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Pilot-Wave Hydrodynamics
John W.M. Bush
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139; email: bush@math.mit.edu

Annu. Rev. Fluid Mech. 2015. 47:269–92

Keywords

First published online as a Review in Advance on
September 10, 2014

walking drops, Faraday waves, quantum analogs

The Annual Review of Fluid Mechanics is online at
ﬂuid.annualreviews.org

Abstract

This article’s doi:
10.1146/annurev-ﬂuid-010814-014506
c 2015 by Annual Reviews.
Copyright All rights reserved

Yves Couder, Emmanuel Fort, and coworkers recently discovered that amillimetric 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 imposedon 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 tode Broglie’s original conception of quantum dynamics, his double-solution
theory, and its relatively recent extensions through researchers in stochastic
electrodynamics.

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1. INTRODUCTION
Long may Louis de Broglie continue to inspire those who suspect that what is proved by impossibility
proofs is lack of imagination.

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J.S. Bell (1982)

Both ﬂuid dynamics and quantum mechanics attempt to describe the motion of particles and waves
in continuous media, albeit on vastly different scales. Given the relative familiarity and accessibilityof ﬂuid systems, it is only natural to draw analogies with them as one attempts to come to grips
with the oddities of quantum mechanics. Newton (1979) described photons skipping through theether like stones on the surface of a pond. Young (1804) demonstrated the wave nature of light
via analogy with ripple tank experiments. Berry et al. (1980) examined plane waves sweeping overa vortex as a hydrodynamic analog of the Aharonov-Bohm effect (Coste et al. 1999). A recent
hydrodynamic analog of the Casimir effect suggests that Faraday waves can mimic the role ofvacuum ﬂuctuations in generating forces between neighboring objects (Denardo et al. 2009). The
theories of ﬂuid and quantum mechanics converge in the realm of superﬂuids (Donnelly 1993)
and Bose-Einstein condensates (Pitaevskii & Stringari 2003). The ﬁrst hydrodynamic analogs ofsingle-particle quantum systems have emerged in the past decade from the Paris laboratory ofYves Couder.
When placed on a vibrating ﬂuid bath, a millimetric drop may interact with the surface insuch a way as to walk steadily across it at a characteristic speed of 1 cm/s, guided or piloted by its
own wave ﬁeld (Couder et al. 2005b, Protière et al. 2006) (Figure 1c). The resulting walkers are
spatially extended objects comprising both the droplet and wave (Eddi et al. 2011b). By virtue of
their spatial delocalization, they exhibit several features previously thought to be peculiar to themicroscopic quantum realm (Bush 2010). Speciﬁcally, the walkers exhibit behavior reminiscent of
single-particle diffraction (Couder & Fort 2006), tunneling (Eddi et al. 2009b), quantized orbits(Fort et al. 2010, Harris & Bush 2014a, Perrard et al. 2014a,b), orbital level splitting (Eddi et al.
2012, Oza et al. 2014a), and spin states (Oza et al. 2014a). Coherent, multimodal statistics havebeen observed to emerge in single-particle diffraction (Couder & Fort 2006), motion in conﬁned
geometries (Harris et al. 2013), orbital motion in a rotating frame (Harris & Bush 2014a), andorbital motion in a central force (Perrard et al. 2014a,b). The basic mechanics and quantum-likefeatures of the walker system are reviewed in Section 2. The theoretical models developed torationalize the walker dynamics are reviewed in Section 3.
Section 4 explores the relation between this ﬂuid system and realist descriptions of quantum
systems, the majority of which are rooted in the hydrodynamic formulation of quantum mechanics (Madelung 1926). The walker system bears a notable resemblance to an early conceptionof relativistic quantum dynamics, Louis de Broglie’s (1926, 1930, 1956, 1987) double-solution

a

b

c

d

Figure 1
Image gallery of (a) Faraday waves, just above threshold; (b) a bouncing drop; (c) a walker; and (d ) a trio of bouncers. Photographs
courtesy of Dan Harris.
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pilot-wave theory, according to which microscopic particles move in resonance with their own
wave ﬁeld. The extensions of de Broglie’s mechanics through researchers in stochastic electrodynamics (SED) (Kracklauer 1992, de la Peña & Cetto 1996, Haisch & Rueda 2000) are also touched
on, and their relation to the walker system is discussed. Although this review is written from theperspective and in the language of a ﬂuid mechanician, my hope is that it will encourage interestin revisiting and reappraising realist quantum theories from the ﬂuid dynamics community andbeyond.

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2. WALKING DROPLETS
The phenomena of interest arise when a millimetric drop is levitated on the surface of a vibrating ﬂuid bath (Walker 1978) (Figure 1), an understanding of which requires familiarity withboth noncoalescence phenomena (Neitzel & Dell’Aversana 2002) and Faraday waves (Miles &Henderson 1990).

2.1. The Faraday Levitator
Casual observation indicates that water drops may skip across the water surface ( Jayaratne &Mason 1964). The drop bounces if the collision time is less than the time taken for the intervening
air layer to drain to the thickness at which coalescence is initiated by the attractive van der Waals
forces acting between the drop and bath. Although this critical thickness generally depends on theliquid and gas phase properties, as well as the system cleanliness, it is typically ∼100 nm (Couder
et al. 2005a; Terwagne et al. 2007, 2009). In the case of rebound, the air layer communicateslubrication stresses (Reynolds 1886) between the drop and bath, causing the drop to deform,
decelerate, and reverse direction (Gilet et al. 2007, Gilet & Bush 2012).
Let us consider a ﬂuid of density ρ, kinematic viscosity ν, and surface tension σ in a horizontal
bath of depth H, driven by a vertical vibration of amplitude A0 and frequency f = ω/(2π ).
The effective gravity in the bath’s frame of reference is g + γ sin ωt, where g is the gravitational
acceleration and γ = A0 ω2 . At low γ , the ﬂuid remains quiescent; however, above the Faraday
threshold γ F , the layer becomes unstable to a ﬁeld of Faraday waves (Faraday 1831) (Figure 1a).
The frequencies of the resulting waves are quantized, being integer multiples of half the driving
frequency (Benjamin & Ursell 1954). Although higher harmonics may be excited, for the silicone
oils used in the walker system, the most unstable waves are subharmonic, with half the frequencyof the imposed vibrational forcing, ω F = ω/2 (Douady 1990, Edwards & Fauve 1994, Kumar &Tuckerman 1994, Kumar 1996). Thus, as γ F is crossed from below, the ﬁrst waves to appear are
subharmonic, with a wavelength λ F = 2π/kF prescribed by the water-wave dispersion relation:
ω2F = ( gkF + σ k3F /ρ) tanh kF H . The experimental system comprises a bath of silicone oil (ν ∼10−100 cS, H ∼ 4−10 mm) subject to vertical vibrations ( f ∼ 20−150 Hz, A0 ∼ 0.1−1 mm)
that induce vibrational accelerations γ ≤ 5g and wavelengths λ F ∼ 3−10 mm. For f ≥ 80 Hz
and H ≥ 7 mm, the Faraday waves are effectively deep-water capillary waves with the dispersion
relation ω2F = σ k3F /ρ.
The walker experiments are performed for γ < γ F , so the ﬂuid interface would remain ﬂat in
the absence of the drop; nevertheless, the Faraday forcing is critical. First, it powers the system,
providing the energy for both the droplet motion and the resulting wave ﬁeld. Second, it preconditions the bath for a monochromatic wave ﬁeld. Eddi et al. (2011b) characterized the form of the
wave ﬁeld generated by particle impact on a vibrating bath (γ < γ F ) using a free-surface synthetic
schlieren technique (Moisy et al. 2009). Following impact, a transient wave propagates away fromthe impact site at a speed of ∼6 cm/s, and in its wake persists a ﬁeld of standing Faraday waves,
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whose longevity depends on the proximity to the Faraday threshold. A demonstration of walking
drops may be simply realized using a petri dish glued onto a woofer from a loudspeaker. Drops ofthe desired size (diameter D ∼ 0.6−1.0 mm) may be produced by breaking the interface with atoothpick. Illuminating with a strobe light makes for a striking demonstration.
Precision experiments require careful leveling and control of the vibrational forcing (Goldman
2002, Harris & Bush 2014b). Any variation from level, nonuniform vibration or spurious resonanceresults in a spatial dependence of the Faraday threshold. The walker dynamics and statistics arehighly sensitive to all the system parameters, drop size, ﬂuid properties, and forcing; consequently,
repeatable experiments require the simultaneous control and documentation of each. Studies of
the walker statistics are typically performed just below the Faraday threshold (Couder & Fort 2006,
Harris et al. 2013, Harris & Bush 2014a, Perrard et al. 2014a,b). The duration of the experiments isthus limited by the constancy of the lab temperature, changes in which alter the Faraday threshold
through the concomitant change in viscosity (Bechhoeffer et al. 1995). The bath should be covered
with a ﬁxed lid to eliminate air currents, a signature of which is curved rather than straight walker
trajectories. Another concern in gathering reliable statistical data is the longevity of the droplet(Terwagne et al. 2007). Even in a controlled environment, rogue coalescence events arise, witha period on the order of an hour, presumably owing to the settling of impurities onto the bath.
Such interruptions make clear the value of a mechanism for creating identical droplets on demand(Yang et al. 1997, Terwagne 2012).
Couder’s group characterized the drop’s bouncing behavior in terms of the drop diameter
D = 2a and dimensionless forcing acceleration γ /g (Protière et al. 2005), with viscosity-frequency
combinations of 50 cS–50 Hz (Protière et al. 2006) and 20 cS–80 Hz (Eddi et al. 2008). Moláček &Bush (2013a,b) and Wind-Willassen et al. (2013) extended these studies to guide their theoreticaldevelopments. They introduced the vibration number, Vi = ω ρa 3 /σ , the relative magnitude of
the forcing frequency and the drop’s natural oscillation frequency, and characterized the droplet
behavior in the Vi − γ /g plane for various viscosity-frequency combinations. Figure 2 presents
a synthesis of their results for the 20 cS–80 Hz combination, which exhibits the most extensive
walking regime. This presentation highlights a key feature of the system: A droplet most readily
bounces or walks when forced at its natural frequency, when Vi ≈ 0.65. Different bouncing states
are denoted by (m, n), where m/f indicates the period of the bouncing mode, during which thedrop contacts the surface n times.
Below the bouncing threshold, the drop coalesces into the bath. Just above it, bouncing arisesin the (1, 1) mode: The drop hits the bath once per driving period. Increasing the acceleration
then leads to a (2, 2) bouncing mode, in which the drop repeats a pair of bounces of unequal
height. For relatively small and large drops, a period-doubling cascade may follow, culminatingin chaotic bouncing or walking. For drops within a limited size range, typically 0.6 mm < D <1.0 mm, a period-doubled (2, 1) bouncing mode emerges and then destabilizes into a (2, 1) walker
at the walking threshold γw . These (2, 1) walkers bounce in synchrony with their monochromatic
Faraday wave ﬁeld, the underlying ﬂuid behaving like a damped oscillator forced at resonance.
However, more complex gaits may also arise within the walking regime, including mode-switching
and chaotic walkers (Wind-Willassen et al. 2013). Section 3 reviews the theoretical rationale forthis behavior.

