Bouncing droplets on a
vibrating fluid bath
Øistein Wind-Willassen (DTU COMPUTE)
Mads Peter Sørensen (DTU COMPUTE)
John Bush (MIT)
Mathematical Colloquium DTU May 8 2013
Thursday, June 13, 13
What is this?
Thursday, June 13, 13
Why is this interesting?
•
•
Thursday, June 13, 13
Mathematics/physics viewpoint:
•
•
Interesting dynamical system
Complicated fluid physics
Philosophical viewpoint:
•
•
Particle-wave duality at macroscopic level
Has features reminiscent of quantum mechanics
This talk
I will focus on the “mode” with which the drop bounces,
and cover the following:
•
•
•
Thursday, June 13, 13
The general physics of the system
•
•
Experimental setup
Theoretical model
The obtained results
Some perspectives
Faraday waves
Thursday, June 13, 13
Faraday waves
Consider a fluid tray oscillating vertically
Thursday, June 13, 13
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
Thursday, June 13, 13
= A! 2
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
⇤
= A! 2
g (t) = g + sin(2⇡f t)
Thursday, June 13, 13
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
⇤
= A! 2
g (t) = g + sin(2⇡f t)
Frequency is constant.
Thursday, June 13, 13
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
⇤
= A! 2
g (t) = g + sin(2⇡f t)
Frequency is constant.
At = F the layer becomes unstable.
Thursday, June 13, 13
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
⇤
= A! 2
g (t) = g + sin(2⇡f t)
Frequency is constant.
At = F the layer becomes unstable.
Standing Faraday waves form, they follow standard dispersion relation.
Thursday, June 13, 13
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
= A! 2
⇤
g (t) = g + sin(2⇡f t)
Frequency is constant.
At = F the layer becomes unstable.
Standing Faraday waves form, they follow standard dispersion relation.
In the experiment:
Thursday, June 13, 13
<
F , i.e. the interface is flat without a droplet!
Faraday waves
Consider a fluid tray oscillating vertically
Amplitude = A
Frequency = f = !/(2⇡)
Forcing =
= A! 2
⇤
g (t) = g + sin(2⇡f t)
Frequency is constant.
At = F the layer becomes unstable.
Standing Faraday waves form, they follow standard dispersion relation.
In the experiment:
Thursday, June 13, 13
<
F , i.e. the interface is flat without a droplet!
The drop locally excites Faraday waves.
Qualitative behaviour
•
•
•
Coalescence for
< B⇡g
Bouncing in place for
< W
B <
Walking horizontally for
< F
W <
Thursday, June 13, 13
Remember:
is the forcing
Qualitative behaviour
•
•
•
Coalescence for
< B⇡g
Bouncing in place for
< W
B <
Walking horizontally for
< F
W <
Remember:
is the forcing
Particle–wave association on a fluid interface
89
1.4
1.2
Int
Drop size
1.0
B = Bouncing
PDB = Period doubling
PDC = Period doubling cascade
W = Walking
Int = Intermittent
F = Faraday waves
D (mm)
W
B
PDB
0.8
0.6
PDC
50cS - 50 Hz[Couder, 2006]
Thursday, June 13, 13
F
0.4
Forcing
0
1
2
3
4
5
Qualitative behaviour
•
•
•
Coalescence for
< B⇡g
Bouncing in place for
< W
B <
Walking horizontally for
< F
W <
Remember:
is the forcing
Particle–wave association on a fluid interface
89
1.4
1.2
Int
Drop size
1.0
B = Bouncing
PDB = Period doubling
PDC = Period doubling cascade
W = Walking
Int = Intermittent
F = Faraday waves
D (mm)
W
B
PDB
0.8
0.6
PDC
50cS - 50 Hz[Couder, 2006]
Thursday, June 13, 13
F
0.4
Forcing
0
1
2
3
4
5
Qualitative behaviour
•
•
•
Coalescence for
< B⇡g
Bouncing in place for
< W
B <
Walking horizontally for
< F
W <
Remember:
is the forcing
Particle–wave association on a fluid interface
89
1.4
1.2
Int
Drop size
1.0
B = Bouncing
PDB = Period doubling
PDC = Period doubling cascade
W = Walking
Int = Intermittent
F = Faraday waves
D (mm)
W
B
PDB
0.8
0.6
PDC
50cS - 50 Hz[Couder, 2006]
Thursday, June 13, 13
F
0.4
Forcing
0
1
2
3
4
5
Qualitative behaviour
•
•
•
Coalescence for
< B⇡g
Bouncing in place for
< W
B <
Walking horizontally for
< F
W <
Remember:
is the forcing
Particle–wave association on a fluid interface
89
1.4
1.2
Int
Drop size
1.0
B = Bouncing
PDB = Period doubling
PDC = Period doubling cascade
W = Walking
Int = Intermittent
F = Faraday waves
D (mm)
W
B
PDB
0.8
0.6
PDC
50cS - 50 Hz[Couder, 2006]
Thursday, June 13, 13
F
0.4
Forcing
0
1
2
3
4
5
Experimental setup
Thursday, June 13, 13
Experimental setup
•
Thursday, June 13, 13
Shaker
Experimental setup
•
•
Thursday, June 13, 13
Shaker
Fluid tray
Experimental setup
•
•
•
Thursday, June 13, 13
Shaker
Fluid tray
High speed camera
Experimental setup
•
•
•
•
Thursday, June 13, 13
Shaker
Fluid tray
High speed camera
CCD camera
Experimental setup
•
•
•
•
•
Thursday, June 13, 13
Shaker
Fluid tray
High speed camera
CCD camera
Lamps with diffusers
Experimental setup
•
•
•
•
•
•
Thursday, June 13, 13
Shaker
Fluid tray
High speed camera
CCD camera
Lamps with diffusers
Computer
Examples - top view
Thursday, June 13, 13
Examples - top view
Thursday, June 13, 13
Examples - side view
Thursday, June 13, 13
Examples - side view
Thursday, June 13, 13
Experimental notation
Thursday, June 13, 13
Experimental notation
•
p
Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 /
- Ratio of forcing frequency to natural oscillation of drop.
