conformally mapped model for accurate simulation of two-dimensional water waves, here with emphasis on mapping boundaries that represent piston- and flap-type wavemakers. With this, a complete numerical representation of wave flumes is provided -- the first of its kind based on conformal mapping. The model is validated both theoretically and experimentally, with special attention devoted to wavemaker characteristics and the generation of spurious waves. It is further demonstrated that the method accurately predicts the spectral evolutions of generated wave fields. The model is computationally efficient, with beyond real-time computation but for the smallest tested periods, making it ideal for numerical wave calibration and for replicating experiments.

Figures

Figures reproduced from arXiv: 2505.13154 by the authors.

Figure 1
Figure 1. Sketch of the Z, Z¯ and Z ¯¯-domains. Fluid velocities are represented using the complex potential w = ϕ + i ψ, ϕ being the fluid velocity potential and ψ its harmonic conjugate, the stream function. By virtue of conformality, these functions remain solutions of the Laplace equation after the mapping w¯(¯z, t) = w[ ¯f(¯z, t), t], w¯¯(z, t ¯¯ ) = ¯w[ ¯¯f(z, t ¯¯ ), t]. (3) In anticipation of wall conditions to come, … view at source ↗
Figure 2
Figure 2. Simplified map setting α = 1—map length varies while depths remain fixed. With the expansion in α, a fixed far wall is achieved by setting α = 1 − [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Preferred map, setting α from (26) and varying D to match the hinge position. Finally, the map must be repositioned such that the waterline touches the paddle face as opposed to the ‘tip’ created by the mirroring seam: ¯f(¯z, t) = ¯f0(¯z − i ∆) + i (H − h). (28) Here, ∆ is the freeboard height (vertical distance to the mirroring seam) when θ = 0, and h = h¯ the mean water depth when θ = 0. The final mapping is the o… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Paddle map (25)–(28) for θ = −30◦ , 30◦ and 30◦ , respectively, with h = 1.0, ∆ = 0.5, d = 0.5 and L = 2.0. Black isocline corresponds to y¯ = 0. 4.2. The background velocity field W¯ ∗ z¯ The background velocity potential is pre-computed according to the wall and floo…
Figure 5
Figure 5. Figure 5: Example simulation with irregular flap wavemaker motion in a small basin [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Validation of the kinematic wavemaker boundary condition. [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Bottom panels: test on phase and group velocities, compared to linear theory [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: The return current set up by stokes drift and mass conservation—plotted are [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Principal and higher-order wave amplitude components of simulation (solid), [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: The spectral content of generated wave trains; modal amplitude contained [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: Cases corresponding to [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Irregular wave time signals measured with wave gauge harp, 90 meters down [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Steeper case 80084; Tp = 2.0 s, Hs = 0.30 m and stabilising damping coeffi￾cients kd = 0.25 kmax, r = 0.25, sampled at different times. 6.3. Energy spectra and wave calibration We now shift our focus to wave spectra (often called power or variance spectra) in the pers…
Figure 14
Figure 14. Figure 14: Full test simulation of case 80060; Tp = 1.5 s, Hs = 0.17 m. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: Power spectral density as measured during calibration procedure. Solid: sim [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: Spatial variation of wave spectrum in test 80063 ( [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]
Figure 17
Figure 17. Figure 17: Power spectrum of steeper case 80084; Tp = 2.0 s, Hs = 0.30 m and stabilising damping coefficients kd = 0.25 kmax, r = 0.25 (see [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]
Figure 18
Figure 18. Figure 18: Wave height distribution of four calibration tests as measured with the centre [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]

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Works this paper leans on

27 extracted references · 10 canonical work pages

  1. [1]

    A. H. Akselsen. A precise conformally mapped method for water waves in complex transient environments. Journal of Computational Physics, page 113848, 2025 a

    work page 2025

  2. [2]

    A. H. Akselsen. Second-order theory for multi-hinged directional wavemakers, 2025 b . URL https://doi.org/10.48550/arXiv.2502.09586. Under review in Coast.\ Eng

    work page Pith review arXiv doi:10.48550/arxiv.2502.09586 2025

  3. [3]

    Bonnefoy, D

    F. Bonnefoy, D. Le Touz \'e , and P. Ferrant. A fully-spectral 3D time-domain model for second-order simulation of wavetank experiments. P art A : F ormulation, implementation and numerical properties. Applied Ocean Research, 28 0 (1): 0 33--43, 2006

    2006

  4. [4]

    Bonnefoy, G

    F. Bonnefoy, G. Ducrozet, D. Le Touz \'e , and P. Ferrant. Time domain simulation of nonlinear water waves using spectral methods. In Advances in numerical simulation of nonlinear water waves, pages 129--164. World Scientific, 2010

    2010

  5. [5]

    Chalikov

    D. Chalikov. Freak waves: Their occurrence and probability. Physics of Fluids, 21 0 (7): 0 076602, 07 2009. ISSN 1070-6631. doi:10.1063/1.3175713. URL https://doi.org/10.1063/1.3175713

    work page doi:10.1063/1.3175713 2009

  6. [6]

