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J. Fluid Mech. (2017), vol. 821, pp. 296–329. c

Cambridge University Press 2017

doi:10.1017/jfm.2017.235

296

Faraday wave–droplet dynamics:

discrete-time analysis

Matthew Durey1,†and Paul A. Milewski1

1Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

(Received 29 August 2016; revised 5 April 2017; accepted 7 April 2017)

A droplet may ‘walk’ across the surface of a vertically vibrating bath of the same

ﬂuid, due to the propulsive interaction with its wave ﬁeld. This hydrodynamic pilot-

wave system exhibits many dynamics previously believed to exist only in the quantum

realm. Starting from ﬁrst principles, we derive a discrete-time ﬂuid model, whereby

the bath–droplet interactions are modelled as instantaneous. By analysing the stability

of the ﬁxed points of the system, we explain the dynamics of a walking droplet

and capture the quantisations for multiple-droplet interactions. Circular orbits in a

harmonic potential are studied, and a double quantisation of chaotic trajectories is

obtained through systematic statistical analysis.

Key words: bifurcations, drops, Faraday waves

1. Introduction

Faraday waves and droplet impact have been separate research areas for much of

the last century. Although Walker (1978) showed that a droplet may ‘ﬂoat’ on a

vertically vibrating bath of ﬂuid, it was not until the last decade that this connection

was re-explored. In 2005, Couder and co-workers showed that for sufﬁciently large,

yet subcritical, vibrations of the liquid bath, a droplet may bounce periodically on

the surface (Couder et al. 2005b). At each impact, a capillary wave propagates away

from the droplet, exciting a ﬁeld of standing Faraday waves in its wake. For larger

forcing, the bouncing destabilises, and the droplet ‘walks’ across the surface of the

bath, propelled at each impact by the slope of its associated wave ﬁeld. As the

forcing vibration increases, so does the decay time of the Faraday waves. This yields

a path ‘memory’ from previous droplet impacts (Eddi et al. 2011), leading to a

macroscopic particle–wave interaction, as previously envisaged as an explanation for

quantum behaviour (de Broglie 1926). This analogy has since been explored through

a remarkable series of experiments, summarised in detail by Bush (2015).

Several quantum analogies have been pursued. A walking droplet passing through

a slit between submerged walls yielded the diffraction and interference patterns for

single- and double-slit experiments respectively (Couder & Fort 2006,2011). This

effect was due to the interaction of the wave ﬁeld with the walls. Furthermore, a

droplet may ‘tunnel’ across the submerged wall separating two deep regions; as the

wall width increases, the tunnelling probability decreases (Eddi et al. 2009). Moreover,

† Email address for correspondence: m.durey@bath.ac.uk

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Faraday wave–droplet dynamics 297

Harris et al. (2013) found that the position of a chaotic walker in a circular corral

exhibits wave-like statistics, whose maxima correspond to the zeros of the fundamental

Faraday mode. This is reminiscent of an electron in a quantum corral.

The interaction between droplets also yields quantum analogues, such as ﬁxed

lattices of bouncing droplets and bound states (Protière, Boudaoud & Couder 2006;

Eddi et al. 2008). At the approach of two walking droplets, the interaction between

wave ﬁelds leads to either scatter or locking in circular orbital motion with quantised

orbit diameters (Couder et al. 2005b). For droplets of different sizes, one droplet

may orbit the other, with complex epicycles emerging (Protière, Bohn & Couder

2008). Two droplets walking in parallel interact through the wave ﬁeld, and stable

transverse oscillations ensue (Protière et al. 2006); these ‘promenade’ modes have

been observed to have quantised average distances between the droplets (Borghesi

et al. 2014).

Further analogues occur when a droplet walks on a rotating bath. Due to the

Coriolis force, the droplet moves in a circular motion in the rotating frame (Fort

et al. 2010). As the forcing vibration is increased, the wave ﬁeld forces the orbit

diameters to be quantised, with a macroscopic analogy to Landau levels (Oza et al.

2014a). In the long-memory limit, more exotic trajectories occur, including drifting,

wobbling and quasi-periodic orbits (Oza et al. 2014b). In particular, the stationary

probability distribution for the droplet position exhibits wave-like statistics, with

maxima at its unstable steady states (Harris & Bush 2014). Two droplets may orbit

each other in the rotating frame, but their orbit diameters exhibit Zeeman-like splitting

depending on whether the orbits are co-/anti-rotational relative to the bath (Eddi et al.

2012).

Circular orbits also exist for a droplet in a harmonic potential (Perrard et al.

2014b), with their convergence explored by Labousse & Perrard (2014). At long

memory, the orbit diameters are quantised (Labousse et al. 2016), and an array of

stable exotic trajectories forms, with a double quantisation in their average radius and

angular momentum (Perrard et al. 2014b). The underlying pivot structure of the wave

ﬁeld governing these trajectories has been explored by Labousse et al. (2014). In the

chaotic regime, the switching time between trajectories is probabilistic (Perrard et al.

2014a).

The above dynamics are governed by a complex set of physical phenomena. For

a bath vibrated sinusoidally with amplitude Aand frequency ω0/(2π), the stability

of the Faraday waves is governed by Γ=Aω2

0/g, which is the ratio of peak forcing

acceleration relative to gravity. In both the inviscid (Benjamin & Ursell 1954) and

viscous (Kumar & Tuckerman 1994; Kumar 1996) cases, a spectral decomposition

yields a system of Mathieu equations, whose stability depends on Γ > 0. In the

dissipative case, the surface destabilises at Γ=ΓF(the Faraday threshold), which

corresponds to the critical wavenumber k=kFand subharmonic waves (relative to

the forcing frequency).

In all non-coalescing states, the droplet and bath remain separated by a thin air

lubrication layer (Walker 1978), where the air slowly escapes (Couder et al. 2005a).

The restoring forces of the wave ﬁeld transmitted through the lubrication layer propel

the droplet back into the air before coalescence, leading to periodic bouncing. The

bouncing threshold ΓBhas been investigated through lubrication theory (Gilet et al.

2008) and a spring model for droplet impact (Hubert et al. 2015).

For Γ > ΓB, a range of bouncing dynamics occur, which destabilise to walking

at Γ=ΓW> ΓB. The vertical dynamics of walkers are frequently observed to be in

subharmonic resonance with the wave ﬁeld, although a range of periodic and chaotic

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298 M. Durey and P. A. Milewski

dynamics exist. Following Gilet & Bush (2009), we distinguish different vertical

dynamics by (m,n), where mis the number of forcing periods and nis the number

of impacts for the dynamics to repeat. The aforementioned subharmonic (2,1)mode

has two distinct energy levels: the lower-energy (2,1)1mode and the higher-energy

(2,1)2mode observed at higher Γ, where the impact durations are much shorter. The

bifurcation to different (m,n)regimes as a function of droplet diameter and Γ /ΓF

was recorded by Protière et al. (2006) and Eddi et al. (2008). However, Moláˇ

cek &

Bush (2013a) showed that the bouncing thresholds for different ﬂuids collapse onto

a single curve if the drops are instead characterised by their dimensionless vibration

number,

Ω=ω0qρR3

0/σ , (1.1)

where R0is the droplet radius, with ﬂuid density ρand surface tension σ. This is

the frequency ratio of bath vibrations to characteristic droplet oscillations. Regime

diagrams in the (Γ, Ω)-plane are found in Wind-Willassen et al. (2013).

Due to the complexity of this system, no uniﬁed model exists to describe all of the

observed dynamics. The ﬁrst simple model captured the qualitative bifurcation from

bouncing to walking through a period-averaged differential equation for the droplet

position, but this was valid only in the low-memory limit and the (2,1)mode (Couder

et al. 2005b; Protière et al. 2006).

Assuming a linear wave ﬁeld, Fort et al. (2010) modelled the wave ﬁeld as a

superposition of exponentially decaying (in time) standing waves centred at each

(instantaneous) impact, with the droplet dynamics restricted to the predominant (2,1)

mode. The wave ﬁeld generated at each impact was the far-ﬁeld approximation to

the Bessel function J0(kFr)with an experimentally observed exponential spatial decay

correction. Although this model numerically veriﬁed the quantised orbits in a rotating

bath, it was not analysed mathematically, not least due to the spatial singularity

centred at each droplet impact.

With no spatial damping correction, Moláˇ

cek & Bush (2013b) coupled the

wave dynamics with a logarithmic spring model for the vertical motion of the

droplet (Moláˇ

cek & Bush 2013a). This model successfully predicts many of the

experimentally observed bouncing and walking (m,n)modes (Wind-Willassen

et al. 2013), but relies on experimentally ﬁtted parameters and is too complex

for mathematical analysis.

