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WATER 9, 1-27, OCTOBER 25, 2017 1
WATER
Determinants of Faraday Wave-Patterns in
Water Samples Oscillated Vertically at a
Range of Frequencies from 50-200 Hz
Merlin Sheldrake* & Rupert Sheldrake
*Corresponding author: merlinsheldrake@gmail.com
Keywords: Faraday wave, pattern formation, vibrational mode, catastrophe theory, bifurcation, analogue model,
morphogenesis, parametric oscillation, nonlinear dynamic system
Received: July 13, 2017; Revised: September 6, 2017; Accepted: September 15, 2017; Published: October 25, 2017;
Available Online: October 25, 2017
DOI: 10.14294/WATER.2017.6
Abstract
The standing wave patterns formed on the
surface of a vertically oscillated uid en-
closed by a container have long been a sub-
ject of fascination, and are known as Fara-
day waves. In circular containers, stable, ra-
dially symmetrical Faraday wave-patterns
are resonant phenomena, and occur at the
vibrational modes where whole numbers of
waves t exactly onto the surface of the uid
sample. These phenomena make excellent
systems for the study of pattern formation
and complex nonlinear dynamics. We pro-
vide a systematic exploration of variables
that affect Faraday wave pattern formation
on water in vertical-walled circular contain-
ers including amplitude, frequency, volume
(or depth), temperature, and atmospheric
pressure. In addition, we developed a novel
method for the quantication of the time
taken for patterns to reach full expression
following the onset of excitation. The exci-
tation frequency and diameter of the con-
tainer were the variables that most strongly
affected pattern morphology. Amplitude
affected the degree to which Faraday wave
patterns were expressed but did not affect
pattern morphology. Volume (depth) and
temperature did not affect overall pattern
morphology but in some cases altered the
time taken for patterns to form. We discuss
our ndings in light of René Thom’s catas-
trophe theory, and the framework of attrac-
tors and basins of attraction. We suggest
that Faraday wave phenomena represent a
convenient and tractable analogue model
system for the study of morphogenesis and
vibrational modal phenomena in dynami-
cal systems in general, examples of which
abound in physical and biological systems.
Introduction
When uid enclosed by a container is sub-
jected to a vertical oscillation, standing
waves arise. These standing waves, which
depend on reections from the edge of the
container, are known as Faraday waves
(Miles and Henderson, 1990). At some fre-
quencies and amplitudes they form highly
ordered patterns, while at others they give
rise to chaotic dynamics (Simonelli and
Gollub, 1989). Faraday wave phenomena
make excellent systems for the study of
pattern formation because of the high de-
gree of control compared with other pat-
tern-forming systems such as convection
or chemical reactions (Huepe et al., 2006),
and the great richness and variety of pat-
terns that are possible (Topaz et al., 2004;
Rajchenbach and Clamond, 2015). Thus,
complex nonlinear phenomena can be
explored using a relatively simple experi-
mental device.
WATER 9, 1-27, OCTOBER 25, 2017 2
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Faraday waves and similar resonant phe-
nomena have been the subject of fascina-
tion for several centuries. The patterns
formed by sound vibrations were described
by Leonardo da Vinci, who found that dust
particles formed mounds and hillocks
when the underlying surface was vibrated
(Winternitz, 1982). Similar phenomena
were experimentally and observationally
researched by Galileo Galilei (1668), Rob-
ert Hooke (Inwood, 2003), Ernst Chladni
(Chladni, 1787; Waller and Chladni, 1961),
Michael Faraday (Faraday, 1831), and Lord
Rayleigh (Rayleigh, 1883; 1887).
Faraday and Rayleigh found that the trans-
verse waves excited on the surface of a ver-
tically vibrated uid oscillated at half the
excitation frequency (Faraday, 1831; Ray-
leigh, 1883). The 2:1 ratio between driving
frequency and oscillatory response is char-
acteristic of parametric resonance (Douady
and Fauve, 1988; Bechhoefer et al., 1995).
A familiar analogy is a child’s swing. If the
swing is given a push when it reaches its
highest point on one side, and then another
push when it reaches the other extreme,
these two pushes will help it complete one
back-and-forth cycle, and the swing will os-
cillate at half the excitation frequency.
Standing waves on the surface of uids are
formed when the driving amplitude ex-
ceeds a critical threshold (Bechhoefer et al.,
1995). Above this point—the so-called Far-
aday instability—there is a bifurcation from
a single state of motion of the free surface of
the liquid to multiple states of motion, such
that the surface of the liquid might be ris-
ing or falling (Miles and Henderson, 1990).
Faraday wave patterns are inuenced both
by properties of the liquid sample, such as
the diameter and shape of the container,
and the viscosity of the uid (intrinsic fac-
tors), and properties of the oscillation such
as excitation frequency and amplitude (ex-
trinsic factors; Abraham, 1976).
Oscillating uid samples in enclosed con-
tainers have particular modes of vibration,
in which whole numbers of waves t exactly
onto the surface (Ball, 2009). These Fara-
day wave modes arise from a combination
of the outgoing waves excited by the driv-
ing frequency and the waves reected from
the vertical wall of the cell, and depend on
the frequency and amplitude of the excita-
tion and the diameter of the cell. They are
resonant patterns in the same sense that
standing waves in wind instruments such
as utes are resonant patterns of vibration.
Faraday waves in uids are also analogous
to the patterns formed by vibrating solid
particles on metal plates described by Ernst
Chladni in 1787 (Chladni, 1787; Waller and
Chladni, 1961). In Chladni’s gures, the pat-
terns are revealed by particles accumulat-
ing along the nodal lines, which are parts of
the plate that are neither moving upwards
nor downwards, and form “a spiderweb of
motionless curves” (Abraham, 1976). By
contrast, in samples of liquid subjected to
vertical vibrations, patterns appear as os-
cillations of standing waves, with any given
point ipping between peak and trough.