Thursday, June 13, 13
Experimental notation
•
p
Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 /
- Ratio of forcing frequency to natural oscillation of drop.
Bouncing “mode” is denoted
(m, n)i
Thursday, June 13, 13
Experimental notation
•
p
Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 /
- Ratio of forcing frequency to natural oscillation of drop.
Bouncing “mode” is denoted
(m, n)i
•
m: number of driving periods
Thursday, June 13, 13
Experimental notation
•
p
Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 /
- Ratio of forcing frequency to natural oscillation of drop.
Bouncing “mode” is denoted
(m, n)i
•
•
m: number of driving periods
n: number of distinct impacts
Thursday, June 13, 13
Experimental notation
•
p
Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 /
- Ratio of forcing frequency to natural oscillation of drop.
Bouncing “mode” is denoted
(m, n)i
•
•
•
m: number of driving periods
n: number of distinct impacts
i: mechanical energy (phase distinction)
Thursday, June 13, 13
Experimental notation
•
p
Proxy for drop size: the “vibration” number ⌦ = 2⇡f ⇢r03 /
- Ratio of forcing frequency to natural oscillation of drop.
Bouncing “mode” is denoted
(m, n)i
•
•
•
m: number of driving periods
n: number of distinct impacts
i: mechanical energy (phase distinction)
(2, 1)

Thursday, June 13, 13
1
(2, 1)



2
7
Theoretical description
Drops bouncing on a vibrating bath
a)
Z=0
Z=−1
b)
ρ, ν
v
Z
g
σ
Z=−1
v
[Molacek & Bush, 2013]
Figure 9. A schematic illustration of our choice of coordinates. The vertical position
drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil
level of the bath (b).
In free flight:
During contact:
mg ⇤ (t) = mz̈
⇤
mg (t) =
1+
c3
ln2
c1 r0
Z
!
Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un
a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2
(2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2)
reduce the amplitude of their smaller bounce until a simple period-doubled bou
mode (2, 1) is reached, and only then commence the period-doubling cascade (2
(4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o
high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe
to the influence of the standing waves created on the bath by previous drop impac
lower frequencies, the Faraday threshold is closer to the period-doubling threshold
the drop impacts create more slowly decaying standing waves on the bath surfa
reducing the relative speed
the drop and bath1at 0
impact, the standing
1 between
0
appear to stabilize the vertical motion, and so delay the period-doubling transitio
The bounds of the frequency range explored were prescribed by experimenta
straints. The presence of the Faraday threshold provides a lower limit on the ra
frequencies over which the period-doubled modes can arise. For example, for 20 c
cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f
Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim
the frequency range is imposed by the finite resolution
of our camera. Since the w
!
region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t
size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in
size measurement leads to increasing relative error in Ω. Similarly, at high freque
becomes increasingly difficult to distinguish between the different bouncing mod
the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which
order 0.1mm for f = 200 Hz.
4⇡⌘r0 c2 (⌫)
2⇡
mz̈ +
Ż + c r Z
c r
3 ln Z
Z
3. Vertical dynamics
3.1. Linear spring model
We proceed by describing the simplest model of the drop’s vertical dynamics, analog
works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi
drop-impactor interaction is described in terms of a linear spring. We nondimensio
the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi
!
variables) and time by the characteristic frequency of drop oscillations ωD = σ
(Rayleigh 1879). We shall always consider the frame of reference fixed relative
shaking platform, and place the origin so that the undisturbed bath surface is at Z
Thursday, June 13, 13
Theoretical description
Drops bouncing on a vibrating bath
a)
Z=0
Z=−1
b)
ρ, ν
v
Z
g
σ
Z=−1
v
[Molacek & Bush, 2013]
Figure 9. A schematic illustration of our choice of coordinates. The vertical position
drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil
level of the bath (b).
