pith. sign in

arxiv: 2605.21940 · v1 · pith:KXP5RRTHnew · submitted 2026-05-21 · ⚛️ physics.flu-dyn

Vertical motion of a periodically driven floating disc

Anand U. Oza , Jack-William Barotta , Eli Silver , Daniel M. Harris This is my paper

Pith reviewed 2026-05-22 03:36 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords floating discvertical dynamicsperiodic forcingwavefieldFredholm integral equationadded masswave dampingeffective stiffness
0
0 comments X

The pith

A linear inviscid wave model predicts the vertical oscillation amplitude of a periodically forced floating disc as a function of driving frequency.

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

The paper establishes a mathematical description of the water waves around a floating disc that is pushed up and down at a steady frequency. It converts the governing partial differential equations into a single integral equation that can be solved numerically to find the disc motion. The resulting amplitude-versus-frequency curve matches laboratory measurements closely. The same solution also supplies the effective added mass, wave damping, and restoring stiffness felt by the disc. These quantities are computed over a range of frequencies and given in closed form for the slow-driving limit.

Core claim

The axisymmetric inviscid wavefield around the disc satisfies a linear elliptic boundary value problem with mixed conditions: no penetration beneath the disc and standard free-surface conditions outside it. This system is recast as a second-kind Fredholm integral equation whose numerical solution yields the disc's vertical displacement amplitude as a function of forcing frequency. The computed amplitude-frequency relation agrees well with experiment. The same solution supplies frequency-dependent added-mass, wave-damping, and effective-spring coefficients, which are also obtained analytically in the low-frequency limit.

What carries the argument

Second-kind Fredholm integral equation obtained by reformulating the linear elliptic boundary-value problem for the axisymmetric velocity potential.

If this is right

  • The amplitude of vertical oscillation is a computable function of forcing frequency that can be read off from the integral-equation solution.
  • Added mass, wave damping and effective stiffness of the disc vary with frequency and approach explicit analytical values at low frequency.
  • The same integral-equation framework supplies the hydrodynamic loads needed for any similar axisymmetric floating body under periodic vertical drive.
  • Agreement between the computed curve and experiment supports the use of inviscid linear theory for this class of problems at moderate amplitudes.

Where Pith is reading between the lines

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

  • The same integral-equation technique could be adapted to predict the response of discs with different radii or densities without new experiments.
  • Frequency-dependent coefficients extracted here might be inserted directly into simple oscillator models for preliminary design of floating sensors or energy harvesters.
  • Relaxing axisymmetry in the integral formulation would allow treatment of tilted or laterally offset forcing while retaining the linear inviscid assumption.

Load-bearing premise

The wavefield stays perfectly axisymmetric and inviscid, obeying linear theory with no-penetration under the disc and free-surface conditions away from it.

What would settle it

Direct measurement of disc vertical position versus time at a forcing frequency where the model predicts a specific amplitude; systematic deviation larger than experimental uncertainty would contradict the central claim.

Figures

Figures reproduced from arXiv: 2605.21940 by Anand U. Oza, Daniel M. Harris, Eli Silver, Jack-William Barotta.