Much of the research on Faraday waves
has focused on attempts to characterize
them mathematically, using theoretical
approaches based on equations stemming
from physical theories, well-reviewed in
Ibrahim (2015). However, the predictions
made by such mathematical models of-
ten disagree with experimental results,
or else make predictions that are only ac-
curate under restricted conditions (Miles
and Henderson, 1990; Bechhoefer et al.,
1995). For example, an approach based on
the Mathieu equations, which are used to
model parametric resonance, was only par-
tially successful in modelling surface waves
because it could not take into account u-
id viscosity (Benjamin and Ursell, 1954;
Bechhoefer et al., 1995). Other investiga-
tions have used the Navier-Stokes equa-
tions (which describe the ow of viscous
substances) to characterize Faraday wave
pattern dynamics (eg. Kumar and Tuck-
erman, 1994). But solving Navier-Stokes
WATER 9, 1-27, OCTOBER 25, 2017 3
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equations for such complex dynamical sys-
tems presents an enormous computational
challenge. Furthermore, even if the equa-
tions can be solved, these models cannot
account for perturbations or disruptions
that the system is likely to encounter in the
real world (Ball, 2009).
These problems with classical mathemati-
cal approaches have led some to adopt a
more phenomenological approach towards
Faraday wave systems, looking at the em-
pirical patterns themselves, as opposed to
the theoretical modes of these patterns. For
the most part, these studies explore vibra-
tional modes and the transitions between
them using analogue stimulation and di-
rect observation. For example, Abraham
(1975) conceived of the experimental Fara-
day wave system as more than a way to ver-
ify the predictions of equation-based mod-
els. Rather, he understood it as a top-down
exploratory process, and described his ex-
perimental system as “an analogue com-
puter simulating the Navier-Stokes equa-
tions” (Abraham, 1975). Another way to
understand this approach is in the context
of Richard Feynman’s call for a “method of
understanding the qualitative content of
equations,” observing that “today we can-
not see that the water ow equations con-
tain such things as the barber pole struc-
ture of turbulence that one sees between
rotating cylinders” (Feynman et al. 1964).
Following this empirical approach, some
researchers have focused on the Faraday
wave patterns that interact with each other
on the boundaries between different modes
(Ciliberto and Gollub, 1984; Simonelli and
Gollub, 1989). Some have used the Faraday
wave system to model the onset of chaotic
behavior at these boundaries, or at high
amplitudes (Tullaro et al., 1989; Gluck-
man et al., 1993). Others have used com-
parable systems to model biological phe-
nomena, most notably the transduction of
sound in the cochlea, which underlies our
own sense of hearing (von Bekesy, 1960).
Experimental work in this area was inu-
enced by Hans Jenny (2001), who investi-
gated the effects of vibrations on the pat-
terns produced by vibrating uids, pastes
and powders.
Faraday wave systems provide an excellent
arena for the physical modelling of preva-
lent theoretical frameworks in the physical
and biological sciences. These include: i)
the widespread mathematical procedure,
the Eigendecomposition of the Laplace op-
erator, which lies at the heart of theories of
heat, light, sound, electricity, magnetism,
gravitation and uid mechanics (Stewart,
1999); ii) the well-known reaction-diffusion
model of Turing (1952), which is based on
standing or oscillatory chemical waves (ie.
peaks and troughs), and underpins much
contemporary understanding of biological
morphogenesis (Ball, 2015); and iii) the ca-
tastrophe theory of René Thom, which pro-
vides a mathematical underpinning for the
modelling of sudden transitions, or catas-
trophes, between alternative states (Thom,
1975), and which has been experimentally
applied to Faraday wave systems by Abra-
ham (1972). Examples of systems that ex-
hibit complex nonlinear dynamics and
modal behavior analogous to those in the
Faraday wave system are widespread and
range from the laser-induced vibrations of
ions in a crystal lattice (Britton et al., 2012)
to cortical activity in the human brain (Ata-
soy et al., 2016).
Despite the practical and theoretical value
of the experimental Faraday wave system to
numerous areas of the physical and life sci-
ences, to our knowledge there are no inves-
tigations into the factors that inuence the
formation of Faraday wave patterns across
a wide frequency range. Some studies have
examined the effects of square, rectangu-
lar and pentagonal as opposed to circular
uid reservoirs (Douady and Fauve, 1988;
Simonelli and Gollub, 1989; Torres et al.,
1995), and some refer to the effect of uid
volume (Henderson and Miles, 1991), tem-
perature (Bechhoefer et al., 1995), topogra-
WATER 9, 1-27, OCTOBER 25, 2017 4
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phy of the reservoir bottom (Kalinichenko
et al., 2015), and liquid purity (Henderson
et al., 1991; Henderson, 1998). However,
these studies do not systematically report
the effects of the major factors determining
overall pattern morphology.
Here we present a systematic exploration of
variables that affect Faraday wave pattern
formation on water in vertical-walled circu-
lar containers. We investigated the effects
of amplitude, frequency, volume (or depth),
temperature, and atmospheric pressure on
the morphology of Faraday wave patterns.
We also developed a novel method for the
quantication of the time taken for patterns
to reach full expression following the onset
of excitation. We studied the spectrum of
vibratory modes across a wide frequency
range, from 50-200 Hz. In addition, we de-
termined the stability boundaries around
three sample frequencies.
Methods
Experimental Apparatus
We generated Faraday waves by vertically
oscillating a liquid sample held in a cylin-
drical container, or visualizing cell, using
sound frequencies. We used the Cyma-
Scope instrument (Sonic Age, Cumbria,
UK). The cell was vertically oscillated using
a voice coil motor (VCM) optimized to op-
erate between 50-200 Hz. Inherent reso-
nances in the electromechanical system of
the VCM were damped internally by enclos-
ing the back pressure of the VCM in an in-
nite bafe arrangement and by inserting a
thermal compressor into the VCM’s circuit.
Resonances were further controlled using
a Klark Technik Square One thirty-band
graphic equalizer (Klark Technik, Kidder-
minster, Worcestershire, UK). A diagram
of the signal path is provided in Figure 1.
