Hostname: page-component-7d8f8d645b-9fg92 Total loading time: 0 Render date: 2023-05-29T00:21:33.044Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory

Published online by Cambridge University Press:  28 June 2013

Jan Moláček
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
John W. M. Bush*
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Email address for correspondence:


We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Benjamin, T. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505515.CrossRefGoogle Scholar
de Broglie, L. 1987 Interpretation of quantum mechanics by the double solution theory. Ann. Fond. Louis de Broglie 12, 123 (English translation).Google Scholar
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. 107, 17 45517 456.CrossRefGoogle Scholar
Chang, E. J. & Maxey, M. R. 1994 Unsteady flow about a sphere at low to moderate Reynolds number. Part 1. Oscillatory motion. J. Fluid Mech. 277, 347379.CrossRefGoogle Scholar
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at macroscopic scale. Phys. Rev. Lett. 97, 154101.CrossRefGoogle ScholarPubMed
Couder, Y., Fort, E., Gautier, C. H. & Boudaoud, A. 2005a 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. 2005b Walking and orbiting droplets. Nature 437, 208.CrossRefGoogle ScholarPubMed
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave-particle association. Phys. Rev. Lett. 102, 240401.CrossRefGoogle ScholarPubMed
Eddi, A., Moukhtar, J., Perrard, S., Fort, E. & Couder, Y. 2012 Level splitting at macroscopic scale. Phys. Rev. Lett. 108, 264503.CrossRefGoogle ScholarPubMed
Eddi, A., Sultan, E., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. 2011 Information stored in Faraday waves: the origin of a path memory. J. Fluid Mech. 674, 433463.CrossRefGoogle Scholar
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.CrossRefGoogle Scholar
Flügge, S. 1959 Handbuch der Physik: Strömungsmechanik I. Springer.Google Scholar
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl Acad. Sci. 107, 17 51517 520.CrossRefGoogle Scholar
Gilet, T. & Bush, J. W. M. 2009 The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167203.CrossRefGoogle Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – I: Motion through a quiescent fluid. Chem. Engng Sci. 22, 637651.CrossRefGoogle Scholar
Harris, D. M., Moukhtar, J., Fort, E., Couder, Y. & Bush, J. W. M. 2013 Wave-like statistics from pilot-wave dynamics in a circular corral. Phys Rev. E (submitted).CrossRefGoogle Scholar
Hartland, S. 1971 The pressure distribution in axisymmetric draining films. J. Colloid Interface Sci. 35, 227237.CrossRefGoogle Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. Math. Phys. Engng Sci. 452, 11131126.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Moláček, J. & Bush, J. W. M. 2012 A quasi-static model of drop impact. Phys. Fluids 24, 127103.CrossRefGoogle Scholar
Moláček, J. & Bush, J. W. M. 2013 Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582611.CrossRefGoogle Scholar
Oza, A. U., Rosales, R. R. & Bush, J. W. M. 2013 A trajectory equation for walking droplets: hydrodynamic pilot-wave theory. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Prosperetti, A. 1976 Viscous effects on small-amplitude surface waves. Phys. Fluids 19, 195203.CrossRefGoogle Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle-wave association on a fluid interface. J. Fluid Mech. 554, 85108.CrossRefGoogle Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17, S3529S3535.Google Scholar
Shirokoff, D. 2013 Bouncing droplets on a billiard table. Chaos 23, 013115.CrossRefGoogle ScholarPubMed
Torby, B. J. 1984 Advanced Dynamics for Engineers. Holt, Rinehart and Winston.Google Scholar
Walker, J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so? The Amateur Scientist, Sci. Am. 238, 151158.Google Scholar
Wind-Willassen, Ø., Molác¨ek, J., Harris, D. M. & Bush, J. W. M 2013 Exotic states of bouncing and walking droplets. Phys. Fluids (submitted).CrossRefGoogle Scholar