pith. sign in

arxiv: 2605.19730 · v1 · pith:YNDQSKMMnew · submitted 2026-05-19 · ⚛️ physics.ao-ph · physics.flu-dyn

Matrix structure and convergence behavior of the matched eigenfunction method for computing heave wave forces on generalized concentric bodies

Yinghui Bimali , Rebecca McCabe , Collin Treacy , Kapil Khanal , En Lo , Maha Haji This is my paper

Pith reviewed 2026-05-20 01:45 UTC · model grok-4.3

classification ⚛️ physics.ao-ph physics.flu-dyn
keywords matched eigenfunction expansionhydrodynamic coefficientswave forcesconcentric cylindersslanted geometriesboundary element methodoffshore structuresconvergence behavior
0
0 comments X

The pith

The matched eigenfunction expansion method computes heave wave forces on slanted concentric cylinders within 5 percent of boundary element results using matrices two orders of magnitude smaller.

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

The paper develops a unifying framework for the matched eigenfunction expansion method that models wave interactions with any number of fixed or heaving annular cylinders whose profiles stay continuous and radially monotonic. It examines the resulting block matrix structure, documents convergence rates for both vertical and slanted shapes, and directly compares accuracy and speed against the boundary element solver Capytaine. A sympathetic reader would care because traditional boundary element calculations create a computational bottleneck when sizing offshore structures, and a faster semi-analytical route would let designers evaluate more shapes in the same time.

Core claim

The central claim is that the matched eigenfunction expansion method supplies a single framework for an arbitrary number of fixed or heaving surface-piercing annular cylinders with continuous and radially monotonic profiles; numerical experiments then show that this framework approximates hydrodynamic coefficients of slanted geometries to within 5 percent of Capytaine even at 15 degrees from vertical, while reaching 2 percent convergence an order of magnitude faster than Capytaine and with a matrix size two orders of magnitude smaller.

What carries the argument

The block matrix structure obtained by dividing the fluid domain into annular regions and matching eigenfunction expansions of the velocity potential at each interface, which organizes the linear system solved for the added mass, radiation damping, and excitation coefficients.

If this is right

  • Hydrodynamic coefficients for a broad range of continuous radially monotonic shapes become computable with far less effort than boundary element methods.
  • Optimization loops that iterate over many offshore structure geometries become practical because each evaluation finishes an order of magnitude sooner.
  • Both fixed and heaving surface-piercing bodies can be treated inside the same matching procedure.
  • Convergence rates are now documented for slanted profiles, giving users a clear rule of thumb for choosing matrix size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The compact matrix size could let the method run inside larger coupled simulations that include multiple structures or time-varying waves.
  • A hybrid scheme that switches to boundary element methods only near sharp features might extend the approach to bodies that violate the monotonic-profile assumption.
  • Insights into the block structure may suggest similar semi-analytical reductions for related problems such as diffraction by arrays of cylinders or motion in irregular waves.

Load-bearing premise

The body profiles must stay continuous and radially monotonic so that the eigenfunction expansions in neighboring annular regions can be matched directly without extra interface conditions or geometric approximations.

What would settle it

A direct numerical comparison on a body whose profile contains a radial discontinuity or reversal would show whether the hydrodynamic coefficients diverge sharply from a high-resolution boundary element reference solution.

Figures

Figures reproduced from arXiv: 2605.19730 by Collin Treacy, En Lo, Kapil Khanal, Maha Haji, Rebecca McCabe, Yinghui Bimali.