The VCM was driven with frequencies de-
ned by a function generator (Aim TTi,
Huntingdon, Cambridgeshire, UK). Unless
otherwise stated, we used sinusoidal oscil-
lations, although we also evaluated the ef-
fect of using square and triangle wave forms
generated by the function generator on pat-
tern formation. The amplitudes of the wave
forms were dened in terms of voltage from
Figure 1. Diagram of the signal path and experimental apparatus.
Camera
LED light ring
MIDI input
Visualising cell
and uid sample
Voice coil
motor (VCM)
Function
generator
30-band graphic
equaliser
Volcano
attenuator
Computer
Amplier
(class A)
Voltmeter
WATER 9, 1-27, OCTOBER 25, 2017 5
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the function generator. The function gen-
erator did not provide sufcient power on
its own and so the signal was amplied us-
ing a Lindell Audio AMPX Class A ampli-
er (Lindell Audio, Phuket, Thailand). To
achieve precise and programmable control
of the amplitude, we included a Volcano at-
tenuator (Sound Sculpture, Bend, OR, USA)
which used a resistor selector circuit con-
trolled by MIDI control change messages
to attenuate the signal to the required am-
plitude. We used as a proxy for amplitude
the magnitude of the signal delivered to the
VCM in millivolts (mV), measured using a
voltmeter (Hioki, Nagano, Japan) connect-
ed across the output terminals of the ampli-
er. Higher excitation frequencies required
greater amplitudes to excite the sample to
the point of Faraday instability and pattern
formation. We found the appropriate am-
plitude by trial and error.
We used cells of three different diameters.
The diameters were 10.00 mm, 24.25 mm,
and 49.50 mm. Unless otherwise stated, all
results presented are obtained using the
medium cell (diameter = 24.25 mm). Cells
were made of fused quartz glass, with sand-
blasted matt black surface on the bottom so
that the light used for visualizing was re-
ected from the surface of the liquid only.
Cells were cleaned using 100% ethanol, and
rinsed with deionized water (resistivity 18
M-ohms). We used the same kind of de-
ionized water for our liquid samples when
performing experimental oscillations. The
cell was illuminated from above by a ring
of LEDs. Wave patterns were imaged from
above using a Canon EOS SLR 7D Mark
II with a Canon 100 mm f2.8 macro lens
(Canon, Tokyo, Japan), shooting at 25
frames per second (fps). The camera was
levelled to ensure that the line of sight was
perpendicular to the surface of the liquid,
thus avoiding parallax errors. The high-
speed video recordings used to ascertain
the dominant frequency of the excited sur-
face ripples were recorded using a Canon
FS7 (Canon, Tokyo, Japan) with the same
lens as above, shooting at 150 fps.
Data Processing and Analysis
Classication of pattern morphol-
ogy: We classied the morphology of the
patterns resulting from the Faraday wave
forms based on fold symmetry. For exam-
ple, a regular six-pointed pattern would be
classied as having six-fold symmetry. This
is a straightforward and inclusive taxonom-
ic criterion that accounts for similarities
and differences in the overall morphology
of patterns.
Evaluation of time to full expression
of pattern (TFE): We dene the TFE as
the time taken for the pattern to reach full
expression following the onset of the exci-
tation frequency. We devised a semi-auto-
mated, objective procedure for dening the
point at which patterns had reached full ex-
pression. Briey, videos were cropped, so
that only the circular surface of the water
sample was visible. Video frames were then
thresholded so that every pixel was either
black or white, and the mean gray value of
each frame calculated (mean value = black
pixels + white pixels / total number of pix-
els). Because the patterns showed up as
light on a dark background (eg. Figure 3),
frames with fuller expression of the pattern
had higher gray values than those where
the pattern was less expressed. The stan-
dard deviation of the mean gray values was
plotted against video frame to produce a
prole plot of the video (Figure 2). We used
the standard deviation of the mean gray
value rather than the mean gray value it-
self because the resulting prole plots were
less noisy. Prole plots clearly indicated the
point at which a stable pattern had formed,
and were used to dene a 95% condence
area around the gray values of the pattern
in full expression. The time at full expres-
sion was dened as the time at which the
gray value of a given frame reached the low-
er 95% condence limit of the gray values
of the pattern in full expression (ie. the time
WATER 9, 1-27, OCTOBER 25, 2017 6
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Figure 2. Method used to evaluate time taken for the pattern to reach full expression
following the onset of the excitation frequency (time to full expression, or TFE). In a,
video frames are displayed, showing the onset of the excitation frequency (i), and the
onset of full expression of the pattern (ii). b is a prole plot showing the standard
deviation of the mean gray value of each video frame as a function of frame number.
The onset of the excitation frequency (i), and the onset of full expression of the pattern
(ii) are marked. Red horizontal lines show the mean (solid line) and 95% condence
area (dashed lines) around the standard deviation of the mean gray values of the
pattern in full expression.
0.25
0.30
0.35
0.40
0.45
0 50 100 150 200 250 300
i
i
ii
ii
a
b
Standard deviation of mean grey value Frame number
Frame number
0
30
60
90
120
150
180
210
240
270
WATER 9, 1-27, OCTOBER 25, 2017 7
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at which there was a statistically signicant
probability—at α = 0.05—that a frame’s
gray value had reached the level of those
where the pattern was fully expressed).
TFE was then calculated by subtracting the
time at onset of excitation from the time at
full expression. Image processing was con-
ducted in R v. 3.1.2 (R Development Core
Team, 2014), using the package EBImage
(Pau et al., 2010). Custom functions were
written to execute the procedures outlined
above.
Effect of Amplitude on Pattern
Formation
To ascertain the effect of amplitude on pat-
tern formation we oscillated a 2.5 ml sample
of water at incrementally increasing ampli-
tudes at each of three sample frequencies,
56, 111 and 180 Hz, replicated three times.