In free flight:
During contact:
mg ⇤ (t) = mz̈
⇤
mg (t) =
kinetic energy (fluid motion in the two
liquids) - this term is the added mass
1+
c3
ln2
c1 r0
Z
!
Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un
a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2
(2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2)
reduce the amplitude of their smaller bounce until a simple period-doubled bou
mode (2, 1) is reached, and only then commence the period-doubling cascade (2
(4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o
high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe
to the influence of the standing waves created on the bath by previous drop impac
lower frequencies, the Faraday threshold is closer to the period-doubling threshold
the drop impacts create more slowly decaying standing waves on the bath surfa
reducing the relative speed
the drop and bath1at 0
impact, the standing
1 between
0
appear to stabilize the vertical motion, and so delay the period-doubling transitio
The bounds of the frequency range explored were prescribed by experimenta
straints. The presence of the Faraday threshold provides a lower limit on the ra
frequencies over which the period-doubled modes can arise. For example, for 20 c
cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f
Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim
the frequency range is imposed by the finite resolution
of our camera. Since the w
!
region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t
size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in
size measurement leads to increasing relative error in Ω. Similarly, at high freque
becomes increasingly difficult to distinguish between the different bouncing mod
the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which
order 0.1mm for f = 200 Hz.
4⇡⌘r0 c2 (⌫)
2⇡
mz̈ +
Ż + c r Z
c r
3 ln Z
Z
3. Vertical dynamics
3.1. Linear spring model
We proceed by describing the simplest model of the drop’s vertical dynamics, analog
works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi
drop-impactor interaction is described in terms of a linear spring. We nondimensio
the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi
!
variables) and time by the characteristic frequency of drop oscillations ωD = σ
(Rayleigh 1879). We shall always consider the frame of reference fixed relative
shaking platform, and place the origin so that the undisturbed bath surface is at Z
Thursday, June 13, 13
Theoretical description
Drops bouncing on a vibrating bath
a)
Z=0
Z=−1
b)
ρ, ν
v
Z
g
σ
Z=−1
v
[Molacek & Bush, 2013]
Figure 9. A schematic illustration of our choice of coordinates. The vertical position
drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil
level of the bath (b).
In free flight:
During contact:
mg ⇤ (t) = mz̈
⇤
mg (t) =
kinetic energy (fluid motion in the two
liquids) - this term is the added mass
1+
c3
ln2
c1 r0
Z
!
Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un
a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2
(2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2)
reduce the amplitude of their smaller bounce until a simple period-doubled bou
mode (2, 1) is reached, and only then commence the period-doubling cascade (2
(4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o
high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe
to the influence of the standing waves created on the bath by previous drop impac
lower frequencies, the Faraday threshold is closer to the period-doubling threshold
the drop impacts create more slowly decaying standing waves on the bath surfa
reducing the relative speed
the drop and bath1at 0
impact, the standing
1 between
0
appear to stabilize the vertical motion, and so delay the period-doubling transitio
The bounds of the frequency range explored were prescribed by experimenta
straints. The presence of the Faraday threshold provides a lower limit on the ra
frequencies over which the period-doubled modes can arise. For example, for 20 c
cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f
Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim
the frequency range is imposed by the finite resolution
of our camera. Since the w
!
region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t
size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in
size measurement leads to increasing relative error in Ω. Similarly, at high freque
becomes increasingly difficult to distinguish between the different bouncing mod
the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which
order 0.1mm for f = 200 Hz.
4⇡⌘r0 c2 (⌫)
2⇡
mz̈ +
Ż + c r Z
c r
3 ln Z
Z
Viscous dissipation
3. Vertical dynamics
3.1. Linear spring model
We proceed by describing the simplest model of the drop’s vertical dynamics, analog
works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi
drop-impactor interaction is described in terms of a linear spring. We nondimensio
the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi
!
variables) and time by the characteristic frequency of drop oscillations ωD = σ
(Rayleigh 1879). We shall always consider the frame of reference fixed relative
shaking platform, and place the origin so that the undisturbed bath surface is at Z
Thursday, June 13, 13
Theoretical description
Drops bouncing on a vibrating bath
a)
Z=0
Z=−1
b)
ρ, ν
v
Z
g
σ
Z=−1
v
[Molacek & Bush, 2013]
Figure 9. A schematic illustration of our choice of coordinates. The vertical position
drop’s centre of mass Z is equal to 0 at the initiation of impact (a), and −1 at the equil
level of the bath (b).
In free flight:
During contact:
mg ⇤ (t) = mz̈
⇤
mg (t) =
kinetic energy (fluid motion in the two
liquids) - this term is the added mass
1+
c3
ln2
c1 r0
Z
!