Figure 1
Figure 1. Figure 1: Side view of concentric cylindrical bodies. 2.1. Linear Hydrodynamics and Eigenfunctions To model the fluid in the internal and external regions, linear potential flow theory is used. This implies small body motion, an inviscid, irrotational, and incompressible fluid, and a wave amplitude-to-wavelength ratio less than unity (Mei et al. 2005). With these 0 X0-3 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Side view of taller and shorter region. will be defined as the taller t and shorter s fluid regions. When considering the vertical boundary dividing the two regions at 𝑟 = 𝑎, there are three different conditions to enforce: 1) the value of the velocity potentials at fluid-fluid boundaries are equal 𝜙 t (𝑎, 𝑧) = 𝜙 s (𝑎, 𝑧) for − ℎ ≤ 𝑧 ≤ −𝑑s , (2.2) 2) the radial fluid velocities at the fluid-fluid boundary … view at source ↗
Figure 3
Figure 3. Figure 3: Left: Comparison of the low frequency approximation, the infinite frequency limit, and standard MEEM (with 𝑁 𝑖𝑚 = 𝑁 𝑒 = 100) for the geometry described in Section 2.7. Right: A comparison of the first four 𝐶 𝑖2 1𝑛 over low frequencies, demonstrating that the behavior of 𝐶 𝑖2 10 eventually dominates as frequency approaches zero for both the real and imaginary parts. This trend is representative of the other… view at source ↗
Figure 4
Figure 4. Figure 4: shows the sparsity pattern of the A, B, and C matrices from Eq. 2.13 for an example configuration consisting of two bi-cylinder bodies with 𝑀 = 4, M1 = {1, 2}, M2 = {3, 4}, 𝑁 𝑖𝑚 = 𝑁 𝑒 = 10, and 𝑑𝑖+1 < 𝑑𝑖 . The A matrix features a block bi-diagonal structure. This can be observed by partitioning A into rectangular blocks such that each block encompasses the full height and half the blockwise width of the A𝑚… view at source ↗
Figure 5
Figure 5. Figure 5: Added mass, radiation damping, excitation magnitude, and excitation phase from MEEM and Capytaine for CorPower-like WEC without slanted portions. 3. Convergence of Hydrodynamic Coefficients Characterizing the convergence behavior of numerical solvers is crucial for ensuring accuracy while avoiding unnecessarily long runtimes. For BEM solvers, a mesh con￾vergence study is typically performed, where the numb… view at source ↗
Figure 6
Figure 6. Figure 6: Left: Added mass and damping calculated for a three body region configuration at 𝑁 𝑖1 = 𝑁 𝑖3 = 𝑁 𝑒 = 200 with region 𝑖2 heaving, for varying 𝑁 𝑖2 . Right: The data at left is transformed to the natural log of the associated error, and fitted to obtain error envelope parameters 𝛼, 𝛽 for each of added mass and damping. 𝜖 in the hydrodynamic coefficients solely based on the geometry and wave conditions being … view at source ↗
Figure 7
Figure 7. Figure 7: Flowchart of term count prediction. A geometry and desired error (green) are passed into the formula (blue) determined by the convergence study, producing a term count recommendation (orange). the region itself, such as its fluid height, radial width, or the ratio of its fluid height to those of its neighbors. Cumulative parameters are overall metrics combining the geometries of other regions, such as tota… view at source ↗
Figure 8
Figure 8. Figure 8: Process of choosing the fitting function for each dimensionless parameter (here ℎ−𝑑2 𝑎2−𝑎1 ), once identified. suggest functional forms for the relationship between the dimensionless parameters and fit parameters ( [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Randomly generated (described in Sec. 3.3) configurations with the target region heaving were fit with the product of the dependencies in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cross-sectional view of a slanted geometry and its equivalent discretized geometry. The close-up shown on the right-hand side shows how a slanted region can be approximated by a finite number of cylindrical rings. The unit vector normal to the horizontal part of all cylindrical rings is ˆ𝑛, while the unit vector normal to the true slanted body in the 𝑚th region is ˆ𝑛𝜁𝑚 [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗
Figure 11
Figure 11. Figure 11: Cross-sectional view of different discretization schemes for approximating a slanted geometry with cylindrical rings. The geometry of the cylindrical rings can be chosen such that their horizontal surfaces are a) within, b) partially within, or c) outside the true slanted body shape. 0 X0-25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the radiated velocity potential computed from MEEM and Capytaine. (a) and (c) show the real part of the potential from MEEM, (b) and (d) show the error in the real potential from MEEM relative to Capytaine, and (e) and (f) show the error in the potential along the slanted and stepped outlines. 0 X0-28 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Convergence of the hydrodynamic coefficients of five cones of varying steepness calculated with MEEM’s step approximation (radius 10, ℎ = 50, 𝜔 = 1, 𝑁 𝑖𝑚 = 𝑁 𝑒 = 400). Subdivisions calculated ranged from 1 to 30. Error was calculated relative to values given by Capytaine. 0.6 0.8 1.0 Added Mass [kg] ×105 MEEM Capytaine Within 5% of Capytaine 0.8 1.0 1.2 Added Mass [kg] ×105 0.4 0.6 0.8 1.0 1.2 1.4 Frequen… view at source ↗
Figure 14
Figure 14. Figure 14: Left: Computed hydrodynamic coefficients for the CorPower-like WEC geometry with a slanted region (the slant intersects the vertical at 𝑑 = 7.13). MEEM approximates the slanted region using 30 subdivisions. Right: Geometry from left plots at 𝜔 = 1. In both cases, MEEM uses 𝑁 𝑖𝑚 = 𝑁 𝑒 = 400 and Capytaine has 5940 panels. not require evaluating Bessel functions with any arguments that were not already evalu… view at source ↗
Figure 15
Figure 15. Figure 15: Left: Distribution of function computation times for the geometry in [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: For the CorPower WEC geometry in [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
read the original abstract