The sample of water was changed between
each trial. We calculated the TFE using the
method described above. We evaluated the
effect of amplitude on the variability of the
TFE by modelling the standard deviation of
the TFE of the three replicates at each am-
plitude as a function of amplitude. We used
linear models and ran separate models for
each of the three frequencies. All models
met assumptions of normality and homo-
geneity of variances.
Effect of Sample Volume (Depth) on
Pattern Formation
To explore the effect of volume on pattern
formation, we oscillated four volumes of
water (1.5 ml, 2.5 ml, 3.5 ml and 4.5 ml) at
each of the three sample frequencies, repli-
cated three times. The sample of water was
changed between each trial. We calculated
the TFE using the method described above,
and evaluated the effect of sample volume
on TFE using linear models (n = 12), run-
ning separate models for each of the three
frequencies. All models met assumptions of
normality and homogeneity of variances.
All statistical analysis was conducted in R
v. 3.1.2 (R Development Core Team, 2014).
To measure evaporative loss of water we
oscillated water for 1 minute, 5 minute and
10 minute periods, and measured the mass
of the sample before and after oscillation.
The 10 minute period substantially exceed-
ed the length of time that any sample was
oscillated in the course of these investiga-
tions.
Effect of Temperature on
Pattern Formation
To investigate the effect of temperature on
pattern formation, we oscillated samples of
water at each of the three sample frequen-
cies at ve temperatures (5oC, 10oC, 15oC,
20oC, 25oC, and 30oC), replicated three
times. The sample of water was changed
between each trial. Temperature was con-
trolled using a thermostatically controlled
incubator (LMS, Sevenoaks, Kent, UK). We
measured the temperature of parallel sam-
ples of water before oscillation to ensure
that the samples had fully equilibrated. Par-
allel samples were used to take temperature
readings to avoid changing the temperature
of the actual sample by the insertion of the
thermometer. We calculated the TFE using
the method described above, and evaluated
the effect of sample temperature on TFE
using linear regression, running separate
models for each of the three frequencies (n
= 18). All models met assumptions of nor-
mality and homogeneity of variances.
Effect of Frequency on Pattern
Formation
To ascertain the effect of frequency on pat-
tern formation we oscillated water samples
of 2.5 mls from 50-200 Hz, increasing in 1
Hz increments, changing the sample of wa-
ter every 10 trials, i.e. every 10 Hz (prelimi-
nary experiments demonstrated that pat-
tern morphology was unaffected by prior
oscillation).
Determination of Stability
Boundaries
In a Faraday wave system, liquid oscillates
WATER 9, 1-27, OCTOBER 25, 2017 8
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in a single stable pattern at certain frequen-
cies when the amplitude is above a critical
threshold (Ibrahim, 2015). These critical
amplitudes vary as a function of excitation
frequency. Plots of critical amplitudes as a
function of excitation frequency are known
as stability curves, and represent the stabil-
ity boundary of the system (Simonelli and
Gollub, 1988; 1989; Douady, 1990; Hender-
son and Miles, 1991). To describe the stabil-
ity boundaries of our system we oscillated
water samples at particular frequencies at
amplitudes well above those required for
pattern formation. We then slowly reduced
the amplitude with a slider until the pattern
disappeared. We dened this value as the
critical amplitude required to sustain a giv-
en Faraday wave pattern. We repeated this
at 0.5 Hz intervals both below and above
each of our three sample frequencies. We
plotted the critical amplitude values as a
function of excitation frequency to describe
stability curves around each of our three
sample frequencies. This approach is simi-
lar to that described in Simonelli and Gol-
lub (1988).
We asked whether reducing the ampli-
tude could shift the pattern to an alterna-
tive form when close to the transition point
between stability curves. To test this we
used two frequencies (115 Hz and 184 Hz)
close to the boundary between two stabil-
ity curves. At these frequencies, one of two
alternative patterns formed unpredictably.
We conducted four trials per sample of wa-
ter, on six samples of water per frequency
(a total of 48 trials), recording which pat-
tern formed. After the pattern was fully ex-
pressed, we slowly reduced the amplitude,
recording any change in the pattern that
occurred.
Effect of Atmospheric Pressure on
Pattern Formation
To investigate the effect of atmospheric
pressure on pattern formation, we excited
samples of water at the two transition fre-
quencies (115 Hz and 184 Hz). We conduct-
ed four trials per sample of water, on six
samples of water per frequency (a total of
48 trials), recording which pattern formed.
This was performed on ve days with dif-
fering atmospheric pressures (ranging
from 987-1021 mm). Between each sample,
the vessel was cleaned with ethanol, rinsed
with deionized water, and dried.
Data were analyzed as the proportion of tri-
als for a given sample forming one or the
other of the patterns using generalized lin-
ear models (GLMs) with atmospheric pres-
sure and temperature as predictors (the
temperature varied only slightly across the
trial days, from 19.1-21.6 oC). The data were
overdispersed (the residual deviance was
greater than the number of degrees of free-
dom), and so quasibinomial error struc-
tures were used (Crawley, 2007). Signi-
cance was assessed using χ2 tests.
Results
Morphology of the Three Sample
Frequencies
In initial tests we explored a wide range of
frequencies, and selected three that gave
repeatable patterns in order to investigate
those factors affecting pattern formation in
more detail. We later returned to a system-
atic examination of the effects of frequency
on pattern over the entire range of our ap-
paratus, from 50-200 Hz, as discussed be-
low.
We chose three frequencies that produced
patterns with different morphologies. At 56
Hz, the pattern showed six-fold symmetry;
at 111 Hz, the pattern showed ten-fold sym-
metry; and at 180 Hz the pattern showed
fourteen-fold symmetry (Figure 3, compos-
ite images). The overall patterns were com-
posites of two alternating phases of oscil-
lation (Figure 3, phases I and II), whereby
peaks became troughs, and troughs peaks.