Fig. 8 shows the first two period-doubling thresholds. Smaller drops (Ω < 0.6) un
a period-doubling cascade, so the first two thresholds correspond to (1, 1) → (2, 2
(2, 2) → (4, 4) transitions. Larger drops (Ω > 0.6) transition from (1, 1) to (2, 2)
reduce the amplitude of their smaller bounce until a simple period-doubled bou
mode (2, 1) is reached, and only then commence the period-doubling cascade (2
(4, 2) → (8, 4) → . . . . Note that the low frequency curves are shifted to the right o
high frequency counterparts (60 Hz curve for 20 cS; 50 − 60 Hz for 50 cS), an effe
to the influence of the standing waves created on the bath by previous drop impac
lower frequencies, the Faraday threshold is closer to the period-doubling threshold
the drop impacts create more slowly decaying standing waves on the bath surfa
reducing the relative speed
the drop and bath1at 0
impact, the standing
1 between
0
appear to stabilize the vertical motion, and so delay the period-doubling transitio
The bounds of the frequency range explored were prescribed by experimenta
straints. The presence of the Faraday threshold provides a lower limit on the ra
frequencies over which the period-doubled modes can arise. For example, for 20 c
cone oil, period-doubling occurs only for Γ > 1.58 (Fig. 8), while ΓF < 1.58 for f
Hz. Thus, for f ≤ 45 Hz, the period-doubling transitions disappear. The upper lim
the frequency range is imposed by the finite resolution
of our camera. Since the w
!
region of ultimate interest is given by Ω = 2πf ρR03 /σ ! 1.5 (see MBII ), the t
size of a walker R0 ∼ f −2/3 . Thus, for higher frequencies, the constant error in
size measurement leads to increasing relative error in Ω. Similarly, at high freque
becomes increasingly difficult to distinguish between the different bouncing mod
the motion itself happens over a distance of at most g(T /2)2 /2 ≤ gf −2 /2, which
order 0.1mm for f = 200 Hz.
4⇡⌘r0 c2 (⌫)
2⇡
mz̈ +
Ż + c r Z
c r
3 ln Z
Z
Viscous dissipation
Nonlinearity of spring
3. Vertical dynamics
3.1. Linear spring model
We proceed by describing the simplest model of the drop’s vertical dynamics, analog
works by Okumura et al. (2003), Gilet & Bush (2009b) and Terwagne (2011), in whi
drop-impactor interaction is described in terms of a linear spring. We nondimensio
the vertical displacement of the drop by its radius (see Table 2 for a list of dimensi
!
variables) and time by the characteristic frequency of drop oscillations ωD = σ
(Rayleigh 1879). We shall always consider the frame of reference fixed relative
shaking platform, and place the origin so that the undisturbed bath surface is at Z
Thursday, June 13, 13
⌦ = 2⇡f
1
0.9
Results
20
cS
80
Hz
p
⇢r03 /
(1,1)
Walking
0.8
(2,1)1
(2,1)2
(4,2)
0.7
(2,2)
1
2
(2,1) & (2,1)
(4,3)
0.6
0.4
C
2
(2,1)
C
2
2.5
(2,1)1
(4,2)
3
2
= A! /g
Thursday, June 13, 13
C
(2,1)2
&
chaos
(4,4)
0.5
0.3
1
(2,1)
3.5
4
The mixed state
Thursday, June 13, 13
The mixed state
Thursday, June 13, 13


Thursday, June 13, 13
The mixed state

The mixed state
11

Thursday, June 13, 13
Perspectives
[Couder,2008]
[Couder,2010]
[Couder,2009]
Thursday, June 13, 13
Perspectives
[Couder,2008]
Analogies with quantum mechanics
[Couder,2010]
[Couder,2009]
Thursday, June 13, 13
Perspectives
[Couder,2008]
Analogies with quantum mechanics
• Single/double slit
FIG. 1. (a), (b) Sketch of the central region of the experimental
cell (seen from above and in a cross section along the y axis,
respectively). An individual trajectory is shown in (a) and the
definitions of !i , yi , and " are given. The width of the slit being
L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the
experiment lit with diffuse light showing the wave pattern as the
walker crosses the aperture. The picture was taken at a time
when the trajectory, initially perpendicular to the aperture, was
deflected by the interference with reflected waves.
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June is
13, also
13
This
observed on individual realizations: with ap-
FIG. 2 (color online). (a) The measured deviations a of successive individual particles as a function of[Couder,2010]
the parameter of
impact Yi (with L=#F ! 2:86). (b) A superposition of three
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions !i ! 90# and Yi ! 0:1.
(c) Experimental histogram of the deviation " as obtained with
NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and
#F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was
improved by taking them into account so that the distributions
correspond to 2NT realizations. The curve is the fit by Eq. (1)
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
(L ! 14:7 mm and #F ! 4:75 mm). The curve is the fit by
Eq. (1) with L=#F ! 2:86.