Figure 1
Figure 1. Figure 1: Schematic of the mathematical problem solved in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) A schematic of the experimental setup. The floating disc is oscillated vertically, with its position tracked by a laser position sensor mounted directly above. (b) A zoomed-in view on the floating disc driven to oscillate via an AC coil and aligned with the center of the bath via a DC coil. (c) Experimental image of a disc of radius 𝑅 = 0.4 cm driven at 30 Hz. (d) Time traces of the vertical position o… view at source ↗
Figure 3
Figure 3. Figure 3: The transmissibility (a) and phase lag (b) of a vertically driven disc, with different values of the fluid kinematic viscosity indicted by the different marker shapes. The insets show experiments with a fluid with kinematic viscosity 𝜈 = 13 cSt, for different values of the driving amplitude as indicated by circular markers of different sizes. The theoretical predictions for the corresponding dimensionless … view at source ↗
Figure 4
Figure 4. Figure 4: (a) Plot of the wavefield Re[𝐻(𝑟)] at the times 𝑡 = [2𝑛𝜋 − arg(𝑍)]/𝜔, at which the disc (red line) is at its maximum amplitude and has zero velocity, 𝜁 = |𝑍| and ¤𝜁 = 0. The parameters are Bo = 3 and Ω = 1. (b) Colormap shows the pressure with the hydrostatic contribution removed, Im[ΩΦ(𝑟, 𝑧)], at the same times. Gray lines show streamlines of the flow. Note that the horizontal axis is different to that in… view at source ↗
Figure 5
Figure 5. Figure 5: (a) The added mass coefficient 𝑀P of a driven disc as a function of the dimensionless frequency, Ω, for the different Bond numbers indicated in the legend. The black curve corresponds to the gravity-wave regime in which surface tension is neglected, Bo = ∞. (b) The dependence of the capillary spring constant 𝑘ST on Ω for different Bo. The inset shows the ratio of 𝑘ST to the static spring constant 𝑘 0 ST de… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Dependence of the maximum oscillation amplitude of the disc [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Physical quantities in the low-frequency limit [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a-f) Transmissibility and phase lag for six different discs, with parameters listed in each plot legend. The mean values of the experimental data across six independent trials are shown as points, with one standard deviation shown as the shaded region between the means of the trials. The corresponding theoretical predictions are shown as the solid curves. The data in panels (a), (c), and (f) are also show… view at source ↗
read the original abstract

We present the results of a combined theoretical and experimental investigation into the vertical dynamics of floating discs subjected to an imposed time-periodic forcing. The axisymmetric and inviscid wavefield is governed by a linear elliptic boundary value problem with mixed boundary conditions, wherein the no-penetration boundary condition is satisfied under the disc while the free surface boundary conditions are enforced away from it. The problem is solved by recasting the system of partial differential equations as a second-kind Fredholm integral equation which is then solved numerically. The solution furnishes a prediction for the dependence of the disc's oscillation amplitude on the forcing frequency, which exhibits excellent agreement with experiments. We interpret our results physically by computing the added mass, wave damping and effective spring coefficients of the disc, both numerically for a range of forcing frequencies and analytically in the low-frequency limit.

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

0 major / 2 minor

Summary. The manuscript presents a combined theoretical and experimental investigation of the vertical dynamics of a floating disc under imposed time-periodic forcing. The axisymmetric inviscid wavefield is governed by a linear elliptic boundary-value problem with mixed boundary conditions (no-penetration under the disc and free-surface conditions away from it). This system is recast as a second-kind Fredholm integral equation and solved numerically to obtain a prediction for the disc's oscillation amplitude as a function of forcing frequency. The predictions exhibit excellent agreement with independent experiments. Hydrodynamic coefficients (added mass, wave damping, and effective spring) are extracted both numerically across a frequency range and analytically in the low-frequency limit.

Significance. If the central results hold, the work supplies a validated, zero-parameter prediction for the amplitude-frequency response of a periodically forced floating disc together with physical insight into the relevant hydrodynamic coefficients. The combination of a standard linear water-wave formulation, numerical solution of the integral equation, independent low-frequency analytics, and direct experimental comparison strengthens the case for the applicability of the inviscid axisymmetric model. Such results are relevant to fluid-structure interaction problems in marine and offshore engineering.