We could see these alternating peaks and
troughs only by using fast shutter speeds;
at the normal shutter speed of 1/30 sec-
WATER 9, 1-27, OCTOBER 25, 2017 9
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Effects of Amplitude
We observed a minimum amplitude below
which no pattern formed, and a maximum
amplitude above which patterns became
overdriven and severely distorted. With-
in the range of amplitudes that resulted
in stable pattern formation, variation of
the excitation amplitude did not alter the
overall morphology of the pattern. For ex-
ample, the six-fold 56 Hz pattern remained
six-fold across an amplitude range of 65
mV to 491 mV. Nonetheless, amplitude did
alter the degree to which patterns were ex-
pressed (Figure 3). At 111 Hz and 180 Hz, at
the highest amplitude resulting in a coher-
ent pattern, the pattern became unstable
and shifted to an alternative pattern. In the
case of 111 Hz, the 10-fold pattern shifted to
ond, the pictures were a composite of the
peak and trough patterns. Thus the six-fold
symmetry we observed at 56 Hz represents
a standing wave pattern of mode three,
with three alternating peaks and troughs,
the ten-fold symmetry at 111 Hz mode ve,
and the fourteen-fold symmetry at 180 Hz
mode seven.
We measured the rate of oscillation by
examining frame-by-frame photographs
from high-speed lms, and found that the
dominant frequency of the Faraday waves
was half the excitation frequency (in other
words, the rst subharmonic of the excita-
tion frequency: f = fo/2) (Francois et al.,
2013). For example, at an excitation fre-
quency of 56 Hz, the frequency of oscilla-
tion of the water sample was 28 Hz. This
conrmed the results in the classic papers
by Faraday (1831) and Rayleigh (1883).
Figure 3. The three sample frequencies resulted
in different pattern morphologies shown as
composite images (left-hand panels), and images
of alternating phases (center and right-hand
panels). 56 Hz resulted in six-fold symmetry, 111
Hz in ten-fold symmetry, and 180 Hz in fourteen-
fold symmetry.
a56 Hz
Composite Phase I Phase II
b111 Hz
c180 Hz
Figure 4. Amplitude altered the degree to
which patterns were expressed, and in all but
two cases did not affect the overall morphology
of the pattern. Patterns formed over a range of
twelve amplitudes are shown for each of the three
sample frequencies. Amplitude is reported beneath
each panel in millivolts (mV). Instances in which
multiple patterns formed at a given amplitude
are marked with horizontal bars and emboldened
millivoltages.
b111 Hz
a56 Hz
c180 Hz
WATER 9, 1-27, OCTOBER 25, 2017 10
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Figure 5. The time taken for patterns to reach full expression after the start of excitation (TFE)
decreased with increasing amplitude (a), as did the variation in TFE between replicates (b-d). In a,
separate curves are plotted for each of the three sample frequencies, with error bars depicting the
standard deviation of the TFE for three replicates at each amplitude. b-d show signicant negative
relationship between the standard deviation of the TFE and amplitude. In b-d, lines are the tted
response of separate linear models for each of the three frequencies, with gray bands depicting 95%
condence intervals. 56 Hz (gray); T1,17 = -2.76; P = 0.01; 111 Hz (maroon); T1,14 = -3.41; P = 0.004; 180
Hz (blue); T1,11 = -2.87, P = 0.02.
0
5
10
15
20
25
30
0 200 400 600 800 1000 1200 1400
Amplitude (mV)
Time to full expression (s)
56 Hz
111 Hz
180 Hz
−4
−2
0
2
100 200 300 400 500 200 400 600 800 300 600 900 1200
Amplitude (mV)
log(standard deviation of TFE)
a
b56 Hz c111 Hz d180 Hz
WATER 9, 1-27, OCTOBER 25, 2017 11
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a 16-fold pattern. In the case of 180 Hz, the
14-fold pattern shifted to an unstable 20-
fold pattern (Figure 3c).
Amplitude altered the time taken for the
pattern to reach full expression follow-
ing the onset of excitation (time to full ex-
pression, or TFE), with higher amplitudes
causing the pattern to form more quickly
until a minimum TFE was reached (Fig-
ure 5). Variation between replicate trials
was greater at lower amplitudes at all three
frequencies (linear regression: 56 Hz; T1,17
= -2.76; P = 0.01; 111 Hz; T1,14 = -3.41; P =
0.004; 180 Hz; T1,11 = -2.87, P = 0.02; Fig-
ure 5).
Effects of Wave Form
The wave form (sine wave, square wave or
triangle wave) of the driving frequency did
not alter the overall morphology of the pat-
tern. For example, the six-fold 56 Hz pat-
tern remained six-fold when oscillated using
sine, square or triangle waves (Figure 6).
Figure 6. The wave form (sine wave, square
wave or triangle wave) of the driving frequency
did not alter the overall morphology of the pattern
at any of the three sample frequencies.
sine square triangle
a56 Hz
b111 Hz
c180 Hz
Effects of Sample Volume and Depth
The sample volume, and thus depth, of os-
cillating water did not alter the overall mor-
phology of the pattern. For example, the
six-fold 56 Hz pattern remained six-fold
across a volume range of 1.5 mls to 4.5 mls
with only marginal changes in the degree to
which the pattern was expressed (Figure 7).
Higher volumes of water increased the time
taken for the pattern to reach full expres-
sion (linear regression: 56 Hz; T1,10 = 2.14;
P = 0.06; 111 Hz; T1,10 = 5.81; P < 0.001; 180
Hz; T1,10 = 5.89, P < 0.001; Figure 8).
Evaporative loss of water during experi-
mental time periods was minimal: over a
ten minute period (longer than any single
water sample was used for in this investi-
gation) a mean of 0.003 g of water (n = 6;
std. dev. = 0.005) was lost from the quartz
cell at standard operating temperature and
pressure. This represents 0.13% of the total
mass of water in the cell.
Although at lower volumes the time taken
for the pattern to reach full expression was
Figure 7. The volume, and thus depth, of
oscillating water had a marginal effect on the
degree to which patterns were expressed, and did
not affect the overall morphology of the pattern.
Patterns formed at four volumes (1.5, 2.5, 3.5 and
4.5 ml) are shown for each of the three sample
frequencies.