To simplify this, Oza, Rosales & Bush (2013) observed that the time scale of

the horizontal motion is much greater than that of the vertical motion in the (2,1)

mode. Under this assumption, they approximated the sum of instantaneous impacts

by a continuous integral, leading to an integro-differential trajectory equation for the

droplet, which records the entire path history of the droplet (unlike the low-memory

limit model of Protière et al. (2006)). This past behaviour can be approximated in

the small-acceleration limit, yielding a hydrodynamic boost factor for the droplet

mass from its wave ﬁeld interaction (Bush, Oza & Moláˇ

cek 2014). By studying

the trajectory equation, analytic expressions are obtained for the bifurcation from

bouncing to walking and the walking speed (Oza et al. 2013), circular orbits in

a rotating frame (Oza et al. 2014a) and circular orbits in a harmonic potential

(Labousse et al. 2016). Advantageously, the linear stability of these dynamics can be

obtained analytically from the trajectory equation.

The above models all have one fundamental shortcoming: they simplify the

complex wave ﬁeld generated by each droplet impact by decoupling the radial

and temporal behaviour. To remedy this, Eddi et al. (2011) modelled the wave

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Faraday wave–droplet dynamics 299

ﬁeld depression around the droplet during impact as a ﬁnite cap, which evolves

under the wave dynamics of Benjamin & Ursell (1954) with a phenomenological

viscous damping factor. As this model includes a range of wavenumbers (rather

than just kF), the experimentally observed spatial damping is automatically captured.

As an alternative to computing the dynamics of many wavenumbers, Moláˇ

cek &

Bush (2013b) (non-rigorously) suggested that the single impact is J0(kFr), spatially

corrected by a radial Gaussian with a linear temporal decay length, and exponentially

decaying in time. On superposition, the exponential spatial decay correction for the

wave ﬁeld of a bouncer is recovered (Damiano et al. 2016).

The approximations in the aforementioned models prevent their usage in two

important situations: complex vertical droplet dynamics (without the restriction to

the (2,1)mode) and the effect of submerged topography. By adopting the theory

of quasi-potential ﬂow (Dias, Dyachenko & Zakharov 2008), coupled with the

logarithmic spring model for the vertical dynamics of the droplet (Moláˇ

cek & Bush

2013a), Milewski et al. (2015) accurately predicted the bifurcations between different

bouncing and walking modes, and the observed exponential spatial damping, and, in

principle, the model may be adapted for any geometry.

The model presented herein considers the accurate wave ﬁeld model of Milewski

et al. (2015) together with the simpliﬁcations of instantaneous and point impacts. In a

sense, this is the opposite limit to the trajectory equation of Oza et al. (2013). In § 2,

we present the wave ﬁeld equations of Milewski et al. (2015) and droplet dynamics of

Moláˇ

cek & Bush (2013b). In § 3, we perform a basis function expansion to collapse

the model to a system of Mathieu differential equations. Assuming instantaneous

impacts, the wave and droplet dynamics are only computed at discrete times, and

the full problem collapses to a discrete map, yielding efﬁcient computation of the

dynamics and deﬁnitive stability results for various states. We capture bouncing

(§ 4) and walking states, and the bifurcation between them (§ 5). We explore the

quantisations of orbiting and promenading pairs, with the ﬁrst investigation of

walking droplet trains (§ 6). Finally, we model circular orbits of a droplet in a

harmonic potential and capture the double quantisation of Perrard et al. (2014b) via

statistical methods in the chaotic regime (§ 7).

2. Model derivation

2.1. Wave dynamics

We employ the governing equations derived by Milewski et al. (2015), who

considered an incompressible viscous ﬂuid in an inﬁnite domain, with a small vortical

boundary layer at the surface. Since a bouncing droplet emits radially symmetric

waves, it is natural to write the spatial system in cylindrical polar coordinates

(r, θ, z). Assuming that the waves and ﬂuid velocity are small with a shallow wave

slope, the velocity potential φ(r, θ , z,t)and wave perturbation η(r, θ, t)at time t>0

satisfy the linearised system

0=∇2

Hφ+φzz,z60,(2.1)

φt=−gΓ(t)η +σ

ρ∇2

Hη+2ν∇2

Hφ−1

ρP(r, θ, t), z=0,(2.2)

ηt=φz+2ν∇2

Hη, z=0,(2.3)

with ∇φ→0and η→0 in the far ﬁeld. In the above, we have constant surface tension

σ, density ρand kinematic viscosity ν, where ∇2

His the horizontal Laplacian.

300 M. Durey and P. A. Milewski

In the vibrating frame, the effective gravity is gΓ(t)≡g(1−Γcos(ω0t)), where

ω0/(2π)is the vibration frequency and Γis the ratio of maximum vibration

acceleration relative to gravity g. As the droplet impacts the surface, the externally

applied pressure on the bath is given by P(r, θ, t), which we prescribe in § 3. Coupled

with droplet dynamics, Milewski et al. (2015) solved this system numerically, but we

make several simplifying assumptions to analyse a wide range of dynamics observed

in an unbounded domain.

2.2. Droplet dynamics

By adapting the work of Moláˇ

cek & Bush (2013b), we model the droplet dynamics as

a rigid sphere of mass munder ballistic motion centred at the point (x,z)=(X(t), Z(t))

in Cartesian coordinates. This neglects droplet deformation, which is a reasonable

assumption for the small droplets considered (Moláˇ

cek & Bush 2013b). During ﬂight,

the droplet experiences an aerodynamical force described by Stokes drag, namely

−νpX0(t)and −νpZ0(t)in the horizontal and vertical directions respectively. Here,

νp=6πR0µair for droplet radius R0and air viscosity µair (Moláˇ

cek & Bush 2013b).

The droplet experiences a force of magnitude f(t)>0 due to the lubrication air

layer during impact ( f=0 during ﬂight). As the droplet radius R0is much smaller

than a typical wavelength λ(R0/λ1), the forces experienced from the wave ﬁeld

interaction are localised to the point x=X(t)on the ﬂuid surface. These are modelled

as an impulsive force f(t)ˆn(X(t), t)and shear forces described by the skidding friction

−c√ρR0/σ f(t)X0(t). Here, ˆn=(−∇η, 1)/(1+|∇η|2)−1/2is the unit normal away from

the ﬂuid surface and cis the dimensionless skidding friction coefﬁcient introduced by

Moláˇ

cek & Bush (2013b) (this is discussed in § 3.3). As the ﬂuid model assumes a

shallow wave slope |∇η| 1, we approximate ˆn∼(−∇η, 1).

For analogies to quantum mechanics, experiments are construed to subject the

horizontal motion of the droplet to an external potential V(X(t), t), such as the

dynamics in a rotating bath (Fort et al. 2010) and a horizontal harmonic potential

well (Perrard et al. 2014b). On combining these forces, conservation of momentum

supplies

mX00(t)+cpρR0/σ f(t)X0(t)+νpX0(t)= −∇V(X(t), t)−f(t)∇η(X(t), t), (2.4)

mZ00(t)+νpZ0(t)=−mgΓ(t)+f(t), (2.5)

written in the vibrating frame. In what follows, we take V≡0 for free walking

dynamics, or V=(1/2)κ|X(t)|2for dynamics in a harmonic potential well with an

adjustable spring constant κ>0. The analysis for other potentials follows akin to this

work.

For simplicity, we assume that the bouncing droplet lies in the prevalent (2,1)mode,

which implies that both Z(t)and f(t)are periodic with subharmonic period T=4π/ω0.

By integrating (2.5) over an impact period and exploiting periodicity, f(t)must satisfy

mgT =Zτ+T

τ

f(t)dt,∀τ>0,(2.6)

as derived by Moláˇ

cek & Bush (2013b). Although we could carefully model the force

f(t)as a response to the impact dynamics (e.g. as a logarithmic spring (Moláˇ

cek &

Bush 2013a)), we use condition (2.6) to prescribe a reasonable choice of f(t)in § 3.2.

Faraday wave–droplet dynamics 301

Variable Value Description

σ2.06 ×10−2kg s−2Surface tension

ρ949 kg m−3Fluid density

ν2×10−5m2s−1Kinematic viscosity (ﬂuid)

µair 1.8×10−5kg m−1s−1Dynamic viscosity (air)

g9.8 m s−2Gravity

ω080 ×2πs−1Vibration frequency (×2π)

R03.8×10−4m Droplet radius

m2.2×10−7kg Droplet mass

νp1.3×10−7kg s−1Stokes drag on droplet

TABLE 1. Fixed variables used in this model.

3. Model reduction

For mathematical analysis, we non-dimensionalise the governing equations (2.1)–

(2.4) and condition (2.6). We rescale the lengths to a typical wavelength λ=0.51 cm

and time to the subharmonic bouncing period T=4π/ω0, with balances f∼mg and

P∼mg/λ2. This yields the dimensionless equations for the waves,

0=∇2

Hφ+φzz,z60,(3.1)

φt=−G˜gΓ(t)η +B∇2

Hη+2∇2

Hφ−MGP(r, θ, t), z=0,(3.2)

ηt=φz+2∇2

Hη, z=0,(3.3)

and droplet dynamics,

X00(t)= −crR

BGf (t)X0(t)− ˜νpX0(t)− ˜κX(t)−Gf (t)∇η(X(t), t), (3.4)

1=Zτ+1

τ

f(t)dt,(3.5)

for any τ > 0. Here, we have deﬁned dimensionless parameters

=νT

λ2,B=σT2

ρλ3,G=gT2

λ,M=m

ρλ3,R=R0

λ,˜νp=νpT

m,˜κ=κT2

m.