minor comments (2)
  1. The abstract states that the numerical solution furnishes a prediction exhibiting 'excellent agreement with experiments,' yet provides no quantitative measures (e.g., RMS error, frequency range, or number of trials) that would allow readers to assess the strength of the validation.
  2. A summary table collecting the frequency-dependent added-mass, damping, and spring coefficients (both numerical and low-frequency analytic) would improve clarity and facilitate comparison with the forced-oscillator model.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the linear wave problem solution, the recognition of its significance for fluid-structure interaction, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper formulates the axisymmetric inviscid wavefield as a standard linear elliptic BVP with mixed boundary conditions, recasts it as a second-kind Fredholm integral equation, and solves numerically to extract hydrodynamic coefficients (added mass, wave damping, effective spring). These coefficients are inserted into the disc's forced-oscillator equation to obtain the amplitude-versus-frequency curve. Low-frequency analytical limits are derived independently. The resulting zero-parameter prediction is compared directly to separate experiments. No fitted parameters are renamed as predictions, no self-citations are load-bearing for the central result, and no step reduces by construction to its own inputs. The chain is externally falsifiable and does not rely on prior author work for uniqueness or ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on standard potential-flow assumptions for water waves; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption The wavefield is axisymmetric and inviscid.
    Explicitly stated as governing the linear elliptic boundary value problem.
  • domain assumption Mixed boundary conditions: no-penetration under the disc and free-surface conditions away from it.
    Defines the elliptic problem solved by the Fredholm integral equation.

pith-pipeline@v0.9.0 · 5674 in / 1170 out tokens · 48599 ms · 2026-05-22T03:36:52.320218+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

246 extracted references · 246 canonical work pages

  1. [1]

    The Exact Solutions of the Problems in Forced Capillary-Gravity Waves Generated by a Plane Wavemaker under

    Nai-Sher Yeh , journal =. The Exact Solutions of the Problems in Forced Capillary-Gravity Waves Generated by a Plane Wavemaker under. 2008 , pages =

    work page 2008

  2. [2]

    A note on the plane vertical wavemaker in the presence of surface tension , author =. Quart. Appl. Math. , volume =. 1991 , pages =

    work page 1991

  3. [3]

    L. M. Hocking , year=. Reflection of capillary-gravity waves , volume=. doi:, journal=

    work page

  4. [4]

    L. M. Hocking , year=. Capillary-gravity waves with boundaries: Three-dimensional effects , volume=. doi:, journal=

    work page

  5. [5]

    G. W. Young and S. H. Davis , year=. A plate oscillating across a liquid interface: effects of contact-angle hysteresis , volume=. doi:, journal=

    work page

  6. [6]

    L. M. Hocking , year=. Capillary-gravity waves produced by a wavemaker , volume=. doi:, journal=

    work page

  7. [7]

    L. M. Hocking , year=. Capillary-gravity waves produced by a heaving body , volume=. doi:, journal=

    work page

  8. [8]

    Physics of Fluids , author=

    Scattering of a capillary–gravity wave by a vertical cylinder , volume=. Physics of Fluids , author=. 1990 , pages=. doi:, number=

    work page 1990

  9. [9]

    Mathematical Proceedings of the Cambridge Philosophical Society , author=

    The effect of a fixed vertical barrier on surface waves in deep water , volume=. Mathematical Proceedings of the Cambridge Philosophical Society , author=. 1947 , pages=. doi:10.1017/S0305004100023604 , number=

    work page doi:10.1017/s0305004100023604 1947

  10. [10]

    R. Holford. On the generation of short two-dimensional surface waves by oscillating surface cylinders of arbitrary cross-section. 1965

    work page 1965

  11. [11]

    Hermans, A. J. , title =. Journal of Engineering Mathematics , year =. doi:10.1007/BF01535270 , url =

    work page doi:10.1007/bf01535270

  12. [12]

    Hermans, A. J. , title =. Journal of Engineering Mathematics , year =. doi:10.1007/BF01535193 , url =

    work page doi:10.1007/bf01535193

  13. [13]

    Ursell , title =

    F. Ursell , title =. The Quarterly Journal of Mechanics and Applied Mathematics , volume =. 1954 , month =. doi:10.1093/qjmam/7.4.427 , url =

    work page doi:10.1093/qjmam/7.4.427 1954

  14. [14]

    I , volume=

    Short surface waves in the presence of a finite dock. I , volume=. Mathematical Proceedings of the Cambridge Philosophical Society , author=. 1964 , pages=. doi:10.1017/S0305004100038433 , number=

    work page doi:10.1017/s0305004100038433 1964

  15. [15]

