# A model for Faraday pilot waves over variable topography

Abstract : Couder et al. (Nature, vol. 437 (7056), 2005, p. 208) discovered that droplets walking on a vibrating bath possess certain features previously thought to be exclusive to quantum systems. These millimetric droplets synchronize with their Faraday wavefield, creating a macroscopic pilot-wave system. In this paper we exploit the fact that the waves generated are nearly monochromatic and propose a hydrodynamic model capable of quantitatively capturing the interaction between bouncing drops and a variable topography. We show that our reduced model is able to reproduce some important experiments involving the drop–topography interaction, such as non-specular reflection and single-slit diffraction.

Faria, L. M. (2017). A model for Faraday pilot waves over variable topography. *Journal of Fluid Mechanics*, *811*, 51-66.

# Non-specular reflection of walking droplets

Abstract : Since their discovery by Yves Couder and Emmanuel Fort, droplets walking on a vibrating liquid bath have attracted considerable attention because they unexpectedly exhibit certain features reminiscent of quantum particles. While the behaviour of walking droplets in unbounded geometries has to a large extent been rationalized theoretically, no such rationale exists for their behaviour in the presence of boundaries, as arises in a number of key quantum analogue systems. We here present the results of a combined experimental and theoretical study of the interaction of walking droplets with a submerged planar barrier. Droplets exhibit non-specular reflection, with a small range of reflection angles that is only weakly dependent on the system parameters, including the angle of incidence. The observed behaviour is captured by simulations based on a theoretical model that treats the boundaries as regions of reduced wave speed, and rationalized in terms of momentum considerations.

Pucci, G., Sáenz, P. J., Faria, L. M., & Bush, J. W. (2016). Non-specular reflection of walking droplets. *J Fluid Mech*, *804*, R3.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2016/09/Pucci-JFM-2016.pdf

# Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction

Abstract : The back-reaction of a radiated wave on the emitting source is a general problem. In the most general case, back-reaction on moving wave sources depends on their whole history. Here we study a model system in which a pointlike source is piloted by its own memory-endowed wave field. Such a situation is implemented experimentally using a self-propelled droplet bouncing on a vertically vibrated liquid bath and driven by the waves it generates along its trajectory. The droplet and its associated wave field form an entity having an intrinsic dual particle-wave character. The wave field encodes in its interference structure the past trajectory of the droplet. In the present article we show that this object can self-organize into a spinning state in which the droplet possesses an orbiting motion without any external interaction. The rotation is driven by the wave-mediated attractive interaction of the droplet with its own past. The resulting “memory force” is investigated and characterized experimentally, numerically, and theoretically. Orbiting with a radius of curvature close to half a wavelength is shown to be a memory-induced dynamical attractor for the droplet’s motion.

Labousse, M., Perrard, S., Couder, Y., & Fort, E. (2016). Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction. *Physical Review E*, *94*(4), 042224.

Available on ResearchGate (Free login required)

# Onset of chaos in orbital pilot-wave dynamics

**Abstract :*** We examine the orbital dynamics of droplets self-propelling along the surface of a **vibrating bath. Circular orbital motion may arise when the walking droplet is subjected to one **of three external force fields, the Coriolis force, a simple harmonic force, and a Coulomb **force. Particular attention is given to a theoretical characterization of the onset of chaos that **accompanies the destabilization of such circular orbits.*

Tambasco, L., Harris, D., Oza, A., Rosales, R., & Bush, J. (2015, November). Onset of chaos in orbital pilot-wave dynamics. In *APS Meeting Abstracts*.

# Surface topography measurements of the bouncing droplet experiment

A technical entry by the american team reproducing the synthetic Schlieren free surface measurment already implemented successfully by the Paris Team.

Damiano, A. P., Brun, P. T., Harris, D. M., Galeano-Rios, C. A., & Bush, J. W. (2016). Surface topography measurements of the bouncing droplet experiment. *Experiments in Fluids*, *57*(10), 163.

# Momentum exchange in the electron double-slit experiment

Yet another report of failure in trying to reproduce the single particle slit diffraction experiment with walking droplets

Batelaan, H., Jones, E., Huang, W. C. W., & Bach, R. (2016). Momentum exchange in the electron double-slit experiment.

Abstract. We provide support for the claim that momentum is conserved for individual events in the electron double slit experiment. The natural consequence is that a physical mechanism is responsible for this momentum exchange, but that even if the fundamental mechanism is known for electron crystal diffraction and the Kapitza–Dirac effect, it is unknown for electron diffraction from nano-fabricated double slits. Work towards a proposed explanation in terms of particle trajectories affected by a vacuum field is discussed. The contentious use of trajectories is discussed within the context of oil droplet analogues of double slit diffraction.

http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1010&context=physicsbatelaan

# Double-slit experiment with single wave-driven particles and its relation to quantum mechanics

Andersen, A., Madsen, J., Reichelt, C., Ahl, S. R., Lautrup, B., Ellegaard, C., … & Bohr, T. (2015). Double-slit experiment with single wave-driven particles and its relation to quantum mechanics. *Physical Review E*, *92*(1), 013006.