Structural survival of offshore structures is crucial for the growing marine economy. Calculating the added mass, radiation damping, and excitation coefficients to quantify wave loads with the traditional boundary element method (BEM) presents a computational bottleneck. The matched eigenfunction expansion method (MEEM), a long-known but rarely-used alternative, offers computational benefits due to its semi-analytical nature. However, previous work fails to directly compare its accuracy and computational performance with BEM, leaving the extent of its utility unknown. Furthermore, the geometry-dependent convergence for cylindrical and slanted geometries has not yet been documented, making the method's practicality for general geometries unclear. This paper presents a unifying MEEM framework for modeling an arbitrary number of fixed or heaving surface-piercing annular cylinders with continuous and radially-monotonic body profiles, and explores the method's block matrix structure, convergence behavior, ability to accurately approximate slanted geometries, and computational advantages over the BEM solver Capytaine. The numerical experiments show that MEEM can compute hydrodynamic coefficients of slanted geometries within 5% of Capytaine, even for angles as steep as 15 degrees from vertical. Finally, MEEM can achieve 2% convergence of its hydrodynamic coefficients an order of magnitude faster than Capytaine with a matrix size two orders of magnitude smaller, making it a computationally effective alternative to traditional BEM solvers. These contributions enable hydrodynamic analysis of a broad range of shapes with increased speed and confidence, paving the way for future optimization studies to yield improved designs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a unifying matched eigenfunction expansion method (MEEM) framework for an arbitrary number of fixed or heaving surface-piercing annular cylinders possessing continuous and radially-monotonic body profiles. It analyzes the resulting block-matrix structure, documents geometry-dependent convergence for both cylindrical and slanted cases, demonstrates approximation of slanted geometries, and reports computational comparisons against the BEM solver Capytaine, claiming 5% agreement on hydrodynamic coefficients for slants up to 15° from vertical together with order-of-magnitude speed-ups at 2% convergence using matrices two orders of magnitude smaller.

