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

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 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

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: 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

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


The dynamics of coughing and sneezing


Respiratory events such as exhalations or  more violent coughs and sneezes are key in transferring respiratory diseases between infectious and susceptible individuals. We present the results of a combined experimental and theoretical investigation of the fluid dynamics of such violent expiratory events. Direct observation reveals that such flows are multiphase turbulent buoyant clouds with suspended droplets of various sizes. Our observations guide the development of an accompanying theoretical model in which pathogen-bearing droplets interact with a turbulent buoyant momentum puff. The  range of validity of our theoretical model is explored experimentally. Our study highlights the importance of the multiphase nature of respiratory clouds in extending the range of respiratory pathogens.
See Press and amusing video.  MIT News piece here.

Biomimicry and the culinary arts

Edible flowers assist in the drinking of fine, palate-cleansing liqueurs.

The group is collaborating with cutting-edge Spanish chef Jose Andres at the margins of science and the culinary arts.Two of our biomimetic inventions are being explored and developed by his ThinkFood group.

Inspired by a family of floating flowers, the edible flower (image at right) provides a mean of drinking small volumes of fluid in an elegant fashion.

Inspired by a means of propulsion used by a class of water-walking insects, cocktail boats (image below) propel themselves by generating a fore-aft chemical gradient.

Both designs developed by Lisa Burton and Nadia Cheng.


The cocktail boat

The aerodynamics of the beautiful game

The dynamics of sports is a subject familiar to many, a rich area of application of  mathematical modeling. The dynamics of footballs in flight has been the subject of  a recent article, and we have a continuing interest in this class of problems through our interaction with Christophe Clanet.

PAPER:  Bush (2013)

PRESS:  Le Progres – Lyons 2009 , MIT News 2014 , America24 , Wall Street Journal

In the video below we see Karl Suabedissan striking two balls, one smooth, the other identical but for an elastic band wrapped around its equator. Both balls are struck with his instep so as to impart a spin that is counterclockwise as viewed from above, so one expects them to bend from right to left. However, the smooth ball bends in precisely the opposite sense. The presence of the surface roughness in the form of the elastic band is sufficient to change the boundary layer on the ball surface from laminar to turbulent, thus restoring the sign of the expected spin-induced lateral force on the ball. It is thus that all footballs have some surface roughness; otherwise, they would bend the wrong way.