1.5 ml 2.5 ml 3.5 ml 4.5 ml
a
56 Hz
b
111 Hz
c
180 Hz
WATER 9, 1-27, OCTOBER 25, 2017 12
WATER
less (Figure 8), volume had minimal effects
on pattern morphology (Figure 7). This in-
dicates that any evaporative loss of water
over the course of our experiments – less
than 0.13% - is unlikely to affect the pattern
formed, since far larger changes – between
-40% and +180% of the standard volume
we used in our experiments – had almost
no effect on pattern morphology (Figure 7).
Effects of Temperature
The temperature of the oscillating water
and cell did not alter the overall morphol-
ogy of the pattern. For example, the six-fold
56 Hz pattern remained six-fold across a
temperature range of 5-30oC with no no-
ticeable differences in the degree to which
the pattern was expressed (Figure 9).
Figure 8. The time taken
for the pattern to reach
full expression after the
start of forced oscillations
(TFE) increased with higher
volumes, and thus depths,
of water. Lines are the
tted response of separate
linear models for each of
the three frequencies, with
gray bands depicting 95%
condence intervals, solid
lines indicating a signicant
relationship (P < 0.05),
and dotted lines indicating
a marginally signicant
relationship (0.5 < P < 0.1).
56 Hz (gray); T1,10 = 2.14; P =
0.06; 111 Hz (maroon); T1,10 =
5.81; P < 0.001; 180 Hz (blue);
T1,10 = 5.89, P < 0.001.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1.5 2.5 3.5 4.5 1.5 2.5 3.5 2.5 3.5 4.5
Volume (ml)
Time to full expression (s)
56 Hz
111 Hz
180 Hz
4.5 1.5
Figure 9. Temperature
did not alter the overall
morphology of the pattern
and had a negligible effect on
the degree to which patterns
were expressed. Patterns
formed at six temperatures
(5oC, 10oC, 15oC, 20oC, 25oC,
and 30oC) are shown for
each of the three sample
frequencies.
5oC 10 oC 15 oC 20 oC 25 oC 30 oC
a56 Hz
b111 Hz
c180 Hz
WATER 9, 1-27, OCTOBER 25, 2017 13
WATER
Temperature had a variable effect on the
time taken for patterns to reach full expres-
sion. At 56 Hz, increasing temperatures
reduced the time taken for the pattern to
reach full expression, although this effect
was not strong (linear regression: T1,16 =
-2.41; P = 0.03; Figure 10). At 111 Hz there
was a non-signicant trend for temperature
to reduce the time taken for the pattern to
reach full expression (T1,16 = -1.81; P = 0.09;
Figure 10). At 180 Hz there was no effect of
temperature on time to full expression (T1,16
= -0.70, P = 0.50; Figure 10).
Effects of Excitation Frequency from
50-200 Hz
Pattern morphology was strongly depen-
dent on the excitation frequency, and var-
ied from two-fold to twenty-fold symmetry
(Figure 11). Different pattern morphologies
appeared in frequency bandwidths, inter-
spersed with frequency bandwidths that
resulted in either no patterns or indistinct
patterns. In other words, pattern morphol-
ogy changed discretely while frequency in-
creased continuously. The spectrum of pat-
tern morphologies was broadly consistent
across the three repetitions, but showed
some variation in the length, and start or
endpoint of any given pattern’s bandwidth
(Figure 11). There was a highly ordered re-
lationship between excitation frequency
and the resulting patterns’ degree of sym-
metry (Figure 12).
Determination of Stability
Boundaries
The critical amplitude required to sustain
a given Faraday wave-pattern varied as a
function of excitation frequency. A lower ex-
citation amplitude was required to sustain
the pattern at the center of the frequency
range at which each pattern was produced
than at the edges of this range. The result-
ing stability curves were thus parabolic, or
trough-shaped, with the middle of the fre-
quency range occurring towards the bot-
tom of the trough (Figure 13). Beyond the
borders of each trough there were either no
patterns or different patterns: the stability
Figure 10. Increasing
temperatures reduced the
time taken for the pattern to
reach full expression (TFE)
at 56 Hz but not at 111 Hz
or 180 Hz. Lines are the
tted response of separate
linear models for each of
the three frequencies, with
gray bands depicting 95%
condence intervals, solid
lines indicating a signicant
relationship (P < 0.05),
and dotted lines indicating
a marginally signicant
relationship (0.5 < P < 0.1).
56 Hz (gray); T1,16 = -2.41;
P = 0.03; 111 Hz (maroon);
T1,16 = -1.81; P = 0.09; 180 Hz
(blue); T1,16 = -0.70, P = 0.50.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
5
10 15 20 25 30
5
10 15 20 25 30
5
10 15 20 25 30
T
empe
r
ature (
o
C)
Time to full
e
xpression (s)
56 Hz
111 Hz
180 Hz
WATER 9, 1-27, OCTOBER 25, 2017 14
WATER
WATER 9, 1-27, OCTOBER 25, 2017 15
WATER
WATER 9, 1-27, OCTOBER 25, 2017 16
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Figure 11. Effect
of excitation
frequency
on pattern
morphology using
the medium cell
(diameter = 24.25
mm) increasing
from 50 Hz to
199 Hz in 1 Hz
increments (10
Hz per row). The
fold symmetry of
each pattern (2-
fold, 4-fold, etc.) is
reported beneath
each panel. Rows
1, 2, and 3 are
three replicates
performed on
different days.
WATER 9, 1-27, OCTOBER 25, 2017 17
WATER
Figure 12. Relationship between the excitation frequency and the fold symmetry of pattern formed
for the medium cell. The three replicates are represented in light blue through dark blue and show
the frequency bandwidths of the patterns. The gray shaded area highlights the patterns of two-fold
symmetry which have poorly dened morphology and represent neither developed patterns nor no
patterns. Data are the same as those presented in Figure 11.
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
Frequency (Hz)
Fold symmetry
Figure 13. Stability curves around
each of the three sample frequencies
as described by the relationship
between excitation frequency and the
critical amplitude required to sustain
a given Faraday wave pattern. A
lower excitation amplitude was
required to sustain the pattern at
the center of the frequency range at
which each pattern was produced
than at the edges of this range.
