Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Achieving Optimal Locomotion using Self-Generated Waves

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Waves alone can propel a vessel at perfect efficiency

desk verdict Clean optimal-control template for wave-driven thrust, with a strong bounded-power result; the acceleration claim in Section 7 needs a formal quasi-static justification. read the letter →

arxiv 2506.11961 v1 pith:C2LARECM submitted 2025-06-13 physics.flu-dyn

D. O'Donovan , M. D. Bustamante , O. Devauchelle , G. P. Benham This is my paper · ORCID
classification physics.flu-dyn MSC 76B1549J20
keywords wave-drivenpropulsionradiationstressshallowwaterwavesoptimalcontrolboundedpowerself-generatedcruisingvelocityFroudenumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks how an oscillating body floating on shallow water should move its pressure field to get the most forward thrust from the waves it generates itself. In the physically motivated case where the time-averaged power injected by the body is bounded, the answer is simple: emit a wave only in the aft direction and none ahead. Under that condition the dimensionless thrust equals the power bound, $\bar F_T = \delta$, so the efficiency defined as thrust divided by power is exactly 1 for every drift velocity $v \ge 0$, in subcritical, critical, and supercritical regimes alike. The paper then couples this optimal thrust to a simple quadratic drag law to show that slowly increasing the power bound accelerates the body from rest to supercritical cruising speeds, with cruising velocity $v_* = \sqrt{2H\delta/(L C_D)}$.

What carries the argument

The load-bearing object is the radiation-stress thrust identity $\bar F_T = \langle \hat Q, \hat h' \rangle = -\tfrac12 [k|\hat h|^2]_-^+$, which equates the thrust to the time-averaged difference of fore-aft wave amplitude squared, derived by multiplying the wave equation by $\hat h'$ and integrating. The optimal-control mechanism is the variational calculus with the power bound $\langle \hat Q, i\hat h\rangle - v\langle \hat Q, \hat h'\rangle \le \delta$, whose Euler-Lagrange condition reduces to a first-order eigenvalue ODE with eigenvalues $\lambda = \pm 1$, selecting a single Doppler-shifted travelling wave. This is what forces the optimal source to radiate only aft, and it is what converts the optimisation into a linear constraint problem with infinitely many solutions, e.g., step-function pressure distributions.

What would settle it

Measure the thrust and the injected power of a real oscillating pressure source on a shallow water tank while resolving the fore and aft wave amplitudes. If the maximum achievable thrust-to-power ratio falls below 1 once surface tension, viscosity, and 2D losses are present, or if the optimal source cannot be made to emit zero wave ahead, the efficiency-1 claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that, in a linear 1D shallow-water model, the thrust from self-generated waves is the fore-aft difference of wave amplitude squared (radiation stress), and when the injected power is bounded the optimal pressure source is a purely aft-travelling wave. This yields dimensionless thrust equal to the power bound, $\bar F_T = \delta$, and formal efficiency $\eta = 1$ for all drift velocities $v \ge 0$, because no power is wasted on a forward wave. The optimum holds across the subcritical, critical, and supercritical regimes, defined by $v = U/c$ with $c = \sqrt{gH}$, and in the supercritical case both waves travel leftward in the body frame yet the aft choice still achieves $\bar F_T = \delta$.

Load-bearing premise

The central claim rests on the linear 1D shallow-water wave model with no surface tension, viscosity, or 2D effects, and the acceleration recipe further assumes that the speed changes slowly enough for the periodic optimal solutions to hold at every instant.

