*Physical Review E*,

*93*(4), 042202.

Gilet, T. (2016). Quantumlike statistics of deterministic wave-particle interactions in a circular cavity. *Physical Review E*, *93*(4), 042202.

A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faradaywaves confined to a circular cavity. Its solutions are investigated theoretically and numerically. The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk and the action per unit mass /m of a quantum particle. The statistics of droplet position and speed are shaped by the cavity eigenmodes, in remarkable agreement with the solution of Schr¨odinger equation for a quantum particle in a similar potential well.

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

Dubertrand, R., Hubert, M., Schlagheck, P., Vandewalle, N., Bastin, T., & Martin, J. (2016). Scattering theory of walking droplets in the presence of obstacles. *arXiv preprint arXiv:1605.02370*.

We aim to describe a droplet bouncing on a vibrating bath. Due to Faraday instability a surface wave is created at each bounce and serves as a pilot wave of the droplet. This leads to so called walking droplets or walkers. Since the seminal experiment by Couder et al [Phys. Rev. Lett. 97, 154101 (2006)] there have been many attempts to accurately reproduce the experimental results. Here we present a simple and highly versatile model inspired from quantum mechanics. We propose to describe the trajectories of a walker using a Green function approach. The Green function is related to Helmholtz equation with Neumann boundary conditions on the

obstacle(s) and outgoing conditions at infinity. For a single slit geometry our model is exactly solvable and reproduces some general features observed experimentally. It

stands for a promising.

Filatov, S. V., Parfenyev, V. M., Vergeles, S. S., Brazhnikov, M. Y., Levchenko, A. A., & Lebedev, V. V. (2016). Nonlinear Generation of Vorticity by Surface Waves. *Physical review letters*, *116*(5), 054501.

We demonstrate that waves excited on a fluid surface produce local surface rotation owing to hydrodynamic nonlinearity. We examine theoretically the effect and obtain an explicit formula for the vertical vorticity in terms of the surface elevation. Our theoretical predictions are confirmed by measurements of surface motion in a cell with water where surface waves are excited by vertical and harmonic shaking the cell. The experimental data are in good agreement with the theoretical predictions. We discuss physical consequences of the effect.

http://ssver.itp.ac.ru/site/publications/Filatov_2016_PRL.pdf

Labousse, M., Oza, A. U., Perrard, S., & Bush, J. W. (2016). Pilot-wave dynamics in a harmonic potential: Quantization and stability of circular orbits.*Physical Review E*, *93*(3), 033122.

“We present the results of a theoretical investigation of the dynamics of a droplet walking on a vibrating fluid bath under the influence of a harmonic potential. The walking droplet’s horizontal motion is described by an integro-differential trajectory equation, which is found to admit steady orbital solutions. Predictions for the dependence of the orbital radius and frequency on the strength of the radial harmonic force field agree favorably with experimental data. The orbital quantization is rationalized through an analysis of the orbital solutions. The predicted dependence of the orbital stability on system parameters is compared with experimental data and the limitations of the model are discussed.”

Milewski, P. A., Galeano-Rios, C. A., Nachbin, A., & Bush, J. W. (2015). Faraday pilot-wave dynamics: modelling and computation. *Journal of Fluid Mechanics*, *778*, 361-388.

A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet’s bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.

Gilet, T. (2014). Dynamics and statistics of wave-particle interactions in a confined geometry. *Physical Review E*, *90*(5), 052917.

A walker is a droplet bouncing on a liquid surface and propelled by the waves that it generates. This macroscopic wave-particle association exhibits behaviors reminiscent of quantum particles. This article presents a toy model of the coupling between a particle and a confined standing wave. The resulting 2D iterated map captures many features of the walker dynamics observed in different configurations of confinement. These features include the time decomposition of the chaotic trajectory in quantized eigenstates, and the particle statistics being shaped by the wave. It shows that deterministic wave-particle coupling expressed in its simplest form can account for some quantumlike behaviors.

https://orbi.ulg.ac.be/bitstream/2268/178755/2/Generic.pdf

Brandenbourger, M., Vandewalle, N., & Dorbolo, S. (2016). Displacement of an Electrically Charged Drop on a Vibrating Bath. *Physical review letters*, *116*(4), 044501.