(3.6a−g)

Typical parameter values from table 1give a reciprocal Reynolds number of ≈0.019,

a Bond number of B≈0.102, G≈1.201, M≈0.0017, R≈0.075 and ˜νp≈0.01.

The droplet radius corresponds to a vibration number of Ω≡4πpR3/B=0.8, which

minimises the walking threshold ΓWfor this ﬂuid (Wind-Willassen et al. 2013). The

dimensionless potential strength ˜κ∼10−2is a free parameter of both the model and

experiments.

As we prescribe the periodic vertical dynamics, we must prescribe the phase shift

βbetween the vertical motion of the droplet and the waves (this is discussed in §3.3).

This alters the dimensionless effective gravity to ˜gΓ(t)=1−Γcos(4πt+β ).

302 M. Durey and P. A. Milewski

3.1. Basis function expansion

The assumption of inﬁnite depth allows us to adapt the ideas of Benjamin & Ursell

(1954), whereby we decompose the wave perturbation ηand velocity potential φin

terms of orthogonal eigenfunctions of Laplace’s equation. Speciﬁcally, we use Bessel

functions Jm(·)to pose the expansions

η(r, θ , t)=∞

X

m=0Z∞

0

k(am(t;k)Φm(r, θ;k)+bm(t;k)Ψm(r, θ ;k)) dk,(3.7)

φ(r, θ , z,t)=∞

X

m=0Z∞

0

kekz(cm(t;k)Φm(r, θ ;k)+dm(t;k)Ψm(r, θ;k)) dk,(3.8)

where Φm(r, θ;k)≡Jm(kr)cos(mθ ) and Ψm(r, θ ;k)≡Jm(kr)sin(mθ) satisfy

∇2

HΦm(r, θ;k)=−k2Φm(r, θ ;k)and ∇2

HΨm(r, θ;k)=−k2Ψm(r, θ ;k). (3.9a,b)

The coefﬁcients am,bm,cmand dmmay be determined on substitution into (3.1)–(3.3).

For clarity, we set b0≡d0≡0 since Ψ0(r, θ ;k)≡0 for all k>0.

By choice of the orthogonal basis functions in (3.8), the continuity equation (2.1)

is automatically satisﬁed. For horizontal droplet position (r, θ) =(rd(t), θd(t)) at time

t>0 and small droplets, we model the droplet impacts at a point. Thus, we prescribe

the pressure

P(r, θ, t)=f(t)1

rδ(r−rd(t))δ(θ −θd(t)), (3.10)

where f(t)is the force applied by the droplet on the surface, which is zero during

droplet ﬂight. By exploiting the closure relation R∞

0krJm(kr)Jm(ξ r)dr=δ(k−ξ ) and

trigonometric orthogonality, we expand

P(r, θ, t)=∞

X

m=0Z∞

0

k(pm(t;k)Φm(r, θ;k)+qm(t;k)Ψm(r, θ ;k)) dk,(3.11)

where

pm(t;k)=1

Wm

f(t)Φm(rd(t), θd(t);k)and qm(t;k)=1

Wm

f(t)Ψm(rd(t), θd(t);k),

(3.12a,b)

with eigenfunction norms Wm=πif m>0 and W0=2π. It should be noted that

q0≡0.

We substitute (3.7)–(3.11) into (3.2)–(3.3) and eliminate cmand dmin favour of

amand bm. By orthogonality, we obtain a system of inhomogeneous damped Mathieu

equations,

Lkam(t;k)=−kMGpm(t;k)and Lkbm(t;k)= −kMGqm(t;k), (3.13a,b)

where

Lk≡∂tt +2γ (k)∂t+(γ 2(k)+ω2(k)−Γ ω2

g(k)cos(4πt+β)) (3.14)

is the wavenumber-dependent homogeneous damped Mathieu differential operator and

γ (k)≡2k2, ω2(k)≡Gk +Bk3, ω2

g(k)≡Gk.(3.15a−c)

The γ2(k)term gives the additional damping from the vortical boundary layer; this

is not present in the work of Eddi et al. (2011), who instead replace with a

phenomenological damping term to match the Faraday threshold.

Faraday wave–droplet dynamics 303

3.2. Instantaneous impacts

To close the system, it remains to prescribe f(t)so that (3.5) is satisﬁed. To alleviate

the difﬁculty of analysing inhomogeneous Mathieu equations, we simply model the

impacts as instantaneous. This gives homogeneous equations for the waves and droplet

during ﬂight, and jump conditions at impact. This is the opposite limit to that of Oza

et al. (2013), where the droplet glides across the surface of the bath with constant

forcing f(t). Assuming that the ﬁrst impact is at t=0, we deﬁne

f(t)=∞

X

n=0

δ(t−tn), (3.16)

where tn≡nfor all non-negative integers nand δ(·)is the Dirac delta function. This

satisﬁes periodicity and the integral condition (3.5). We now exploit properties of δ(·)

to replace the inhomogeneities in (3.13) with jump conditions.

For physical consistency, we assume that the droplet position X(t)and wave

amplitudes (am(t;k)and bm(t;k)) are continuous across all impacts. We denote jumps

[Q(tn)]+

−≡Q(t+

n)−Q(t−

n)for any function of time Q(t), where t+

nand t−

nare the right

and left limits of tnrespectively. We now integrate the governing equations (3.13)

over t∈[t−

n,t+

n]. For all n>0, the ﬁrst equation gives

Zt+

n

t−

n

Lkam(t;k)dt=−k

Wm

MG Zt+

n

t−

n

δ(t−tn)Φm(rd(t), θd(t);k)dt.(3.17)

As the droplet position is assumed to be continuous, the sifting property of δ(·)can

be applied to the right-hand side. We use the continuity of am(t;k)to simplify the

left-hand side. Hence, (3.13) supply jump conditions

[a0

m(tn;k)]+

−=−Pm(k)Φm(rd(tn), θd(tn);k), (3.18)

[b0

m(tn;k)]+

−=−Pm(k)Ψm(rd(tn), θd(tn);k), (3.19)

where a prime denotes the partial derivative with respect to tand Pm(k)=kMG/Wm.

For the droplet dynamics (3.4), we ﬁrst consider a single impact at t=t?. Hence,

X00(t)+cpR/BGδ(t−t?)X0(t)+ ˜νpX0(t)+ ˜κX(t)=−Gδ(t−t?)∇η(X(t), t). (3.20)

For c=0, we may proceed as above to ﬁnd the jump in X0(t)at t=t?. The case

c>0 is more delicate; the sifting property cannot be applied as X0(t)is discontinuous

at t=t?. Following the method of Catllá et al. (2008), we replace δ(t−t?)with

δε(t−t?)≡ε−1ϕ((t−t?)ε−1), ∀ε > 0,(3.21)

where ϕ(τ ) >0 for all τ∈Rand R∞

−∞ ϕ(τ ) dτ=1. Hence, the functions δε(·)→δ(·)

pointwise as ε→0, except at the jump discontinuity in δ(·). This is an appealing

formulation, as, physically, no impact is actually instantaneous – it just occurs over a

much faster time scale than the dynamics of the rest of the system.

The idea is to ﬁnd a solution to (3.20) when t>t?with δ(·)replaced by δε(·),

and then consider the limit as ε→0. This determines X0(t+

?). However, for t<t?,

(3.20) can be solved directly without approximation (as the δ(·)terms vanish), which

supplies X0(t−

?). Hence, the jump [X0(t?)]+

−is obtained, which is independent of ϕ(·).

By generalising the proof of Catllá et al. (2008) (as outlined in appendix A) and

extending to periodic impacts, we obtain the jump condition shown in (3.27) below.

304 M. Durey and P. A. Milewski

3.3. Model summary

We have the dimensionless system

0=Lkam(t;k), ∀t6= tn,(3.22)

0=Lkbm(t;k), ∀t6= tn,(3.23)

0=X00(t)+ ˜νpX0(t)+ ˜κX(t), ∀t6=tn,(3.24)

[a0

m(tn;k)]+

−= −Pm(k)Φm(X(tn);k), (3.25)

[b0

m(tn;k)]+

−= −Pm(k)Ψm(X(tn);k), (3.26)

[X0(tn)]+

−= −F(c) 1

crB

R

∇η(X(tn), tn)+X0(t−

n)!,(3.27)

where Pm(k)=kMG/Wmand F(c)=1−exp(−cG√R/B). One should note the abuse

of notation Φm(X(tn);k)≡Φm(rd(tn), θd(tn);k)(and similarly for Ψm). The system of

jump conditions is self-consistent, with both ηand Xcontinuous across impacts.

This model has two undeﬁned parameters: the skidding friction cand the phase

shift β. From the theoretical calculations of Moláˇ

cek & Bush (2013b), c≈0.3, but the

authors consider c∈[0.17,0.33]for simulations. Oza et al. (2013) use the lower bound

c=0.17 in the stroboscopic approximation model, which has the opposite impact

duration limit to our model. Hence, it is natural to choose the upper bound c=0.33,

which is ﬁxed throughout the paper. The phase shift βbetween bath vibrations and

droplet impacts for periodic states arises naturally in models where the droplet vertical

dynamics are explicitly modelled (Moláˇ

cek & Bush 2013b; Milewski et al. 2015).