Editorial extensions

If this is right

  • In the bounded-power case, the optimal time-averaged thrust is exactly equal to the power bound, $\bar F_T = \delta$, at every drift velocity, so propulsive efficiency is 1 across subcritical, critical, and supercritical motion.
  • The optimal pressure source radiates no wave ahead of the body: all injected power goes into the aft wave, and this remains true for every value of the drift velocity.
  • Modulating the power bound $\delta$ changes the cruising velocity according to $v_* = \sqrt{2H\delta/(L C_D)}$, so a slowly accelerating body can be driven from rest to supercritical speeds by increasing the injected power.
  • In the bounded-norm case, there are resonant velocities where the forward wave vanishes and efficiency reaches 1, but away from these resonances efficiency drops, so the bounded-power case is the one that guarantees optimal efficiency continuously.
  • There are infinitely many optimal pressure distributions satisfying the bounded-power conditions, including simple step functions, so the optimal strategy is a family rather than a unique waveform.

Reading between the lines

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

  • The 'no forward wave' principle suggests a general design rule for wave-propelled craft: regardless of speed, shape and oscillate the body so that all emitted wave energy is directed opposite to the direction of travel.
  • The same optimal-control template, with a Galilean transform and Doppler-shifted wavenumbers, might be applied to other linear wave-bearing media; a plausible inference is that efficiency-1 optimality extends to any energy-conserving linear wave equation without dissipation.
  • Adding surface tension, viscosity, or finite-depth dispersion would likely break the exact efficiency-1 result, so a testable extension is to compute the optimal fore-aft wave balance for gravity-capillary waves and check how much the aft-only rule changes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies optimal wave-driven propulsion in a 1D shallow-water model where a periodically oscillating pressure source acts as the body. The authors derive expressions for time-averaged thrust and power directly from the wave equation, then solve two constrained optimal-control problems (bounded norm and bounded power) for the start-up, subcritical, critical, and supercritical velocity regimes. The main analytical result is that in the bounded-power case, the optimal thrust equals the power bound (F_T = delta), yielding efficiency eta = 1 for all v > 0. Using a quadratic drag law, the paper derives a cruising velocity v* = sqrt(2H delta/(L C_D)) and argues that slow modulation of the power bound accelerates the body from rest to supercritical velocities. Numerical validation with Ipopt/JuMP is reported for the optimality conditions and thrust values.

Significance. If the results stand, the paper provides a clean analytical template for optimal wave-driven propulsion in a simplified setting, with self-contained derivations of thrust and power from the wave equation rather than imported formulas. The numerical validation is a genuine strength: the analytical optima are checked against an independent discretized optimization, with relative errors mostly below 0.1%, and the authors provide a repository for the code. The bounded-power result that an aft-travelling wave achieves F_T = delta and eta = 1 across all velocity regimes is elegant and intuitively appealing. However, the quantitative reach is limited by the acknowledged idealizations (shallow water, no surface tension, no viscosity, 1D), and the prefactor discrepancy with the classical radiation-stress formula (1.1) means the dimensional cruising-velocity prediction (7.4) should be treated as model-internal unless that factor is resolved.