In this work, the manipulation of an electrically charged droplet bouncing on a vertically vibrated, bath is investigated. When a horizontal, uniform and static electric eld is applied to it, a motion is induced. The droplet is accelerated when the droplet is small. On the other hand, large droplets appear to move with a constant speed that depends linearly on the applied electrical eld. In the latter regime, high speed imaging of one bounce reveals that the droplet experiences an acceleration due to the electrical force during the ight and decelerates to zero when interacting with the surface of the bath. Thus, the droplet moves with a constant average speed on a large time scale. We propose a criterion based on the force necessary to move a charged droplet at the surface of the

bath to discriminate between constant speed and accelerated droplet regimes.

http://orbi.ulg.ac.be/bitstream/2268/194253/1/PhysRevLett.116.044501.pdf

Molteni, D., Vitanza, E., & Battaglia, O. R. (2016). Smoothed Particles Hydrodynamics numerical simulations of droplets walking on viscous vibrating fluid. *arXiv preprint arXiv:1601.05017*.

**Abstract :**

“We study the phenomenon of the “walking droplet”, by means of numerical fluid dynamics simulations using a standard version of the Smoothed Particle Hydrodynamics method. The phenomenon occurs when a millimetric drop is released on the surface of an oil of the same composition contained in a container subjected to vertical oscillations of frequency and amplitude close to the Faraday instability threshold. At appropriate values of the parameters of the system under study, the liquid drop jumps permanently on the surface of the vibrating fluid forming a localized wave-particle system, reminding the behavior of a wave particle quantum system as suggested by de Broglie. In the simulations, the drop and the wave travel at nearly constant speed, as observed in experiments. In our study we made relevant simplifying assumptions, however we observe that the wave-drop coupling is easily obtained. This fact suggests that the phenomenon may occur in many contexts and opens the possibility to study the phenomenon in an extremely wide range of physical configurations.”

Bacot, V., Labousse, M., Eddi, A., Fink, M., & Fort, E. (2015). Revisiting time reversal and holography with spacetime transformations. *arXiv preprint arXiv:1510.01277*.

Wave control is usually performed by spatially engineering the properties of a medium. Because time and space play similar roles in wave propagation, manipulating time boundaries provides a complementary approach. Here, we experimentally demonstrate the relevance of this concept by introducing instantaneous time mirrors. We show with water waves that a sudden change of the effective gravity generates time-reversed waves that refocus at the source. We generalize this concept for all kinds of waves introducing a universal framework which explains the effect of any time disruption on wave propagation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire space at the time disruption. The time-reversed waves originate from these “Cauchy sources” which are the counterpart of Huygens virtual sources on a time boundary. It allows us to revisit the holographic method and introduce a new approach for wave control.

Filoux, B., Hubert, M., Schlagheck, P., & Vandewalle, N. (2015). Waveguides for walking droplets. *arXiv preprint arXiv:1507.08228*.

When gently placing a droplet onto a vertically vibrated bath, a drop can bounce permanently. Upon increasing the forcing acceleration, the droplet is propelled by the wave it generates and becomes a walker with a well dened speed. We investigate the connement of a walker in different rectangular cavities, used as waveguides for the Faraday waves emitted by successive droplet bounces. By studying the walker velocities, we discover that 1d connement is optimal for narrow channels. We also propose an analogy with waveguide models based on the observation of the Faraday instability within the channels.

Bush, J. W. (2015). Pilot-wave hydrodynamics. *Annual Review of Fluid Mechanics*, *47*, 269-292.

Yves Couder, Emmanuel Fort, and coworkers recently discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. This article reviews experimental evidence indicating that the walking droplets exhibit certain features previously thought to be exclusive to the microscopic, quantum realm. It then reviews theoretical descriptions of this hydrodynamic pilot-wave system that yield insight into the origins of its quantumlike behavior. Quantization arises from the dynamic constraint imposed on the droplet by its pilot-wave field, and multimodal statistics appear to be a feature of chaotic pilot-wave dynamics. I attempt to assess the potential and limitations of this hydrodynamic system as a quantum analog. This fluid system is compared to quantum pilot-wave theories, shown to be markedly different from Bohmian mechanics and more closely related to de Broglie’s original conception of quantum dynamics, his double-solution theory, and its relatively recent extensions through researchers in stochastic electrodynamics.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Bush-AnnRev2015.pdf

Richardson, C. D., Schlagheck, P., Martin, J., Vandewalle, N., & Bastin, T. (2014). On the analogy of quantum wave-particle duality with bouncing droplets.*arXiv preprint arXiv:1410.1373*.

