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Faraday wave-droplet dynamics: Discrete-time analysis


Abstract and Figures

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.
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J. Fluid Mech. (2017), vol. 821, pp. 296–329. c
Cambridge University Press 2017
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,
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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.
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.
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
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 , 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
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
ρP(r, θ, t), z=0,(2.2)
Hη, z=0,(2.3)
with φ0and η0 in the far field. In the above, we have constant surface tension
σ, density ρand kinematic viscosity ν, where 2
His the horizontal Laplacian.
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300 M. Durey and P. A. Milewski
In the vibrating frame, the effective gravity is gΓ(t)g(1Γcos0t)), 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
cρR0f(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
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
mX00(t)+cpρR0f(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 V0 for free walking
dynamics, or V=(1/2|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
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=4π0.
By integrating (2.5) over an impact period and exploiting periodicity, f(t)must satisfy
mgT =Zτ+T
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.
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Faraday wave–droplet dynamics 301
Variable Value Description
σ2.06 ×102kg s2Surface tension
ρ949 kg m3Fluid density
ν2×105m2s1Kinematic viscosity (fluid)
µair 1.8×105kg m1s1Dynamic viscosity (air)
g9.8 m s2Gravity
ω080 ×2πs1Vibration frequency (×2π)
R03.8×104m Droplet radius
m2.2×107kg Droplet mass
νp1.3×107kg s1Stokes 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=4π0, with balances fmg and
Pmg/λ2. This yields the dimensionless equations for the waves,
HφMGP(r, θ, t), z=0,(3.2)
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)
for any τ > 0. Here, we have defined dimensionless parameters
Typical parameter values from table 1give a reciprocal Reynolds number of 0.019,
a Bond number of B0.102, G1.201, M0.0017, R0.075 and ˜νp0.01.
The droplet radius corresponds to a vibration number of 4πpR3/B=0.8, which
minimises the walking threshold ΓWfor this fluid (Wind-Willassen et al. 2013). The
dimensionless potential strength ˜κ102is a free parameter of both the model and
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)=1Γcos(4πt+β ).
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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)=
k(am(t;km(r, θ;k)+bm(t;km(r, θ ;k)) dk,(3.7)
φ(r, θ , z,t)=
kekz(cm(t;km(r, θ ;k)+dm(t;km(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
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 b0d00 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(rrd(t))δθd(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 r)dr=δ(kξ ) and
trigonometric orthogonality, we expand
P(r, θ, t)=
k(pm(t;km(r, θ;k)+qm(t;km(r, θ ;k)) dk,(3.11)
f(tm(rd(t), θd(t);k)and qm(t;k)=1
f(tm(rd(t), θd(t);k),
with eigenfunction norms Wm=πif m>0 and W0=2π. It should be noted that
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
Lkam(t;k)=kMGpm(t;k)and Lkbm(t;k)= kMGqm(t;k), (3.13a,b)
Lktt +2γ (k)∂t+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
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.
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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
δ(ttn), (3.16)
where tnnfor 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
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]. For all n>0, the first equation gives
MG Zt+
δ(ttnm(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
=Pm(km(rd(tn), θd(tn);k), (3.18)
=Pm(km(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(tt?)X0(t)+ ˜νpX0(t)+ ˜κX(t)=Gδ(tt?)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(tt?)with
δε(tt?)ε1ϕ((tt?1), ε > 0,(3.21)
where ϕ(τ ) >0 for all τRand R
−∞ ϕ(τ ) dτ=1. Hence, the functions δε(·)δ(·)
pointwise as ε0, 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 ε0. 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.
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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)
= −Pm(km(X(tn);k), (3.25)
= −Pm(km(X(tn);k), (3.26)
= −F(c) 1
η(X(tn), tn)+X0(t
where Pm(k)=kMG/Wmand F(c)=1exp(cGR/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), c0.3, 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
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, λF1,
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
n;k), a0
n;k), . . .)T, and similarly for bn(k)and b0
n(k). We also write the
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Faraday wave–droplet dynamics 305
eigenfunctions as vectors Φ(·;k)=0(·; 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
Mk(Γ ) =m11;k)Im12 ;k)I
m21;k)Im22 ;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 XnX(tn)and velocity X0
(i) Use Xnand X0
nto compute Xn+1and ˜
X0=F(˜κ) Xn
(ii) Update wave amplitudes including jump conditions k>0 (similarly for bn(k)):
n+1(k)=Mk(Γ ) an(k)
(iii) Apply droplet jump conditions:
n+1=(1F(c)) ˜
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 mN. For any impact time tn, the wavenumbers are generally peaked
around k=kF, with the peak becoming narrower as ΓΓ
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 δk0.2 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 <zm+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 /ΓF0.81 but m?50 for Γ /ΓF0.95. 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
We typically simulate 1000 impacts per second on a standard desktop machine using
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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 (X0) with radially symmetric wave
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)for all nand all xR2. By orthogonality, the wave amplitudes must satisfy
a0(tn+1;k)=a0(tn;k), (4.2)
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
I2m11;k)m12 ;k)
m21;k)m22 ;k)a0(0;k)
where I2R2×2is 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
bm0 for all m>1, and ˆ
b00. We then consider small
X(t)+(t), am(t;k)= ˆam(t;k)+ ˜am(t;k), bm(t;k)=ˆ
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
= −Pm(k)Φm(0;k)T(tn), (4.6)
= −Pm(k)Ψm(0;k)T(tn), (4.7)
= −F(c) 0(t
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
bm(tn;k)Ψm(0;k)) dk.(4.9)
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Faraday wave–droplet dynamics 307
Equations (4.6)–(4.8) are simplified considerably by noting that
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
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, bm0, m.