Significance. If the accuracy and performance claims survive clarification of the geometric-proxy issue, the work supplies a semi-analytical, matrix-structured alternative to BEM that could materially accelerate hydrodynamic-coefficient evaluation for generalized offshore geometries and thereby support subsequent optimization studies.

major comments (2)
  1. [Numerical experiments / slanted-geometry results] Numerical experiments (results section on slanted geometries): the headline claim that MEEM reproduces Capytaine coefficients within 5% for 15° slants rests on replacing the true linear slant with a continuous radially-monotonic profile for the MEEM computation. It is not stated whether the Capytaine reference solutions are obtained on the identical monotonic proxy or on the exact slanted geometry; without this clarification the reported tolerance conflates profile-approximation error with MEEM truncation and interface-matching error, weakening attribution of the observed accuracy to the method itself.
  2. [Convergence behavior and performance comparison] Convergence and performance subsection: the statements that 2% convergence is reached an order of magnitude faster with a matrix two orders of magnitude smaller lack explicit truncation criteria (number of retained eigenfunctions per region), definition of matrix size (total unknowns versus per-annulus blocks), Capytaine mesh resolution and convergence tolerance, and precise error-bar construction. These omissions make the quantitative speed-up claims difficult to reproduce or compare directly.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'generalized concentric bodies' is immediately qualified by the continuous radially-monotonic restriction; a single clarifying sentence would prevent readers from over-generalizing the scope.
  2. [Throughout] Notation: ensure that the block-matrix partitioning and the matching conditions at vertical interfaces are denoted consistently between the theoretical derivation and the numerical implementation sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help strengthen the clarity and reproducibility of the work. We address each major comment below and will incorporate revisions to resolve the identified ambiguities.

read point-by-point responses

  1. Referee: Numerical experiments (results section on slanted geometries): the headline claim that MEEM reproduces Capytaine coefficients within 5% for 15° slants rests on replacing the true linear slant with a continuous radially-monotonic profile for the MEEM computation. It is not stated whether the Capytaine reference solutions are obtained on the identical monotonic proxy or on the exact slanted geometry; without this clarification the reported tolerance conflates profile-approximation error with MEEM truncation and interface-matching error, weakening attribution of the observed accuracy to the method itself.

    Authors: We agree that explicit clarification is required. In the numerical experiments, Capytaine reference solutions were computed on the exact linear-slant geometry, while MEEM was applied to a continuous radially-monotonic profile chosen to approximate the slant. The reported 5% agreement therefore represents the combined effect of profile approximation and MEEM truncation/interface-matching error. Because the manuscript positions MEEM as a method for monotonic profiles that can approximate slanted bodies, this comparison is intentional; however, we will revise the text to state the distinction unambiguously so that readers can correctly attribute the observed accuracy. revision: yes

  2. Referee: Convergence and performance subsection: the statements that 2% convergence is reached an order of magnitude faster with a matrix two orders of magnitude smaller lack explicit truncation criteria (number of retained eigenfunctions per region), definition of matrix size (total unknowns versus per-annulus blocks), Capytaine mesh resolution and convergence tolerance, and precise error-bar construction. These omissions make the quantitative speed-up claims difficult to reproduce or compare directly.

    Authors: We accept that the current presentation lacks sufficient detail for direct reproduction. In the revised manuscript we will add: (i) the truncation criterion (number of retained eigenfunctions retained in each fluid region), (ii) an explicit definition of matrix size as the total number of unknowns in the assembled block system, (iii) the Capytaine mesh density and solver tolerance employed for the reference solutions, and (iv) the precise procedure used to construct the error bars shown in the convergence plots. These additions will allow readers to verify the reported order-of-magnitude speed-up at 2% convergence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; quantitative claims rest on external Capytaine benchmarks

full rationale

The paper defines a MEEM framework for continuous radially-monotonic annular bodies, derives the block-matrix structure and interface matching conditions from standard eigenfunction expansions in annular regions, and validates all accuracy (within 5% for 15° slants) and convergence (2% with order-of-magnitude speed-up) claims through direct numerical comparison to the independent Capytaine BEM solver. No fitted parameters are renamed as predictions, no self-citation chain is load-bearing for the central results, and the monotonic-profile assumption is stated explicitly as a modeling precondition rather than derived from the outputs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard potential-flow assumptions and eigenfunction matching at cylindrical interfaces; no new free parameters, ad-hoc constants, or postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption The velocity potential in each fluid region can be expanded in a complete set of eigenfunctions that satisfy the linearized free-surface and bottom boundary conditions.
    Standard assumption in linear water-wave theory for axisymmetric geometries.

