Comptes Rendus
Bouncing drops, memory
Resonant interactions in bouncing droplet chains
Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 573-589.

In a pioneering series of experiments, Yves Couder, Emmanuel Fort and coworkers demonstrated that droplets bouncing on the surface of a vertically vibrating fluid bath exhibit phenomena reminiscent of those observed in the microscopic quantum realm. Inspired by this discovery, we here conduct a theoretical and numerical investigation into the structure and dynamics of one-dimensional chains of bouncing droplets. We demonstrate that such chains undergo an oscillatory instability as the system’s wave-induced memory is increased progressively. The predicted oscillation frequency compares well with previously reported experimental data. We then investigate the resonant oscillations excited in the chain when the drop at one end is subjected to periodic forcing in the horizontal direction. At relatively high memory, the drops may oscillate with an amplitude larger than that prescribed, suggesting that the drops effectively extract energy from the collective wave field. We also find that dynamic stabilization of new bouncing states can be achieved by forcing the chain at high frequency. Generally, our work provides insight into the collective behavior of particles interacting through long-range and temporally nonlocal forces.

Supplementary Materials:
Supplementary material for this article is supplied as a separate file:

Published online:
DOI: 10.5802/crmeca.30
Keywords: Pilot-wave hydrodynamics, Walking droplets, Nonlinear dynamics, Drop interactions, Collective dynamics, Non-equilibrium systems
Lauren Barnes 1; Giuseppe Pucci 2; Anand U. Oza 3

1 Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
2 Université de Rennes 1, CNRS, IPR (Institut de Physique de Rennes) UMR 6251, F-35000 Rennes, France
3 Department of Mathematical Sciences & Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Lauren Barnes and Giuseppe Pucci and Anand U. Oza},
     title = {Resonant interactions in bouncing droplet chains},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {573--589},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {6-7},
     year = {2020},
     doi = {10.5802/crmeca.30},
     language = {en},
AU  - Lauren Barnes
AU  - Giuseppe Pucci
AU  - Anand U. Oza
TI  - Resonant interactions in bouncing droplet chains
JO  - Comptes Rendus. Mécanique
PY  - 2020
SP  - 573
EP  - 589
VL  - 348
IS  - 6-7
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.30
LA  - en
ID  - CRMECA_2020__348_6-7_573_0
ER  - 
%0 Journal Article
%A Lauren Barnes
%A Giuseppe Pucci
%A Anand U. Oza
%T Resonant interactions in bouncing droplet chains
%J Comptes Rendus. Mécanique
%D 2020
%P 573-589
%V 348
%N 6-7
%I Académie des sciences, Paris
%R 10.5802/crmeca.30
%G en
%F CRMECA_2020__348_6-7_573_0
Lauren Barnes; Giuseppe Pucci; Anand U. Oza. Resonant interactions in bouncing droplet chains. Comptes Rendus. Mécanique, Volume 348 (2020) no. 6-7, pp. 573-589. doi : 10.5802/crmeca.30.

[1] J. Walker Drops of liquid can be made to float on the liquid. What enables them to do so?, Sci. Am., Volume 238 (1978) (151)

[2] Y. Couder; E. Fort; C.-H. Gautier; A. Boudaoud From bouncing to floating: noncoalescence of drops on a fluid bath, Phys. Rev. Lett., Volume 94 (2005) (177801) | DOI

[3] Y. Couder; S. Protière; E. Fort; A. Boudaoud Walking and orbiting droplets, Nature, Volume 437 (2005) (208) | DOI

[4] S. Protière; A. Boudaoud; Y. Couder Particle-wave association on a fluid interface, J. Fluid Mech., Volume 554 (2006), pp. 85-108 | DOI | MR | Zbl

[5] A. Eddi; E. Fort; F. Moisy; Y. Couder Unpredictable tunneling of a classical wave-particle association, Phys. Rev. Lett., Volume 102 (2009) (240401) | DOI

[6] E. Fort; A. Eddi; J. Moukhtar; A. Boudaoud; Y. Couder Path-memory induced quantization of classical orbits, Proc. Natl. Acad. Sci., Volume 107 (2010) no. 41, pp. 17515-17520 | DOI

[7] D. M. Harris; J. W. M. Bush Drops walking in a rotating frame: From quantized orbits to multimodal statistics, J. Fluid Mech., Volume 739 (2014), pp. 444-464 | DOI

[8] S. Perrard; M. Labousse; M. Miskin; E. Fort; Y. Couder Self-organization into quantized eigenstates of a classical wave-driven particle, Nat. Commun., Volume 5 (2014) (3219) | DOI