We denote 0 < δxX(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
(1)pJp(kδx)ap(tn;k), (5.1)
(Jmp(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)
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)+ ˜ν1
For X0(t+
n+1)=V0, the jump condition (3.27) and (5.6) give the requirement
η(X(tn+1), tn+1). (5.7)
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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
where Φ1(0;k)=(k/2,0)T. From the x-components of (5.5) and (5.8), δxmust
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
0 A(k;δx)Mk(Γ )a0(k)
0(k)= 0
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 Ψm0 along the x-axis,
so the linearised jumps for the perturbation coefficients ˜
bmwill be zero for an in-
line perturbation. Furthermore, xΨm0 (for all m) along the x-axis, so the ˜
do not contribute to the linearised droplet perturbation jump condition. Hence, the ˜
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.
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Faraday wave–droplet dynamics 309
0.80 0.85 0.90 0.95 1.00 0.78 0.79 0.80 0.81
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 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)
m(t;k)+· ·· ,(5.12)
for all m>0. By the Frobenius series for Bessel functions, we have
δx2+Ox4), (5.13a)
δx3+Ox5), (5.13b)
δx3+Ox5). (5.14a,b)
These expansions are valid for k/2=O(1). As the least-damped wavenumber kF
satisfies kF/2π=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
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310 M. Durey and P. A. Milewski
differential equations with initial and jump conditions is obtained, all of the general
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)
m0 for m>1. This is verified by noting
that H(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
0=P0(k), (5.16ac)
0(0;k), J(1)
1=P1(k) (k/2). (5.17ac)
Equating δxfrom the walking condition (5.9), we require ΓWsuch that
Therefore, (5.16)–(5.18) is a closed system for a(0)
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)
20, so
0(t;k), B(1)
1(t;k), B(2)
0(0;k), J(2)
The δx2term from (5.9) completes the system with
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)
00 on (0,1). By (5.19), this is achieved if
and only if Γ1=0. With these vanishing terms, equation (5.20) becomes
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Faraday wave–droplet dynamics 311
yielding a(2)
10 on (0,1), which satisfies (5.21). Neglecting terms of ox2), we have
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(|xX(tn)|,0), where
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
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
n=0¯η(|xX(tn)|,ttn)H(ttn), (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
Me(Γ ) =Td(Γ )
1Γ /ΓF
,where Td(Γ ) =16(ω2(kc(Γ )) +γ2(kc(Γ )) +4π2)
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 kcF)=kFfor a range of Γ,
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312 M. Durey and P. A. Milewski
–2–4–6 0 2 4 6 8 10–8–10 1.5
0.84 0.88 0.92 0.96 1.00
25 50 75 1000
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(r2)/rfor parameters 1, 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.
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Faraday wave–droplet dynamics 313
–2–46 02468–8 0.90
0 0.02 0.04 0.06
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(kkF), 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
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314 M. Durey and P. A. Milewski
–3 0 2–2
–3 02–2
–3 02–2
–3 02–2
–3 02–2
–3 02–2
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
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Faraday wave–droplet dynamics 315
Stability type Line Description
Stable Solid black U=
Unstable (real) Dashed black UR6=
Unstable (c.c.) Solid grey µU, µ C\Rand ¯µU
TABLE 2. Stability types of the stability transition matrix Twith complex eigenvalue
spectrum σ (T). We define the set of unstable eigenvalues U= {µ:µσ(T), |µ|>1},
where Uis empty only when the system is stable.
in-phase interactions, the jump condition (3.25) becomes
Φm(X(p)(tn);k), (6.1)
and similarly for b0
m. For antiphase impacts, additional jumps occur at t=tn+1/2tn+
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)=θnnδθ (by rotational invariance). The droplet
motion is piecewise linear, so rd(t)6=Rdfor t6= tn.
As the implicit second droplet has angular component n+π)at time tn, (6.1)
= −Pm(km(X(tn);k)(1+(1)m), (6.2)
= −Pm(km(X(tn);k)(1+(1)m). (6.3)
For modd, amand bmare never excited during the orbit, giving ambm0. 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
n+1;k)=D(mδθ )c0
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 ) (1F(c))e−˜νpI2)(Dθ ) I2)X(tn)=F(c)
D(δθ )(η)n,(6.6)
where (η)nη(X(tn), tn), and (η)n+1=Dθ )(η)ndue to the rotation.
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316 M. Durey and P. A. Milewski
0.85 0.90 0.95 1.00
1–1 0
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 <zm+1).
The orbit diameters D2Rdare 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 MkFF)as ρ1exp1)=1 and ρ2exp2). Periodicity
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Faraday wave–droplet dynamics 317
(I2MkFF)) a0(0;kF)
0(0;kF)= 0
where (I2MkFF)) 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
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 n+1/2+π)respectively, where
tn+1/2tn+1/2 and θn+1/2θn+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
=Pm(k)(1)mΦm(Rd, θn+1/2;k), (6.8)
=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 7π+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 D2δydepends weakly on
Γ, which was not reported in Borghesi et al. (2014). Instead, the authors stated
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318 M. Durey and P. A. Milewski
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
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 δxx1. Grey and
white backgrounds denote in-phase and antiphase dynamics respectively. The thin grey
lines connect points of equal ˜
ΓΓ /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,
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Faraday wave–droplet dynamics 319
8 101214161820
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.
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
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 Dδs. 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
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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.
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
D(δθ ) e−˜νpI2X(tn)+(Dθ ) eµ+I2)(D(δθ ) eµI2)X(tn)
µ+µD(δθ )(η)n,(7.1)