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Information stored in Faraday waves: the origin of a path memory

Published online by Cambridge University Press:  25 March 2011

ANTONIN EDDI*
Affiliation:
Laboratoire Matières et Systèmes Complexes, Université Paris Diderot and CNRS, UMR 7057, Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
ERIC SULTAN
Affiliation:
Laboratoire FAST, Université Pierre et Marie Curie, Université Paris-Sud and CNRS, UMR 7608, Bâtiment 502, Campus Universitaire, 91405 Orsay, France
JULIEN MOUKHTAR
Affiliation:
Laboratoire Matières et Systèmes Complexes, Université Paris Diderot and CNRS, UMR 7057, Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
EMMANUEL FORT
Affiliation:
Institut Langevin, ESPCI ParisTech, Université Paris Diderot and CNRS, UMR 7587, 10 rue Vauquelin, 75231 Paris CEDEX 05, France
MAURICE ROSSI
Affiliation:
Institut Jean Le Rond D'Alembert, Université Pierre et Marie Curie and CNRS, UMR 7190, 4 Place Jussieu, 75252 Paris CEDEX 05, France
YVES COUDER
Affiliation:
Laboratoire Matières et Systèmes Complexes, Université Paris Diderot and CNRS, UMR 7057, Bâtiment Condorcet, 10 rue Alice Domon et Léonie Duquet, 75013 Paris, France
*
Email address for correspondence: antonin.eddi@univ-paris-diderot.fr