    Chalikov

    D. Chalikov. Numerical modeling of sea waves. Izvestiya, Atmospheric and Oceanic Physics, 56: 0 312--323, 2020

    2020

  7. [7]

    Chalikov and D

    D. Chalikov and D. Sheinin. Numerical modeling of surface waves based on principal equations of potential wave dynamics. US Department of Commerce, National Oceanic and Atmospheric Administration National Weather Service, 1996

    1996

  8. [8]

    Chalikov and D

    D. Chalikov and D. Sheinin. Modeling extreme waves based on equations of potential flow with a free surface. Journal of Computational Physics, 210 0 (1): 0 247--273, 2005

    2005

Show all 27 references
  1. [9]

    D. V. Chalikov. Numerical modeling of sea waves. Springer, 2016

    2016

  2. [10]

    Clamond and D

    D. Clamond and D. Dutykh. Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth. Journal of Fluid Mechanics, 844: 0 491--518, 2018

    2018

  3. [11]

    Ducrozet, F

    G. Ducrozet, F. Bonnefoy, N. Mori, M. Fink, and A. Chabchoub. Experimental reconstruction of extreme sea waves by time reversal principle. Journal of Fluid Mechanics, 884: 0 A20, 2020. doi:10.1017/jfm.2019.939

    2020 doi

  4. [12]

    A. I. Dyachenko, E. A. Kuznetsov, M. Spector, and V. E. Zakharov. Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Physics Letters A, 221 0 (1-2): 0 73--79, 1996

    1996

  5. [13]

    S. A. Dyachenko. On the dynamics of a free surface of an ideal fluid in a bounded domain in the presence of surface tension. Journal of Fluid Mechanics, 860: 0 408--418, 2019. doi:10.1017/jfm.2018.885

    2019 doi

  6. [14]

    Gramstad, T

    O. Gramstad, T. B. Johannessen, and G. Lian. Long-term analysis of wave-induced loads using high order spectral method and direct sampling of extreme wave events. Marine Structures, 91: 0 103473, 2023

    2023

  7. [15]

    Houtani, T

    H. Houtani, T. Waseda, W. Fujimoto, K. Kiyomatsu, and K. Tanizawa. Generation of a spatially periodic directional wave field in a rectangular wave basin based on higher-order spectral simulation. Ocean Engineering, 169: 0 428--441, 2018

    2018

  8. [16]

    H. Lamb. Hydrodynamics, 1932

    1932

  9. [17]

    M. S. Longuet-Higgins and O. M. Phillips. Phase velocity effects in tertiary wave interactions. Journal of Fluid Mechanics, 12 0 (3): 0 333--336, 1962

    1962

  10. [18]

    E. P. Mansard and E. Funke. The measurement of incident and reflected spectra using a least squares method. In Coastal Engineering 1980, pages 154--172. 1980

    1980

  11. [19]

    McLean, Y

    J. McLean, Y. Ma, D. Martin, P. Saffman, and H. Yuen. Three-dimensional instability of finite-amplitude water waves. Physical Review Letters, 46 0 (13): 0 817, 1981

    1981

  12. [20]

    Milne-Thomson

    L. Milne-Thomson. Theoretical hydrodynamics, 1962

    1962

  13. [21]

    V. Ruban. Water waves over a strongly undulating bottom. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 70 0 (6): 0 066302, 2004

    2004

  14. [22]

    V. P. Ruban. Water waves over a time-dependent bottom: Exact description for 2d potential flows. Physics Letters A, 340 0 (1-4): 0 194--200, 2005

    2005

  15. [23]

    H. A. Sch \"a ffer. Second-order wavemaker theory for irregular waves. Ocean Engineering, 23 0 (1): 0 47--88, 1996. doi:10.1016/0029-8018(95)00013-B

    1996 doi

  16. [24]

    Viotti, D

    C. Viotti, D. Dutykh, and F. Dias. The conformal-mapping method for surface gravity waves in the presence of variable bathymetry and mean current. Procedia IUTAM, 11: 0 110--118, 2014

    2014

  17. [25]

    B. J. West, K. A. Brueckner, R. S. Janda, D. M. Milder, and R. L. Milton. A new numerical method for surface hydrodynamics. Journal of Geophysical Research: Oceans, 92 0 (C11): 0 11803--11824, 1987

    1987

  18. [26]

    V. E. Zakharov, A. I. Dyachenko, and O. A. Vasilyev. New method for numerical simulation of a nonstationary potential flow of incompressible fluid with a free surface. European Journal of Mechanics-B/Fluids, 21 0 (3): 0 283--291, 2002

    2002

  19. [27]

    V. E. Zakharov, A. I. Dyachenko, and A. O. Prokofiev. Freak waves as nonlinear stage of stokes wave modulation instability. European Journal of Mechanics-B/Fluids, 25 0 (5): 0 677--692, 2006

    2006

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Reviewed August 15, 2026 · model on record in the stance chip above.