    Meylan , year=

    Michael H. Meylan , year=. Time-dependent linear water-wave scattering in two dimensions by a generalized eigenfunction expansion , volume=. doi:10.1017/S002211200900723X , journal=

    work page doi:10.1017/s002211200900723x

  16. [16]

    Propulsion and interaction of wave-propelled interfacial particles , author =. Phys. Rev. Fluids , volume =. 2025 , month =. doi:10.1103/353x-p2dx , url =

    work page doi:10.1103/353x-p2dx 2025

  17. [17]

    Report 1661, Department of the Navy, David W

    The impulse response function and ship motion , author=. Report 1661, Department of the Navy, David W. Taylor Model Basin, Hydromechanics Laboratory, Research and Development Report, October 1962 , year=

    work page 1962

  18. [18]

    and Prakash, Manu and Chan, Brian and Bush, John W

    Hu, David L. and Prakash, Manu and Chan, Brian and Bush, John W. M. , title =. Experiments in Fluids , year =. doi:10.1007/s00348-007-0339-6 , url =

    work page doi:10.1007/s00348-007-0339-6

  19. [19]

    Catalytic nanomotors for environmental monitoring and water remediation

    Soler, Lluís and Sánchez, Samuel. Catalytic nanomotors for environmental monitoring and water remediation. Nanoscale. 2014. doi:10.1039/C4NR01321B

    work page doi:10.1039/c4nr01321b 2014

  20. [20]

    Timm and Saeed Jafari Kang and Jonathan P

    Mitchel L. Timm and Saeed Jafari Kang and Jonathan P. Rothstein and Hassan Masoud , title =. 2021 , publisher =. doi:10.1088/1748-3190/ac253c , url =

    work page doi:10.1088/1748-3190/ac253c 2021

  21. [21]

    Advanced Materials Interfaces , volume =

    Feldmann, David and Das, Rakesh and Pinchasik, Bat-El , title =. Advanced Materials Interfaces , volume =. doi:https://doi.org/10.1002/admi.202001300 , url =

    work page doi:10.1002/admi.202001300

  22. [22]

    , title =

    Chen, Yufeng and Doshi, Neel and Goldberg, Benjamin and Wang, Hongqiang and Wood, Robert J. , title =. Nature Communications , year =. doi:10.1038/s41467-018-04855-9 , url =

    work page doi:10.1038/s41467-018-04855-9

  23. [23]

    Jablonski and Robert J

    Je-Sung Koh and Eunjin Yang and Gwang-Pil Jung and Sun-Pill Jung and Jae Hak Son and Sang-Im Lee and Piotr G. Jablonski and Robert J. Wood and Ho-Young Kim and Kyu-Jin Cho , title =. Science , volume =. 2015 , doi =

    work page 2015

  24. [24]

    Taylor and Metin Sitti , title =

    Onur Ozcan and Han Wang and Jonathan D. Taylor and Metin Sitti , title =. International Journal of Advanced Robotic Systems , volume =. 2014 , doi =

    work page 2014

  25. [25]

    and McIver, M

    McIver, P. and McIver, M. and Zhang, J. , year=. Excitation of trapped water waves by the forced motion of structures , volume=. doi:10.1017/S0022112003005949 , journal=

    work page doi:10.1017/s0022112003005949

  26. [26]

    and Meylan, Michael H

    Fitzgerald, Colm J. and Meylan, Michael H. , year=. Generalized eigenfunction method for floating bodies , volume=. doi:10.1017/S0022112010005653 , journal=

    work page doi:10.1017/s0022112010005653

  27. [27]

    Journal of Fluid Mechanics , author=

    The transient motion of a floating body , volume=. Journal of Fluid Mechanics , author=. 1970 , pages=. doi:10.1017/S0022112070001842 , number=

    work page doi:10.1017/s0022112070001842 1970

  28. [28]