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2.2. New Classical Behavior
A critical feature of the walker system is that of path memory (Eddi et al. 2011b). The walker
receives a lateral force at impact that depends on the local slope of the interface (Protière et al.
2006), whose form depends in turn on the previous impacts (Eddi et al. 2011b). In the low-memory
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1.1

)1
(2 , 1Chaos
1.0

Coalescence
(2 ,

0.8

Walk
Bounceos(2, 1) 1
Ch a

0.7(1, 1)

Vi(4 ,3)(2, 2)(2, 1) 2
,2

)

0.6
0.5

(4

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(4

,2

)

0.9

2
1)

0.4

Chaos
0.3
0.2

(1, 1) 2(1, 1)1
0

Chaos

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

γ /g
Figure 2
Regime diagram
indicating the dependence of the drop’s bouncing behavior on the driving acceleration, γ /g, and vibration number,
Vi = ω/ σ/ρa 3 . The ﬂuid bath is 20-cS silicone oil driven at f = ω/2π = 80 Hz. The associated Faraday threshold is γ F /g = 4.2. A
drop in the (m, n)i mode bounces n times in m forcing periods, with the integer subscript i ordering multiple (m, n) states according totheir total mechanical energy, with i = 1 being the lowest. Colors indicate predicted bouncing and walking modes; data indicate
measured thresholds between them (Moláček & Bush 2013b, Wind-Willassen et al. 2013).

limit, the waves are quickly damped, and the walker feels only the wave from its most recent impact.
At high memory, the waves are relatively persistent, and the walker’s trajectory depends on its
distant past. Although this physical picture is quantiﬁed by the theoretical developments describedin Section 3, it sufﬁces now to say that the quantum-like features of the walkers emerge at highmemory, as arises when the Faraday threshold is approached.
2.2.1. Single-particle diffraction. The ﬁrst evidence of the walker’s quantum-like behavior wasprovided by Couder & Fort’s (2006) study of walker diffraction through slits. In their single-slit
study, walkers were directed, one at a time, toward a slit in a barrier (Figure 3). When walkers
passed through the slit, they were diverted from their original path owing to the interaction
between their pilot-wave ﬁeld and the barrier. The authors directed 125 walkers toward the slit
with uniform spatial density at normal incidence, providing the best approximation to an incident
plane wave. The walker diffraction was quantiﬁed by the deﬂection angle α in the far ﬁeld. The
histogram for α was wavelike, with three peaks, and roughly consistent with the amplitude expectedfrom the interference of a monochromatic wave with wavelength λ F (Figure 3b). A qualitatively
similar distribution was obtained with their simulations. This system represents a hydrodynamic
analog of the single-photon diffraction experiments that constituted the doctoral thesis of G.I.
Taylor (1909).
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40

a

16

b
α

14

30

L

c

d

12

N(α)

N(α)

10
20

8
6

10

4

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2
0
–90

–60

–30

0

α (°)

30

60

90

0
–90

–60

–30

0

α (°)

30

60

90

Figure 3
Single-particle diffraction. (a) A walker passing through a single slit. (b) The histogram for the deﬂection angle α from the single-slit
experiments. (c) The histogram for α from the double-slit experiments. Figure adapted with permission from Couder & Fort (2006),
courtesy of Yves Couder.

Couder & Fort’s (2006) second study examined a hydrodynamic analog of the double-slit experiment with photons or electrons (Davisson & Germer 1927, Bach et al. 2013), the inscrutabilityof which is widely accepted (Feynman et al. 1964). A wavelike histogram for α emerged, roughly
consistent with the amplitude of a monochromatic wave of wavelength λ F diffracted by the double
slit. The proposed mechanism is as follows. While the walker passes through one slit or the other,its guiding wave passes through both; thus, the walker effectively feels the second slit by virtue
of its pilot wave. The feasibility of such an interaction, which relies on the spatial delocalizationof the walker, is readily checked by comparing the decay time of the pilot wave to the walker’stranslation time.
Couder & Fort (2006) point out that, if one could not directly observe the walker, one mightinfer an uncertainty principle in position and momentum by virtue of the diffraction of the pilot
wave. Thus, if an observer cannot ascertain the crossing point, he or she would be unable topredict the walker’s ﬁnal trajectory, hence the resulting uncertainty in momentum. An additional
uncertainty might be inferred if one could not resolve the walker’s bouncing phase, speciﬁcallywhen it is in contact with the bath and when it is in free ﬂight. Given the complex interaction
between the walker and its wave ﬁeld as it passes through the slit, one expects its path to dependon this bouncing phase. Such a dependence is suggested by the apparent independence of α on the
crossing point. The experiments of Couder & Fort (2006) indicate that the pilot-wave dynamics
within the slit is sufﬁciently complex to be chaotic, with the deﬂection angle α sensitive to boththe crossing point and the bouncing phase.
2.2.2. Tunneling. Eddi et al. (2009b) examined the interaction of a walker with barriers overlaid by a thin ﬂuid layer. The incidence of the pilot-wave ﬁeld on the barrier generates partial
reﬂection and an evanescent tail that decays across the barrier. The reﬂected wave typically causes
the reﬂection of an approaching walker; however, the drop-wave-barrier interaction occasionallypermits the droplet to tunnel across. The tunneling probability decreases exponentially with thebarrier width and increases as the Faraday threshold is approached. Once again, the quantum-likebehavior (Gamow 1928) results from the pilot-wave dynamics, and the unpredictability is rooted
in the complex interaction between the droplet and its spatially extended guiding wave.
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mm/s
30

a

b

25

20

15

1

y/R

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0.5

10

0
5

–0.5
–1.0
–1.0

–0.5

0

0.5

1.0

0

x/R
Figure 4
A walker in a circular corral of radius R = 14.3 mm. (a) Trajectories of increasing length in the high-memory regime are color coded
according to droplet speed. (b) The histogram of the walking droplet’s position corresponds roughly to the amplitude of the corral’s
Faraday wave mode. Figure adapted with permission from Harris et al. (2013). Copyright 2013, AIP Publishing LLC.

2.2.3. Motion in confined geometries. The second experimental arrangement in which a coherent statistical behavior was reported was for walkers in conﬁned geometries, speciﬁcally circular corrals (Harris & Bush 2013, Harris et al. 2013) (see Supplemental Video 1; follow the
Supplemental Material link from the Annual Reviews home page at http://www.
annualreviews.org). At low memory, the walkers loop around the walls of the corral. As the memory is increased, progressively more complex orbits arise: wobbling circular orbits, drifting ellipticalorbits, or epicycles. At very high memory, the trajectories become complex and presumably chaotic
owing to the complexity of the wave ﬁeld resulting from reﬂections off the boundaries (Figure 4a).
The histogram presented in Figure 4b indicates the emergence of a coherent statistical behavior
from this chaotic pilot-wave dynamics. The probability of ﬁnding a walker at a given point in
the corral is roughly prescribed by the amplitude of the Faraday wave mode of the cavity at theprescribed forcing frequency. As was the case for single-particle diffraction (Couder & Fort 2006),
the histogram has approximately the same wavelength as the pilot wave but a form prescribed by
the system geometry. Because the corral was tuned to ﬁt a single Faraday mode, the statistics mightthus have been anticipated with no knowledge of the dynamics other than the guiding wavelength
λ F and the corral geometry. The observed statistical behavior is thus roughly analogous to thatreported in the quantum corral experiments (Crommie et al. 1993a,b), in which the density ofelectrons trapped on a copper substrate was found to have a wavelike pattern with the de Broglie
wavelength, λdB , and a form prescribed by the corral shape (Fiete & Heller 2003).

Supplemental Material

2.2.4. Motion in a rotating frame. An object moving horizontally with uniform speed u0 ina frame rotating about a vertical axis with angular frequency executes an inertial orbit of radius ri = u 0 /2 , at which the inertial force mu 20 /ri is balanced by the Coriolis force 2m u 0 .
Fort et al. (2010) examined walkers on a rotating, vibrating bath and characterized the dependence of the orbital radii ro on . In the low-memory regime, the walker’s orbital radii decreasewww.annualreviews.org • Pilot-Wave Hydrodynamics

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Theoretical predictions
Linearly stable
Linearly unstable
Linearly unstable oscillatory

Pure inertial orbits
3.0

a

b

c

2.5

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ro /λ F

2.0
1.5
1.0
0.5
0

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

2ΩλF /uo
Figure 5
Measured orbital radius ro as a function of the bath’s rotation rate (Harris & Bush 2014a): (a) low memory (γ /γ F = 0.822),(b) intermediate memory (γ /γ F = 0.954), and (c) high memory (γ /γ F = 0.971). The dashed curve in panel a indicates pure inertial
orbits, ri = u 0 /2 , the offset from which is prescribed by the droplet’s hydrodynamic boost factor: ro = γ B ri (Bush et al. 2014). The
solid curves are the theoretical predictions of Oza et al. (2014a). The blue segments are linearly stable, the red linearly unstable, and thedark yellow linearly unstable oscillatory branches. Data on the unstable dark yellow branches indicate the nonlinear stability of the
wobbling states. Error bars denote wobbling amplitudes. Figure adapted with permission from Oza et al. (2014a).