[Couder,2009]
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!:
(1)
!
! $L sin"=#F !
!
This amplitude of diffraction of a plane wave turns out to
provide an approximate fit for these histograms. Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
become smaller.
[Couder,2008]
The possible orbits being discrete, we label
the plateaus by an
order n, with n ¼ 0 corresponding to the tightest orbit observed at
large Ω. Each plateau crosses the continuous short path-memory
curve given by Eq. 3 at a value Ω0n of the angular velocity (Fig. 2B).
At these crossings, the observed orbits have the same radius
exp
0
0
R0n ¼ Rexp
n ðΩn Þ ¼ RC ðΩn Þ they would have had with short-term
memory. A plateau extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
A
FIG. 1. (a), (b) Sketch of the central region of the experimental
cell (seen from above and in a cross section along the y axis,
respectively). An individual trajectory is shown in (a) and the
definitions of !i , yi , and " are given. The width of the slit being
L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the
experiment lit with diffuse light showing the wave pattern as the
walker crosses the aperture. The picture was taken at a time
when the trajectory, initially perpendicular to the aperture, was
deflected by the interference with reflected waves.
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June is
13, also
13
This
observed on individual realizations: with ap-
FIG. 2 (color online). (a) The measured deviations a of sucB
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Yi (with L=#F ! 2:86). (b) A superposition of three
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions !i ! 90# and Yi ! 0:1.
(c) Experimental histogram of the deviation " as obtained with
NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and
#F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was
improved by taking them into account so that the distributions
correspond to 2NT realizations. The curve is the fit by Eq. (1)
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
(L ! 14:7 mm and #F ! 4:75 mm). The curve is the fit by
Eq. (1) with L=#F ! 2:86.
C
[Couder,2009]
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!:
(1)
!
! $L sin"=#F !
!
This amplitude of diffraction of a plane wave turns out to
provide an approximate fit for these histograms. Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
become smaller.
[Couder,2008]
The possible orbits being discrete, we label
the plateaus by an
order n, with n ¼ 0 corresponding to the tightest orbit observed at
large Ω. Each plateau crosses the continuous short path-memory
curve given by Eq. 3 at a value Ω0n of the angular velocity (Fig. 2B).
At these crossings, the observed orbits have the same radius
exp
0
0
R0n ¼ Rexp
n ðΩn Þ ¼ RC ðΩn Þ they would have had with short-term
memory. A plateau extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
FIG. 2 (color online).
• Harmonic oscillator
A
FIG. 1. (a), (b) Sketch of the central region of the experimental
cell (seen from above and in a cross section along the y axis,
respectively). An individual trajectory is shown in (a) and the
definitions of !i , yi , and " are given. The width of the slit being
L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the
experiment lit with diffuse light showing the wave pattern as the
walker crosses the aperture. The picture was taken at a time
when the trajectory, initially perpendicular to the aperture, was
deflected by the interference with reflected waves.
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June is
13, also
13
This
observed on individual realizations: with ap-
(a) The measured deviations a of sucB
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Yi (with L=#F ! 2:86). (b) A superposition of three
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions !i ! 90# and Yi ! 0:1.
(c) Experimental histogram of the deviation " as obtained with
NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and
#F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was
improved by taking them into account so that the distributions
correspond to 2NT realizations. The curve is the fit by Eq. (1)
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
(L ! 14:7 mm and #F ! 4:75 mm). The curve is the fit by
Eq. (1) with L=#F ! 2:86.
C
[Couder,2009]
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!:
(1)
!
! $L sin"=#F !
!
This amplitude of diffraction of a plane wave turns out to
provide an approximate fit for these histograms. Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
become smaller.
[Couder,2008]
The possible orbits being discrete, we label
the plateaus by an
order n, with n ¼ 0 corresponding to the tightest orbit observed at
large Ω. Each plateau crosses the continuous short path-memory
curve given by Eq. 3 at a value Ω0n of the angular velocity (Fig. 2B).
At these crossings, the observed orbits have the same radius
exp
0
0
R0n ¼ Rexp
n ðΩn Þ ¼ RC ðΩn Þ they would have had with short-term
memory. A plateau extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator
A
B
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Y (with L=# ! 2:86). (b) A superposition of three
• Lattice structures
i
F
FIG. 1. (a), (b) Sketch of the central region of the experimental
cell (seen from above and in a cross section along the y axis,
respectively). An individual trajectory is shown in (a) and the
definitions of !i , yi , and " are given. The width of the slit being
L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the
experiment lit with diffuse light showing the wave pattern as the
walker crosses the aperture. The picture was taken at a time
when the trajectory, initially perpendicular to the aperture, was
deflected by the interference with reflected waves.