    Communications on Pure and Applied Mathematics , volume =

    Rubin, Hanan , title =. Communications on Pure and Applied Mathematics , volume =. doi:https://doi.org/10.1002/cpa.3160070205 , url =

    work page doi:10.1002/cpa.3160070205

  29. [29]

    MacCamy, R. C. , title =. Journal of Ship Research , volume =. 1961 , issn =. doi:10.5957/jsr.1961.5.4.34 , url =

    work page doi:10.5957/jsr.1961.5.4.34 1961

  30. [30]

    and MacCamy, Richard C

    Heins, Albert E. and MacCamy, Richard C. , title =. Zeitschrift für angewandte Mathematik und Physik (. 1960 , volume =. doi:10.1007/BF01602674 , url =

    work page doi:10.1007/bf01602674 1960

  31. [31]

    Peters, A. S. and Stoker, J. J. , title =. Communications on Pure and Applied Mathematics , volume =. doi:https://doi.org/10.1002/cpa.3160100307 , url =

    work page doi:10.1002/cpa.3160100307

  32. [32]

    MacCamy, R. C. , title =. Archive for Rational Mechanics and Analysis , year =. doi:10.1007/BF00277434 , url =

    work page doi:10.1007/bf00277434

  33. [33]

    Journal of Fluid Mechanics , author=

    The pitching motion of a circular disk , volume=. Journal of Fluid Mechanics , author=. 1963 , pages=. doi:10.1017/S0022112063001543 , number=

    work page doi:10.1017/s0022112063001543 1963

  34. [34]

    J. C. Murray , journal =. On the linear capillary gravity wave problem , volume =

    work page

  35. [35]

    Mathematical Proceedings of the Cambridge Philosophical Society , author=

    The influence of surface tension on the reflection of water waves by a plane vertical barrier , volume=. Mathematical Proceedings of the Cambridge Philosophical Society , author=. 1968 , pages=. doi:10.1017/S0305004100043504 , number=

    work page doi:10.1017/s0305004100043504 1968

  36. [36]

    Mathematical Proceedings of the Cambridge Philosophical Society , author=

    The effect of surface tension on the waves produced by a heaving circular cylinder , volume=. Mathematical Proceedings of the Cambridge Philosophical Society , author=. 1968 , pages=. doi:10.1017/S030500410004353X , number=

    work page doi:10.1017/s030500410004353x 1968

  37. [37]

    Journal of Fluid Mechanics , author=

    Gravity wave damping of hydrostatic oscillations for a buoyant disk , volume=. Journal of Fluid Mechanics , author=. 1968 , pages=. doi:10.1017/S0022112068000170 , number=

    work page doi:10.1017/s0022112068000170 1968

  38. [38]

    Davis, A. M. J. , title =. Mathematika , volume =. doi:https://doi.org/10.1112/S0025579300008342 , url =

    work page doi:10.1112/s0025579300008342

  39. [39]

    Journal of Fluid Mechanics , author=

    On the harmonic oscillations of a rigid body on a free surface , volume=. Journal of Fluid Mechanics , author=. 1965 , pages=. doi:10.1017/S0022112065000253 , number=

    work page doi:10.1017/s0022112065000253 1965

  40. [40]

    The Motion of Floating Bodies

    Wehausen, J V. The Motion of Floating Bodies. Annual Review of Fluid Mechanics. 1971. doi:https://doi.org/10.1146/annurev.fl.03.010171.001321

    work page doi:10.1146/annurev.fl.03.010171.001321 1971

  41. [41]

    Communications on Pure and Applied Mathematics , volume =

    John, Fritz , title =. Communications on Pure and Applied Mathematics , volume =. doi:https://doi.org/10.1002/cpa.3160020102 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.3160020102 , year =

    work page doi:10.1002/cpa.3160020102

  42. [42]

    Annals of PDE , year =

    David Lannes , title =. Annals of PDE , year =. doi:10.1007/s40818-017-0029-5 , url =

    work page doi:10.1007/s40818-017-0029-5

  43. [43]