monotonically with the rotation rate according to ro = c o ri , where c o ≈ 1.3 is a ﬁtting parameter. At high memory, ro no longer varies continuously with : Instead, certain orbital radii are
forbidden. Orbital quantization then emerges through the walker’s interaction with its own wake
(Figure 5).
Because the Coriolis force 2mẋ p ×  acts on a mass m in a rotating frame as does the Lorentz
force q ẋ p × B on a charge q in a uniform magnetic ﬁeld B (Weinstein & Pounder 1945), Fortet al. (2010) proposed a correspondence between the quantized inertial orbits of walkers and theLandau levels of electrons. In the high-memory regime, the orbital radii are roughly quantizedon the Faraday wavelength, just as Landau levels are quantized on the de Broglie wavelength.
Eddi et al. (2012) further explored this correspondence by examining the inﬂuence of rotation ona pair of orbiting droplets. They found that the orbital radii increased or decreased according towhether the pair rotated in the same or opposite sense relative to the bath, indicating an analogy
with Zeeman splitting of atomic energy levels.
The rotating system was revisited by Harris & Bush (2014a), who demonstrated that orbital
quantization arises only for a limited range of forcing accelerations. The emergence of orbital
quantization was detailed (Figure 5), along with the relatively rich behavior arising at higher
memory. As the memory is increased progressively, the orbital states become unstable, giving way
to wobbling orbital motions and then more complex periodic or aperiodic trajectories. In the highmemory limit, irregular chaotic trajectories emerge as the walker drifts between unstable orbits(Figure 6). The histogram of the trajectory’s radius of curvature is characterized by a multimodal
form, with peaks corresponding to the radii of the unstable orbits, which suggests their persistent
dynamical inﬂuence.
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3.5

a

c

3.0

R/λ F

2.5
2.0
1.5

λF

0.5

3.0

0

b

2.5

0.970

0.975

0.980

2.0

0.990

0.995

0

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50

Time (s)

60

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2

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2

n=5

n=4

n=3

80

n=2

200

n=1

160
n=4

N 600

1.0

400

d

n=3

1,200

1.5

0.5

0.985

γ/γF
n=4

R/λ F

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1.0

3

Figure 6
Chaotic walking in a frame rotating at = 0.79 rad/s. (a) Chaotic trajectory at high memory, γ /γ F = 0.990. (b) Time trace of theradius of curvature R of the trajectory shown in panel a. (c) Probability distribution of R as a function of the vibrational forcing γ /γ F .
The brightness of a segment corresponds to the relative probability within a single column. (d ) Histograms of the local radius ofcurvature along the three vertical columns indicated in panel c at γ /γ F = 0.975 (dark yellow), 0.985 (red ), and 0.990 (blue). The graydashed vertical lines represent the zeros of the Bessel function J0 (k F r). Figure adapted with permission from Harris & Bush (2014a).

2.2.5. Motion in a central force. Couder’s group has succeeded in encapsulating ferroﬂuids
with silicone oil. The resulting two-component drops can walk just as do their homogeneous
counterparts. By applying a vertical magnetic ﬁeld with a radial gradient, Perrard et al. (2014a,b)
examined the dynamics of walkers in the central force ﬁeld resulting from a harmonic potential.
In addition to circular orbits, more complex orbital states arose, including ovals, lemniscates, andtrefoils, all of which may either be stationary or precess azimuthally. The authors characterized
the resulting family of complex orbits in terms of the mean radius R̄ and angular momentumL̄ z (Figure 7). The results indicate that the dynamic constraint imposed on the walker by its
pilot-wave ﬁeld imposes on its orbital states a double quantization in the mean energy and angularmomentum.
Perrard et al. (2014b) presented a theoretical rationale for the quantized orbital states (oreigenstates) in terms of the energy landscape, speciﬁcally the wave ﬁeld generated by the orbiting
walker (see also M. Labousse, S. Perrard, Y. Couder & E. Fort, submitted manuscript). When thewalker did not settle onto a pure eigenstate, it oscillated between nearby eigenstates, for example,
an oval and a lemniscate. Thus, as was the case in the rotating walker system (Harris & Bush2014a), complex trajectories emerged in the high-memory limit: The walker drifted between thesystem’s unstable eigenstates, the result being multimodal statistics (Perrard et al. 2014a).
2.2.6. Bound states and other oddities. Pairs of bouncing drops may interact through their wave
ﬁelds to form bound states (Protière et al. 2005). Likewise, crystalline structures may form fromaggregates of nearly identical bouncers (Figure 1d ). Lieber et al. (2007) reported both stationary
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2.5

(4, –2)(4, –4)(4, 2)(4, 0)

2.0

R(3, 3)(3, 1)(3, –1)

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1.0(2, 2)(2, 0)(2, –2)0.5(1, 1)(1, –1)
0

–2

–1

0

1

2

LZ
Figure 7
Double quantization for walkers in a central force
 ﬁeld. Observed orbits include circles, lemniscates, dumbbells, and trefoils. Orbits areclassiﬁed according to their mean radius R̄ = R2 /λ F and angular momentum L̄ z = L z /(mλ F u). Figure adapted with permission
from Perrard et al. (2014b).

and spontaneously spinning lattices. Eddi et al. (2009a) achieved 8 of the 11 Archimedean tilingsof the plane with arrays of bouncers, the stability of which was also examined (Eddi et al. 2011a).
For pairs of bouncers of different size, the asymmetry of the resulting wave ﬁeld may lead to
self-propelling, ratcheting states (Eddi et al. 2008). Dynamic bound states may also arise fromthe interaction of walking droplets, their form depending on both the size and bouncing phaseof the walkers (Protière et al. 2005). Protière et al. (2008) explored the exotic orbits arising fromthe interaction of walkers of different size. Pairs of identical walkers may scatter, lock into orbit,
or translate together side by side in the promenade mode, in which the distance between them
ﬂuctuates periodically (C. Borghesi, J. Moukhtar, M. Labousse, A. Eddi, Y. Couder & E. Fort,
submitted manuscript).
Gilet et al. (2008) and Dorbolo et al. (2008) examined the bouncing of relatively large oildrops (D ∼ 1.5 mm, ν < 100 cS) on a highly viscous bath (ν = 1,000 cS). Levitation criteria
were deduced, and nonaxisymmetric modes of droplet vibration excited, leading to propulsion
via tumbling. More complex ﬂuid topologies have also been explored. Terwagne et al. (2010)
examined the bouncing of a two-phase droplet, elucidating the criterion for emulsiﬁcation. Gieret al. (2012) studied the levitation of polymeric droplets. Pucci et al. (2011, 2013) examined the
dynamics of a ﬂoating lens of alcohol on a vibrating bath of relatively viscous oil. When the lensbecomes unstable to Faraday waves, the resulting radiation pressure may deform the lens, leading
to striking static and dynamic forms.

3. THEORETICAL DEVELOPMENTS: AN EXPOSEDVARIABLE THEORY
The charm of the hydrodynamic pilot-wave system is its accessibility: All the properties of thewalker, the droplet and its guiding wave ﬁeld, are visible. That is not to say that it is simply
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described: The walker system is remarkably subtle, with its dynamical and statistical behavior
strongly dependent on all the system parameters.

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3.1. Bouncing Mechanics
Considerable effort has been devoted to elucidating the high–Reynolds number (Re = Ua/ν)impact of millimetric drops on solids (de Gennes et al. 2002, Yarin 2006). At small Bond number
Bo = ρga 2 /σ , surface tension dominates gravity, so a drop remains nearly spherical unless distorted
by impact. For low–Weber number impacts, W e = ρU 2 a 3 /σ  1, the distortion is weak, andthe drop behaves roughly like a linear spring with a spring constant proportional to σ (Okumura
et al. 2003): During impact, kinetic energy is converted to surface energy, and then back to kinetic
1. Moláček & Bush (2012) demonstrated
energy, with only a small viscous loss provided Re
that for the parameter regime of walkers (Re ∼ 20, Bo ∼ 0.1, W e ∼ 0.1), the distortion induced
by impact on a rigid substrate may be described in terms of a family of quasi-static forms, and thecollision dynamics in terms of a logarithmic spring.
The walkers are millimetric, and the principal force imparted to them during impact on thebath is associated with curvature rather than hydrostatic pressure. Thus, the dominant physics
of impact is captured through the consideration of drop impact on a soap ﬁlm, a conﬁguration
examined by Gilet & Bush (2009a,b), who demonstrated that the soap ﬁlm acts on the drop like alinear spring, with a spring constant proportional to σ . Their investigation of a drop on a vibrating
ﬁlm revealed and rationalized a number of complex bouncing states. Drop impact on a vibrating
ﬂuid bath is complicated by the inﬂuence of the bath’s inertia (Prosperetti & Oguz 1993).
To describe the impact of a bouncing drop on a vibrating liquid bath, Moláček & Bush (2013a,b)developed a hierarchy of models of increasing complexity, building on their model of impact on arigid substrate (Moláček & Bush 2012) through consideration of the bath deformation. A logarithmic spring model again emerged, their model incorporating the measured logarithmic dependenceof the coefﬁcient of restitution CR and contact time TC on We . Although thus semiempirical, theresulting model has no free parameters. It correctly predicts the bouncing threshold, as well as
the thresholds between the more energetic bouncing states (Figure 2). The bouncer must satisfy
a resonance condition. For the (1, 1) bouncer, this is simply stated: The impact time plus thetime of ﬂight must equal the Faraday period. As γ increases progressively, the phase of impact
changes continuously until the resonance condition can no longer be satisﬁed without the bouncer
changing modes. For a description of the low-energy bouncing states, it was adequate to assume
that the interface recovered to horizontal before the next impact. Modeling the more vigorous
bouncing and walking states required consideration of the wave ﬁeld.
Moláček & Bush (2013b) calculated the form of the wave ﬁeld generated for the small-drop
parameter regime of interest. Because a drop in the period-doubled bouncing state lands on the
crest of its wave (Protière et al. 2006), it may be destabilized into a walking state. Modeling theresulting horizontal dynamics required the dependence of the coefﬁcient of tangential restitution
CT on We , which was again deduced empirically. The resulting model successfully rationalized
the observed walking thresholds (Figure 2), as well as the dependence of the walking speed on thesystem parameters (Moláček & Bush 2013b). Their study underscored the subtle role of impact
phase on the walker dynamics: As the forcing acceleration is increased progressively, the impact
phase shifts, and the walker may switch from one walking mode to another.
Terwagne et al. (2013) developed a relatively simple mass-spring-dashpot model of a droplet
bouncing on a bath. Wind-Willassen et al. (2013) built upon the models of Moláček & Bush(2013a,b) to present the most complete study to date. New complex bouncing modes and different
gaits arising within the walking regime were reported and rationalized (Figure 2). The dynamics ofwww.annualreviews.org • Pilot-Wave Hydrodynamics

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droplets on a vibrating bath in the parameter regime relevant to walkers is now well characterized
and understood. Existing regime diagrams provide a valuable resource for benchmarking newexperimental studies.