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions !i ! 90# and Yi ! 0:1.
(c) Experimental histogram of the deviation " as obtained with
NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and
#F ! 6:95 mm). Since each trajectory has a symmetrical counterpart with respect to the axis of the aperture, the statistic was
improved by taking them into account so that the distributions
correspond to 2NT realizations. The curve is the fit by Eq. (1)
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
(L ! 14:7 mm and
A. Eddi#et
Fal.! 4:75 mm). The curve is the fit by
Eq. (1) with L=#F ! 2:86.
C
[Couder,2009]
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June is
13, also
13
This
observed on individual realizations: with ap-
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!:
(1)
!
! $L sin"=#F !
!
Fig. 1: The crystalline organization of a large number of
droplets
= 61) with a triangular
This amplitude
of(Ndiffraction
of lattice.
a plane wave turns out to
provide an approximate fit for these histograms. Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
comes shorter as the wall thickness is decreased
[Fig. 4(c)].
become smaller.
[Couder,2008]
possible
orbits
being discrete, we label
the plateaus by an
Ultimately the limit cycle is never reachedThe
[Fig.
4(d)]
[10–
order n, with n ¼ 0 corresponding to the tightest orbit observed at
12]. The trajectories inside the cavity and
the escapes are
large Ω. Each plateau crosses the continuous short path-memory
correlated. As long as the droplet follows
limit
curvethe
given
by Eq.cycle,
3 at a value Ω0n of the angular velocity (Fig. 2B).
At thesebecome
crossings, disthe observed orbits have the same radius
no escape is observed. When the trajectories
exp
exp
0
0
Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term
ordered, the collisions with the walls show
a larger
variety
memory.
A plateau
extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator
A
B
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Y (with L=# ! 2:86). (b) A superposition of three
• Lattice structures
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions ! ! 90 and Y ! 0:1.
effects
• FIG.Tunneling
1. (a), (b) Sketch of the central region of the experimental
i
F
i
#
i
(c) Experimental histogram of the deviation " as obtained with
cell (seen from above and in a cross section along the y axis,
NT ! 125 single walkers with L=#F ! 2:11 (L ! 14:7 mm and
respectively). An individual trajectory is shown in (a) and the
#F ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of !i , yi , and " are given. The width of the slit being
terpart with respect to the axis of the aperture, the statistic was
L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the
improved by taking them into account so that the distributions
experiment lit with diffuse light showing the wave pattern as the
correspond to 2NT realizations. The curve is the fit by Eq. (1)
walker crosses the aperture. The picture was taken at a time
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
when the trajectory, initially perpendicular to the aperture, was
(L ! 14:7 mm and
A. Eddi#et
Fal.! 4:75 mm). The curve is the fit by
[Couder,2008]
deflected by the interference with reflected waves.
Eq. (1) with L=#F ! 2:86.
C
[Couder,2009]
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle !i and by its normalized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June is
13, also
13
This
observed on individual realizations: with ap-
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
FIG.
5
(color
online).
(a) Sup
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!
:
(1)
!
!
!
!
walker
velocity
V ¼ 12 mm
$Lof
sin"=#
F
a barrier with e ¼ 3 mm leadin
Fig. 1: The crystalline organization of a large number of
droplets
= 61) with a triangular
This amplitude
of(Ndiffraction
of lattice.
a plane wave turns out to
offit10forconsecutive
collisions
of
provide an approximate
these histograms.
Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
comes shorter as the wall thickness is decreased
[Fig. 4(c)].
become smaller.
[Couder,2008]
possible
orbits
being discrete, we label
the plateaus by an
Ultimately the limit cycle is never reachedThe
[Fig.
4(d)]
[10–
order n, with n ¼ 0 corresponding to the tightest orbit observed at
12]. The trajectories inside the cavity and
the escapes are
large Ω. Each plateau crosses the continuous short path-memory
correlated. As long as the droplet follows
limit
curvethe
given
by Eq.cycle,
3 at a value Ω0n of the angular velocity (Fig. 2B).
At thesebecome
crossings, disthe observed orbits have the same radius
no escape is observed. When the trajectories
exp
exp
0
0
Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term
ordered, the collisions with the walls show
a larger
variety
memory.
A plateau
extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator
A
B
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Y (with L=# ! 2:86). (b) A superposition of three
• Lattice structures
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions ! ! 90 and Y ! 0:1.
effects
• FIG.Tunneling
1. (a), (b) Sketch of the central region of the experimental
(c) Experimental histogram of the deviation " as obtained with
cell (seen from above and in a cross section along the y axis,
N ! 125 single walkers with L=# ! 2:11 (L ! 14:7 mm and
Wave-like
statistics in confined geometries
• respectively).