    Hulme, A. , year=. The wave forces acting on a floating hemisphere undergoing forced periodic oscillations , volume=. doi:10.1017/S0022112082001980 , journal=

    work page doi:10.1017/s0022112082001980

  44. [44]

    Proceedings of the Royal Society of London

    Havelock, Thomas Henry , title =. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , volume =. 1955 , doi =

    work page 1955

  45. [45]

    and Anathakrishnan, P

    Yeung, Ronald W. and Anathakrishnan, P. , year=. Viscosity and Surface-Tension Effects on Wave Generation by a Translating Body , volume=. doi:10.1023/A:1004291021985 , journal=

    work page doi:10.1023/a:1004291021985

  46. [46]

    and Anathakrishnan, P

    Yeung, Ronald W. and Anathakrishnan, P. , year=. Oscillation of a floating body in a viscous fluid , volume=. doi:10.1007/BF00043236 , journal=

    work page doi:10.1007/bf00043236

  47. [47]

    Miles, John W. , year=. On surface-wave forcing by a circular disk , volume=. doi:10.1017/S0022112087000302 , journal=

    work page doi:10.1017/s0022112087000302

  48. [48]

    J. C. Murray , title =. 1975 , volume =. doi:10.1007/BF01174022 , url =

    work page doi:10.1007/bf01174022 1975

  49. [49]

    Cohl, H. S. , journal =. Portent of. 2003 , page =

    work page 2003

  50. [50]

    Cohl and Joel E

    Howard S. Cohl and Joel E. Tohline , title =. 1999 , publisher =. doi:10.1086/308062 , url =

    work page doi:10.1086/308062 1999

  51. [51]

    Synchronization and self-assembly of free capillary spinners , author =. Phys. Rev. E , volume =. 2025 , month =. doi:10.1103/PhysRevE.111.035104 , url =

    work page doi:10.1103/physreve.111.035104 2025

  52. [52]

    Journal of Mechanical Science and Technology , volume =

    Bio-inspired micro/mini propulsion at air-water interface: A review , author =. Journal of Mechanical Science and Technology , volume =. doi:10.1007/s12206-012-1002-6 , url =

    work page doi:10.1007/s12206-012-1002-6

  53. [53]

    Journal of Fluid Mechanics , volume =

    Dynamics of gravity-capillary solitary waves in deep water , author =. Journal of Fluid Mechanics , volume =

    work page

  54. [54]

    Stability of periodic waves of finite amplitude on the surface of a deep fluid , author =. S. J. Appl. Mech. Tech. Phys. , volume =

    work page

  55. [55]

    , journal =

    Hooshanginejad, Alireza and Barotta, Jack-William and Spradlin, Victoria and Pucci, Giuseppe and Hunt, Robert and Harris, Daniel M. , journal =. Interactions and pattern formation in a macroscopic magnetocapillary. 2024 , doi =

    work page 2024

  56. [56]

    2022 , publisher =

    Eugene Rhee and Robert Hunt and Stuart J Thomson and Daniel M Harris , title =. 2022 , publisher =. doi:10.1088/1748-3190/ac78b6 , url =

    work page doi:10.1088/1748-3190/ac78b6 2022

  57. [57]

    Synchronization of wave-propelled capillary spinners , author =. Phys. Rev. E , volume =. 2025 , doi =

    work page 2025

  58. [58]

    Communications Physics , volume =

    Bidirectional wave-propelled capillary spinners , author =. Communications Physics , volume =. 2023 , doi =

    work page 2023

  59. [59]

    and Devauchelle, Olivier and Thomson, Stuart J

    Benham, Graham P. and Devauchelle, Olivier and Thomson, Stuart J. , year=. On wave-driven propulsion , volume=. doi:10.1017/jfm.2024.352 , journal=

    work page doi:10.1017/jfm.2024.352 2024

  60. [60]

    2023 , howpublished =

    Wave field of capillary surfers , author =. 2023 , howpublished =

    work page 2023

  61. [61]

