Non-specular reflection of walking droplets

A walking droplet reflects off a submerged barrier.

While the behavior 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.

See paper: Pucci, Saenz, Faria & Bush (JFM, 2016)  pdf

Circular orbits in a harmonic potential

 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.

See paper here:  Labousse, M., Oza, A.U., Perrard, S. and Bush, J.W.M. (2016)

Visualizing pilot-wave phenomena: the $60 rig

The $60 rig, driven by your cell phone.

The reflection of an object can be distorted by undulations of the reflector, be it a funhouse mirror or a fluid surface. Painters and photographers have long exploited this effect, for example, in imaging scenery distorted by ripples on a lake. Here, we use this phenomenon to visualize micrometric surface waves generatedas a millimetric droplet bounces on the surface of a vibrating fluid bath. This system, discovered a decade ago (Couder et al. 2005 ), is of current interest as a hydrodynamic quantum analog; specifically, the walking droplets exhibit several features reminiscent of quantum particles (Bush, ARFM, 2015). We present a simple and inexpensive experimental device that allows one to see many striking pilot-wave phenomena. It is our hope that this will be of interest as a high school physics classroom demonstration.

See paper here:  Harris, D.M., Quintela, J., Prost, V., Brun, P.-T. and Bush, J.W.M. (2016)  pdf

See the related Gallery of Fluid Motion Winner:  Brun, P.-T., Harris, D.M., Prost, V., Quintela, J. and Bush, J.W.M. (2016) pdf   (Link to video from pdf).


The new wave of pilot-wave theory

a) Faraday waves excited above threshold. A millimetric drop b) bounces and c-d) walks over the vibrating bath. Strobed images show e) a walker and f) an orbiting pair.

A decade ago, Yves Couder and Emmanuel Fort discovered that a millimeter-sized droplet may propel itself along the surface of a vibrating fluid bath by virtue of a resonant interaction with its own wave field, and that these walking droplets exhibit several features reminiscent of quantum systems. We here describe the walking-droplet system and, where possible, provide rationale for its quantum-like features. Further, we discuss the physical analogy between this hydrodynamic system and its closest relations in quantum theory, Louis de Broglie’s pilot-wave theory and its modern extensions.

See paper: Bush, Physics Today (2015)

Pilot-wave dynamics: modeling & computation

The evolution of the computed wave field with increasing memory.

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 a 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.

See paper here:  Milewski, Galeano-Rios, Nachbin and Bush (2015)


Pilot-wave hydrodynamics: A review

Walking in color. Photo credit: Dan Harris


Yves Couder and Emmanuel Fort 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 quantum-like 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.

See paper:    Bush (2015)

Select Press:  Quanta,  MIT News,  Wired

How to build a better drop generator

A lattice of droplets generated by our droplet generator.


We present the design of a piezoelectric droplet-on-demand generator capable of producing droplets of highly repeatable size ranging from 0.5 to 1.4 mm in diameter. The generator is low cost, simple to fabricate, and suitable for hydrodynamic quantum analog experiments. We demonstrate the manner in which droplet diameter can be controlled through variation of the piezoelectric driving waveform parameters, outlet pressure, and nozzle diameter.

See paper here:  Harris, Liu & Bush (2015)

How to build a better shaker

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 improve- ment 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.

See paper:  Harris & Bush (2015)

The hydrodynamic boost factor of walking drops

A droplet bouncing on the free surface. Image: Dan Harris.


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.

See paper:  Bush, Oza & Molacek (2014)


Walkers in a rotating frame: Exotic orbits


Exotic quasi-periodic orbits arising for a walker in a rotating frame.

We present the results of a numerical investigation of droplets walking on a rotating vibrating fluid bath. The drop’s trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. As the forcing acceleration is progressively increased, stable circular orbits give way to wobbling orbits, which are succeeded in turn by instabilities of the orbital center characterized by steady drifting then discrete leaping. In the limit of large vibrational forcing, the walker’s trajectory becomes chaotic, but its statistical behavior reflects the influence of the unstable orbital solutions. The study results in a complete regime diagram that summarizes the dependence of the walker’s behavior on the system parameters. Our predictions compare favorably to the experimental observations of Harris and Bush (JFM, 2014).

See paper:  Oza, Wind-Willassen, Harris, Rosales & Bush (2014) 



Walkers in a rotating frame: Orbital stability

The wave field generated by a droplet (black dot) executing an inertial orbit (dashed circle).


We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath’s rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.

See paper:  Oza, Harris, Rosales & Bush (2014)

Walkers in a rotating frame: Experiments

We present the results of an experimental investigation of a droplet walking on the surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating that the stable quantized orbits reported by Fort et al. (2010) arise only for a finite range of vibrational forcing, above which complex trajectories with multimodal statistics arise. We first present a detailed characterization of the emergence of orbital quantization, and then examine the system behaviour at higher driving amplitudes. As the vibrational forcing is increased progressively, stable circular orbits are succeeded by wobbling orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude about a stationary centre is punctuated by the orbital centre leaping approximately half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular trajectories emerge, characterized by a multimodal probability distribution that reflects the persistent dynamic influence of the unstable orbital states.