pith-pipeline@v0.9.0 · 5829 in / 1308 out tokens · 61272 ms · 2026-05-20T01:45:10.260961+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    Michelén and Ancellin, Matthieu and Haji, Maha , month = jan, year =

    Khanal, Kapil and Ströfer, Carlos A. Michelén and Ancellin, Matthieu and Haji, Maha , month = jan, year =. Fully. doi:10.48550/arXiv.2501.06988 , abstract =

    work page doi:10.48550/arxiv.2501.06988

  2. [2]

    Open-source toolbox for semi-analytical hydrodynamic coefficients via the matched eigenfunction expansion method , url =

    McCabe, Rebecca and Khanal, Kapil and Haji, Maha , month = aug, year =. Open-source toolbox for semi-analytical hydrodynamic coefficients via the matched eigenfunction expansion method , url =. doi:10.5281/zenodo.14504017 , abstract =

    work page doi:10.5281/zenodo.14504017

  3. [3]

    Best, Hope and Khanal, Kapil and McCabe, Rebecca and Jiang, Ruiyang and Treacy, Collin and Haji, Maha , year =

    work page

  4. [4]

    , month = jan, year =

    Chau, Fun Pang and Yeung, Ronald W. , month = jan, year =. Inertia and. 25th

    work page

  5. [5]

    Development,

    McCabe, Rebecca and Dietrich, Madison and Haji, Maha , year =. Development,

    work page

  6. [6]

    Theoretical

    Nguyen, Nhu , month = jun, year =. Theoretical. doi:10.1115/OMAE2024-128018 , abstract =

    work page doi:10.1115/omae2024-128018

  7. [7]

    Applied Ocean Research , author =

    Theoretical modeling of a bottom-raised oscillating surge wave energy converter structural loadings and power performances , volume =. Applied Ocean Research , author =. 2024 , keywords =. doi:10.1016/j.apor.2024.104031 , abstract =

    work page doi:10.1016/j.apor.2024.104031 2024

  8. [8]

    A survey of

    Hiptmair, Ralf and Moiola, Andrea and Perugia, Ilaria , year =. A survey of. Building bridges: connections and challenges in modern approaches to numerical

    work page

  9. [9]

    Renewable Energy , author =

    A novel two-objective optimization computational framework for a two-body heaving wave energy converter , volume =. Renewable Energy , author =. 2022 , keywords =. doi:10.1016/j.renene.2022.04.006 , abstract =

    work page doi:10.1016/j.renene.2022.04.006 2022

  10. [10]

    Advances in Engineering Software , author =

    Trefftz method: an overview , volume =. Advances in Engineering Software , author =. 1995 , keywords =. doi:10.1016/0965-9978(95)00067-4 , abstract =

    work page doi:10.1016/0965-9978(95)00067-4 1995

  11. [11]

    Seah, Robert K. M. and Yeung, Ronald W. , month = apr, year =. Symmetric

    work page

  12. [12]

    Numerical

    Quarteroni, Alfio and Sacco, Riccardo and Saleri, Fausto , month = oct, year =. Numerical

    work page

  13. [13]

    Fuchs, R. A. and MacCamy, R. C. , month = dec, year =. Wave forces on piles: a diffraction theory , shorttitle =

    work page

  14. [14]

    Proceedings of the Royal Society of London

    The forces on a circular cylinder submerged in a uniform stream , volume =. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences , author =. 1936 , pages =. doi:10.1098/rspa.1936.0212 , abstract =

    work page doi:10.1098/rspa.1936.0212 1936

  15. [15]