major comments (3)
  1. [Section 7, Eq. (7.4)] The claim that modulating the power bound delta accelerates the body from rest to supercritical velocities is not supported by the analysis. Equation (7.4) is an algebraic balance F_T = F_D, which defines the terminal cruising velocity for a fixed delta, not an equation of motion. To demonstrate acceleration, the manuscript must state and use the body momentum balance m dv/dt = F_T(v,delta) - F_D(v) and show that a slowly varying delta(t) tracks the instantaneous equilibrium v*(delta(t)). This requires an additional timescale separation, essentially m/(rho L^2 c) being much longer than the modulation timescale; the stated condition \ddot{U} << omega c does not by itself ensure that the body velocity follows v*(delta). Without this step, the central demonstration in Section 7 is incomplete.
  2. [Section 7, Eqs. (4.3) and (4.5)] The quasi-static use of the periodic optimal solutions during acceleration needs a rigorous error bound or a multiple-timescale derivation. The Galilean transformation (4.2) maps between inertial frames and is only valid for constant v; when v = v(t), the wave equation in the moving frame contains additional terms involving dv/dt, and the Sommerfeld boundary conditions (4.5) are not exact. The condition \ddot{U} << omega c is asserted but never derived from the transformed PDE. A concrete test would be to substitute v(t) = v0 + epsilon t into the original wave equation and estimate the size of the neglected terms relative to the leading-order optimal solution. As written, the paper does not establish that the periodic optimal solutions remain valid instantaneously during the proposed acceleration protocol.
  3. [Section 2, Eqs. (1.1), (2.10), and (7.2)-(7.4)] The prefactor discrepancy between the derived thrust (2.10) and the classical radiation-stress formula (1.1) is acknowledged but then set aside. This is acceptable for the dimensionless optimization results, but it is load-bearing for the dimensional cruising-velocity prediction in Section 7: if the physically correct thrust is the classical 3/4 rho g [h^2] result rather than the 1/2 [k|h|^2] version, then v* in (7.4) changes by a factor sqrt(3/2). The manuscript should either incorporate the correct prefactor into the dimensional force balance or explicitly state that (7.4) is only a model-internal estimate, not a quantitative physical prediction.
minor comments (4)
  1. [Section 3.1, Eq. (3.5)] The text reports a relative error of 0.003% for 'lambda|D_L|^2' but the surrounding equations and text elsewhere refer to lambda^2|D_L|^2; please check the exponent and make the notation consistent.
  2. [Section 3.2, Eq. (3.17)] The step-function ansatz is presented as representative, but the manuscript does not explain how it was chosen among the infinite family of optimal sources; a brief remark that the optimality conditions are underdetermined would clarify that the shown solution is one example, not a unique optimum.
  3. [Section 6, Eq. (6.7)] The resonance condition is written for all integers n in Z\{0}, but the subcritical resonances are only physical when 0 < n < l/pi, while supercritical resonances require n < 0; the statement 'for all possible n' is slightly misleading and could be rephrased to specify the allowed ranges.
  4. [Section 7, Fig. 8] The caption for Figure 8 does not explain the colour scheme beyond 'blue' and 'red'; it would help to state explicitly that the first and second halves of the period are plotted in blue and red, respectively.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: thrust and power are derived from the wave equation, and the bounded-power optimum is a genuine variational result; Section 7's quasi-static extension is a stated assumption, not a circular prediction.

full rationale

The paper's central derivation chain is self-contained and does not reduce to its inputs by construction. The thrust F_T = <Q,h'> and the power Pow = <Q,i h> - v<Q,h'> are derived from the 1D wave equation (2.3) through the identities (2.10) and (A.16), rather than being imported as assumed inputs. The bounded-power optimal condition is obtained from variational calculus in Appendix B.2, which produces the eigenvalue problem (B.21) and the two solutions lambda = -1 and lambda = 1; selecting the purely aft-travelling wave (C_R = 0) is a derived extremality condition, not an ansatz dressed as a result. Substituting that condition into the independently defined thrust and power expressions gives F_T = delta and eta = 1, which is algebra following from a genuine optimality condition rather than a definitional tautology. The numerical Ipopt/JuMP solutions independently reproduce the analytical conditions to small relative errors, supporting internal consistency. The only self-citations, to Benham et al. (2022) and Benham et al. (2024), are contextual or concern a simplifying drag-model choice ('Similar to Benham et al. (2024), we will neglect oscillatory drag'), and neither is load-bearing for the main derivation. Section 7 reuses the paper's own periodic optimal solutions to argue that slow modulation of the power bound changes the cruising velocity, but this is explicitly flagged as a quasi-static assumption ('A dimensional acceleration such that U_ddot << omega c permits the use of the previous results'; Section 8: 'A small acceleration U_ddot << omega c is required to maintain the periodic assumption'). The cruising-velocity relation (7.4) is a steady force balance, not a derived acceleration law, so the dynamical extrapolation is limited but not circular. Overall, no prediction in the paper is equivalent to a fitted parameter or to a self-citation chain, and the bounded-power eta = 1 result follows from the derived optimality structure rather than from the problem setup by definition.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claims rest on the linear shallow-water wave model, the periodic single-frequency ansatz, radiation boundary conditions, and a Galilean frame change. No new particles, forces, or conserved quantities are introduced. The free parameters are normalization bounds (epsilon, delta) and an unmeasured drag coefficient C_D; the latter makes Section 7's cruising velocity a scaling demonstration. The step-function source used for validation is an ansatz, not a fitted object.