[9] S. Perrard; M. Labousse; E. Fort; Y. Couder Chaos driven by interfering memory, Phys. Rev. Lett., Volume 113 (2014) (104101) | DOI

[10] S. Perrard; M. Labousse Transition to chaos in wave memory dynamics in a harmonic well: Deterministic and noise-driven behavior, Chaos, Volume 28 (2018) (096109) | DOI | MR

[11] A. Eddi; J. Moukhtar; S. Perrard; E. Fort; Y. Couder Level splitting at a macroscopic scale, Phys. Rev. Lett., Volume 108 (2012) (264503) | DOI

[12] D. M. Harris; J. Moukhtar; E. Fort; Y. Couder; J. W. M. Bush Wavelike statistics from pilot-wave dynamics in a circular corral, Phys. Rev. E, Volume 88 (2013) (011001)

[13] T. Cristea-Platon; P. J. Sáenz; J. W. M. Bush Walking droplets in a circular corral: Quantisation and chaos, Chaos, Volume 28 (2018) (096116) | DOI

[14] P. J. Sáenz; T. Cristea-Platon; J. W. M. Bush Statistical projection effects in a hydrodynamic pilot-wave system, Nat. Phys., Volume 14 (2018), pp. 315-319 | DOI

[15] P. J. Sáenz; T. Cristea-Platon; J. W. M. Bush A hydrodynamic analog of Friedel oscillations, Sci. Adv., Volume 6 (2020) (20)

[16] R. Valani; A. C. Slim; T. Simula Hong-Ou-Mandel-like two-drop correlations, Chaos, Volume 28 (2018) (096104) | DOI

[17] Y. Couder; E. Fort Single particle diffraction and interference at a macroscopic scale, Phys. Rev. Lett., Volume 97 (2006) (154101) | DOI

[18] A. Andersen; J. Madsen; C. Reichelt; S. R. Ahl; B. Lautrup; C. Ellegaard; M. T. Levinsen; T. Bohr Double-slit experiment with single wave-driven particles and its relation to quantum mechanics, Phys. Rev. E, Volume 92 (2015) (013006) | DOI

[19] T. Bohr; A. Andersen; B. Lautrup Bouncing droplets, pilot-waves, and quantum mechanics, Recent Advances in Fluid Dynamics with Environmental Applications (J. Klapp; L. D. G. Sigalotti; A. Medina; A. López; G. Ruiz-Chavarría, eds.), Springer International Publishing, Switzerland, 2016, pp. 335-349 | DOI

[20] R. Dubertrand; M. Hubert; P. Schlagheck; N. Vandewalle; T. Bastin; J. Martin Scattering theory of walking droplets in the presence of obstacles, New J. Phys., Volume 18 (2016) (113037) | DOI

[21] G. Pucci; D. M. Harris; L. M. Faria; J. W. M. Bush Walking droplets interacting with single and double slits, J. Fluid Mech., Volume 835 (2018), pp. 1136-1156 | DOI

[22] M. Rode; J. Madsen; A. Andersen Wave fields in double-slit experiments with wave-driven droplets, Phys. Rev. Fluids, Volume 4 (2019) (104801) | DOI

[23] C. Ellegaard; M. T. Levinsen “Interaction of wave-driven particles with slit structures”, preprint, arXiv:2005.12335 (2020)

[24] Y. Couder; E. Fort Probabilities and trajectories in a classical wave-particle duality, J. Phys.: Conf. Ser., Volume 361 (2012) (012001)

[25] J. W. M. Bush Pilot-wave hydrodynamics, Ann. Rev. Fluid Mech., Volume 47 (2015), pp. 269-292 | DOI | MR

[26] J. W. M. Bush The new wave of pilot-wave theory, Phys. Today, Volume 68 (2015) no. 8, pp. 47-53 | DOI

[27] A. Eddi; E. Sultan; J. Moukhtar; E. Fort; M. Rossi; Y. Couder Information stored in Faraday waves: the origin of a path memory, J. Fluid Mech., Volume 674 (2011), pp. 433-463 | DOI | MR | Zbl

[28] A. Eddi; A. Decelle; E. Fort; Y. Couder Archimedean lattices in the bound states of wave interacting particles, Europhys. Lett., Volume 87 (2009) (56002) | DOI

[29] S. Protière; Y. Couder; E. Fort; A. Boudaoud The self-organization of capillary wave sources, J. Phys.: Condens. Matter, Volume 17 (2005) (3529)