As we restrict the droplet to periodic impacts, we must choose β. We focus on the

prevalent (2,1)2walking mode, where βhas a typical value of β=π(Milewski

et al. 2015), which is ﬁxed henceforth. For later works, cand βmay be tuned when

comparing with experimental data.

3.4. Faraday instability

Following Milewski et al. (2015) to determine the subharmonic Faraday instability,

we look for subharmonic solutions to Lka(t;k)=0 of the form a(t;k)=Acos(2πt)+

Bsin(2πt). After substitution, higher-order frequencies are neglected, which is

equivalent to truncating the Hill matrix. For a non-trivial system, we require

(w2(k)+γ2(k)−4π2)2+16π2γ2(k)−1

4w4

g(k)Γ 2=0.(3.28)

The function Γ=Γ (k)is globally minimised at k=kF, where for Γ < Γ (kF)≡ΓFthe

subharmonic solutions are stable for any k>0. By our dimensionless scaling, λF≈1,

where kF=2π/λF. Although it is known that ΓFobtained in this model is not accurate

when compared with experiments (Milewski et al. 2015), it is usual to use Γ /ΓFas

a controlling parameter (Eddi et al. 2011). In fact, we show in § 5.3 that the wave

ﬁeld temporal decay rate may be written as a function of Γ /ΓF, which justiﬁes our

approach for a theoretical investigation.

3.5. Discrete-time model

As (3.22)–(3.27) form a homogeneous system with jump conditions, it is natural to

reformulate as a discrete-time system. We denote an(k)≡(a0(tn;k), a1(tn;k), . . .)Tand

a0

n(k)≡(a0

0(t+

n;k), a0

1(t+

n;k), . . .)T, and similarly for bn(k)and b0

n(k). We also write the

Faraday wave–droplet dynamics 305

eigenfunctions as vectors Φ(·;k)=(Φ0(·; k), Φ1(·; k), . . .)T, and similarly for Ψ(·;k),

with P(k)a diagonal matrix with elements Pm(k). By periodicity of the Mathieu

operator Lk, we numerically construct the principal fundamental matrix Mk(Γ ) over

the interval (0,1). As Lkis independent of the Bessel order, Mkhas block diagonal

form

Mk(Γ ) =m11(Γ ;k)Im12 (Γ ;k)I

m21(Γ ;k)Im22 (Γ ;k)I,(3.29)

where Iis the inﬁnite identity matrix. The principal fundamental matrix F(˜κ)

corresponding to the droplet dynamics is constructed by analytically solving (3.24).

We reformulate (3.22)–(3.27) as an efﬁcient one-step map with stages for droplet

position Xn≡X(tn)and velocity X0

n≡X0(t+

n).

(i) Use Xnand X0

nto compute Xn+1and ˜

X0≡X0(t−

n+1):

Xn+1

˜

X0=F(˜κ) Xn

X0

n.(3.30)

(ii) Update wave amplitudes including jump conditions ∀k>0 (similarly for bn(k)):

an+1(k)

a0

n+1(k)=Mk(Γ ) an(k)

a0

n(k)−0

P(k)Φ(Xn+1;k).(3.31)

(iii) Apply droplet jump conditions:

X0

n+1=(1−F(c)) ˜

X0−F(c)

crB

R

∇η(Xn+1,tn+1). (3.32)

3.6. Numerical implementation

The remainder of this work involves simulating and ﬁnding ﬁxed points of the system

(3.30)–(3.32), both of which require suitable truncations in the wavenumber k>0 and

Bessel mode m∈N. For any impact time tn, the wavenumbers are generally peaked

around k=kF, with the peak becoming narrower as Γ→Γ−

F, which can be analysed

from Floquet analysis of the damped Mathieu equation. This determines the reﬁnement

and truncation in k, which is successively reduced until the change in the numerical

solution of the ﬁxed points becomes negligible. Away from the Faraday threshold, a

reasonably coarse mesh is sufﬁcient, with δk∼0.2 and k∈[kF/2,3kF/2], where kFis

a mesh point. Integrals over k(e.g. for ﬁnding ∇η) are evaluated using the trapezium

rule, which is well suited to capturing the peaked behaviour in k.

Truncation of the Bessel modes follows from the asymptotic behaviour of Bessel

functions, namely Jm(z)∼(1/m!)(z/2)mfor 0 <z√m+1. Hence, for orbital

solutions or simulations with a central force, the maximum radial extent of the droplet

can be estimated, which provides a good guide for the truncation m∈{0,1, . . . , m?}.

For walking dynamics, the Floquet exponents provide an estimate of how the temporal

decay affects the number of past impacts that contribute to the current wave ﬁeld

(this is the system ‘memory’, as discussed in § 5.3), where we typically have m?=15

for Γ /ΓF≈0.81 but m?≈50 for Γ /ΓF≈0.95. The accuracy of this truncation can

be easily checked a posteriori, with m?increased until there is a negligible change

in the numerical solution.

For simulation efﬁciency, Bessel functions are only evaluated once per impact

period, with derivatives calculated using the identity J0

m(z)=(1/2)(Jm−1(z)−Jm+1(z)).

We typically simulate 1000 impacts per second on a standard desktop machine using

MATLAB.

306 M. Durey and P. A. Milewski

4. Bouncing states

We now ﬁnd bouncing states of (3.22)–(3.27) with ˜κ=0. By translational invariance,

we assume that the drop bounces at the origin (X≡0) with radially symmetric wave

ﬁeld

η(r,t)=Z∞

0

ka0(t;k)J0(kr)dk.(4.1)

This ensures that the droplet receives no horizontal kick at impact, and so remains a

bouncer. We demand a periodic wave ﬁeld, with η(x,tn+1)=η(x,tn)and ηt(x,t+

n+1)=

ηt(x,t+

n)for all nand all x∈R2. By orthogonality, the wave amplitudes must satisfy

a0(tn+1;k)=a0(tn;k), (4.2)

a0

0(t+

n+1;k)=a0

0(t+

n;k), (4.3)

for all tnand ∀k>0. By periodicity, it is sufﬁcient to consider the interval t∈ [0,1].

By considering (3.31) and the form of Mk(Γ ), it remains to solve the linear system

I2−m11(Γ ;k)m12 (Γ ;k)

m21(Γ ;k)m22 (Γ ;k)a0(0;k)

a0

0(0+;k)=0

−P0(k),(4.4)

where I2∈R2×2is the identity matrix and we used Φ0(0;k)=1, ∀k.

4.1. Stability analysis

To determine the walking threshold Γ=ΓW, we perform linear stability analysis of

the periodic bouncing system. The aim is to construct a one-step matrix map T(Γ )

for the perturbed system, where stability is determined by the spectral radius ρ(T).

We denote the steady state by X=ˆ

X,am= ˆamand bm=ˆ

bm, where, for bouncing at

the origin, ˆ

X≡0,ˆam≡ˆ

bm≡0 for all m>1, and ˆ

b0≡0. We then consider small

perturbations

X(t)=ˆ

X(t)+(t), am(t;k)= ˆam(t;k)+ ˜am(t;k), bm(t;k)=ˆ

bm(t;k)+˜

bm(t;k),

(4.5a−c)

where we assume that |˜am/ˆa0|∼|˜

bm/ˆa0| ∼ |||| 1. By including an explicit

perturbation to the wave ﬁeld, we consider a more general perturbation than Oza

et al. (2013).

The system is linearised via the jump conditions (3.25)–(3.27), giving

[˜a0

m(tn;k)]+

−= −Pm(k)∇Φm(0;k)T(tn), (4.6)

[˜

b0

m(tn;k)]+

−= −Pm(k)∇Ψm(0;k)T(tn), (4.7)

[0(tn)]+

−= −F(c) 0(t−

n)+1

crB

R(H(Γ )(tn)+∇˜η(0,tn))!.(4.8)

The Hessian matrix of the steady-state wave ﬁeld at droplet impact is H(Γ ) and

∇˜η(0,tn)=∞

X

m=0Z∞

0

k(˜am(tn;k)∇Φm(0;k)+˜

bm(tn;k)∇Ψm(0;k)) dk.(4.9)

Faraday wave–droplet dynamics 307

Equations (4.6)–(4.8) are simpliﬁed considerably by noting that

∇Φ1(0;k)=(k/2,0)T,∇Ψ1(0;k)=(0,k/2)T,∇Φm(0;k)≡∇Ψm(0;k)≡0,

(4.10a−c)

for m6=1. Hence, for m6= 1, the jump conditions for ˜am,˜

bmand are all independent

of each other (to O(||||)). Therefore, the perturbations ˜am,˜

bm(m6=1) each decouple

from the system. As they are unexcited solutions to the damped Mathieu equation with

Γ < ΓF, they are stable and hence are neglected from the stability analysis.