An individual trajectory is shown in (a) and the
i
F
i
#
i
3
T
F
#F ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of !i , yi , and " are given. The width of the slit being
terpart with respect to the axis of the aperture, the statistic was
L, the parameter of impact is Yi' ! yi =L. (c) A photograph of the
improved by taking them into account so that the distributions
experiment lit with diffuse light showing the wave pattern as the
correspond to 2NT realizations. The curve is the fit by Eq. (1)
walker crosses the aperture. The picture was taken at a time
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
when the trajectory, initially perpendicular to the aperture, was
(L ! 14:7 mm and
A. Eddi#et
Fal.! 4:75 mm). The curve is the fit by
[Couder,2008]
deflected by the interference withc reflected waves.
Eq. (1) with L=#F ! 2:86.
b
C
[Harris,2013]
[Couder,2009]
a 1
30
0.5
y/R
20
0
10
−0.5
−1
−1
0
−0.5
0
x/R
0.5
1
4000
3000
n/r
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle
!i and by its normalA
B
ized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June(a)is
13,
13
FIG. 3.This
(color)
Trajectories
of a droplet of diameter
= 0.67 mm walkingrealizations:
in a circular corral with with
radius 14.3apmm and
also
observed
on Dindividual
2000
1000
0
velocity difference
from mean (%)
50
0
−50
0
0.2
0.4
0.6
r/R
0.8
1
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
FIG.
5
(color
online).
(a) Sup
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!
:
(1)
!
!
!
!
walker
velocity
V ¼ 12 mm
$Lof
sin"=#
F
a barrier with e ¼ 3 mm leadin
Fig. 1: The crystalline organization of a large number of
droplets
= 61) with a triangular
This amplitude
of(Ndiffraction
of lattice.
a plane wave turns out to
offit10forconsecutive
collisions
of
provide an approximate
these histograms.
Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
comes shorter as the wall thickness is decreased
[Fig. 4(c)].
become smaller.
[Couder,2008]
possible
orbits
being discrete, we label
the plateaus by an
Ultimately the limit cycle is never reachedThe
[Fig.
4(d)]
[10–
order n, with n ¼ 0 corresponding to the tightest orbit observed at
12]. The trajectories inside the cavity and
the escapes are
large Ω. Each plateau crosses the continuous short path-memory
correlated. As long as the droplet follows
limit
curvethe
given
by Eq.cycle,
3 at a value Ω0n of the angular velocity (Fig. 2B).
At thesebecome
crossings, disthe observed orbits have the same radius
no escape is observed. When the trajectories
exp
exp
0
0
Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term
ordered, the collisions with the walls show
a larger
variety
memory.
A plateau
extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator
A
B
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Y (with L=# ! 2:86). (b) A superposition of three
• Lattice structures
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions ! ! 90 and Y ! 0:1.
effects
• FIG.Tunneling
1. (a), (b) Sketch of the central region of the experimental
(c) Experimental histogram of the deviation " as obtained with
cell (seen from above and in a cross section along the y axis,
N ! 125 single walkers with L=# ! 2:11 (L ! 14:7 mm and
Wave-like
statistics in confined geometries
• respectively).
An individual trajectory is shown in (a) and the
# ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of ! , y , and " are given. The width of the slit being
terpart with respect to the axis of the aperture, the statistic was
• L,Scattering?
the parameter of impact is Y ! y =L. (c) A photograph of the
i
F
i
#
i
3
a 1
30
i
0.5
T
F
F
i
i'
20
i
y/R
improved by taking them into account so that the distributions
experiment lit with diffuse light showing the wave pattern as the
correspond to 2NT realizations. The curve is the fit by Eq. (1)
walker crosses the aperture. The picture was taken at a time
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
when the trajectory, initially perpendicular to the aperture, was
(L ! 14:7 mm and
A. Eddi#et
Fal.! 4:75 mm). The curve is the fit by
[Couder,2008]
deflected by the interference withc reflected waves.
Eq. (1) with L=#F ! 2:86.
b
C
[Harris,2013]
[Couder,2009]
0
10
−0.5
−1
−1
0
−0.5
0
x/R
0.5
1
4000
3000
n/r
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle
!i and by its normalA
B
ized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June(a)is
13,
13
FIG. 3.This
(color)
Trajectories
of a droplet of diameter
= 0.67 mm walkingrealizations:
in a circular corral with with
radius 14.3apmm and
also
observed
on Dindividual
2000
1000
0
velocity difference
from mean (%)
50
0
−50
0
0.2
0.4
0.6
r/R
0.8
1
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
FIG.
5
(color
online).
(a) Sup
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!
:
(1)
!
!
!
!
walker
velocity
V ¼ 12 mm
$Lof
sin"=#
F
a barrier with e ¼ 3 mm leadin
Fig. 1: The crystalline organization of a large number of
droplets
= 61) with a triangular
This amplitude
of(Ndiffraction
of lattice.
a plane wave turns out to
offit10forconsecutive
collisions
of
provide an approximate
these histograms.