ABSTRACT :

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. **97**, 154101 (2006)] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

http://sci-hub.bz/10.1103/physreve.92.013006#

# New reference website from Paris Couder & Fort Team

The Paris team led by Yves Couder & Emmanuel Fort has published online a wonderful reference website, with many new videos

# Shedding light on pilot-wave phenomena

Brun, P. T., Harris, D. M., Prost, V., Quintela, J., & Bush, J. W. (2016). Shedding light on pilot-wave phenomena. *Physical Review Fluids*, *1*(5), 050510.

ABSTRACT

This paper is associated with a video winner of a 2015 APS/DFD Gallery of Fluid Motion Award. The original video is available from the Gallery of Fluid Motion,

# Neimark–Sacker bifurcation and evidence of chaos in a discrete dynamical model of walkers

Rahman, A., & Blackmore, D. (2015). Neimark–Sacker bifurcation and evidence of chaos in a discrete dynamical model of walkers. *arXiv preprint arXiv:1507.08057*.

ABSTRACT

Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integro-differential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show evidence of exotic dynamics such as period doubling bifurcations, Neimark–Sacker (N–S) bifurcations, and even chaos. For example, in [Gilet, PRE 2014], based on simulations Gilet conjectured the existence of a supercritical N-S bifurcation as the damping factor in his one-dimensional path model. We prove Gilet’s conjecture and more; in fact, both supercritical and subcritical (N-S) bifurcations are produced by separately varying the damping factor and wave-particle coupling for all eigenmode shapes. Then we compare our theoretical results with some previous and new numerical simulations, and find complete qualitative agreement. Furthermore, evidence of chaos is shown by numerically studying a global bifurcation.

# Wave-Based Turing Machine: Time Reversal and Information Erasing

Perrard, S., Fort, E., & Couder, Y. (2016). Wave-Based Turing Machine: Time Reversal and Information Erasing. *Physical Review Letters*, *117*(9), 094502.

ABSTRACT

The investigation of dynamical systems has revealed a deep-rooted difference between waves and objects regarding temporal reversibility and particlelike objects. In nondissipative chaos, the dynamic of waves always remains time reversible, unlike that of particles. Here, we explore the dynamics of a wave-particle entity. It consists in a drop bouncing on a vibrated liquid bath, self-propelled and piloted by the surface waves it generates. This walker, in which there is an information exchange between the particle and the wave, can be analyzed in terms of a Turing machine with waves as the information repository. The experiments reveal that in this system, the drop can read information backwards while erasing it. The drop can thus backtrack on its previous trajectory. A transient temporal reversibility, restricted to the drop motion, is obtained in spite of the system being both dissipative and chaotic.

Available at Researchgate (requires free login)

# Transition Orbits of Walking Droplets

Parker, J. (2015). *Transition Orbits of Walking Droplets* (Doctoral dissertation, California Polytechnic State University, San Luis Obispo).

“It was recently discovered that millimeter-sized droplets bouncing on the surface of an oscillating bath of the same fluid can couple with the surface waves it produces and begin walking across the fluid bath. These walkers have been shown to behave similarly to quantum particles; a few examples include single-particle diffraction, tunneling, and quantized orbits. Such behavior occurs because the drop and surface waves depend on each other to exist, making this the first and only known macroscopic pilot-wave system. In this paper, the quantized orbits between two identical drops are explored. By sending a perturbation to a pair of orbiting walkers, the orbit can be disrupted and transition to a new orbit. The numerical results of such transitions are analyzed and discussed.”

TWO interesting things in this small report :

1 – Influence of apparatus temperature on the faraday thresold

2 – sending a plane wave towards a 2-droplet orbiting system can cause a shift in the orbit quantization

# emergence of statistical pattern in a 1D cavity

On this video you will see how a walking droplet in a small 1D cavity moves “randomly” if the memory of the system is high enough (ie if the forcing is strong enough, but still below the Faraday Thresold)

And how a statistical pattern emerges with time

# Quantumlike statistics of deterministic waveparticle interactions in a circular cavity

*Physical Review E*,

*93*(4), 042202.

# Retro-influence of a square cavity on a circular standing wave

Read More# Square cavity

Read More# Rainbow colored dotwave

# Dotwave picture on Phys.org

Russian Physicians from Moscow Institute of Physics and Technology have chosen a dotwave.org picture to illustrate an article published on phys.org concerning their latest paper,

Des physiciens russes de l’institut de physique et de technologie de Moscou ont choisi une de mes photos pour illustrer un résumé d’un de leur papier sur les ondes de Faraday publié sur phys.org

# Faraday wave lattice as an elastic metamaterial

Domino, L., Tarpin, M., Patinet, S., & Eddi, A. (2016). Faraday wave lattice as an elastic metamaterial. *arXiv preprint arXiv:1601.08024*.

(Also on PhysRev E.)

Metamaterials enable the emergence of novel physical properties due to the existence of an underlying sub-wavelength structure. Here, we use the Faraday instability to shape the uid-air interface with a regular pattern. This pattern undergoes an oscillating secondary instability and exhibits spontaneous vibrations that are analogous to transverse elastic waves. By locally forcing these waves, we fully characterize their dispersion relation and show that a Faraday pattern presents an

effective shear elasticity. We propose a physical mechanism combining surface tension with the Faraday structured interface that quantitatively predicts the elastic wave phase speed, revealing that the liquid interface behaves as an elastic metamaterial.

http://arxiv.org/pdf/1601.08024.pdf