We explore the hydrodynamic analogues of quantum wave-particle duality in the context of a bouncing droplet system which we model in such a way as to promote comparisons to the de Broglie-Bohm interpretation of quantum mechanics. Through numerical means we obtain single-slit difraction and double-slit interference patterns that strongly resemble those reported in experiment and that re ect a striking resemblance to quantum diraction and interference on a phenomenological level. We, however, identify evident dierences from quantum mechanics which arise from the governing equations at the fundamental level.

http://arxiv.org/pdf/1410.1373.pdf

Filoux, B., Hubert, M., & Vandewalle, N. (2015). Strings of droplets propelled by coherent waves. *arXiv preprint arXiv:1504.00484*.

Bouncing walking droplets possess fascinating properties due to their peculiar wave/particle interaction. In order to study such walkers in a 1d system, we considered the case of a few droplets in an annular cavity. We show that, in this geometry, they spontaneously form a string of synchronized bouncing droplets that share a common coherent wave propelling the group at a speed faster

than single walkers. The formation of this coherent wave and the collective droplet behaviors are captured by a model, which sheds a new light on droplet/wave interactions.

Borghesi, C., Moukhtar, J., Labousse, M., Eddi, A., Fort, E., & Couder, Y. (2014). Interaction of two walkers: Wave-mediated energy and force. *Physical Review E*, *90*(6), 063017.

A bouncing droplet, self-propelled by its interaction with the waves it generates, forms a classical wave-particle association called a “walker.” Previous works have demonstrated that the dynamics of a single walker is driven by its global surface wave field that retains information on its past trajectory. Here, we investigate the energy stored in this wave field for two coupled walkers and how it conveys an interaction between them. For this purpose, we characterize experimentally the “promenade modes” where two walkers are bound, and propagate together. Their possible binding distances take discrete values, and the velocity of the pair depends on their mutual binding. The mean parallel motion can be either rectilinear or oscillating. The experimental results are recovered analytically with a simple theoretical framework. A relation between the kinetic energy of the droplets and the total energy of the standing waves is established.

Labousse, M., Perrard, S., Couder, Y., & Fort, E. (2014). Build-up of macroscopic eigenstates in a memory-based constrained system. *New Journal of Physics*, *16*(11), 113027.

A bouncing drop and its associated accompanying wave forms a walker. Based on previous works, we show in this article that it is possible to formulate a simple theoretical framework for the walker dynamics. It relies on a time scale decomposition corresponding to the effects successively generated when the memory effects increase. While the short time scale effect is simply responsible for the walkerʼs propulsion, the intermediate scale generates spontaneously pivotal structures endowed with angular momentum. At an even larger memory

scale, if the walker is spatially confined, the pivots become the building blocks of a self-organization into a global structure. This new theoretical framework is applied in the presence of an external harmonic potential, and reveals the underlying mechanisms leading to the emergence of the macroscopic spatial organization reported by Perrard et al (2014 Nature Commun. 5 3219).

https://hal-univ-artois.archives-ouvertes.fr/hal-01084731/document

Shats, M., Xia, H., & Punzmann, H. (2012). Parametrically excited water surface ripples as ensembles of oscillons. *Physical review letters*, *108*(3), 034502.

We show that ripples on the surface of deep water which are driven parametrically by monochromatic vertical vibration represent ensembles of oscillating solitons, or quasi-particles, rather than

waves. Horizontal mobility of oscillons determines the broadening of spectral lines and transitions from chaos to regular patterns. It is found that microscopic additions of proteins to water dra-

matically aect the oscillon mobility and drive transitions from chaos to order. The shape of the oscillons in physical space determines the shape of the frequency spectra of the surface ripple.

http://people.physics.anu.edu.au/~mgs112/papers/2012_PRL_Shats_oscillons.pdf

Perrard, S. (2014). *Une mémoire Ondulatoire: états propres, chaos et probabilités* (Doctoral dissertation, Paris 7)