3.2. Trajectory Equation

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Protière et al. (2006) developed the ﬁrst theoretical description of walking droplets. The model
captures the essential physics of the problem, speciﬁcally the propulsion of the drop by virtue of aresonant interaction with a monochromatic wave ﬁeld centered on the drop and generated at each
impact. It reproduced several qualitative features of droplet behavior, including the bifurcationfrom bouncing to walking and the increase of walking speed with forcing acceleration. However,
the model incorporated only the wave ﬁeld of the most recent impact. A similar low-memory
approximation was used to model walkers in corrals (Shirikoff 2013). Variable memory has been
incorporated in Emmanuel Fort’s numerical model, which has captured many features of thewalker system, including single-particle diffraction (Couder & Fort 2006), quantized orbits (Fortet al. 2010), and doubly quantized orbits (Perrard et al. 2014a,b). This model provides a valuable
exploratory tool that illustrates the critical role of memory on the walker dynamics; however, it
does not capture the detailed dependence of the system on the ﬂuid parameters.
Moláček & Bush (2013b) developed a hydrodynamically consistent equation of motion for adrop of mass m at a position x p (t) walking in resonance with its wave ﬁeld of shape h(x, t). By time
averaging over the bouncing period, they eliminated the vertical dynamics from consideration andso deduced the trajectory equation:mẍ p + Dẋ p = −mg∇h(x p , t).

(1)

The second term represents the time-averaged drag resulting from both ﬂight and impact, with
D being a drag coefﬁcient that depends on the system parameters in a known fashion (Moláček &Bush 2013b). The third term is the propulsive wave force resulting from the drop’s landing on
an inclined surface. As h(x, t) depends on the history of the drop, so too does the wave force. The
surface wave generated by a single impact may be approximated as a monochromatic radial Bessel
function of the ﬁrst kind, J0 (kF r) (Eddi et al. 2011b, Moláček & Bush 2013a, Oza et al. 2013), so
the interface height generated by all previous bounces required for this time-averaged model is


t/T F

h(x, t) = A

J0 (kF |x − x p (nT F )|)e −(t−nT F )/(T F M ) .

(2)

n=−∞

The dimensionless memory parameter M = T d /[TF (1 − γ /γ F )] depends on the Faraday
period TF , the decay time of unforced waves Td , and the proximity to the Faraday threshold γ F(Eddi et al. 2011b). The wave amplitude A likewise depends in a known fashion on the systemparameters, including the impact phase (Moláček & Bush 2013b). The wave force becomesincreasingly important at high memory, as γ → γ F , when the waves are relatively persistent.
With a wave ﬁeld in the form of the discrete sum (Equation 2), the trajectory equation shownin Equation 1 is difﬁcult to analyze. Oza et al. (2013) demonstrated that, as the vertical dynamics
is fast relative to the horizontal dynamics, one may approximate this sum by an integral. The
resulting stroboscopic approximation yields the dimensionless trajectory equation
 t
J1 (|x p (t) − x p (s )|)2(x p (t) − x p (s ))e −(t−s ) d s .
(3)
κ0 (1 − )ẍ p + ẋ p =(1 − )2 −∞ |x p (t) − x p (s )|This formulation includes two dimensionless parameters. The ﬁrst, = (γ − γW )/(γ F − γW ),indicates the vigor of the forcing, the relative distance from the walking ( = 0) and Faraday
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( = 1) thresholds.The second contains all the ﬂuid parameters appearing in Equations 1 and
2: κ0 = (m/D)3/2 kF g A/2T F . The nonlocal form of the dynamics is apparent: The wave force
acting on the walker is a function of its past, speciﬁcally its path, along which it lays down a ﬁeldof Faraday waves.
The trajectory equation shown in Equation 3 describes the dynamics evident when the systemis strobed at ω F , when the walker appears to be pushed along continuously by its pilot-wave ﬁeld
(Harris & Bush 2013) (see Supplemental Video 1). Through its neglect of the vertical dynamics
and impact phase, it is relatively amenable to analysis. Oza et al. (2013) showed that the stationary
bouncing state ẋ p = 0 is stable until the walking threshold γw , beyond which steady walking
solutions arise. The resulting theoretical predictions for both the walking threshold γw and thedependence of the walker speed on γ for γ > γw adequately describe the existing data. The only
ﬁtting parameter in this strobed model is the impact phase, reasonable bounds on which are known
from experiments (Moláček & Bush 2013b).
The trajectory equation shown in Equation 3 also yields insight into the stability of the walking
state, the robustness of which is not obvious, given that the drop is riding the crest of its wave
ﬁeld. Linear stability analysis demonstrates that the walker is stable to colinear perturbations but
is neutrally stable to transverse perturbations: If perturbed laterally from rectilinear motion, it will
simply change course and proceed in a new direction (Oza et al. 2013). This result is consistent
with the skittishness of the walker, the sensitivity of its path to perturbation, and its evidently
chaotic motion in complex geometries, including corrals and slits.

Supplemental Material

3.3. Orbital Dynamics
The strobed model of Oza et al. (2013) has also allowed for an assessment of the stability of thequantized orbits in a rotating frame (Fort et al. 2010, Harris & Bush 2014a). To describe thisconﬁguration, the trajectory equation shown in Equation 3 is augmented by the Coriolis force.
Circular orbital solutions with constant radius ro and frequency ωo are sought, and their linear
stability assessed (Oza et al. 2014a). Figure 5 illustrates the predicted dependence of the orbital
radius on the rotation rate for three memory values. The predicted stability of orbits along thesolution curves is indicated by their color. At low memory (Figure 5a), all circular orbits arestable, with orbital radii decreasing monotonically with . At intermediate memory (Figure 5b),
two unstable solution branches arise, corresponding to forbidden radii, the origins of the orbital
quantization. At high memory (Figure 5c), the solution curve has linearly stable, linearly unstable,
and linearly unstable oscillatory branches.
Numerical simulation of the trajectory equation has revealed the behavior of the walkers in thelinearly unstable regions (Oza et al. 2014b). Figure 8 summarizes the dependence of the orbital
dynamics on γ /γ F and the initial orbital radius ro . As γ /γ F is increased progressively, the stable
circular orbits are typically succeeded by wobbling orbits with, in turn, stationary and translating
orbital centers. In the high-memory limit, all but the smallest orbits become unstable, and chaotic
trajectories arise. Although the trajectories are irregular, they exhibit coherent statistics. As in theexperiments of Harris & Bush (2014a) (Figure 6), the walker has a tendency to follow arcs withradii of curvature corresponding to the unstable orbital states; thus, the histograms for the radiusof curvature exhibit peaks at these radii, the number and relative magnitudes of which dependon the system memory. A coherent multimodal statistics thus emerges owing to the persistent
dynamical inﬂuence of the unstable orbital states. These recent studies elucidate the origins of
two key quantum-like behaviors, quantization and multimodal statistics, in orbital pilot-wave
dynamics (Fort et al. 2010; Harris & Bush 2014a; Oza et al. 2014a,b; Perrard et al. 2014a,b;
M. Labousse, S. Perrard, Y. Couder & E. Fort, submitted manuscript).
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0.98
0.97
0.96
0.95

CircleDrifting

Jump up
Jump down

Wobbleand leap2ω wobble

Quasi-periodic

3ω wobble

ChaoticC

B

γ/γF 0.94

n=0

n=1

n=2

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0.93
0.92
0.91
0.90

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

r0 /λF
Figure 8
Numerical simulations of walker trajectories in a rotating frame indicate the dependence of their form on the vibrational forcing γ /γ F
and the initial orbital radius ro . Blue regions denote stable orbital regions, the boundaries of which are deﬁned by the linear stability
analysis of Oza et al. (2014a). Traverses B and C correspond to the solution curves illustrated in Figure 5b,c. Trajectories are color
coded according to the legend. Note the windows of periodicity within the chaotic regime. Figure adapted with permission from Oza
et al. (2014b). Copyright 2014, AIP Publishing LLC.

Another intriguing feature of the rotating system revealed through the analysis of Oza et al.
(2014a) is the possibility of hydrodynamic spin states. As memory increases, the solution curves
touch down on the = 0 axis (Figure 5c), indicating the existence of orbital solutions even in theabsence of rotation. The wave force generated by the walker is then sufﬁcient to balance the radial
inertial force and so sustain its circular motion, a physical picture reminiscent of classical modelsof the electron (Schrödinger 1930, Burinskii 2008). When rotation is applied, as in the two-walker
orbits of Eddi et al. (2012), a hydrodynamic analog of Zeeman splitting occurs. Although suchhydrodynamic spin states are unstable in the parameter regime explored in our experiments, they
are stable in other regions of the (κ0 , ) parameter space accessible in our more general pilot-wave
framework (Equation 3).
An important conclusion of Harris & Bush (2014a) is that the statistics are extremely sensitive
to all of the ﬂuid parameters, their form changing drastically as γ /γ F changes by as little as 1%(Figure 6d ). Although the studies of single-particle diffraction (Couder & Fort 2006) and corrals(Harris et al. 2013) suggested that a unique Schrödinger-like wave function emerges in the highmemory limit (Figures 3 and 4), this relatively systematic study suggests otherwise. Rather, it isanticipated that a systematic reexamination of the diffraction and corral experiments will reveal
multimodal statistics whose form will be strongly dependent on the system parameters.

3.4. Future Directions
The strobed model (Oza et al. 2013) successfully describes the motion of a single walker in unbounded domains. Having been benchmarked against the experiments of Harris & Bush (2014a),
it is currently being applied to describe walker motion in central force ﬁelds, including harmonic and Coulomb potentials. A salient question is why the double quantization evident in the
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harmonic potential (Perrard et al. 2014a,b; M. Labousse, S. Perrard, Y. Couder & E. Fort, submitted manuscript) was not apparent in the rotating system (Harris & Bush 2014a, Oza et al.
2014a,b). The principal shortcoming of the strobed model is that the wave ﬁeld (Equation 2) does
not incorporate the transient wave emitted during impact (Eddi et al. 2011b) and so cannot reliably
capture the interaction of walkers with barriers (Eddi et al. 2009b, Harris et al. 2013, Carmigniani
et al. 2014) or other walkers (Protière et al. 2005, 2008; C. Borghesi, J. Moukhtar, M. Labousse,
A. Eddi, Y. Couder & E. Fort, submitted manuscript). This shortcoming is being addressed in
the next generation of pilot-wave models (P. Milewski, C. Galeano-Rios, A. Nachbin & J.W.M.
Bush, submitted manuscript).
The theoretical developments have provided a more general framework for studying pilot-wave
dynamics. In the stroboscopic model (Equation 3), only two dimensionless parameters appear. We
have seen that certain quantum-like features, such as spin states, are unstable in the parameter
regime accessible to this hydrodynamic pilot-wave system but are stable in others. The question
then arises: For what values of (κ0 , ) does this generalized pilot-wave system exhibit behavior
most reminiscent of quantum mechanics?
Fort & Couder (2013) took another intriguing step away from this hydrodynamic pilot-wave
system and introduced the concept of an inertial walker. Whereas the real walker lays down a ﬁeldof waves along its path, the inertial walker’s wave ﬁeld continues at the walker’s velocity at thetime of emission. For a discrete set of orbits, nonradiative resonant wave modes are excited; whencircular, these orbits satisfy the Bohr-Sommerfeld quantization condition.
It is also interesting to consider how the walker motion might be described if the system wereobserved from above with no knowledge of the vibrational forcing, the bouncing, or the wave ﬁeld.
Labousse & Perrard (2014) highlighted the non-Hamiltonian features of the walker, describing
its motion at low memory in terms of a Rayleigh oscillator. For a certain class of motions, Bushet al. (2014) demonstrated that the inﬂuence of the propulsive wave ﬁeld and dissipation cancel
to leading order, and the walker motion may be described in terms of the inviscid dynamics of aparticle whose mass depends on its speed. They thus computed a hydrodynamic boost factor γ B
for the walker in terms of the system parameters: The effective inertial mass of a walker is γ B m.
This perspective rationalizes the anomalously large orbital radii evident in Figure 5a in termsof the walker’s wave-induced added mass. Its value in more general settings is currently being
explored.