Dotted vertical lines show the three
sample frequencies used in this study.
Values are the critical amplitude
required to elicit a Faraday wave
pattern. Filled symbols represent
values at which characteristic
Faraday wave patterns formed
(lled circles = 6-fold pattern; lled
triangles = 10-fold pattern; lled
squares = 14-fold pattern). Open
symbols represent frequencies at
which other patterns formed (open
triangles = 16-fold pattern; open
squares = 20-fold pattern). The
patterns themselves are shown
beneath the graph and the range of
frequencies at which they formed is
indicated by gray bars.
100
200
300
400
500
54 56 108 110 112 114 116 176 178 180 182 184 186
Frequency (Hz)
Minimum amplitude required to sustain pattern (mV)
10-fold6-fold 16-fold 14-fold 20-fold
●
●●
●●
●
●
●
●
WATER 9, 1-27, OCTOBER 25, 2017 18
WATER
curve for 56 Hz did not intersect with an
adjacent pattern, while the stability curve
for 111 Hz and 180 Hz intersected with ad-
jacent patterns at 113 Hz and 183 Hz re-
spectively (Figure 13). In the regions close
to the points of intersection between adja-
cent troughs, patterns were often unstable.
For example at 184 Hz, a 14-fold pattern
formed to start with, and then gave way to
an unstable 20-fold pattern (Figure 14).
In regions close to the points of intersection
between different troughs, patterns some-
times shifted to alternative patterns when
the amplitude was slowly reduced. At 115
Hz, the initial pattern (10-fold or 16-fold)
transitioned to the alternative pattern in
37.5% of trials (standard deviation = 20%).
At 184 Hz, the initial pattern (14-fold or
26-fold) transitioned to an alternative pat-
tern in 25.2% of trials (standard deviation
= 20.2%).
Effect of Atmospheric Pressure on
Pattern Formation
At 115 Hz, higher atmospheric pressures
increased the probability that the higher-
order pattern would form (16-fold versus
10-fold: χ2 = 8.0, P = 0.005; Figure 15). At
184 Hz higher atmospheric pressures mar-
ginally increased the probability that the
Figure 14. Montage of video frames showing the instability between alternative Faraday wave
patterns at 184 Hz. The 14-fold pattern was replaced by an unstable 26-fold pattern.
WATER 9, 1-27, OCTOBER 25, 2017 19
WATER
higher-order pattern would form (26-fold
versus 14-fold: χ2 = 4.7, P = 0.03; Figure
15). There was no effect of temperature on
the probability that the alternative pattern
would form at either frequency (115 Hz: χ2 =
0.43, P = 0.51; 184 Hz: χ2 = 0.92, P = 0.34).
Effects of Cell Diameter
Cell diameter had a pronounced effect on
pattern morphology. Using the small cell
(diameter = 10.00 mm) we obtained regu-
lar patterns across the 50-200 Hz range,
which differed from the patterns obtained
using the medium cell (diameter = 24.25
mm). A spectrum of pattern morphologies
for the smaller cell is presented in Figure 16.
As was the case for the medium cell, there
was a highly ordered relationship between
excitation frequency and the resulting de-
gree of symmetry of the pattern (Figure 17).
Using the large cell we obtained indistinct
patterns. At frequencies below ~65 Hz pat-
terns displayed classiable morphologies,
although were somewhat unstable and ir-
regular. At frequencies above ~65 Hz, pat-
Figure 15. Atmospheric pressure increased the probability of the higher order pattern forming when
samples of water were oscillated at frequencies close to the point of intersection between adjacent
stability troughs. At 115 Hz either 10-fold or 14-fold patterns formed. At 184 Hz either 14-fold or 26-
fold patterns formed. Values are the probability that the higher order pattern would form, and lines
are tted values of separate generalized linear models for each of the two frequencies, with gray bands
depicting 95% condence intervals. 115 Hz: χ2 = 8.0, P = 0.005; 184 Hz: χ2 = 4.7, P = 0.03).
(b) 184 Hz(a) 115 Hz
0.00
0.25
0.50
0.75
1.00
990 1000 1010 1020 990 1000 1010 1020
Pressure (mm)
Proportion of trials forming higher-fold pattern
terns became more unstable and indistinct,
and we were unable to classify them accord-
ing to fold symmetry. An example is shown
in Figure 18.
Discussion
Determinants of Faraday Wave
Pattern Morphology
As Abraham (1976) pointed out, Faraday
wave pattern morphology depends on i) in-
trinsic controls, which are properties of the
liquid, such as its dimensions, viscosity and
elasticity; and ii) extrinsic controls, which
are properties of the driving oscillations,
such as excitation frequency or amplitude.
We conrm that the morphologies of Fara-
day wave patterns in water are dependent
on both intrinsic and extrinsic controls,
most notably: i) the frequency of the forced
oscillation (extrinsic), and ii) on the diam-
eter of the uid reservoir (intrinsic). On
the whole, none of the other variables ex-
amined, whether intrinsic (uid volume/
WATER 9, 1-27, OCTOBER 25, 2017 20
WATER
Figure 16. Effect of
excitation frequency on
pattern morphology using
the small cell (diameter
= 10.00 mm) increasing
from 50 Hz to 199 Hz in 1
Hz increments (10 Hz per
row). The fold symmetry
of each pattern (2-fold,
4-fold, etc.) is reported
beneath each panel.
WATER 9, 1-27, OCTOBER 25, 2017 21
WATER
2
3
4
5
6
7
8
9
10
50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
Frequency (Hz)
Fold symmetry
Figure 17. Relationship between the excitation frequency and the fold symmetry of pattern formed for
the small cell. The gray shaded area highlights the patterns of two-fold symmetry which have poorly
dened morphology and represent neither developed patterns nor no patterns. Data are the same as
those presented in Figure 16.
a45 Hz
c90 Hz d150 Hz
b63 Hz
Figure 18. Patterns formed in the large cell (diameter = 49.50 mm)
were irregular and somewhat unstable. At frequencies below
~65 Hz (a and b), pattern morphologies could be classied according
to their fold symmetry. At frequencies above ~65 Hz (c and d),
pattern morphologies could not be classied.