[30] S. I. Lieber; M. C. Hendershott; A. Pattanaporkratana; J. E. Maclennan Self-organization of bouncing oil drops: two dimensional lattices and spinning clusters, Phys. Rev. E, Volume 75 (2007) (56308)

[31] A. Eddi; D. Terwagne; E. Fort; Y. Couder Wave propelled ratchets and drifting rafts, Europhys. Lett., Volume 82 (2008) (44001) | DOI

[32] A. Eddi; A. Boudaoud; Y. Couder Oscillating instability in bouncing drop crystals, Euro. Phys. Lett., Volume 94 (2011) (20004) | DOI

[33] C. Kittel Introduction to Solid State Physics, Vol. 8, Wiley, New York, 1976

[34] P. J. Sáenz; G. Pucci; A. Goujon; T. Cristea-Platon; J. Dunkel; J. W. M. Bush Spin lattices of walking droplets, Phys. Rev. Fluids, Volume 3 (2018) (100508)

[35] P. J. Sáenz; G. Pucci; S. E. Turton; A. Goujon; R. R. Rosales; J. Dunkel; J. W. M. Bush “Emergent order in hydrodynamic spin lattices”, (2020) (submitted)

[36] B. Filoux; M. Hubert; N. Vandewalle Strings of droplets propelled by coherent waves, Phys. Rev. E, Volume 92 (2015) 041004(R) | DOI

[37] A. Rahman Standard map-like models for single and multiple walkers in an annular cavity, Chaos, Volume 28 (2018) (096102) | DOI | MR | Zbl

[38] S. J. Thomson; M. M. P. Couchman; J. W. M. Bush Collective vibrations of confined levitating droplets, Phys. Rev. Fluids, Volume 5 (2020) (083601)

[39] S. J. Thomson; M. Durey; R. R. Rosales Collective vibrations of a hydrodynamic active lattice, Proc. R. Soc. A, Volume 476 (2020) no. 2239 | MR

[40] M. M. P. Couchman; J. W. M. Bush “Free rings of bouncing droplets: stability and dynamics”, J. Fluid Mech. (2020) (in press)

[41] P. L. Kapitza Dynamical stability of a pendulum when its point of suspension vibrates, Collected Papers of P. L. Kapitza, Vol. 2 (D. Ter Haar, ed.), Pergamon, Oxford, UK, 1965, pp. 714-725

[42] P. L. Kapitza Pendulum with a vibrating suspension, Collected Papers of P. L. Kapitza, Vol. 2 (D. Ter Haar, ed.), Pergamon, Oxford, UK, 1965, pp. 726-737

[43] L. Gammaitoni; P. Hänggi; P. Jung; F. Marchesoni Stochastic resonance, Rev. Mod. Phys., Volume 70 (1998) no. 1, pp. 223-287 | DOI

[44] P. Hänggi Driven quantum systems, Quantum Transport and Dissipation (T. Dittrich; P. Hänggi; G.-L. Ingold; G. Schön; W. Zwerger, eds.), Wiley-VCH, New York, 1998, pp. 249-286

[45] M. Bukov; L. D’Aless; A. Polkovnikov Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering, Adv. Phys., Volume 64 (2015) no. 2, pp. 139-226 | DOI

[46] F. L. Traversa; M. Di Ventra; F. Bonani Generalized Floquet theory: Application to dynamical systems with memory and Bloch’s theorem for nonlocal potentials, Phys. Rev. Lett., Volume 110 (2013) (170602)

[47] S. Perrard (Une mémoire ondulatoire: Etats propres, Chaos et Probabilités. PhD thesis, Université Paris Diderot, 2014, p. 182-183)

[48] M. Faraday On the forms and states of fluids on vibrating elastic surfaces, Phil. Trans. R. Soc. Lond., Volume 121 (1831), pp. 319-340

[49] T. Gilet; J. W. M. Bush The fluid trampoline: droplets bouncing on a soap film, J. Fluid Mech., Volume 625 (2009), pp. 167-203 | DOI | MR | Zbl

[50] T. B. Benjamin; F. Ursell The stability of the plane free surface of a liquid in vertical periodic motion, Proc. R. Soc. Lond. A, Volume 225 (1954), pp. 505-515 | MR | Zbl

[51] K. Kumar; L. S. Tuckerman Parametric instability of the interface between two fluids, J. Fluid Mech., Volume 279 (1994), pp. 49-68 | DOI | MR | Zbl

[52] K. Kumar Linear theory of Faraday instability in viscous liquids, Proc. R. Soc. A, Volume 452 (1996), pp. 1113-1126 | MR | Zbl