4. RELATION TO QUANTUM MECHANICS
Realist interpretations of quantum mechanics assert that there is a real dynamics underlying thestatistical description provided by standard quantum theory; thus, microscopic particles follow
trajectories as do their classical counterparts (Bohr 1935, Einstein et al. 1935). This perspective
suffered an early setback when von Neumann (1932) published a proof that erroneously suggested
the impossibility of such a dynamics. It was over 30 years before Bell (1966) discredited von
Neumann’s and subsequent impossibility proofs. On such matters, I am inclined to give Bell(1987) the ﬁnal word, to entertain the possibility of a real quantum dynamics underlying the wellestablished statistical theory. It is then only natural to ask what relation this hydrodynamic system
might have to existing realist models of quantum dynamics.

4.1. The Madelung Transformation
The Madelung (1926) transformation allows the linear Schrödinger equation (LSE) to be recast inhydrodynamic form. Let us consider the standard quantum mechanical description of a particle ofwww.annualreviews.org • Pilot-Wave Hydrodynamics

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mass m conﬁned to a plane in the presence of a potential V(x). Expressing the probability functionin polar form, (x, t) = R(x, t)e i S(x,t)/ h̄ , and substituting into the LSE, we obtain

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Dρq+ ρq ∇ · vq = 0,
Dt

Supplemental Material

(4a)

∂S 1
+ m vq2 + Q + V = 0,
(4b)
∂t
2
where vq (x, t) = ∇ S(x, t)/m is the quantum velocity of probability, S(x, t) is the action,ρq (x, t) = R2 is the probability density, and Q(x, t) = −(h̄ 2 / 2m)(1/R)∇ 2 R is the quantum potential. Equation 4a indicates the conservation of probability, and Equation 4b corresponds tothe quantum Hamilton-Jacobi equation. Taking the gradient of the latter yields the evolution
equation for vq :Dvq= −∇ Q − ∇V .
(5)m
Dt
From the perspective of the ﬂuid mechanician, the Madelung formulation (Equation 4) has two
attractive features (Spiegel 1980). First, Planck’s constant h̄ appears only once, as the coefﬁcienton the quantum potential. Second, the system corresponds to that describing the evolution of ashallow, inviscid ﬂuid layer if one associates ρq with the ﬂuid depth and vq with the depth-averaged
ﬂuid velocity. When Q is linearized about a state of uniform ρq , it assumes the form of the curvaturepressure in shallow-water hydrodynamics (Bühler 2010). One thus sees the relation between h̄ inquantum statistics and surface tension σ in shallow-water hydrodynamics.
The Madelung transformation simply indicates that one can model quantum statistics hydrodynamically. For example, the statistics reported in the quantum corral experiments (Crommie et al.
1993a,b) might in principle be modeled with a Faraday wave system (see Supplemental Video 1)(Harris et al. 2013, Harris & Bush 2013). Speciﬁcally, provided the corrals can be tuned to ﬁt asingle mode, given the energy of the particle E = h̄ω, one can deduce the de Broglie wavelength,
λdB = 2π/k, from the quantum dispersion relation ω = h̄k2 /2m, from which one might anticipate the probability distribution. Similarly, given the driving frequency, we know the Faraday√
3/2
wavelength from the water-wave dispersion relation (e.g., ω F = σ/ρ kF for deep-water capillary waves), on the basis of which we might deduce the cavity’s Faraday wave mode (Figure 4b).
However, the corral experiments (Harris et al. 2013) indicate that one can also predict the statistics of walkers hydrodynamically; speciﬁcally, the histogram of a walker in a circular cavity isprescribed by the cavity’s Faraday wave mode. Thus, in certain parameter regimes, walkers andquantum particles display similar statistical behavior, with the ﬂuid properties (σ, ρ) standing
in for (h̄, m). What then can we make of the relation between walker motion and quantum
dynamics?

4.2. Pilot-Wave Theories
The ﬁrst pilot-wave theory was proposed by de Broglie (1923), who envisaged microscopic
particles being guided by an accompanying wave ﬁeld, pushed in a direction perpendicular tosurfaces of constant phase (de Broglie 1926, 1930; Bacchiagaluppi & Valentini 2009). His original
conception, his “double-solution theory” (de Broglie 1956), involved two waves, a real pilot wave
centered on the particle and the statistical wave predicted by standard quantum theory. He asserted
that quantum mechanics was intrinsically relativistic and proposed that the pilot wave originates
in internal particle oscillations at the Compton frequency, ωc = mc 2 / h̄, at which rest mass energy
is exchanged with wave energy. He proposed that the guiding wave ﬁeld evolves according to
the Klein-Gordon equation and consists of a monochromatic wave ﬁeld in the particle’s frame of
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reference. The de Broglie relation, p = h̄k, then relates the particle momentum to the de Broglie
wavelength, λdB = 2π/k. Finally, he stressed the importance of the harmony of phases, by which
the particle’s internal vibration, seen as that of a clock, stays in phase with its guiding wave (deBroglie 1930, 1987). Thus, according to his conception, the wave and particle maintain a state ofresonance.
Bohm (1952a,b) proposed a single-wave pilot-wave theory, according to which the guiding
wave is precisely equivalent to the solution of Schrödinger’s equation, and the particle velocity
is identical to the quantum velocity of probability vq . Provided one takes an ensemble of initialconditions consistent with a solution to the LSE, the predictions of Bohmian mechanics are identical to those of standard quantum mechanics (Keller 1953). The result represented an important
counterexample to the impossibility proofs of von Neumann (1932) and others that held sway
at the time. Bohmian mechanics has since been developed substantially (Holland 1993, Towler
2009, Durr et al. 2012) and over time has been conﬂated with de Broglie’s theory into what isnow commonly referred to as the de Broglie–Bohm theory. The walking drop system suggests thepotential value of disentangling these two pilot-wave theories.
The walker system is closer to de Broglie’s double-solution theory than to Bohmian mechanics (Couder & Fort 2012). In the double-solution theory, as in this hydrodynamic system, thestatistical wave must be augmented by the real pilot wave: The standard wave–particle duality
must be extended to the wave–particle–pilot-wave trinity. The distinction is particularly evident
in the corral geometry. According to Bohmian mechanics, the guiding wave is the statistical wave.
In the walker system, conversely, the drop is piloted by the instantaneous wave ﬁeld within thecorral, whose form is relatively complex (Harris et al. 2013). Although both the real and statistical waves are characterized by the Faraday wavelength, the former is effectively centered on
the particle and the latter is prescribed by the corral geometry (Figure 4b). The marked difference between the walker system and Bohmian mechanics is also evident in the diffraction
experiments (Couder & Fort 2006). In quantum single-particle diffraction, Bohmian trajectories,
as guided by the symmetric statistical wave, are necessarily symmetric: No particles cross thecenterline (Philippidis et al. 1979). Conversely, the walker is guided by the complex pilot wave
arising as it enters the gap. Its trajectory is complex, chaotic, and decidedly asymmetric: Walkers
often cross the centerline. Although analog Bohmian trajectories could be deﬁned for walkers
on the basis of the statistics, they would simply represent ensemble averages of the real chaotic
trajectories.
Table 1 compares de Broglie’s double-solution theory and the walker system. Both systems
are characterized by three timescales: a short time associated with a high-frequency oscillation,
an intermediate time associated with the pilot-wave dynamics, and a long time over which thecoherent statistics emerge (Harris & Bush 2013) (see Supplemental Video 1). Both systems
are characterized by a single length scale that appears in both the dynamics and the statistics.
The walker diffraction and corral experiments provide the ﬁrst evidence that a particle-centeredpilot-wave dynamics may give rise to a statistical wave with the same wavelength, but with a form
prescribed by the bounding geometry.
De Broglie (1956) speciﬁed neither the physical origin nor the geometric form of his pilot wave.
He proposed that it was linearly related to the statistical wave in the far ﬁeld but was nonlinear (inan unspeciﬁed fashion) in the vicinity of the particle (de Broglie 1987). As he saw the particle as
the source of the ﬁeld, he invoked a singularity at the particle position. Three key features of thewalker system are absent from de Broglie’s conception. First, the walker interacts with an existingﬁeld (the interface) and so need not represent a singularity. Second, its monochromatic pilot-wave
ﬁeld is generated by parametric forcing. Third, the relation between the real and statistical waves
emerges through chaotic pilot-wave dynamics.
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Table 1 A comparison among the walking droplet system, de Broglie’s double-solution pilot-wave
theory (de Broglie 1956, 1987), and its extension to stochastic electrodynamics (SED) (Kracklauer
1992, de la Peña & Cetto 1996, Haisch & Rueda 2000)

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Walkersde Broglie

SED pilot wave

Pilot wave

Faraday capillary

Unspeciﬁed

Electromagnetic (EM)Driving

Bath vibration

Internal clock

Vacuum ﬂuctuations

Spectrum

Monochromatic

Monochromatic

Broad

Trigger

Bouncing

Zitterbewegung

Zitterbewegung

Trigger frequencyωFωc = mc 2 / h̄ωc = mc 2 / h̄Energetics

GPE ↔ wave

mc 2 ↔ h̄ω

mc 2 ↔ EM

Resonance

Droplet—wave

Harmony of phases

Unspeciﬁed

Dispersion ω(k)

ω2F ≈ σ k3 /ρω2 = ωc2 + c 2 k2

ω = ckCarrier λ

λF

λdB

λc

Statistical λ

λF

λdB

λdB

In the walker system, energy is exchanged at ωF between the drop’s gravitational potential energy (GPE) and the capillary
Faraday wave ﬁeld. Zitterbewegung denotes particle oscillations at the Compton frequency ωc .