    Theory of the synchronous motion of an array of floating flap gates oscillating wave surge converter , volume =

    Michele, Simone and Sammarco, Paolo and d’Errico, Michele , month = aug, year =. Theory of the synchronous motion of an array of floating flap gates oscillating wave surge converter , volume =. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , publisher =. doi:10.1098/rspa.2016.0174 , abstract =

    work page doi:10.1098/rspa.2016.0174 2016

  16. [16]

    and Yue, Dick K.-P

    Edwards, Emma C. and Yue, Dick K.-P. , month = feb, year =. Optimisation of the geometry of axisymmetric point-absorber wave energy converters , volume =. Journal of Fluid Mechanics , publisher =. doi:10.1017/jfm.2021.993 , abstract =

    work page doi:10.1017/jfm.2021.993 2021

  17. [17]

    Garrett, C. J. R. , month = sep, year =. Bottomless harbours , volume =. Journal of Fluid Mechanics , publisher =. doi:10.1017/S0022112070002495 , abstract =

    work page doi:10.1017/s0022112070002495

  18. [18]

    , year =

    Chatjigeorgiou, Ioannis K. , year =. Analytical. doi:10.1017/9781316838983 , abstract =

    work page doi:10.1017/9781316838983

  19. [19]

    Applied Ocean Research , publisher =

    Reflection of oblique ocean water waves by a vertical porous structure placed on a multi-step impermeable bottom , volume =. Applied Ocean Research , publisher =. 2014 , pages =. doi:10.1016/j.apor.2014.07.001 , abstract =

    work page doi:10.1016/j.apor.2014.07.001 2014

  20. [20]

    Garrett, C. J. R. , month = mar, year =. Wave forces on a circular dock , volume =. Journal of Fluid Mechanics , publisher =. doi:10.1017/S0022112071000430 , abstract =

    work page doi:10.1017/s0022112071000430

  21. [21]

    Marine Structures , author =

    Second-order hydrodynamic effects on an arrangement of two concentric truncated vertical cylinders , volume =. Marine Structures , author =. 2009 , keywords =. doi:10.1016/j.marstruc.2008.12.003 , abstract =

    work page doi:10.1016/j.marstruc.2008.12.003 2009

  22. [22]

    Ocean Engineering , author =

    Hydrodynamic characteristics of floating toroidal bodies , volume =. Ocean Engineering , author =. 1997 , pages =. doi:10.1016/S0029-8018(96)00020-0 , abstract =

    work page doi:10.1016/s0029-8018(96)00020-0 1997

  23. [23]

    China Ocean Engineering , author =

    Hydrodynamic analysis and shape optimization for vertical axisymmetric wave energy converters , volume =. China Ocean Engineering , author =. 2016 , keywords =. doi:10.1007/s13344-016-0062-2 , abstract =

    work page doi:10.1007/s13344-016-0062-2 2016

  24. [24]

    Ocean Engineering , author =

    Behaviour of vertical bodies of revolution in waves , volume =. Ocean Engineering , author =. 1986 , pages =. doi:10.1016/0029-8018(86)90037-5 , abstract =

    work page doi:10.1016/0029-8018(86)90037-5 1986

  25. [25]

    Coastal Engineering , author =

    Analysis of orthogonal wave reflection by a caisson with open front chamber filled with sloping rubble mound , volume =. Coastal Engineering , author =. 2014 , keywords =. doi:10.1016/j.coastaleng.2014.05.002 , abstract =

    work page doi:10.1016/j.coastaleng.2014.05.002 2014

  26. [26]

    Applied Ocean Research , author =

    Added mass and damping of a vertical cylinder in finite-depth waters , volume =. Applied Ocean Research , author =. 1981 , pages =. doi:10.1016/0141-1187(81)90101-2 , abstract =

    work page doi:10.1016/0141-1187(81)90101-2 1981

  27. [27]