Using the linearised jump conditions with matrices Mk(Γ ) and F(0), we construct

a discrete-time linear map for all variables, given by the matrix T(Γ ). The system is

neutrally stable if the spectral radius ρ(T)=1, and unstable if ρ (T) > 1. The walking

threshold ΓWis the largest Γsuch that ρ(T)=1. By the translational invariance of the

system, there always exists an eigenvalue µof Tsuch that µ=1, which prevents us

from obtaining asymptotically stable solutions (ρ(T) < 1)in the absence of a central

force.

5. Steady walking states

By extension, we now ﬁnd steady walking states and analyse their stability. By

rotational and translational invariance, we consider steady walking along the x-axis

in the direction of increasing x, with X(t0)=0(t0=0). By symmetry, bm≡0, ∀m.

We denote 0 < δx≡X(tn+1)−X(tn)for all tn, where X(t)=(X(t), 0). For the wave

ﬁeld to follow the droplet between impacts, Graf’s addition theorem (Abramowitz &

Stegun 1964) supplies the requirement

a0(tn+1;k)=∞

X

p=0

(−1)pJp(kδx)ap(tn;k), (5.1)

∀m>1:am(tn+1;k)=∞

X

p=0

(Jm−p(kδx)+(−1)pJm+p(kδx))ap(tn;k), (5.2)

and similarly for the ﬁrst time derivatives. Hence, we have a discrete-time map

an+1(k)=A(k;δx)an(k), (5.3)

a0

n+1(k)=A(k;δx)a0

n(k), (5.4)

where A(k;δx)is an inﬁnite matrix given by (5.1)–(5.2). As A(k;0)=I, (5.3)–(5.4)

simplify to the periodicity requirement (4.2)–(4.3) for a bouncing droplet.

For steady walking, we require X0(t+

n)=V0≡(V,0)for all tn, for some unknown

V>0. Solving (3.24) with ˜κ=0 for t∈[ tn,tn+1)gives

X(tn+1)=X(tn)+ ˜ν−1

p(1−e−˜νp)V0,(5.5)

X0(t−

n+1)=e−˜νpV0.(5.6)

For X0(t+

n+1)=V0, the jump condition (3.27) and (5.6) give the requirement

V0=(1−F(c))e−˜νpV0−F(c)

crB

R

∇η(X(tn+1), tn+1). (5.7)

308 M. Durey and P. A. Milewski

By the assumed periodicity of the wave ﬁeld when centred at the droplet, ∇η(X(tn), tn)

is constant for all tn. Hence, using (4.10) to simplify the wave ﬁeld gradient, we

require

V0=−1

1−e−˜νp(1−F(c))

F(c)

crB

RZ∞

0

ka1(0;k)∇Φ1(0;k)dk,(5.8)

where ∇Φ1(0;k)=(k/2,0)T. From the x-components of (5.5) and (5.8), δxmust

satisfy

δx=−˜ν−1

p(1−e−˜νp)

1−e−˜νp(1−F(c))

F(c)

crB

RZ∞

0

k2

2a1(0;k)dk.(5.9)

To progress, we use the discrete map for the wave amplitudes (3.31). By periodicity

and translational invariance, it is sufﬁcient to consider t∈ [0,1]. Conditions (5.3)–(5.4)

are thus equivalent to solving the linear system

A(k;δx)0

0 A(k;δx)−Mk(Γ )a0(k)

a0

0(k)=− 0

P(k)Φ([δx,0]T;k),(5.10)

for all k>0, where δxis determined by (5.9). The block matrix form of this system

allows for quick numerical solution, with the walking speed δxshown in ﬁgure 1(a).

To shed light on this bifurcation, we consider the average energy of the wave ﬁeld

across one period. As derived in appendix B, we compute the change in energy due

to the existence of the waves formed by the droplet. This cannot be computed in the

models of Moláˇ

cek & Bush (2013b) and Oza et al. (2013), as the single-wavenumber

(kF) approximation gives insufﬁcient decay at inﬁnity. The dimensionless energy

perturbation Eas given in (B 7) is shown in ﬁgure 1(b), demonstrating that the

wave ﬁeld of a walker requires less energy than that of the corresponding unstable

bouncer. We neglect the horizontal kinetic energy of the droplet as it is signiﬁcantly

smaller than the wave ﬁeld energy for all walking speeds. Furthermore, the assumed

periodicity of Z(t)gives a constant droplet vertical energy (which balances the wave

ﬁeld energy) for both bouncing and walking.

5.1. Stability analysis

The stability analysis of a walker is similar to that of a bouncer and therefore we

do not give the details. The main difference is that we use Graf’s addition theorem

(Abramowitz & Stegun 1964) to map the wave amplitude perturbation variables so

that they are centred on the steady-state droplet position at each impact.

Following Oza et al. (2013), we consider general perturbations, and perturbations

to the droplet in the direction of motion. In the former case, the walking is unstable.

Physically, this corresponds to a new walker forming after an initial transient, but in

a new direction. The latter case is achieved by noting that Ψm≡0 along the x-axis,

so the linearised jumps for the perturbation coefﬁcients ˜

bmwill be zero for an in-

line perturbation. Furthermore, ∂xΨm≡0 (for all m) along the x-axis, so the ˜

bmterms

do not contribute to the linearised droplet perturbation jump condition. Hence, the ˜

bm

terms decouple from the perturbed system and can be neglected as they are stable

solutions to the Mathieu equation. In this case, the walker is neutrally stable, where

an eigenvalue of 1 always exists due to the translational invariance of the problem.

This agrees with experiments and also the model of Oza et al. (2013), whose analysis

is restricted to lateral and in-line perturbations.

Faraday wave–droplet dynamics 309

0

0.02

0.04

0.06

0.08

0.10

0.80 0.85 0.90 0.95 1.00 0.78 0.79 0.80 0.81

1.0

1.1

1.2

1.3

(a)(b)

E

FIGURE 1. (a) Average walking speed δxfor complete numerical method (black) and

approximation valid only for 0 < δx1 (grey). By the non-dimensionalisation, δxis the

average walking speed relative to the phase speed of the waves. (b) The average wave

ﬁeld energy for a bouncer (black) and walker (grey), with a bifurcation at Γ=ΓW.

5.2. Slow-walking analysis

As observed by Protière et al. (2006), there is a supercritical bifurcation at the

walking threshold, where the walking speed grows like (Γ −ΓW)1/2. This behaviour

has been captured in the trajectory equation of Oza et al. (2013). We explore the

slow-walking speed in terms of the distance between impacts δx, which is the average

walking speed in dimensionless variables. For 0 < δx1, we consider asymptotic

expansions of the form

Γ=ΓW+δxΓ1+δx2Γ2+· ·· ,(5.11)

am(t;k)=a(0)

m(t;k)+δxa(1)

m(t;k)+δx2a(2)

m(t;k)+· ·· ,(5.12)

for all m>0. By the Frobenius series for Bessel functions, we have

J0(kδx)∼1−k

22

δx2+O(δx4), (5.13a)

J1(kδx)∼k

2δx−1

2k

23

δx3+O(δx5), (5.13b)

J2(kδx)∼1

2k

22

δx2+O(δx)4,J3(kδx)∼1

6k

23

δx3+O(δx5). (5.14a,b)

These expansions are valid for k/2=O(1). As the least-damped wavenumber kF

satisﬁes kF/2≈π=O(1), and wavenumbers far away from kFcorrespond to negligible

wave amplitudes, this assumption is valid. We prove that Γ1=0 and verify numerically

that Γ2>0, which gives the desired bifurcation.

Substituting the asymptotic expansions into the map (5.1)–(5.2), the Mathieu

equation (3.22) and jump condition (3.25), and equating powers of δx, a system of

310 M. Durey and P. A. Milewski

differential equations with initial and jump conditions is obtained, all of the general

form

LΓW

ka(t;k)=H,a(1;k)=a(0;k)+B,a0(1+;k)=a0(0+;k)+B0,

[a0(1;k)]+

−=J.)(5.15)

Here, H,B,B0and Jare inhomogeneities, and LΓW

kis the damped Mathieu

differential operator with Γ=ΓW. By (5.3)–(5.4), the map for the wave amplitude

derivatives is the same as that for the wave amplitudes themselves. Hence, all terms

in Bof the form a(0;k)are replaced by a0(0+;k)in B0, so for notational efﬁciency,

we do not state B0explicitly.

First, we note that for δx=0, we have a bouncer at the walking threshold, so the

wave ﬁeld is radially symmetric, giving a(0)

m≡0 for m>1. This is veriﬁed by noting

that H(0)

m=B(0)

m=J(0)

m=0 for all m>1, as the only order-1 term in the Bessel

expansions is in J0. This yields a periodic unexcited solution to the damped Mathieu

equation, which must be the zero solution.