    Harris , title =

    Alireza Hooshanginejad and Victoria Spradlin and Jack-William Barotta and Robert Hunt and Giuseppe Pucci and Daniel M. Harris , title =

    work page

  62. [62]

    Theoretical modeling of capillary surfer interactions on a vibrating fluid bath , author =. Phys. Rev. Fluids , volume =. 2023 , publisher =. doi:10.1103/PhysRevFluids.8.114001 , url =

    work page doi:10.1103/physrevfluids.8.114001 2023

  63. [63]

    Journal of Fluid Mechanics , volume=

    Capillary-scale solid rebounds: experiments, modelling and simulations , author=. Journal of Fluid Mechanics , volume=. 2021 , publisher=

    work page 2021

  64. [64]

    Journal of Fluid Mechanics , volume=

    Inertio-capillary rebound of a droplet impacting a fluid bath , author=. Journal of Fluid Mechanics , volume=. 2023 , publisher=

    work page 2023

  65. [65]

    Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume =

    Abramowitz, Milton and Stegun, Irene A , editor =. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume =

    work page

  66. [66]

    A. P. Prudnikov and Y. A. Brychkov and O. I. Marichev and R. H. Romer , publisher =. Integrals and Series Vol. 2: Special Functions , year =

    work page

  67. [67]

    Capillary surfers: Wave-driven particles at a vibrating fluid interface , author =. Phys. Rev. Fluids , volume =. 2023 , publisher =. doi:10.1103/PhysRevFluids.8.L112001 , url =

    work page doi:10.1103/physrevfluids.8.l112001 2023

  68. [68]

    Douglas Schwarz , title =

    work page

  69. [69]

    NIST Digital Library of Mathematical Functions

    work page

  70. [70]

    Lamb , publisher =

    H. Lamb , publisher =. Hydrodynamics , year =

    work page

  71. [71]

    Vella and L

    D. Vella and L. Mahadevan , journal=. The ``

    work page

  72. [72]

    Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions , author=. Phys. Lett. A , volume=

    work page

  73. [73]

    On the forms and states assumed by fluids in contact with elastic vibrating surfaces , author=. Philos. Trans. R. Soc. London , volume=

    work page

  74. [74]

    Communications on Pure and Applied Mathematics , volume =

    John, Fritz , title =. Communications on Pure and Applied Mathematics , volume =. doi:https://doi.org/10.1002/cpa.3160030106 , url =

    work page doi:10.1002/cpa.3160030106

  75. [75]

    and Taylor, Geoffrey Ingram , title =

    Ursell, F. and Taylor, Geoffrey Ingram , title =. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences , volume =. 1953 , doi =

    work page 1953

  76. [76]

    Leppington, F. G. , year=. On the radiation and scattering of short surface waves. Journal of Fluid Mechanics , publisher=. doi:10.1017/S0022112072002216 , number=

    work page doi:10.1017/s0022112072002216

  77. [77]

    Leppington, F. G. , year=. On the radiation and scattering of short surface waves. Journal of Fluid Mechanics , publisher=. doi:10.1017/S0022112073001461 , number=

    work page doi:10.1017/s0022112073001461

  78. [78]

    and Linder, Vincent and Sia, Samuel K

    Chin, Curtis D. and Linder, Vincent and Sia, Samuel K. Lab-on-a-chip devices for global health: Past studies and future opportunities. Lab Chip. 2007. doi:10.1039/B611455E

    work page doi:10.1039/b611455e 2007

  79. [79]

    1981 , issn =

    The interaction of colloidal particles collected at fluid interfaces , journal =. 1981 , issn =. doi:https://doi.org/10.1016/0021-9797(81)90092-8 , url =

    work page doi:10.1016/0021-9797(81)90092-8 1981

  80. [80]

    1992 , issn =

    Capillary meniscus interaction between colloidal particles attached to a liquid—fluid interface , journal =. 1992 , issn =. doi:https://doi.org/10.1016/0021-9797(92)90239-I , url =

    work page doi:10.1016/0021-9797(92)90239-i 1992

Showing first 80 references.