    Ocean Engineering , author =

    A novel solution to the second-order wave radiation force on an oscillating truncated cylinder based on the application of control surfaces , volume =. Ocean Engineering , author =. 2020 , keywords =. doi:10.1016/j.oceaneng.2020.107278 , abstract =

    work page doi:10.1016/j.oceaneng.2020.107278 2020

  28. [28]

    Applied Ocean Research , author =

    The added mass of bodies heaving at low frequency in water of finite depth , volume =. Applied Ocean Research , author =. 1991 , pages =. doi:10.1016/S0141-1187(05)80036-7 , abstract =

    work page doi:10.1016/s0141-1187(05)80036-7 1991

  29. [29]

    IEEE Journal of Oceanic Engineering , author =

    Hydrodynamic. IEEE Journal of Oceanic Engineering , author =. 2015 , note =. doi:10.1109/JOE.2014.2344951 , abstract =

    work page doi:10.1109/joe.2014.2344951 2015

  30. [30]

    Renewable Energy , author =

    Performance assessment of a. Renewable Energy , author =. 2024 , keywords =. doi:10.1016/j.renene.2024.121547 , abstract =

    work page doi:10.1016/j.renene.2024.121547 2024

  31. [31]

    European Journal of Mechanics - B/Fluids , author =

    Hydrodynamics of the oscillating wave surge converter in the open ocean , volume =. European Journal of Mechanics - B/Fluids , author =. 2013 , keywords =. doi:10.1016/j.euromechflu.2013.01.007 , abstract =

    work page doi:10.1016/j.euromechflu.2013.01.007 2013

  32. [32]

    Applied Ocean Research , author =

    Hydrodynamic coefficients in heave of two concentric surface-piercing truncated circular cylinders , volume =. Applied Ocean Research , author =. 2004 , keywords =. doi:10.1016/j.apor.2005.03.002 , abstract =

    work page doi:10.1016/j.apor.2005.03.002 2004

  33. [33]

    , year =

    Chau, Fun Pang and Yeung, Ronald W. , year =. Inertia,. doi:10.1115/OMAE2012-83987 , abstract =

    work page doi:10.1115/omae2012-83987

  34. [34]

    , month = jan, year =

    Chau, Fun Pang and Yeung, Ronald W. , month = jan, year =. Inertia and

    work page

  35. [35]

    2005 , publisher=

    Theory and applications of ocean surface waves: Part 1: linear aspects , author=. 2005 , publisher=

    work page 2005

  36. [36]

    IET Renewable Power Generation , volume=

    Energy-maximising moment-based constrained optimal control of ocean wave energy farms , author=. IET Renewable Power Generation , volume=. 2021 , publisher=

    work page 2021

  37. [37]

    2018 , publisher=

    Analytical methods in marine hydrodynamics , author=. 2018 , publisher=

    work page 2018

  38. [38]

    International Conference on Offshore Mechanics and Arctic Engineering , volume=

    Inertia, damping, and wave excitation of heaving coaxial cylinders , author=. International Conference on Offshore Mechanics and Arctic Engineering , volume=. 2012 , organization=

    work page 2012

  39. [39]

    2018 , publisher=

    Marine hydrodynamics , author=. 2018 , publisher=

    work page 2018

  40. [40]

    Abramowitz and I

    M. Abramowitz and I. Stegun , title =

    work page

  41. [41]

    Barker and D

    T. Barker and D. G. Schaeffer and P. Boh\'orquez and J. M. N. T. Gray , title =. J. Fluid. Mech. , year =

    work page

  42. [42]

    Batchelor , title =

    G.K. Batchelor , title =. J. Fluid Mech. , year =

    work page

  43. [43]

    A. V. Briukhanov and S. S. Grigorian and S. M. Miagkov and M. Y. Plam and I. E. Shurova and M. E. Eglit and Y. L. Yakimov , title =. Physics of Snow and Ice, Proceedings of the International Conference on Low Temperature Science , volume =

    work page

  44. [44]