**Résumé** : Une goutte rebondissant sur un bain de liquide en vibration verticale peut se mettre spontanément en mouvement, sous l’action des ondes qu’elle a elle-même générées. Celles ci, appelées ondes de Faraday sont entretenues par la vibration du bain durant un temps de mémoire qui peut être contrôlé expérimentalement. Le champ d’ondes stationnaires généré par la goutte contient ainsi dans ses motifs d’interférence une mémoire de la trajectoire précédemment suivie. L’entité résultante appelée marcheur est caractérisée par cette interaction entre la goutte et les ondes qui l’entourent, via la mémoire de chemin. Cette thèse est consacrée à l’étude expérimentale et théorique de cette mémoire de chemin. Dans ce but, une goutte de liquide encapsulant un volume de ferrofluide est piégée dans un puits de potentiel harmonique d’origine magnétique. La goutte sera ainsi amenée à interagir avec les ondes qu’elle a précédemment générées. Ce confinement induit un processus d’auto-organisation entre la goutte et l’onde sous-jacente qui mène à des comportements de type ondulatoire pour une particule. Les notions de quantifications ou de probabilité de mesure d’un état propre peuvent ainsi être appliquées au cas d’un marcheur. Ces comportements révèlent que le marcheur est un exemple d’objet étendu en temps qui ne peut être réduit à une approximation ponctuelle rappelant, dans un tout autre contexte, la théorie de l’onde pilote développée par de Broglie au début du XXème siècle

A droplet bouncing on a vertically vibrated liquid bath can be self-propelled by the surface waves it generates. Theses Faraday waves are sustained by the vertical bath vibration for a memory time which can be tuned experimentally. The wave field thus contains in its interference pattern a memory of the past-trajectory. The resulting entity called a walker is characterized by the interaction between the drop and its surrounding waves through this path-memory. This thesis is devoted to an experimental and theoretical investigation of such a wave-mediated path-memory. For this purpose a bouncing drop is magnetically loaded with a droplet of ferrofluid and can then be trapped in an harmonie well. The drop is thus forced to interact with its own path. The confinement induces a self-organization process between the particle and its wave packet, leading to wave-type behavior for a particle. Notions such quantization or probability of measuring an eigenstate can thus be used for the walker dynamics description. These features originate from the temporal coherence of the walker’ s dynamics. In that sense, the walker is an entity extended in time, we cannot reduce to a point-like approximation. It reminds us, in another context, the pilot wave theory developped by de Broglie at the beginning of the XXst century.

Bush, J. W., Oza, A. U., & Moláček, J. (2014). The wave-induced added mass of walking droplets. *Journal of Fluid Mechanics*, *755*, R7.

It has recently been demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic realm. The walker, consisting of a droplet plus its guiding wavefield, is a spatially extended object. We here examine the dependence of the walker mass and momentum on its velocity. Doing so indicates that, when the walker’s time scale of acceleration is long relative to the wave decay time, its dynamics may be described in terms of the mechanics of a particle with a speed-dependent mass and a nonlinear drag force that drives it towards a fixed speed. Drawing an analogy with relativistic mechanics, we define a hydrodynamic boost factor for the walkers. This perspective provides a new rationale for the anomalous orbital radii reported in recent studies

Harris, D. M., & Bush, J. W. (2015). Generating uniaxial vibration with an electrodynamic shaker and external air bearing. *Journal of Sound and Vibration*,*334*, 255-269.

Electrodynamic shakers are widely used in experimental investigations of vibrated fluids and granular materials. However, they are plagued by undesirable internal resonances that can significantly impact the quality of vibration. In this work, we measure the performance of a typical shaker and characterize the influence that a payload has on its performance. We present the details of an improved vibration system based on a concept developed by Goldman (2002) [1] which consists of a typical electrodynamic shaker with an external linear air bearing to more effectively constrain the vibration to a single axis. The principal components and design criteria for such a system are discussed. Measurements characterizing the performance of the system demonstrate considerable improvement over the unmodified test shaker. In particular, the maximum inhomogeneity of the vertical vibration amplitude is reduced from approximately 10 percent to 0.1 percent; moreover, transverse vibrations were effectively eliminated.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2015/01/Harris-Shaker.pdf

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