4.3. Stochastic Electrodynamics
Soon after its inception (Bohm 1952a,b), Bohmian mechanics was extended to incorporate theinﬂuence of a stochastic subquantum realm (Bohm & Vigier 1954). According to this physicalpicture, stochastic particle motion underlies the quantum velocity of probability vq , the Madelung
ﬂow, just as Brownian motion underlies the streamlines within a ﬂuid ﬂow. This perspective,
also pursued by de Broglie (1964) in his later years, led to a considerable literature (Goldstein
1987, Chebotarev 2000). Of particular interest is the result of Nelson (1966), who derived theLSE from consideration of a stochastic classical system, namely the Brownian motion of a mass
m with diffusion coefﬁcient h̄/(2m). A recent review of the successes and failures of the resulting
stochastic interpretation of quantum mechanics is presented by Nelson (2012).
Stochastic electrodynamics (SED) posits a zero-point ﬁeld (ZPF), electromagnetic vacuum
ﬂuctuations with an energy spectral form: U (ω) = h̄ω/2 (Boyer 2011). Notably, these ﬂuctuations
represent a natural means of introducing Planck’s constant h̄ into a classical theory. SED has
provided rationale for the Casimir effect, the stable ground state of the hydrogen atom (Cole &Zhou 2003), and the blackbody radiation spectrum (Boyer 2010). Surdin (1974) proposed the
ZPF as a source of stochasticity in the subquantum realm. The resulting physical picture is being
advanced: Grossing et al. (2012a,b) have taken inspiration from the bouncing droplets, referringto microscopic particles as bouncers.
De la Peña & Cetto (1996) suggested that vacuum ﬂuctuations may generate a quantum particle’s zitterbewegung (Schrödinger 1930, Hestenes 1990), speciﬁcally oscillations at the Compton
frequency ωc = mc 2 / h̄. They further suggested that, as a particle translates, this vibrationalmotion interacts selectively with waves in the ambient vacuum ﬁeld, a resonant interaction that
ampliﬁes an electromagnetic pilot wave. According to this physical picture, the carrier wave has
the Compton wavelength λc = 2π h̄/mc , and the de Broglie wavelength prescribes its modulation(Kracklauer 1992, 1999). This conception of a particle as a ZPF-driven oscillating charge witha resonance at ωc has been further explored by Haisch & Rueda (2000), who suggested that it
may offer insight into the origins of inertial mass, thereby linking the ZPF with the quantum
wave nature of matter and relativistic mechanics (Haisch et al. 2001, Rueda & Haisch 2005). The
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similarities between the resulting physical picture and the walker system are intriguing: The vacuum ﬂuctuations would play the role of the vibrating bath in powering the system, with theparticle’s zitterbewegung that of the bouncing drop in triggering the pilot wave. This SED-based
physical picture is also included in Table 1.

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5. DISCUSSION
The hydrodynamic pilot-wave system discovered by Yves Couder, Emmanuel Fort, and coworkers is a fascinating new dynamical system in which a drop self-propels by virtue of a resonantinteraction with its own wave ﬁeld. It is the ﬁrst macroscopic realization of a pilot-wave systemof the form envisioned by de Broglie in his double-solution theory, and has extended the rangeof behaviors accessible to classical systems. It exhibits features reminiscent of quantum singleparticle diffraction, tunneling, singly and doubly quantized orbits, orbital level splitting, spin, andmultimodal statistics, all of which may be rationalized in terms of pilot-wave dynamics. Willful
misinterpretation based on either intrusive measurement or impaired observation might lead tothe inference of uncertainty relations and hydrodynamic boost factors. Likewise, exclusion principles might be inferred; for example, certain multiple-walker orbital states are forbidden by virtueof the fact that walkers must share the same wave ﬁeld.
What features are critical to the quantum-like behavior? Quantization emerges from the dynamic constraint imposed on the walker by its monochromatic pilot-wave ﬁeld. For example,
quantized orbits may be understood in terms of the interaction of the walker with its own wake:
Quantization emerges only when the wave ﬁeld lasts longer than the orbital period. The dispersion
relation of the wave ﬁeld is not important. What is important is that the wave be monochromatic,hence the critical importance of the Faraday forcing and the resonance between the bouncing drop
and its pilot wave. Studies of orbital motion suggest that multimodal statistics are a generic featureof chaotic pilot-wave dynamics: Unstable eigenstates serve as attractors that leave an imprint on thestatistics. The general validity of this perspective needs to be explored through further examination of walkers in corrals (T. Gilet, submitted manuscript), single-particle diffraction, and entirely
new geometries. Characterizing the transitions to chaos in this new pilot-wave system poses an
exciting challenge to the dynamical systems community. The inﬂuence of stochastic forcing on
this chaotic pilot-wave dynamics is also of considerable interest. Such a forcing may arise naturally in the laboratory through the inﬂuence of air currents or when the drop is in a chaotic
walking state, and might be imposed in a controlled fashion through noise in the vibrational
forcing.
The hydrodynamic system has numerous limitations as a quantum analog; however, some
may be circumvented. Although the experiments can capture only two-dimensional pilot-wave
dynamics, the attendant theoretical developments can be generalized to three dimensions. The
wavelength of the walker’s pilot wave is imposed by the frequency of the vibrational forcing andis independent of its speed; conversely, in quantum mechanics, the de Broglie relation, p = h̄k,dictates a speed-dependent de Broglie wavelength. Such a dependence could be incorporated into
existing pilot-wave models. Finally, we have seen that intrusive measurement, a deﬁning featureof quantum mechanics, can be artiﬁcially imposed on the walker system. The question remains
open as to whether some combination of intrusive measurement and chaotic pilot-wave dynamics
might give rise to a hydrodynamic analog of entanglement.
Bell (1987) championed the de Broglie–Bohm pilot-wave theory on the grounds that it presents
a conceptual framework for a mechanistic understanding of quantum dynamics, a framework thata purely statistical theory cannot possibly provide. Whatever its shortcomings and limitations,the new physical picture suggested by the walkers, that of a relatively complex chaotic pilot-wave
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dynamics, also has this appealing feature. Moreover, it has precursors in a continuous arc ofliterature that leads back to de Broglie and Einstein, both of whom sought to reconcile quantum
mechanics and relativity through consideration of the wave nature of matter (Bohm & Hiley 1982,
Chebotarev 2000). The relation among this physical picture, quantum chaos (Gutzwiller 1990),
and quantum ﬁeld theory (Weinberg 1995) is left to the interested reader.
And lest the longevity of the quantum paradoxes be mistaken for their insurmountability,
ﬂuid mechanics has a cautionary tale to tell. In 1749, d’Alembert’s paradox indicated that an
object moving through an inviscid ﬂuid experiences no drag, a prediction that was clearly atodds with experiments on high–Reynolds number gas ﬂows. The result was a longstanding rift
between experimentalists and theorists: For much of the nineteenth century, the former worked
on phenomena that could not be explained, and the latter on those that could not be observed(Lighthill 1956). D’Alembert’s paradox stood for over 150 years, until Prandtl’s developments(Anderson 2005) allowed for the resolution of the dynamics on the hitherto hidden scale of theviscous boundary layer.
Finally, as concerns my alignment vis-à-vis quantum interpretations, I remain steadfastly agnostic; however, if forced to choose, I would be inclined to back, by virtue of its inclusivity, thelogical extension of the Many-Worlds interpretation (Everett 1957), the Many-Many-Worlds
interpretation, according to which each quantum interpretation is realized in some edition of themultimultiverse, and there is even one world in which there is only one world, a world in whichquantum statistics are underlaid by chaotic pilot-wave dynamics, there is no philosophical schism
between large and small, and beables be.

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FL47CH12-BushDISCLOSURE STATEMENT
The author is not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS
I gratefully acknowledge the ﬁnancial support of the NSF through grants CBET-0966452 andCMMI-1333242, the MIT-France and MIT-Brazil Programs, and the CNPq–Science Without
Borders Program. I have greatly beneﬁted from conversations with Yves Couder, Emmanuel Fort,
and Ruben Rosales. I also thank Dan Harris, Jan Moláček, Anand Oza, and Leo Zhou for valuablecomments on and assistance with the article.

LITERATURE CITED
Anderson JD. 2005. Ludwig Prandtl’s boundary layer. Phys. Today 58:45–48
Bacchiagaluppi G, Valentini A. 2009. Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference.
Cambridge, UK: Cambridge Univ. Press
Bach R, Pope D, Liou S, Batelaan H. 2013. Controlled double-slit electron diffraction. New J. Phys. 15:033018
Bechhoeffer J, Ego V, Manneville S, Johnson B. 1995. An experimental study of the onset of parametrically
pumped surface waves in viscous ﬂuids. J. Fluid Mech. 288:325–50
Bell JS. 1966. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38:447–52
Bell JS. 1982. On the impossible pilot wave. Found. Phys. 12:989–99
Bell JS. 1987. Speakable and Unspeakable in Quantum Mechanics. Cambridge, UK: Cambridge Univ. Press
Benjamin TB, Ursell F. 1954. The stability of the plane free surface of a liquid in vertical periodic motion.
Proc. R. Soc. Lond. A 225:505–15
288

Bush

Annu. Rev. Fluid Mech. 2015.47:269-292. Downloaded from www.annualreviews.org
Access provided by Massachusetts Institute of Technology (MIT) on 01/07/15. For personal use only.