To ﬁnd ΓW, we ﬁrst note that a(0)

0and a(1)

1are governed by inhomogeneities

H(0)

0=0,B(0)

0=0,J(0)

0=−P0(k), (5.16a−c)

H(1)

1=0,B(1)

1=ka(0)

0(0;k), J(1)

1=−P1(k) (k/2). (5.17a−c)

Equating δxfrom the walking condition (5.9), we require ΓWsuch that

1=−˜ν−1

p(1−e−˜νp)

1−e−˜νp(1−F(c))

F(c)

crB

RZ∞

0

k2

2a(1)

1(0;k)dk.(5.18)

Therefore, (5.16)–(5.18) is a closed system for a(0)

0,a(1)

1and ΓW. This nonlinear

analysis gives a consistent walking threshold ΓWto the linear stability analysis in

§§ 4.1 and 5.1.

Given the solution to the above system, we may obtain equations for Γ1, which is

coupled with equations for a(1)

0and a(2)

1. By vanishing inhomogeneities, a(1)

2≡0, so

H(1)

0=Γ1ω2

g(k)cos(4πt+β)a(0)

0(t;k), B(1)

0=0,J(1)

0=0,(5.19a−c)

H(2)

1=Γ1ω2

g(k)cos(4πt+β)a(1)

1(t;k), B(2)

1=ka(1)

0(0;k), J(2)

1=0.(5.20a−c)

The δx2term from (5.9) completes the system with

0=Z∞

0

k2

2a(2)

1(0;k)dk.(5.21)

We now exploit the directional symmetry of the steady walker to show that Γ1=0.

The map (5.1)–(5.2) also holds for δx<0, which corresponds to a droplet walking

in the negative x-direction. Since a0is the coefﬁcient of the radially symmetric J0(kr)

basis function, a0must be even in δxfor the wave ﬁeld to be the same for both

walking directions. Hence, we require a(1)

0≡0 on (0,1). By (5.19), this is achieved if

and only if Γ1=0. With these vanishing terms, equation (5.20) becomes

H(2)

1=0,B(2)

1=0,J(2)

1=0,(5.22a−c)

Faraday wave–droplet dynamics 311

yielding a(2)

1≡0 on (0,1), which satisﬁes (5.21). Neglecting terms of o(δx2), we have

δx=Γ−1/2

2pΓ−ΓW,(5.23)

which is valid for 0 < δ x1. Equations for determining Γ2>0 are given in appendix

C. The solution is shown in ﬁgure 1(a) (grey curve).

5.3. Wave ﬁeld of a single impact

Previous models for the wave ﬁeld have superposed given radially symmetric sources

centred at each impact with exponential temporal decay (Fort et al. 2010; Eddi et al.

2011; Moláˇ

cek & Bush 2013b; Oza et al. 2013). In our model, the jump conditions

(3.25)–(3.26) at time tnare in fact equivalent to [η(x,tn)]+

−= ¯η(|x−X(tn)|,0), where

¯η(r,t)=Z∞

0

k¯a(t;k)J0(kr)dk.(5.24)

For all k>0, ¯a(t;k)is governed by Lk¯a=0 for t>0, with initial conditions

¯a(0;k)=0,¯a0(0;k)= −P0(k), (5.25a,b)

with P0(k)=kMG/(2π). This is quickly veriﬁed using Graf’s addition theorem

(Abramowitz & Stegun 1964). As ¯η(r,0)=0 for all r, the wave ﬁeld is continuous

across the droplet impact, which is not true in the aforementioned models. Although

¯ηt(r,0)= ∞ for all r>0 (from the δ(·)-function forcing), the rapid temporal decay

of ¯a(t;k)for large kgives a ﬁnite solution for any t>0. Furthermore, ¯ηdepends

only on the current position of the droplet and is independent of its velocity and path

memory.

The development of this wave ﬁeld over time is shown in ﬁgure 2(a). A decaying

capillary wave propagates away from the droplet, which excites a ﬁeld of standing

Faraday waves. The wave ﬁeld generated is similar to that of Milewski et al. (2015)

near the origin, but differs slightly at the wave front over time. This dissimilarity

may be explained partially by the assumption of instantaneous point impacts and the

truncation of the wavenumbers. As the wave speed is much larger than the walking

speed, this difference has a negligible effect on the resulting dynamics.

By superposition of such wave ﬁelds, we may now express the wave ﬁeld ηby

η(x,t)=btc

X

n=0¯η(|x−X(tn)|,t−tn)H(t−tn), (5.26)

where H(·)is the Heaviside function. Following Moláˇ

cek & Bush (2013b), we deﬁne

the dimensionless memory Me(Γ ), which is the number of impacts until ¯ηbecomes

exponentially small. Using the calculations of Milewski et al. (2015), for 0 < ΓF−

Γ1,

Me(Γ ) =Td(Γ )

1−Γ /ΓF

,where Td(Γ ) =16(ω2(kc(Γ )) +γ2(kc(Γ )) +4π2)

G2Γ2

F

,(5.27)

is the dimensionless decay rate of the waves (ΓF=5.13 for this ﬂuid model). The

critical wavenumber kc(Γ) is the least-damped wavenumber given Γ. This dependence

is weak, with a 1 % variation in kc(Γ ) away from kc(ΓF)=kFfor a range of Γ,

312 M. Durey and P. A. Milewski

t

–2–4–6 0 2 4 6 8 10–8–10 1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.84 0.88 0.92 0.96 1.00

1.5

2.5

3.5

4.5

25 50 75 1000

(a)(b)(c)

FIGURE 2. (a) The wave ﬁeld of a single impact at times 1, . . . , 8, which are normalised

to have the same amplitude at the origin. A damped travelling capillary wave propagates

from the impact, exciting a standing ﬁeld of Faraday waves in its wake (Γ /ΓF=0.994).

(b) Spatial decay length ldof a walker as a function of Γ /ΓFand memory Me(inset (c)).

resulting in a small variation to Td(Γ ) (typically Td(Γ ) ≈0.6). This form is slightly

more sophisticated than that of Moláˇ

cek & Bush (2013b), where Tdis assumed to be

constant. From numerical veriﬁcation, this is also a reasonable approximation for the

range of Γconsidered.

From ﬁgure 2(a), the wave ﬁeld approaches a Bessel-like function as time increases.

As kFis the least-damped wavenumber, ¯a(t;k)is peaked about k=kFfor large t.

Hence, the integral for ¯η(r,t)may be approximated by ¯η(r,t)=a0(t;kF)J0(kFr)for

large t. The function a0(t;kF)has a Floquet exponent with real part −1/Me, where

the underlying periodic behaviour is well approximated by a sinusoid. This long-time

approximation is used by Moláˇ

cek & Bush (2013b) and Oza et al. (2013), and

prevents the occurrence of a travelling capillary wave formed at each impact and a

Doppler shift. As our model generates a similar wave ﬁeld to that of Milewski et al.

(2015), we refer the reader to their paper for a comprehensive comparison between

wave ﬁeld models.

The wavenumbers near to kFalso prove to be signiﬁcant in leading to the

experimentally observed spatial damping (Eddi et al. 2011), and are computed

numerically Milewski et al. (2015). We repeat their statistical calculation for our

model, with the resulting exponential decay length ldshown in ﬁgure 2(b). Speciﬁcally,

we compute the wave ﬁeld ηfor a walker, and ﬁnd the extrema to |η|in the direction

perpendicular to travel. As the far-ﬁeld spatial decay of a Bessel function is ∼1/√r,

we ﬁt a nonlinear model of the form θ1exp(−r/θ2)/√rfor parameters (θ1, θ2), where

ld=θ2, which is an approximate envelope for the wave ﬁeld. Typically, we consider

r∈(0,10], which gives enough points for a reasonable statistical ﬁt. As this approach

contains some numerical error, the resulting solution is not smooth.

The spatial decay length ldincreases with Γ, and grows linearly with memory as

Me→ ∞. However, due to the ﬁxed impact phase, we do not obtain jumps in the

decay length as the impacts switch between (m,n)modes (Milewski et al. 2015). As

Eddi et al. (2011) and Borghesi et al. (2014) used ﬁxed dimensionless values of ld=

1.6 and 2.5 respectively, they did not capture this intrinsic change in the wave ﬁeld

as Γvaries.

Faraday wave–droplet dynamics 313

0

2

4

6

8

–4

–2

–2–4–6 02468–8 0.90

0.95

1.00

1.05

1.10

0 0.02 0.04 0.06

(a)(b)

FIGURE 3. Evidence of a Doppler effect. (a) Wave in direction of travel (black) and

transverse wave (grey) recentred to have the same peaks. The arrow determines the

direction of travel, where Γ /ΓF=0.9. Waves are elongated behind the droplet and

compressed ahead. (b) Measured wavelengths behind (+)and ahead (×)of the droplet.

The grey lines are the theoretical predictions made by Eddi et al. (2011).

5.4. Doppler effect

As shown by Eddi et al. (2011), a Doppler effect is observed in the wave ﬁeld

of the walker; the waves ahead of the droplet are compressed, yet are elongated

behind. This phenomenon was observed in the simulations of Milewski et al. (2015).

Moreover, the authors proved that this cannot occur for wave amplitudes of the form

a(t;k)=α(t)δ(k−kF), which was used in Oza et al. (2013). However, as we maintain

a signiﬁcant range of kin our numerical solution, we observe a Doppler effect far

from the droplet (ﬁgure 3). Our model exhibits a smooth yet nonlinear change in

the wavelength due to the Doppler effect, which differs from the linear prediction of

Eddi et al. (2011).