free parameters (3)
  • norm bound epsilon = 1 (dimensionless)
    Chosen to compare analytical and numerical results in the bounded-norm problem (Sec. 3.1). It is a normalization, not fitted to data, and the paper notes thrust grows with l.
  • power bound delta = 1 for main results; general delta in Sec. 7
    Chosen for the analytical-numerical comparison; in Section 7 delta is the control knob of the cruising-velocity relation v* = sqrt(2H*delta/(L*C_D)), so it is an input parameter rather than a fitted constant.
  • drag coefficient C_D = unspecified (general)
    Introduced in Eq. (7.1) through a simple quadratic drag law to define a cruising velocity. Its value is not measured or fitted, so the v* formula is a scaling demonstration, not a quantitative prediction.
assumptions (7)
  • domain assumption Linear shallow-water wave equation (2.1) obtained from the Euler equations at leading order in aspect ratio epsilon = H/L_w << 1, with h << H and no surface tension, viscosity, or vortices.
    Appendix A.1. This is the physics foundation of the whole paper; all thrust and power results inherit it. It transfers poorly to gravity-capillary WDP relevant to honeybees and water striders, as the discussion acknowledges.
  • domain assumption Source and wave field are strictly periodic with a single frequency omega (Eq. 2.4); quasi-periodic modulation in Section 7 requires acceleration U_ddot << omega*c.
    Periodicity reduces the PDE to an ODE and justifies time-averaging. The applicability of the instantaneous optimal solutions during acceleration is asserted, not derived.
  • domain assumption Sommerfeld radiation boundary conditions (2.7): waves only radiate away from the source; in the supercritical case both radiated waves are left-traveling in the moving frame (6.1)-(6.2).
    These boundary conditions close the Green's function solution and are reasonable for an isolated body, but they exclude reflected or incident waves.
  • standard math Galilean transformation (4.2) maps the lab frame to the body frame, with wave speed c constant and the medium at rest in the lab frame.
    Invoked in Section 4 to handle moving sources. It is standard kinematics, but it assumes no background current and constant depth.
  • domain assumption Thrust is identified with the radiation-stress difference (1/2)[k|h|^2] (Eq. 2.10), and the 3/2 prefactor difference relative to the Longuet-Higgins-Stewart formula (1.1) is set aside as irrelevant.
    Section 2. The absolute thrust scale, and hence the dimensional cruising velocity v* in Section 7, depends on this choice, so the prefactor is not fully irrelevant.
  • ad hoc to paper Simple quadratic drag law F_D = (1/2)*C_D*rho*U^2*L (7.1), with no wave drag, form drag, or oscillatory components.
    Introduced in Section 7 to generate a cruising velocity. The authors acknowledge it is a simplification; the v* ~ sqrt(delta) scaling is a consequence of this assumed law, not a measured relation.
  • ad hoc to paper The step-function source (3.17) is representative of the optimal solutions in the bounded-power case.
    Used in Sections 3.2 and 4.2 for numerical validation. Since (4.18) leaves an infinite-dimensional solution set, the chosen Q is illustrative, and the authors state other optima exist.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Achieving Optimal Locomotion using Self-Generated Waves." pith.science (2026). https://pith.science/paper/C2LARECM

@misc{pith2026250611961,
  author       = {Pith},
  title        = {Pith review of: Achieving Optimal Locomotion using Self-Generated Waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C2LARECM}},
  note         = {Machine review of arXiv:2506.11961}
}
abstract