FL47CH12-Bush

ARI

1 December 2014

20:26

Berry MV, Chambers RG, Large MD, Upstill C, Walmsley JC. 1980. Wavefront dislocations in the AharanovBohm effect and its water wave analogue. Eur. J. Phys. 1:154–62
Bohm D. 1952a. A suggested interpretation of the quantum theory in terms of hidden variables, I. Phys. Rev.
85:166–79
Bohm D. 1952b. A suggested interpretation of the quantum theory in terms of hidden variables, II. Phys. Rev.
85:180–93
Bohm D, Vigier JP. 1954. Model of the causal interpretation of quantum theory in terms of a ﬂuid withirregular ﬂuctuations. Phys. Rev. 96:208–16
Bohm D, Hiley BJ. 1982. The de Broglie pilot wave theory and the further development of new insights arisingout of it. Found. Phys. 12:1001–16
Bohr N. 1935. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev.
48:696–702
Boyer TH. 2010. Derivation of the Planck spectrum for relativistic classical scalar radiation from thermal
equilibrium in an accelerating frame. Phys. Rev. D 81:105024
Boyer TH. 2011. Any classical description of nature requires classical electromagnetic zero-point radiation.
Am. J. Phys. 79:1163–67
Bühler O. 2010. Wave-vortex interactions in ﬂuids and superﬂuids. Annu. Rev. Fluid Mech. 42:205–28
Burinskii A. 2008. The Dirac-Kerr-Newman electron. Gravit. Cosmol. 14:109–22
Bush JWM. 2010. Quantum mechanics writ large. Proc. Natl. Acad. Sci. USA 107:17455–56
Bush JWM, Oza A, Moláček J. 2014. The wave-induced added mass of walking droplets. J. Fluid Mech. 755:R7.
doi: 10.1017/jfm.2014.459
Carmigniani R, Lapointe S, Symon S, McKeon BJ. 2014. Inﬂuence of a local change of depth on the behaviorof walking oil drops. Exp. Thermal Fluid Sci. 54:237–46
Chebotarev L. 2000. Introduction: the de Broglie-Bohm-Vigier approach in quantum mechanics. In JeanPierre Vigier and the Stochastic Interpretation of Quantum Mechanics, ed. S Jeffers, B Lehnert, N Abramson,
L Chebotarev, pp. 1–18. Montreal: Apeiron
Cole DC, Zhou Y. 2003. Quantum mechanical ground state of hydrogen obtained from classical electrodynamics. Phys. Lett. A 317:14–20
Coste C, Lund F, Umeki M. 1999. Scattering of dislocated wave fronts by vertical vorticity and the AharonovBohm effect. I. Shallow water. Phys. Rev. E 60:4908–16
Couder Y, Fort E. 2006. Single particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett.
97:154101
Couder Y, Fort E. 2012. Probabilities and trajectories in a classical wave-particle duality. J. Phys. Conf. Ser.
361:012001
Couder Y, Fort E, Gautier C-H, Boudaoud A. 2005a. From bouncing to ﬂoating: noncoalescence of drops ona ﬂuid bath. Phys. Rev. Lett. 94:177801
Couder Y, Protière S, Fort E, Boudaoud A. 2005b. Walking and orbiting droplets. Nature 437:208
Crommie MF, Lutz CP, Eigler DM. 1993a. Conﬁnement of electrons to quantum corrals on a metal surface.
Science 262:218–20
Crommie MF, Lutz CP, Eigler DM. 1993b. Imaging standing waves in a two-dimensional electron gas. Nature
363:524–27
Davisson C, Germer LH. 1927. The scattering of electrons by a single crystal of nickel. Nature 119:558–60
de Broglie L. 1923. Ondes et quanta. C. R. 177:507–10
de Broglie L. 1926. Ondes et mouvements. Paris: Gautier-Villars
de Broglie L. 1930. An Introduction to the Study of Wave Mechanics. London: Methuen & Co.
de Broglie L. 1956. Une interprétation causale et nonlinéaire de la Mechanique ondulatoire: la théorie de la double
solution. Paris: Gautier-Villars
de Broglie L. 1964. La thermodynamique cachée des particules. Ann. Inst. Henri Poincaré 1:1–19
de Broglie L. 1987. Interpretation of quantum mechanics by the double solution theory. Ann. Fond. Louis
Broglie 12:1–23
de Gennes PG, Brochard-Wyart F, Quéré D. 2002. Gouttes, bulles, perles et ondes. Paris: Belinde la Peña L, Cetto AM. 1996. The Quantum Dice: An Introduction to Stochastic Electrodynamics. Dordrecht:
Kluwer Acad.
www.annualreviews.org • Pilot-Wave Hydrodynamics

289

ARI

1 December 2014

20:26

Denardo BC, Puda JJ, Larraza A. 2009. A water wave analog of the Casimir effect. Am. J. Phys. 77:1095–101
Donnelly RJ. 1993. Quantized vortices and turbulence in helium II. Annu. Rev. Fluid Mech. 25:327–71
Dorbolo S, Terwagne D, Vandewalle N, Gilet T. 2008. Resonant and rolling droplets. New J. Phys. 10:113021
Douady S. 1990. Experimental study of the Faraday instability. J. Fluid Mech. 221:383–409
Durr D, Goldstein S, Zanghi N. 2012. Quantum Physics Without Quantum Philosophy. New York: Springer
Eddi A, Boudaoud A, Couder Y. 2011a. Oscillating instability in bouncing drop crystals. Europhys. Lett.
94:20004
Eddi A, Decelle A, Fort E, Couder Y. 2009a. Archimedean lattices in the bound states of wave interacting
particles. Europhys. Lett. 87:56002
Eddi A, Fort E, Moisy F, Couder Y. 2009b. Unpredictable tunneling of a classical wave-particle association.
Phys. Rev. Lett. 102:240401
Eddi A, Moukhtar J, Perrard J, Fort E, Counder Y. 2012. Level splitting at a macroscopic scale. Phys. Rev.
Lett. 108:264503
Eddi A, Sultan E, Moukhtar J, Fort E, Rossi M, Couder Y. 2011b. Information stored in Faraday waves: theorigin of path memory. J. Fluid Mech. 674:433–63
Eddi A, Terwagne D, Fort E, Couder Y. 2008. Wave propelled ratchets and drifting rafts. Europhys. Lett.
82:44001
Edwards WS, Fauve S. 1994. Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278:123–48
Einstein A, Podolsky B, Rosen N. 1935. Can quantum-mechanical description of physical reality be considered
complete? Phys. Rev. 47:777–80
Everett H. 1957. Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29:454–62
Faraday M. 1831. On the forms and states of ﬂuids on vibrating elastic surfaces. Philos. Trans. R. Soc. Lond.
121:319–40
Feynman RP, Leighton RB, Sands M. 1964. The Feynman Lectures on Physics. New York: Addison-Wesley
Fiete GA, Heller EJ. 2003. Theory of quantum corrals and quantum mirages. Rev. Mod. Phys. 75:933–48
Fort E, Couder Y. 2013. Trajectory eigenmodes of an orbiting wave source. Europhys. Lett. 102:16005
Fort E, Eddi A, Boudaoud A, Moukhtar J, Couder Y. 2010. Path-memory induced quantization of classicalorbits. Proc. Natl. Acad. Sci. USA 107:17515–20
Gamow G. 1928. The theory of nuclear disintegration. Nature 122:805–7
Gier S, Dorbolo S, Terwagne D, Vandewalle N, Wagner C. 2012. Bouncing of polymeric droplets on liquidinterfaces. Phys. Rev. E 86:066314
Gilet T, Bush JWM. 2009a. Chaotic bouncing of a drop on a soap ﬁlm. Phys. Rev. Lett. 102:014501
Gilet T, Bush JWM. 2009b. The ﬂuid trampoline: droplets bouncing on a soap ﬁlm. J. Fluid Mech. 625:167–203
Gilet T, Bush JWM. 2012. Droplets bouncing on a wet, inclined surface. Phys. Fluids 24:122103
Gilet T, Terwagne D, Vandewalle N, Dorbolo S. 2008. Dynamics of a bouncing droplet onto a verticallyvibrated interface. Phys. Rev. Lett. 100:167802
Gilet T, Vandewalle N, Dorbolo S. 2007. Controlling the partial coalescence of a droplet on a verticallyvibrated bath. Phys. Rev. E 76:035302
Goldman DI. 2002. Pattern formation and ﬂuidization in vibrated granular layers, and grain dynamics and jammingin a water ﬂuidized bed. PhD Thesis, Univ. Tex., Austin
Goldstein S. 1987. Stochastic mechanics and quantum theory. J. Stat. Phys. 47:645–67
Grossing G, Fussy S, Mesa Pascasio J, Schwabl H. 2012a. An explanation of interference effects in the double
slit experiment: classical trajectories plus ballistic diffusion caused by zero-point ﬂuctuations. Ann. Phys.
327:421–27
Grossing G, Fussy S, Mesa Pascasio J, Schwabl H. 2012b. The quantum as an emergent system. J. Phys. Conf.
Ser. 361:021008
Gutzwiller MC. 1990. Chaos in Classical and Quantum Mechanics. Berlin: Springer-Verlag
Haisch B, Rueda A. 2000. On the relation between a zero-point-ﬁeld-induced inertial effect and the Einstein–de Broglie formula. Phys. Rev. A 268:224–27
Haisch B, Rueda A, Dobyns Y. 2001. Inertial mass and the quantum vacuum ﬁelds. Phys. Rev. A 268:224–27
Harris DM, Bush JWM. 2013. The pilot-wave dynamics of walking droplets. Phys. Fluids 25:091112
Harris DM, Bush JWM. 2014a. Droplets walking in a rotating frame: from quantized orbits to multimodal
statistics. J. Fluid Mech. 739:444–64

Annu. Rev. Fluid Mech. 2015.47:269-292. Downloaded from www.annualreviews.org
Access provided by Massachusetts Institute of Technology (MIT) on 01/07/15. For personal use only.

FL47CH12-Bush

290

Bush

Annu. Rev. Fluid Mech. 2015.47:269-292. Downloaded from www.annualreviews.org
Access provided by Massachusetts Institute of Technology (MIT) on 01/07/15. For personal use only.