Abstract

On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely. When approaching the Faraday instability onset, the droplet couples to the wave it generates and starts propagating horizontally. The resulting wave–particle association, called a walker, was shown previously to have remarkable dynamical properties, reminiscent of quantum behaviours. In the present article, the nature of a walker's wave field is investigated experimentally, numerically and theoretically. It is shown to result from the superposition of waves emitted by the droplet collisions with the interface. A single impact is studied experimentally and in a fluid mechanics theoretical approach. It is shown that each shock emits a radial travelling wave, leaving behind a localized mode of slowly decaying Faraday standing waves. As it moves, the walker keeps generating waves and the global structure of the wave field results from the linear superposition of the waves generated along the recent trajectory. For rectilinear trajectories, this results in a Fresnel interference pattern of the global wave field. Since the droplet moves due to its interaction with the distorted interface, this means that it is guided by a pilot wave that contains a path memory. Through this wave-mediated memory, the past as well as the environment determines the walker's present motion.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Aranson, I. S., Gorshkov, K. A., Lomov, A. S. & Rabinovich, M. I. 1990 Stable particle-like solutions of multidimensional nonlinear fields. Physica D 43, 435453.CrossRefGoogle Scholar
Bach, G. A., Koch, D. L. & Gopinath, A. 2004 Coalescence and bouncing of small aerosol droplets. J. Fluid Mech. 518, 157185.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of a plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505515.Google Scholar
Bohm, D. 1952 A suggested interpretation of the quantum theory in terms of hidden variables. Phys. Rev. 85, 166179.CrossRefGoogle Scholar
Born, M. & Wolf, E. 1959 Principles of Optics. Cambridge University Press.Google Scholar
de Broglie, L. 1926 Ondes et mouvements. Gautier Villars.Google Scholar
Burke, J. & Knobloch, E. 2006 Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211.CrossRefGoogle ScholarPubMed
Ciliberto, S. & Gollub, J. P. 1984 Pattern competition leads to chaos. Phys. Rev. Lett. 52, 922925.CrossRefGoogle Scholar
Cornu, A. 1874 Méthode nouvelle pour la discussion des problèmes de diffraction dans le cas d'une onde cylindrique. J. Phys. 3, 44.Google Scholar
Couder, Y. 2000 Viscous fingering as an archetype for growth patterns. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffat, H. K. & Worster, M. G.), pp. 53104. Cambridge University Press.Google Scholar
Couder, Y. & Fort, E. 2006 Single particle diffraction and interferences at macroscopic scale. Phys. Rev. Lett. 97, 154101.CrossRefGoogle ScholarPubMed
Couder, Y., Fort, E., Gautier, C. H. & Boudaoud, A. 2005 a From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.CrossRefGoogle ScholarPubMed
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005 b Dynamical phenomena: walking and orbiting droplets. Nature 437, 208.CrossRefGoogle ScholarPubMed
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.CrossRefGoogle Scholar
Douady, S. & Couder, Y. 1996 Phyllotaxis as a dynamical self-organizing process, the simulation of the transient regimes of ontogeny. J. Theor. Biol. 178, 295312.CrossRefGoogle Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Europhys. Lett. 6, 221226.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102, 240401.CrossRefGoogle ScholarPubMed
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.CrossRefGoogle Scholar
Faraday, M. 1831 On the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 52, 299.Google Scholar
Fauve, S. 1998 Pattern forming instabilities. In Hydrodynamics and Nonlinear Instabilities (ed. Godreche, C. & Manneville, P.), pp. 387491. Cambridge University Press.CrossRefGoogle Scholar
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl Acad. Sci. USA 107 (41), 1751517520.CrossRefGoogle Scholar
Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., Booth, M. & Rossi, F. 2009 GNU Scientific Library Reference Manual, 3rd edn. Network Theory Ltd.Google Scholar
Gilet, T. & Bush, J. W. M. 2009 a Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.CrossRefGoogle ScholarPubMed
Gilet, T. & Bush, J. W. M. 2009 b The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167203.CrossRefGoogle Scholar
Goldman, T., Livne, A. & Fineberg, J. 2010 Acquisition of inertia by a moving crack. Phys. Rev. Lett. 104, 114301.CrossRefGoogle ScholarPubMed
Gopinath, A. & Koch, D. L. 2001 Dynamics of droplet rebound from a weakly deformable gas–liquid interface. Phys. Fluids 13, 3526.CrossRefGoogle Scholar
Gopinath, A. & Koch, D. L. 2002 Collision and rebound of small droplets in an incompressible continuum gas. J. Fluid Mech. 454, 145201.CrossRefGoogle Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.CrossRefGoogle Scholar
Huygens, C. 1690 Traité de la lumière. Van der Aa.Google Scholar
Keolian, R., Turkevich, L. A., Putterman, J., Rudnick, I. & Rudnick, J. A. 1984 Subharmonic sequences in the Faraday experiment: departures from period doubling. Phys. Rev. Lett. 47, 11331136.CrossRefGoogle Scholar
Kolodner, P., Bensimon, D. & Surko, C. M. 1988 Traveling-wave convection in an annulus. Phys. Rev. Lett. 60, 17231726.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.CrossRefGoogle Scholar
Liehr, A. W., Moskalenko, A. S., Astrov Yu., A., Bode, M. & Purwins, H.-G. 2004 Rotating bound states of dissipative solitons in systems of reaction–diffusion type. Eur. Phys. J. B 37, 199204.CrossRefGoogle Scholar
Lioubashevski, O. & Fineberg, J. 2001 Shock wave criterion for propagating solitary states in driven surface waves. Phys. Rev. E 63, 035302.CrossRefGoogle ScholarPubMed
Moisy, F., Rabaud, M. & Salsac, K. 2009 A synthetic Schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46, 10211036.CrossRefGoogle Scholar
Pieranski, P. 1983 Jumping particle model: period doubling cascade in an experimental system. J. Phys. (Paris) 44, 573578.CrossRefGoogle Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85108.CrossRefGoogle Scholar
Ramazza, P. L., Benkler, E., Bortolozzo, U., Boccaletti, S., Ducci, S. & Arecchi, F. T. 2002 Tailoring the profile and interactions of optical localized structures. Phys. Rev. E 65, 066204.CrossRefGoogle ScholarPubMed
Thual, O. & Fauve, S. 1988 Localized structures generated by subcritical instabilities. J. Phys. 49, 18291833.CrossRefGoogle Scholar
Tsimring, L. S. & Aranson, I. S. 1997 Localized and cellular patterns in a vibrated granular layer. Phys. Rev. Lett. 79, 213216.CrossRefGoogle Scholar
Umbanhowar, P. B., Melo, F. & Swinney, H. L. 1996 Localized excitations in a vertically vibrated granular layer. Nature 382, 793796.CrossRefGoogle Scholar
Vandewalle, N., Terwagne, D., Mulleners, K., Gilet, T. & Dorbolo, S. 2008 Dynamics of a bouncing droplet onto a vertically vibrated surface. Phys. Rev. Lett. 100, 167802.Google Scholar