    Brownell and L.K

    C.J. Brownell and L.K. Su , title =. AIAA Paper 2004-2335 , year =

    work page 2004

  45. [45]

    Brownell and L.K

    C.J. Brownell and L.K. Su , title =. AIAA Paper 2007-1314 , year =

    work page 2007

  46. [46]

    Dennis , title =

    S.C.R. Dennis , title =. Ninth Intl Conf. on Numerical Methods in Fluid Dynamics , publisher =. 1985 , editor =

    work page 1985

  47. [47]

    Formation of levees, troughs and elevated channels by avalanches on erodible slopes , author =. J. Fluid Mech. , volume =

    work page

  48. [48]

    Hwang and E.O

    L.-S. Hwang and E.O. Tuck , title =. J. Fluid Mech. , year =

    work page

  49. [49]

    and Saut, Jean Claude , title =

    Joseph, Daniel D. and Saut, Jean Claude , title =. Theoretical and Computational Fluid Dynamics , publisher=

    work page

  50. [50]

    Worster , title =

    M.G. Worster , title =. Interactive dynamics of convection and solidification , publisher =. 1992 , editor =

    work page 1992

  51. [51]

    Koch , title =

    W. Koch , title =. J. Sound Vib. , year =

    work page

  52. [52]

    Lee , title =

    J.-J. Lee , title =. J. Fluid Mech. , year =

    work page

  53. [53]

    Linton and D.V

    C.M. Linton and D.V. Evans , title =. Phil.\ Trans.\ R. Soc.\ Lond. , year =

    work page

  54. [54]

    Martin , title =

    P.A. Martin , title =. 1980 , journal =

    work page 1980

  55. [55]

    Rogallo , title =

    R.S. Rogallo , title =

    work page

  56. [56]

    Ursell , title =

    F. Ursell , title =. 1950 , journal =

    work page 1950

  57. [57]

    On the oscillations Near and at resonance in open pipes , year =

    L. On the oscillations Near and at resonance in open pipes , year =. J. Engng Maths , volume =

    work page

  58. [58]

    Miller , title =

    P.L. Miller , title =. 1991 , address =

    work page 1991

  59. [59]

    Camera Calibration Toolbox for Matlab , howpublished =

    Bouguet, J.-Y , year=. Camera Calibration Toolbox for Matlab , howpublished =

    work page

  60. [60]

    Sustainability , volume=

    Techno-economic related metrics for a wave energy converters feasibility assessment , author=. Sustainability , volume=. 2016 , publisher=

    work page 2016

  61. [61]

    IEEE Transactions on Sustainable Energy , volume=

    Control co-design of power take-off systems for wave energy converters using WecOptTool , author=. IEEE Transactions on Sustainable Energy , volume=. 2023 , publisher=

    work page 2023

  62. [62]

    2020 International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM) , pages=

    Super capacitor energy storage system design for wave energy converter demonstration , author=. 2020 International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM) , pages=. 2020 , organization=

    work page 2020

  63. [63]

    International Marine Energy Journal , volume=

    Degrees of freedom effects on a laboratory scale WEC point absorber , author=. International Marine Energy Journal , volume=. 2025 , url=

    work page 2025

  64. [64]

    2014 , journal=

    Methodology for design and economic analysis of marine energy conversion (MEC) technologies , author=. 2014 , journal=

    work page 2014

  65. [65]

    Ocean engineering , volume=

    Hydrodynamic characteristics of floating toroidal bodies , author=. Ocean engineering , volume=. 1997 , publisher=

    work page 1997

  66. [66]

    International Conference on Offshore Mechanics and Arctic Engineering , volume=

    Hydrodynamic analysis of a vertical axisymmetric oscillating water column device floating in finite depth waters , author=. International Conference on Offshore Mechanics and Arctic Engineering , volume=. 2012 , organization=

    work page 2012