FL47CH12-Bush

ARI

1 December 2014

20:26

Harris DM, Bush JWM. 2014b. Generating uniaxial vibration with an electrodynamic shaker and external air
bearing. J. Sound Vib. In press
Harris DM, Moukhtar J, Fort E, Couder Y, Bush JWM. 2013. Wavelike statistics from pilot-wave dynamics
in a circular corral. Phys. Rev. E 88:011001
Hestenes D. 1990. The zitterbewegung interpretation of quantum mechanics. Found. Phys. 20:1213–32
Holland PR. 1993. The Quantum Theory of Motion: An Account of the De Broglie–Bohm Causal Interpretation of
Quantum Mechanics. Cambridge, UK: Cambridge Univ. Press
Jayaratne OW, Mason BJ. 1964. The coalescence and bouncing of water drops at an air-water interface. Proc.
R. Soc. Lond. A 280:545–65
Keller J. 1953. Bohm’s interpretation of the quantum theory in terms of “hidden” variables. Phys. Rev. 89:1040–
41
Kracklauer AF. 1992. An intuitive paradigm for quantum mechanics. Phys. Essays 5:226–34
Kracklauer AF. 1999. Pilot wave steerage: a mechanism and test. Found. Phys. Lett. 12:441–53
Kumar K. 1996. Parametric instability of the interface between two ﬂuids. Proc. R. Soc. Lond. A 452:1113–26
Kumar K, Tuckerman LS. 1994. Parametric instability of the interface between two ﬂuids. J. Fluid Mech.
279:49–68
Labousse M, Perrard S. 2014. Non-Hamiltonian features of a classical pilot-wave dynamics. Phys. Rev. E
90:022913
Lieber SI, Hendershott MC, Pattanaporkratana A, Maclennan JE. 2007. Self-organization of bouncing oil
drops: two-dimensional lattices and spinning clusters. Phys. Rev. E 75:056308
Lighthill J. 1956. Physics of gas ﬂow at very high speeds. Nature 178:343–45
Madelung E. 1926. Quantentheorie in Hydrodynamischen form. Z. Phys. 40:322–26
Miles J, Henderson D. 1990. Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22:143–65
Moisy F, Rabaud M, Salsac K. 2009. A synthetic Schlieren method for the measurement of the topography ofa liquid interface. Exp. Fluids 46:1021–36
Moláček J, Bush JWM. 2012. A quasi-static model of drop impact. Phys. Fluids 24:127103
Moláček J, Bush JWM. 2013a. Droplets bouncing on a vibrating ﬂuid bath. J. Fluid Mech. 727:582–611
Moláček J, Bush JWM. 2013b. Droplets walking on a vibrating ﬂuid bath: towards a hydrodynamic pilot-wave
theory. J. Fluid Mech. 727:612–47
Neitzel GP, Dell’Aversana P. 2002. Noncoalescence and nonwetting behavior of liquids. Annu. Rev. Fluid
Mech. 34:267–89
Nelson E. 1966. Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 150:1079–85
Nelson E. 2012. Review of stochastic mechanics. J. Phys. Conf. Ser. 361:012011
Newton I. 1979. Opticks: Or a Treatise of the Reﬂections, Refractions, Inﬂections and Colours of Light. Mineola, NY:
Dover
Okumura K, Chevy F, Richard D, Quéré D, Clanet C. 2003. Water spring: a model for bouncing drops.
Europhys. Lett. 62:237–43
Oza A, Harris DM, Rosales RR, Bush JWM. 2014a. Pilot-wave dynamics in a rotating frame: on the emergenceof orbital quantization. J. Fluid Mech. 744:404–29
Oza A, Rosales RR, Bush JWM. 2013. A trajectory equation for walking droplets: a hydrodynamic pilot-wave
theory. J. Fluid Mech. 737:552–70
Oza A, Wind-Willassen Ø, Harris DM, Rosales RR, Bush JWM. 2014b. Pilot-wave hydrodynamics in arotating frame: exotic orbits. Phys. Fluids 26:082101
Perrard S, Labousse M, Fort E, Couder Y. 2014a. Chaos driven by interfering memory. Phys. Rev. Lett.
113:104101
Perrard S, Labousse M, Miskin M, Fort E, Couder Y. 2014b. Self-organization into quantized eigenstates ofa classical wave-driven particle. Nat. Commun. 5:3219
Philippidis C, Dewdney C, Hiley BJ. 1979. Quantum interference and the quantum potential. Nuovo Cimento
52B:15–28
Pitaevskii L, Stringari S. 2003. Bose-Einstein Condensation. New York: Oxford Univ. Press
Prosperetti A, Oguz HN. 1993. The impact of drops on liquid surfaces and the underwater noise of rain.
Annu. Rev. Fluid Mech. 25:577–602
www.annualreviews.org • Pilot-Wave Hydrodynamics

291

ARI

1 December 2014

20:26

Protière S, Bohn S, Couder Y. 2008. Exotic orbits of two interacting wave sources. Phys. Rev. E 78:036204
Protière S, Boudaoud A, Couder Y. 2006. Particle wave association on a ﬂuid interface. J. Fluid Mech. 554:85–
108
Protière S, Couder Y, Fort E, Boudaoud A. 2005. The self-organization of capillary wave sources. J. Phys.
Condens. Matter 17:S3529–35
Pucci G, Ben Amar M, Couder Y. 2013. Faraday instability in ﬂoating liquid lenses: the spontaneous mutual
adaptation due to radiation pressure. J. Fluid Mech. 725:402–27
Pucci G, Fort E, Ben Amar M, Couder Y. 2011. Mutual adaptation of a Faraday instability pattern with its
ﬂexible boundaries in ﬂoating ﬂuid drops. Phys. Rev. Lett. 106:024503
Reynolds O. 1886. On the theory of lubrication. Philos. Trans. R. Soc. Lond. A 177:157–234
Rueda A, Haisch B. 2005. Gravity and the quantum vacuum inertia hypothesis. Ann. Phys. 14:479–98
Schrödinger E. 1930. Uber die kraftefreie Bewegung in der relativistischen Quantenmechanik. Sitz. Preuss.
Akad. Wiss. Phys. Math. Kl. 24:418–28
Shirikoff D. 2013. Bouncing droplets on a billiard table. Chaos 23:013115
Spiegel E. 1980. Fluid dynamical form of the linear and nonlinear Schrödinger equations. Physica D 1:236–40
Surdin M. 1974. L’éctrodynamique stochastique et l’interprétation de la Mécanique Quantique. C. R. Acad.
Sci. Paris B 278:881–83
Taylor GI. 1909. Interference fringes with feeble light. Proc. Camb. Philos. Soc. 15:114–15
Terwagne D. 2012. Bouncing droplets, the role of deformations. PhD Thesis, Univ. Liège
Terwagne D, Gilet T, Vandewalle N, Dorbolo S. 2009. Metastable bouncing droplets. Phys. Fluids 21:054103
Terwagne D, Gilet T, Vandewalle N, Dorbolo S. 2010. From a bouncing compound drop to a double emulsion.
Langmuir 26:11680–85
Terwagne D, Ludewig F, Vandewalle N, Dorbolo S. 2013. The role of droplet deformations in the bouncing
droplet dynamics. Phys. Fluids 25:122101
Terwagne D, Vandewalle N, Dorbolo S. 2007. Lifetime of a bouncing droplet. Phys. Rev. E 76:056311
Towler M. 2009. De Broglie-Bohm pilot-wave theory and the foundations of quantum mechanics: a graduate lecture
course. Univ. Camb. http://www.tcm.phy.cam.ac.uk/∼mdt26/pilot_waves.html
von Neumann J. 1932. Mathematische Grundlagen der Quantenmechanik. Berlin: Springer
Walker J. 1978. Drops of liquid can be made to ﬂoat on the liquid. What enables them to do so? Sci. Am.
238:151–58
Weinberg S. 1995. The Quantum Theory of Fields, Vol. I: Foundations. Cambridge, UK: Cambridge Univ. Press
Weinstein A, Pounder JR. 1945. An electromagnetic analogy in mechanics. Am. Math. Mon. 52:434–38
Wind-Willassen Ø, Moláček J, Harris DM, Bush JWM. 2013. Exotic states of bouncing and walking droplets.
Phys. Fluids 25:082002
Yang AL, Chien W, King M, Grosshandler WL. 1997. A simple piezoelectric droplet generator. Exp. Fluids
23:445–47
Yarin AL. 2006. Drop impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech.
38:159–92
Young T. 1804. The Bakerian Lecture: experiments and calculations relative to physical optics. Philos. Trans.
R. Soc. Lond. 94:1–16

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Annu. Rev. Fluid Mech. 2015.47:269-292. Downloaded from www.annualreviews.org
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Contents

Annual Review of
Fluid Mechanics
Volume 47, 2015

Fluid Mechanics in Sommerfeld’s School
Michael Eckert p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1
Discrete Element Method Simulations for Complex Granular Flows
Yu Guo and Jennifer Sinclair Curtis p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p21
Modeling the Rheology of Polymer Melts and Solutions
R.G. Larson and Priyanka S. Desai p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p47
Liquid Transfer in Printing Processes: Liquid Bridges with Moving
Contact Lines
Satish Kumar p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p67
Dissipation in Turbulent Flows
J. Christos Vassilicos p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p95
Floating Versus Sinking
Dominic Vella p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 115
Langrangian Coherent Structures
George Haller p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 137
Flows Driven by Libration, Precession, and Tides
Michael Le Bars, David Cébron, and Patrice Le Gal p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 163
Fountains in Industry and Nature
G.R. Hunt and H.C. Burridge p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 195
Acoustic Remote Sensing
David R. Dowling and Karim G. Sabra p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 221
Coalescence of Drops
H. Pirouz Kavehpour p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 245
Pilot-Wave Hydrodynamics
John W.M. Bush p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 269
Ignition, Liftoff, and Extinction of Gaseous Diffusion Flames
Amable Liñán, Marcos Vera, and Antonio L. Sánchez p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 293
The Clinical Assessment of Intraventricular Flows
Javier Bermejo, Pablo Martı́nez-Legazpi, and Juan C. del Álamo p p p p p p p p p p p p p p p p p p p p p p 315
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Green Algae as Model Organisms for Biological Fluid Dynamics
Raymond E. Goldstein p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 343
Fluid Mechanics of Blood Clot Formation
Aaron L. Fogelson and Keith B. Neeves p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 377
Generation of Microbubbles with Applications to Industry and Medicine
Javier Rodrı́guez-Rodrı́guez, Alejandro Sevilla, Carlos Martı́nez-Bazán,
and José Manuel Gordillo p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 405
Annu. Rev. Fluid Mech. 2015.47:269-292. Downloaded from www.annualreviews.org
Access provided by Massachusetts Institute of Technology (MIT) on 01/07/15. For personal use only.

Beneath Our Feet: Strategies for Locomotion in Granular Media
A.E. Hosoi and Daniel I. Goldman p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 431
Sports Ballistics
Christophe Clanet p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 455
Dynamic Stall in Pitching Airfoils: Aerodynamic Damping andCompressibility Effects
Thomas C. Corke and Flint O. Thomas p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 479
Ocean Spray
Fabrice Veron p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 507
Stability of Constrained Capillary Surfaces
J.B. Bostwick and P.H. Steen p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 539
Mixing and Transport in Coastal River Plumes
Alexander R. Horner-Devine, Robert D. Hetland, and Daniel G. MacDonald p p p p p p p p p 569
IndexesCumulative Index of Contributing Authors, Volumes 1–47 p p p p p p p p p p p p p p p p p p p p p p p p p p p p 595
Cumulative Index of Article Titles, Volumes 1–47 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 605
ErrataAn online log of corrections to Annual Review of Fluid Mechanics articles may be foundat http://www.annualreviews.org/errata/ﬂuidvi

Contents