An oscillating body floating at the water surface produces a wave-field of self-generated waves. When the oscillation induces a difference in fore-aft wave amplitude squared, these self-generated waves can be used as a mechanism to propel the body horizontally across the surface (Longuet-Higgins and Stewart 1964). The optimisation of this wave-driven propulsion (WDP) is the interest of this work. To study the conditions necessary to produce optimal thrust we will utilise a shallow water set-up where a periodically oscillating pressure source acts as the body. In this framework, an expression for the thrust is derived by relation to the aforementioned difference in fore-aft amplitude squared. The conditions on the source for maximal thrust are explored both analytically and numerically in two optimal control problems. The first case is where a bound is imposed on the norm of the control function to regularise it. Secondly, a more physically motivated case is studied where the power injected by the source is bounded. The body is permitted to have a drift velocity $U$. When scaled with the wave speed $c$, the dimensionless velocity $v=U/c$ divides the study into subcritical, critical and supercritical regimes and the optimal conditions are presented for each. The result in the bounded power case is then used to demonstrate how the modulation of power injected can slowly change the cruising velocity from rest to supercritical velocities.

Figures

Figures reproduced from arXiv: 2506.11961 by the authors.

Figure 1
Figure 1. Diagram of the set-up where there is a pressure source [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) An example of a source, 𝑄ˆ (𝑥), that results in the optimal thrust under the bounded norm constraint where 𝑣 = 0 and 𝑙 = 3𝜋/2. (b) The corresponding wave-field, ℎˆ(𝑥) under the same conditions [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Plot of the time-averaged thrust 𝐹¯ 𝑇 vs the dimensionless length scale 𝑙 resulting from the bounded norm optimisation on start-up 𝑣 ≈ 0. It can also be seen that as 𝑙 increases, the dimensional thrust tends to scale with 𝑙 2 which is the same behaviour found in (3.9) for 𝑙 ≫ 1 and numerical expressions was found to be 0.003% for 𝜆|𝐷L| 2 and 0.014% for 𝜆 2 |𝐷R| 2 . Given the aforementioned arbitrary choice of phase,… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (a) Resulting plots for the real and imaginary parts of the source [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: (a) Comparison between the analytically and numerically calculated left and (b) right [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: (a) Contour plot where colour corresponds to the magnitude of the force. For a given [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: (a) Time-averaged thrust over velocities from [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: (a) Demonstration of the one period of oscillation where [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Propulsion and interaction of wave-propelled interfacial particles

    physics.flu-dyn 2025-06 accept novelty 3.0 of 10

    A review of wave-propelled interfacial particles, describing how self-generated capillary waves drive motion and long-range interactions.

Reference graph

Works this paper leans on

24 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    , Alventosa, L.F.L

    Barotta, J.W , Thomson, S.J. , Alventosa, L.F.L. , Lewis, M. & Harris, D.M. 2023 Bidirectional wave-propelled capillary spinners . Commun. Physics 6 (1), 87

    work page 2023

  4. [4]

    , Devauchelle, O

    Benham, G.P. , Devauchelle, O. , Morris, S.W. & Neufeld, J.A. 2022 Gunwale bobbing . Phys. Rev. Fluids 7 (7), 074804

    work page 2022

  5. [5]

    , Devauchelle, O

    Benham, G.P. , Devauchelle, O. & Thomson, S.J. 2024 On wave-driven propulsion . J. Fluid Mech. 987 , A44

    work page 2024

  6. [6]

    & Hu, D.L

    Bush, J.W.M. & Hu, D.L. 2006 Walking on water: Biolocomotion at the interface . Annu. Rev. Fluid Mech. 38 , 339--369

    work page 2006

  7. [7]

    , Yrke, A

    Bøckmann, E. , Yrke, A. & Steen, S. 2018 Fuel savings for a general cargo ship employing retractable bow foils . App. Ocean Research 76 , 1--10

    work page 2018

  8. [8]

    2007 Impulsive fluid forcing and water strider locomotion

    Bühler, O. 2007 Impulsive fluid forcing and water strider locomotion . J. Fluid Mech. 573 , 211–236

    work page 2007

Show all 24 references
  1. [9]