Hence, the
ulus of the walkers velocity V W ðΩÞ is observed as the orbits
comes shorter as the wall thickness is decreased
[Fig. 4(c)].
become smaller.
[Couder,2008]
possible
orbits
being discrete, we label
the plateaus by an
Ultimately the limit cycle is never reachedThe
[Fig.
4(d)]
[10–
order n, with n ¼ 0 corresponding to the tightest orbit observed at
12]. The trajectories inside the cavity and
the escapes are
large Ω. Each plateau crosses the continuous short path-memory
correlated. As long as the droplet follows
limit
curvethe
given
by Eq.cycle,
3 at a value Ω0n of the angular velocity (Fig. 2B).
At thesebecome
crossings, disthe observed orbits have the same radius
no escape is observed. When the trajectories
exp
exp
0
0
Rn ¼ Rn ðΩn Þ ¼ RC ðΩ0n Þ they would have had with short-term
ordered, the collisions with the walls show
a larger
variety
memory.
A plateau
extends on both sides of the curve given by
relation 3 between two limiting values Ω−n < Ωn < Ωþ
n . A hysteresis is observed: The abrupt transition from one plateau to the
Perspectives
Analogies with quantum mechanics
• Single/double slit
• Charged particle in magnetic field
FIG. 2 (color online). (a) The measured deviations a of suc• Harmonic oscillator
A
B
cessive individual particles as a function of[Couder,2010]
the parameter of
impact Y (with L=# ! 2:86). (b) A superposition of three
• Lattice structures
different trajectories of the same droplet passing 3 times through
the slit with similar initial conditions ! ! 90 and Y ! 0:1.
effects
• FIG.Tunneling
1. (a), (b) Sketch of the central region of the experimental
(c) Experimental histogram of the deviation " as obtained with
cell (seen from above and in a cross section along the y axis,
N ! 125 single walkers with L=# ! 2:11 (L ! 14:7 mm and
Wave-like
statistics in confined geometries
• respectively).
An individual trajectory is shown in (a) and the
# ! 6:95 mm). Since each trajectory has a symmetrical coundefinitions of ! , y , and " are given. The width of the slit being
terpart with respect to the axis of the aperture, the statistic was
• L,Scattering?
the parameter of impact is Y ! y =L. (c) A photograph of the
improved by taking them into account so that the distributions
experiment lit with diffuse light showing the wave pattern as the
Entanglement?
correspond to 2N realizations. The curve is the fit by Eq. (1)
• walker
crosses the aperture. The picture was taken at a time
i
F
i
#
i
3
a 1
30
i
y/R
0.5
F
F
i
i'
0
T
20
i
10
−0.5
T
with L=#F ! 1:96. (d) Histogram obtained with L=#F ! 3:1
when the trajectory, initially perpendicular to the aperture, was
(L ! 14:7 mm and
A. Eddi#et
Fal.! 4:75 mm). The curve is the fit by
[Couder,2008]
deflected by the interference withc reflected waves.
Eq. (1) with L=#F ! 2:86.
b
C
[Harris,2013]
[Couder,2009]
−1
−1
0
−0.5
0
x/R
0.5
1
4000
3000
n/r
ments, having the same walker cross the slit, with various
initial motions. As shown on Fig. 1(a) each initial trajectory is defined by its incidence angle
!i and by its normalA
B
ized impact parameter in the slit Yi ! yi =L
("0:5 < Yi < 0:5) [Fig. 1(a)]. The walkers were always
started far from the screen and only those impinging perpendicularly on the slit (!I ! 90# ) were retained.
The plot "$Yi % [Fig. 2(a)] shows that there is no simple
relation between the deviation and the parameter of impact.
Thursday,
June(a)is
13,
13
FIG. 3.This
(color)
Trajectories
of a droplet of diameter
= 0.67 mm walkingrealizations:
in a circular corral with with
radius 14.3apmm and
also
observed
on Dindividual
2000
1000
0
velocity difference
from mean (%)
50
0
−50
0
0.2
0.4
0.6
r/R
0.8
1
#F =L and large amplitude lateral lobes are clearly observed. Hence, on Figs. 2(c) and 2(d) the general shapes
of the histograms are compared with the amplitude diffraction pattern of a plane wave passing through a slit:
!
!
!
!
FIG.
5
(color
online).
(a) Sup
!
!
sin$$L
sin"=#
%
!
!
F
!
!
f$"% ! A!
!
:
(1)
!
!
!
!
walker
velocity
V ¼ 12 mm
$Lof
sin"=#
F
a barrier with e ¼ 3 mm leadin
Fig. 1: The crystalline organization of a large number of
droplets
= 61) with a triangular
This amplitude
of(Ndiffraction
of lattice.
a plane wave turns out to
offit10forconsecutive
collisions
of
provide an approximate
these histograms.
Hence, the