[53] J. Moláček; J. W. M. Bush Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory, J. Fluid Mech., Volume 727 (2013), pp. 612-647 | DOI | Zbl

[54] A. U. Oza; R. R. Rosales; J. W. M. Bush A trajectory equation for walking droplets: hydrodynamic pilot-wave theory, J. Fluid Mech., Volume 737 (2013), pp. 552-570 | DOI | MR | Zbl

[55] A. Prosperetti Viscous effects on small-amplitude surface waves, Phys. Fluids, Volume 19 (1976) no. 2, pp. 195-203 | DOI | MR | Zbl

[56] A. P. Damiano; P.-T. Brun; D. M. Harris; C. A. Galeano-Rios; J. W. M. Bush Surface topography measurements of the bouncing droplet experiment, Exp. Fluids, Volume 57 (2016) (163)

[57] P. A. Milewski; C. Galeano-Rios; A. Nachbin; J. W. M. Bush Faraday pilot-wave dynamics: modelling and computation, J. Fluid Mech., Volume 778 (2015), pp. 361-388 | DOI | MR | Zbl

[58] S. E. Turton; M. M. P. Couchman; J. W. M. Bush A review of the theoretical modeling of walking droplets: towards a generalized pilot-wave framework, Chaos, Volume 28 (2018) (096111)

[59] L. Tadrist; J.-B. Shim; T. Gilet; P. Schlagheck Faraday instability and subthreshold Faraday waves: surface waves emitted by walkers, J. Fluid Mech., Volume 848 (2018), pp. 906-945 | DOI | MR | Zbl

[60] M. M. P. Couchman; S. E. Turton; J. W. M. Bush Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs, J. Fluid Mech., Volume 871 (2019), pp. 212-243 | DOI | Zbl

[61] A. U. Oza; E. Siéfert; D. M. Harris; J. Moláček; J. W. M. Bush Orbiting pairs of walking droplets: Dynamics and stability, Phys. Rev. Fluids, Volume 2 (2017) (053601)

[62] J. Arbelaiz; A. U. Oza; J. W. M. Bush Promenading pairs of walking droplets: Dynamics and stability, Phys. Rev. Fluids, Volume 3 (2018) (013604) | DOI

[63] Ø. Wind-Willassen; J. Moláček; D. M. Harris; J. W. M. Bush Exotic states of bouncing and walking droplets, Phys. Fluids, Volume 25 (2013) (082002) | DOI | Zbl

[64] A. U. Oza; Ø. Wind-Willassen; D. M Harris; R. R. Rosales; J. W. M. Bush Pilot-wave hydrodynamics in a rotating frame: Exotic orbits, Phys. Fluids, Volume 26 (2014) (082101) | Zbl

[65] N. B. Budanur; M. Fleury State space geometry of the chaotic pilot-wave hydrodynamics, Chaos, Volume 29 (2019) (013122) | MR | Zbl

[66] L. D. Landau; E. M. Lifshitz Course of Theoretical Physics. Vol. 1: Mechanics, Butterworth-Heinemann, Oxford, UK, 1976

[67] C. O. Reinhold; J. Burgdörfer; M. T. Frey; F. B. Dunning Dynamic stabilization of the periodically kicked Rydberg atom, Phys. Rev. Lett., Volume 79 (1997) (26)

[68] T. M. Hoang; C. S. Gerving; B. J. Land; M. Anquez; C. D. Hamley; M. S. Chapman Dynamic stabilization of a quantum many-body spin system, Phys. Rev. Lett., Volume 111 (2013) (090403)

[69] S. S. Rao Mechanical Vibrations, Addison Wesley, Harlow, Essex, UK, 1995 | Zbl

[70] C. Borghesi; J. Moukhtar; M. Labousse; A. Eddi; E. Fort; Y. Couder Interaction of two walkers: Wave-mediated energy and force, Phys. Rev. E, Volume 90 (2014) (063017) | DOI

[71] T. Beatus; T. Tlusty; B.-Z. Roy Phonons in a one-dimensional microfluidic crystal, Nat. Phys., Volume 2 (2006) no. 11, pp. 743-748 | DOI

Cited by Sources:

Articles of potential interest

Walkers in a wave field with memory

Olivier Devauchelle; Éric Lajeunesse; François James; ...

C. R. Méca (2020)

Hydrodynamic quantum field theory: the free particle

Yuval Dagan; John W. M. Bush

C. R. Méca (2020)

Terra Bulla, the influence of Yves Couder on the emerging domain of arts and physics sciences

Jean-Marc Chomaz

C. R. Méca (2020)