5.5. Summary of steady walking dynamics

To conclude this section, a bouncing droplet destabilises via a pitchfork bifurcation to

a steady walker, whose lower-energy wave ﬁeld automatically captures the exponential

spatial damping correction observed in experiments (Eddi et al. 2011). The jump

conditions are equivalent to adding a radially symmetric propagating wave ﬁeld ¯η

at each impact, whose temporal equidistant superposition yields a Doppler shift,

with a weakly nonlinear dependence on δx. Furthermore, the temporal decay of

the system may be described by its memory Me(Γ ), which allows for comparison

between previous models (Fort et al. 2010; Moláˇ

cek & Bush 2013b). The wave ﬁelds

corresponding to the dynamics of a walker are easily computed and are shown in

ﬁgures 4(a) and 4(b) for two different walking speeds. For large Γ, an interference

pattern forms behind the droplet (Eddi et al. 2011).

6. Multiple-droplet interactions

In this section, we analyse three experimentally observed conﬁgurations for multiple

droplets: orbiting pairs, side-by-side walkers (promenades) and trains of walkers. For

Ndroplets X(1)(t), . . . , X(N)(t), each droplet is governed by (3.24) during ﬂight

314 M. Durey and P. A. Milewski

13–1

–1

0

1

2

3

–2

–3 0 2–2

–1

0

1

2

3

–2

–3 02–2

–1

0

1

2

3

–2

–3 02–2

–1

0

1

2

3

–2

–3 02–2

–1

0

1

2

3

–2

–3 02–2

–1

0

1

2

3

–2

–3 02–2

6

0

2

4

–2

6

0

2

4

–2

–4

0

2

4

–2

6

8

–4

–6

0

2

4

–2

6

8

–4

–6

0

2

4

–2

6

–4

–6

–8

0

2

4

–2

6

8

–4

(a)(b)

(c)(d)

(e)(f)

FIGURE 4. Wave ﬁelds corresponding to different steady-state dynamics at droplet impact.

The droplet is denoted by a blue circle for in-phase impacts and a red circle when in

ﬂight for antiphase dynamics. Walking droplets for (a)δx=0.065 with Γ /ΓF=0.9 and

(b)δx=0.08 with Γ /ΓF=0.96. Two anticlockwise orbiting droplets with Γ /ΓF=0.91

for (c) in-phase and (d) antiphase impacts. A single droplet orbiting anticlockwise under

a central force for Γ /ΓF=0.975 with orbit radius (e)Rd=0.45 and ( f)Rd=0.95.

(with ˜κ=0) and the jump condition (3.27) at impact. The wave amplitudes are

still modelled by the damped Mathieu equations (3.22)–(3.23) during ﬂight, but for

Faraday wave–droplet dynamics 315

Stability type Line Description

Stable Solid black U=∅

Unstable (real) Dashed black U∩R6= ∅

Unstable (c.c.) Solid grey ∀µ∈U, µ ∈C\Rand ¯µ∈U

TABLE 2. Stability types of the stability transition matrix Twith complex eigenvalue

spectrum σ (T). We deﬁne the set of unstable eigenvalues U= {µ:µ∈σ(T), |µ|>1},

where Uis empty only when the system is stable.

in-phase interactions, the jump condition (3.25) becomes

[a0

m(tn;k)]+

−=−Pm(k)

N

X

p=1

Φm(X(p)(tn);k), (6.1)

and similarly for b0

m. For antiphase impacts, additional jumps occur at t=tn+1/2≡tn+

1/2. In the bifurcation diagrams that follow in §§ 6and 7, we identify three stability

types, as described in table 2. A pair of unstable complex conjugate (c.c.) eigenvalues

often allows the system to destabilise slowly to a new oscillatory stable regime.

6.1. In-phase orbiting

We consider two orbiting droplets about the origin; both droplets impact simultaneously

on a radius Rd, with the droplets and wave ﬁeld rotating by an angle δθ > 0 between

impacts. Moreover, we pose that the angular difference between X(1)(tn)and X(2)(tn)

is πfor all tn. This allows us to model X(1)(t)explicitly, with the contribution of

X(2)(tn)to the wave amplitude jump conditions treated implicitly. For notational

convenience, we write X(t)≡X(1)(t), where X(tn)has radial component rd(tn)=Rd

and angular component θd(tn)=θn≡nδθ (by rotational invariance). The droplet

motion is piecewise linear, so rd(t)6=Rdfor t6= tn.

As the implicit second droplet has angular component (θn+π)at time tn, (6.1)

gives

[a0

m(tn;k)]+

−= −Pm(k)Φm(X(tn);k)(1+(−1)m), (6.2)

[b0

m(tn;k)]+

−= −Pm(k)Ψm(X(tn);k)(1+(−1)m). (6.3)

For modd, amand bmare never excited during the orbit, giving am≡bm≡0. This

ensures that the wave ﬁeld with respect to each droplet is the same.

Denoting cm(t;k)=(am(t;k), bm(t;k))T, the wave ﬁeld rotates with the droplets if

cm(tn+1;k)=D(mδθ )cm(tn;k), c0

m(t+

n+1;k)=D(mδθ )c0

m(t+

n;k), (6.4a,b)

where D(ϕ) ∈R2×2is the rotation matrix for angle ϕ. The droplet rotation requires

X(tn+1)=D(δθ )X(tn), X0(t+

n+1)=D(δθ )X0(t+

n). (6.5a,b)

As with steady walking, we solve the droplet dynamics (3.24)–(3.27) with ˜κ=0 for

t∈[tn,tn+1)subject to (6.5), and eliminate X0(t+

n)in favour of X(tn). This gives

(D(δθ ) −(1−F(c))e−˜νpI2)(D(δθ ) −I2)X(tn)=−F(c)

c

1−e−˜νp

˜νprB

R

D(δθ )(∇η)n,(6.6)

where (∇η)n≡∇η(X(tn), tn), and (∇η)n+1=D(δθ )(∇η)ndue to the rotation.

316 M. Durey and P. A. Milewski

0.85 0.90 0.95 1.00

0.5

0

1.0

1.5

2.0

2.5

3.0

3.5

1–1 0

0

1

D

(a)(b)

FIGURE 5. (a) Bifurcation diagram for the diameters Dof in-phase In(light grey

background) and antiphase An(white background) orbiting pairs. The dark grey regions

give the bounds for two droplets in a stable wobbly orbit. This has an upper bound given

by the unstable upper branch. The curves correspond to stability types in table 2. (b)

Example wobbly in-phase anticlockwise orbiters with Γ /ΓF=0.83, which corresponds to

the black cross in (a).

We need to ﬁnd Rdand δθ such that the wave ﬁeld amplitudes governed by (3.22)–

(3.23) with jump conditions (6.2)–(6.3) satisfy the rotation conditions (6.4) and droplet

condition (6.6). By periodicity, it is sufﬁcient to solve for n=0 (so t∈ [0,1]), where

X(t0)=(Rd,0)T. This is formulated as many 4 ×4 linear systems subject to (6.6), akin

to ﬁnding the walking states. An example in-phase wave ﬁeld is given in ﬁgure 4(c).

The stability analysis is similar to that of the walking states with two independent

droplet perturbations, although we rotate the domain by δθ between iterations so

that the steady-state droplet always lies at (r, θ) =(Rd,0)in the current domain.

It should be noted that the wave amplitudes amand bmare required for all m(not

just even indices), and the system is truncated for sufﬁciently large mgiven Rd, by

the shape of the large-order Bessel functions near zero (it should be recalled that

Jm(z)∼(1/m!)(z/2)mfor 0 <z√m+1).

The orbit diameters D≡2Rdare shown in ﬁgure 5(a), where curves with a light

grey background correspond to in-phase orbiters. The diameters of the stable solutions

are in good agreement with the experimental values recorded by Couder et al. (2005b),

namely Dn=(n− ¯), where ¯≈0.2 and nis an integer. However, the co-existence

of stable n=1,2,3 orbits occurs only for a limited range of Γ. This system also

contains wobbly orbits, which appear when the steady orbit destabilises via a complex

conjugate pair of eigenvalues. Due to nonlinear effects, the droplets remain locked in

a periodic wobbly orbit (see ﬁgure 5(b)). The minimum and maximum distances apart

reached by the droplets are indicated by the dark grey regions in ﬁgure 5(a), and are

obtained by simulating away from the steady states. We observe that the maximum

distance apart is bounded above by the corresponding unstable upper branch to each

solution, which sheds some light on why wobbly solutions only exist over a limited

range of Γ.

For further insight, we consider the solution at Γ=ΓFwith wavenumber k=kF.