See paper  here:  Harris & Bush (2014)

Hydrodynamic pilot-wave theory

The nature of the wave-particle coupling. As the system memory becomes more pronounced, the wave field more intense, the drop moves down its associated wave field and walks faster.


We present the results of a theoretical investigation of droplets bouncing on a vertically vibrating fluid bath. An integro-differential equation for the horizontal motion of the drop is developed by approximating the drop as a continuous moving source of standing waves. We demonstrate that, as the forcing acceleration is increased, the bouncing state destabilizes into steady horizontal motion along a straight line, a walking state, via a su- percritical pitchfork bifurcation. Predictions for the dependence of the walking threshold and drop speed on the system parameters compare favorably with experimental data. By considering the stability of the walking state, we show that the drop is stable to perturbations in the direction of motion and neutrally stable to lateral perturbations. This result lends insight into the possibility of chaotic dynamics emerging when droplets walk in complex geometries.


See paper:  Oza, Rosales & Bush (2013).

Pilot-wave dynamics of walking drops

We present here a videographic description of the pilot-wave hydrodynamics arising when a fluid drop walks on a vibrating fluid bath.

See paper: Harris & Bush (2013)


Pilot-wave dynamics in a circular corral

The trajectory of a droplet walking in a circular corral, color coded according to speed. Note the correlation between position and speed, which results in the wavelike statistics.

The probability distribution of a droplet walking in a circular corral, which is well described by the corral's Faraday wave mode.


Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.

See papers: Harris, Moukhtar, Fort, Couder and Bush (2013) ,  Harris & Bush (2013)

Select Press:  MIT News  , Tracinski Letter

Exotic states of bouncing and walking drops


The most complete regime diagram to date, indicating the observed (data) and predicted (curves) dependence of the bouncing and walking modes on the vibration number and vibrational forcing.

We present the results of an integrated experimental and theoretical investigation of droplets bouncing on a vibrating fluid bath. A comprehensive series of experiments provides the most detailed characterisation to date of the system’s dependence on fluid properties, droplet size and vibrational forcing. A number of new bouncing and walking states are reported, including complex periodic and aperiodic motions. Particular attention is given to the rst characterisation of the dierent gaits arising within the walking regime. In addition to complex periodic walkers and limping droplets, we highlight a previously unreported mixed state, in which the droplet switches periodically between two distinct walking modes. Our experiments are complemented by a theoretical study based on our previous developments [J. Fluid Mech., 727, 582-611 (2013)], [J. Fluid Mech., 727, 612-647 (2013)], which provides a basis for rationalising all observed bouncing and walking states.

See paper:  Wind-Willassen, Molacek, Harris and Bush (2013)

Select Press:  American Institute of Physics

Droplets walking on a vibrating fluid bath

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 crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measured 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 dynamics, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes, the critical requirement being that they achieve resonance with their 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.

The results of this paper form the basis of the first rational hydrodynamic pilot-wave theory.

See paper:   Molacek & Bush (2013).

Select Press:  Inside Science


Droplets bouncing on a vibrating fluid bath

The dependence of the bouncing state on drop size and vibrational forcing.

We present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically, the drop size,  driving acceleration and driving frequency. As the acceleration is increased, depending on drop size, we observe the transition from coalescing to hovering or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the driving frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence-bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and hovering, for identical system parameters. In the former state the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against data. We first model the drop-bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

See paper:  Molacek & Bush (2013).


Drop impact on a nonwetting surface

We develop a conceptually simple theoretical model of non-wetting drop impact on a rigid surface at small Weber numbers. Flat and curved impactor surfaces are considered, and the influence of surface curvature is elucidated. Particular attention is given to characterizing the contact time of the impact and the coefficient of restitution, the goal being to provide a reasonable estimate for these two parameters with the simplest model possible. Approximating the shape of the drop during impact as quasi-static allows us to derive the governing differential equation for the droplet motion from a Lagrangian. Predictions of the resulting model are shown to compare favorably with previously reported experimental results.

This study represents an important prerequisite to a theoretical description of drops bouncing and walking on a fluid surface.

See paper:  Molacek & Bush (2012)


Drops bouncing on a vibrating soap film

A millimetric drop bounces on a soap film.

We examine the dynamics arising when a water droplet bounces on a horizontal soap film suspended on a vertically oscillating circular frame. A variety of simple and complex periodic bouncing states are observed, in addition to multiperiodicity and period-doubling transitions to chaos. The system is simply and accurately modeled by a single ordinary differential equation, the numerical solution of which captures all the essential features of the observed behavior. Iterative maps and bifurcation diagrams indicate that the system exhibits all the features of a classic low-dimensional chaotic oscillator.

See Gilet & Bush (2009a) and Gilet & Bush (2009b) .

Our analysis of the dynamics is the first step towards a rational description of droplets bouncing and walking on a vibrating fluid bath.

SELECT PRESS:    MIT News  ,  Scientific American ,  Nature Physics: News and Views , Science News

Different bouncing states for a droplet on a vibrating film.