    Ship Technology Research 44 , 32--36

    Doctors, L.J 1997 Optimal pressure distributions for river-based air-cushion vehicles . Ship Technology Research 44 , 32--36

    1997

  2. [10]

    & Sharma, S.D

    Doctors, L.J. & Sharma, S.D. 1972 The wave resistance of an air-cushion vehicle in steady and accelerated motion . J. Ship Research 16 (04), 248--260

    1972

  3. [11]

    , Carmigniani, R

    Dode, A. , Carmigniani, R. , Cohen, C. , Clanet, C. & Bocquet, L. 2022 Wave drag during an unsteady motion . J. Fluid Mech. 951 , A15

    2022

  4. [12]

    & Feng, J.J

    Gao, P. & Feng, J.J. 2011 A numerical investigation of the propulsion of water walkers . J. Fluid Mech. 668 , 363–383

    2011

  5. [13]

    , Pucci, G

    Ho, I. , Pucci, G. , Oza, A.U. & Harris, D.M. 2023 Capillary surfers: Wave-driven particles at a vibrating fluid interface . Phys. Rev. Fluids 8 , L112001

    2023

  6. [14]

    , Chan, B

    Hu, D.L. , Chan, B. & Bush, J.W.M. 2003 The hydrodynamics of water strider locomotion . Nature 424 (6949), 663–666

    2003

  7. [15]

    & Stewart, R.W

    Longuet-Higgins, M.S. & Stewart, R.W. 1964 Radiation stresses in water waves; a physical discussion, with applications . Deep Sea Research and Oceanographic Abstracts 11 (4), 529--562

    1964

  8. [16]

    Longuet-Higgins , M. S. 1977 The Mean Forces Exerted by Waves on Floating or Submerged Bodies with Applications to Sand Bars and Wave Power Machines . Proc. R. Soc. Lon. A 352 (1671), 463--480

    1977

  9. [17]

    , Dowson, O

    Lubin, M. , Dowson, O. , Dias Garcia , F. , Huchette, J. , Legat, B. & Vielma, J.P. 2023 JuMP 1.0: R ecent improvements to a modeling language for mathematical optimization . Math. Prog. Comp. 15 , 581--589

    2023

  10. [18]

    , Ben Amar, M

    Pucci, G. , Ben Amar, M. & Couder, Y. 2015 Faraday instability in floating drops . Phys. of Fluids 27 (9), 091107

    2015

  11. [19]

    , Hunt, R

    Rhee, E. , Hunt, R. , Thomson, S. & Harris, D.M. 2022 Surferbot: a wave-propelled aquatic vibrobot . Bioinspir. and Biomim. 17 (5), 055001

    2022

  12. [20]

    & Gharib, M

    Roh, C. & Gharib, M. 2019 Honeybees use their wings for water surface locomotion . Proc. Natl Acad. Sci. USA 116 (49), 24446--24451

    2019

  13. [21]

    , Arutkin, M

    Steinmann, T. , Arutkin, M. , Cochard, P. , Raphaël, E. , Casas, J. & Benzaquen, M. 2018 Unsteady wave pattern generation by water striders . J. Fluid Mech. 848 , 370–387

    2018

  14. [22]

    , Cribellier, A

    Steinmann, T. , Cribellier, A. & Casas, J. 2021 Singularity of the water strider propulsion mechanisms . J. Fluid Mech. 915 , A118

    2021

  15. [23]

    Tuck, E. O. & Lazauskas, L. 1998 Optimum hull spacing of a family of multihulls . Ship Technol. Res.-Schiffstechnik 45 (4), 180

    1998

  16. [24]

    , Jia, L

    Yuan, Z.M , Chen, M. , Jia, L. , Ji, C. & Incecik, A. 2021 Wave-riding and wave-passing by ducklings in formation swimming . J. Fluid Mech. 928 , R2

    2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.