The Floquet exponents are µ1=0 and µ2= −2γ (kF), giving the eigenvalues of the

Mathieu fundamental matrix MkF(ΓF)as ρ1≡exp(µ1)=1 and ρ2≡exp(µ2). Periodicity

Faraday wave–droplet dynamics 317

requires

(I2−MkF(ΓF)) a0(0;kF)

a0

0(0;kF)=− 0

2P0(kF)J0(kFRd),(6.7)

where (I2−MkF(ΓF)) is singular. A candidate radius Rdsatisﬁes J0(kFRd)=0, with a

corresponding wave amplitude vector from the matrix null space. In fact, such radii

satisfy the full problem and correspond to the lower branches in ﬁgure 5(a) as Γ→

ΓF.

6.2. Antiphase orbiting

For antiphase orbiting of two droplets, we impose that X(1)(tn)and X(2)(tn+1/2)both

lie on the radius Rdwith angular components θnand (θn+1/2+π)respectively, where

tn+1/2≡tn+1/2 and θn+1/2≡θn+δθ /2.We construct the wave ﬁeld to rotate by

π+δθ /2 in half of an impact period, which ensures that the wave ﬁeld is the same

with respect to each droplet at impact. The antiphase jump conditions are

[a0

m(tn+1/2;k)]+

−=−Pm(k)(−1)mΦm(Rd, θn+1/2;k), (6.8)

[b0

m(tn+1/2;k)]+

−=−Pm(k)(−1)mΨm(Rd, θn+1/2;k). (6.9)

Hence, both odd and even mare required for antiphase orbiters.

It remains to ﬁnd Rdand δθ such that (6.6) is satisﬁed. This is coupled with

Mathieu equations (3.22)–(3.23), jump conditions (6.8)–(6.9) and wave amplitude

maps (6.4), but with tn+17→ tn+1/2and δθ 7→ π+δθ /2. An example wave ﬁeld is

shown in ﬁgure 4(d). The stability analysis requires a two-stage transition map to

account for the antiphase impacts.

The quantisation of antiphase orbiters is given by the regions of white background

shown in ﬁgure 5(a). Although the reported orbit quantisation D0

n=(n+1/2− ¯)

is obtained (Couder et al. 2005b), the range of stable solutions with Γis limited,

particularly for the smallest orbit. We note that the stability and existence of solutions

are dependent on the skidding friction cand impact phase β, which are both ﬁxed for

this theoretical investigation. We expect that these results will be particularly sensitive

to the impact phase.

6.3. Promenade pairs

For simplicity, we only analyse rectilinear promenade pairs. To exploit the symmetry

of two in-phase droplets, we ﬁx the droplets to walk in parallel along lines y=±δy

in the direction of increasing x. To ﬁnd δy, we require ∂yη=0 at each drop, and

the walking speed δxis given by an equation analogous to (5.9). This couples with

the wave amplitude maps (5.3)–(5.4) and multiple-droplet jump conditions (6.1), with

X(1)(tn)=(nδx, δy)and X(2)(tn)=(nδx,−δy). The stability analysis follows similarly

to the walking case, yet the two droplets have independent perturbations.

The extension to antiphase promenaders is analogous to that for orbiters. In half of

an impact period, we require the wave ﬁeld to be translated δx/2 along the x-axis and

reﬂected about y=0. This ensures that each droplet receives the same kick at impact.

Results are shown in ﬁgure 6(a), where the grey and white backgrounds correspond

to in-phase and antiphase dynamics, respectively. We ﬁrst observe that the speed of

a promenading pair is less than that of a single walker, as reported by Borghesi

et al. (2014). The quantised distance between droplets D≡2δydepends weakly on

Γ, which was not reported in Borghesi et al. (2014). Instead, the authors stated

318 M. Durey and P. A. Milewski

0.88

0.92

0.96

1.00

1.00

1.02

1.04

0.96

0.98

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1.5 2.0 2.5 3.0 3.5 4.0 5.04.5

DD

(a)(b)

FIGURE 6. Bifurcation diagram for (a) promenade pairs and (b) two-droplet trains as

a function of distance apart Dand speed relative to a single walker δx/δx1. Grey and

white backgrounds denote in-phase and antiphase dynamics respectively. The thin grey

lines connect points of equal ˜

Γ≡Γ /ΓF. The thick lines correspond to stability types in

table 2.

that Dhas approximate in-phase values of 0.6,1.6,2.6, . . . and antiphase values of

1.1,2.1,3.1, . . ., which is consistent with our ﬁndings.

Crucially, we note that straight-line promenade pairs are unstable for all quantisations

and all values of Γ. When there is an oscillatory instability (with no real eigenvalue),

we expect the droplets to become bound in parallel motion, but with a transverse

oscillation, as reported in experiments (Protière et al. 2006; Borghesi et al. 2014).

This is demonstrated in ﬁgure 7, where we simulate from the unstable steady state

(with no real unstable eigenvalues), and nonlinear effects keep the droplets in a

steady oscillatory bound state. Such a result was speculated on by Borghesi et al.

(2014), but is hard to observe experimentally due to the conﬁnes of a ﬁnite domain.

The size of these oscillations depends on the skidding friction cand phase shift β

free parameters, which would both be tuned for a more careful comparison with

experimental data. Larger cincreases the solution space for oscillatory pairs, and the

transverse extent of the oscillations is reduced.

To shed light on the instability of straight-line promenaders, we construct a new

wave ﬁeld based on the superposition of two parallel walkers (from § 5), and compare

the average wave ﬁeld energy over one impact period (appendix B). The distance

between each walker is given by the continuous parameter D. The antiphase case

requires a mid-period impact for the second droplet; this presents a minor difﬁculty

as we do not know the speed of the newly constructed walking pair, so the impact

location is unknown (it should be recalled that a promenading pair travels slower than

a single walker). For ease, we assume that the pair of walkers travel at the same speed

as a single walker, which determines the impact location. We note that any error for

this impact is small, and as the energy is a quadratic function of the wave amplitudes,

this gives a negligible change to the energy.

Results of this calculation are shown in ﬁgure 8. Intuitively, we expect the energy

of the walking pair to be minimised at the same distance as that of the promenading

pair. However, the walking pair do not obey the condition ∂yη=0 at each droplet,

Faraday wave–droplet dynamics 319

8 101214161820

–1

0

1

0.04 0.06 0.08 0.10 0.12

FIGURE 7. Example transition from an unstable straight-line promenade (starting at x=0)

to a stable oscillating promenade, with the speed given by the grey scale bar.

1023

1

2

3

4

D

FIGURE 8. Wave ﬁeld energy E/EW(relative to a single walker) for two in-phase (black)

and antiphase (grey) parallel walkers at a distance Dapart for different values of Γ. The

larger values of Γgive the most extreme variations in energy (Γ /ΓF=0.88,0.9, . . . , 0.98).

The thick black lines give the energy of the corresponding quantised parallel promenade

solutions. The required interaction energy for the promenade mode partially explains its

instability.

which is required for a parallel procession. Hence, additional energy must be stored

in the wave ﬁeld in the case of the quantised promenaders, whose energy range (with

Γ) is denoted by the thick black lines in ﬁgure 8. The difference in energy decreases

as Dincreases due to the spatial decay of the wave ﬁeld of the walker. Physically, the

wave ﬁeld would tend to adopt the lowest-energy state of two walkers; however, as

this motion cannot be rectilinear (since ∂yη6= 0 at each droplet impact), a transverse

oscillation ensues.

6.4. Droplet trains

Trains of droplets may be studied in a similar fashion to promenades, except that

all droplets lie on the x-axis and are spaced δsapart. This requires constant ∂xηat

each droplet, with the in-phase and antiphase results shown in ﬁgure 6(b), where

we denote D≡δs. For sufﬁciently large Γ, two droplets may walk faster than a

single droplet, which was also observed for experiments in a conﬁned annulus (Filoux,

Hubert & Vandewalle 2015). The stability types reported in ﬁgure 6(b) are for general

perturbations, although the droplet trains are neutrally stable in the direction of travel

320 M. Durey and P. A. Milewski

(like a single walking droplet). Interestingly, the presence of a second droplet stabilises

the system to general perturbations for some antiphase trains. A possible extension is

to consider trains of multiple droplets, as considered in an annulus by Filoux et al.

(2015).

7. Droplet dynamics under a central force

When a droplet is subjected to a central force (˜κ > 0 in (3.24)), a range of new

dynamics occur. For increasing Γ, a bouncing droplet may destabilise to a circular

orbit, which itself destabilises to a variety of trajectories whose average radius and

angular momentum are quantised (Perrard et al. 2014b). Analysis for circular orbits

is analogous to § 6.1, but the more complex dynamics are explored numerically.

7.1. Circular orbits

For a single droplet in a circular orbit, the wave amplitudes must satisfy the rotational

maps (6.4a,b), and are governed by (3.22)–(3.23) with jump conditions (3.25)–(3.26).

However, as ˜κ > 0, the droplet rotation condition (6.6) becomes

F(c)µ+eµ+−µ−eµ−

µ+−µ−

D(δθ ) −e−˜νpI2X(tn)+(D(δθ ) −eµ+I2)(D(δθ ) −eµ−I2)X(tn)

=−F(c)

crB

Reµ+−eµ−

µ+−µ−D(δθ )(∇η)n,(7.1)

where

µ±≡1

2(−˜νp±q˜ν2

p−4˜κ). (7.2)