WATER 9, 1-27, OCTOBER 25, 2017 22
WATER
depth, and temperature) or extrinsic (am-
plitude, and wave form) caused changes in
pattern morphology, although amplitude
altered the degree to which the pattern was
expressed (Figure 4), and amplitude, vol-
ume, and to a small extent temperature al-
tered the time taken for the pattern to form
(Figures 4, 7, 9). The only exceptions oc-
curred when water samples were oscillated
at 111 Hz and 180 Hz at very high ampli-
tudes, which caused the Faraday wave pat-
terns to jump to alternative patterns (Fig-
ure 4), and at transition frequencies (115
and 184 Hz) when reductions in amplitude
brought about transitions between alterna-
tive patterns. We discuss this below.
Effect of Frequency on Pattern
Morphology
The relationship between excitation fre-
quency and the resulting Faraday wave pat-
tern was discontinuous: when we increased
the frequency continuously, the Faraday
wave patterns changed discretely, jump-
ing or wobbling between alternative forms,
or disappearing altogether at a particular
cutoff frequency (Figures 10-11; Abraham,
1976). A familiar analogy is the tuning of a
radio. As the tuning frequency is increased
continuously, there is a discontinuous se-
ries of broadcasts that are tuned into, each
a resonant band with a certain bandwidth.
The discontinuous relationship between
frequency and Faraday wave pattern dem-
onstrates that the Faraday wave patterns,
or attractors—the mathematical descrip-
tion of a Faraday wave pattern (Abraham,
1976)—are discrete and cannot change
continuously as the excitation frequency is
changed. The bandwidth, or range of fre-
quencies that give rise to a given pattern
may thus be thought of as the basin of at-
traction—the set of initial frequencies that
lead asymptotically to the attractor (Sim-
onelli and Gollub, 1988). The basin bound-
aries, or frequency values at which a small
change in the control parameters elicit a
signicant change in form, may be thought
as catastrophic points in the sense of
Abraham (1972) and René Thom (1975).
Catastrophes may be of different types:
Abraham (1972) classes the transition be-
tween patterns into “jump” (Hopf-Tak-
ens) or “wobble” (Dufng-Zeeman) type
catastrophes, both of which we observed
(a “jump” from 14-fold to 26-fold is shown
in Figure 14, frames 0-93; a “wobble” is
shown between alternative forms of a 26-
fold pattern in Figure 14, frames 93-165).
Thus, the spectrum of patterns formed
across the frequency range (Figure 11) are
best thought of as describing basins of at-
traction, separated from each other by ca-
tastrophes.
There was a clear systematic relationship
between excitation frequency and the fold
symmetry of the pattern formed (Figure
12, 17). We observed a similar ascending
series in both the medium and the small
cell, which suggests that our ndings de-
scribe a generalisable relationship be-
tween excitation frequency and the fold
symmetry of Faraday wave patterns for a
given diameter of oscillating reservoir. If
formalised, this relationship could pro-
vide the basis for predicting pattern mor-
phology from excitation frequency and
vice versa (for a given liquid medium and
reservoir diameter). A similar relationship
has been described by Telfer (Web ref. 1).
While we offer no formalisation of this re-
lationship here, it seems probable that the
repetition of different patterns describes
harmonics and subharmonics of the ex-
citation frequency, and their relationship
with the diameter of the vessel and intrin-
sic properties of the liquid medium.
We limit the enquiry presented here to de-
ionized water. Nonetheless, we know from
our own investigations and from the liter-
ature (Henderson et al., 1991; Henderson,
1998) that pattern morphology is strongly
dependent on the properties of the oscil-
lating uid. For example, at a given excita-
tion frequency and cell diameter, we found
WATER 9, 1-27, OCTOBER 25, 2017 23
WATER
patterns of different fold symmetry were
formed in water, n-butanol and 1-pentanol
(data not shown). Further work is needed
to identify whether liquids with different
physical properties show systematic rela-
tionships similar to the one described here,
between excitation frequency and fold sym-
metry (Figure 12 and 17).
We observed that the amplitude, or energy,
required to sustain a given pattern was de-
pendent on frequency: patterns required
the lowest amplitude towards the center
of their frequency bandwidths, or basins
of attraction. At the edges of the frequency
bandwidths, greater amplitudes were re-
quired to sustain the patterns (Figure 13).
Ciliberto and Gollub (1984) described a
similar phenomenon, with amplitudes re-
quired to sustain a Faraday wave pattern
rising at frequencies towards the edge of a
pattern’s bandwidth. This relationship may
be thought of as a resonance curve, a com-
mon way to describe the resonant response
of a system (Siebert, 1986).
From another point of view, this relation-
ship between the minimum amplitude re-
quired to sustain a given pattern resembles
the energy landscape approach used in the
study of complex chemical systems such as
proteins (Wales, 2003). Both consider the
energy of the system (in this case, the mini-
mum amplitude required to sustain a pat-
tern) as a function of form or conguration
(in this case the Faraday wave pattern). Ac-
cording to energy landscape approaches,
the conguration, or form, of a molecule is
funnelled towards a minimum energy state,
which can be reached by multiple pathways
and intermediate forms (Leopold et al.,
1992), analogous to a ball or other such ob-
ject rolling to the bottom of a valley, and
reaching the bottom of the valley no matter
which point on the sloped sides of the val-
ley it is released from. Forms, or congura-
tions higher up the energy “funnel” are less
stable than those towards the bottom of the
funnel. Indeed, we found that the Faraday
wave patterns displayed greater stability at
the vertex of the parabola than higher up
the curve (Figure 14), as did Ciliberto and
Gollub (1984).
At the boundaries between adjacent ba-
sins of attraction, we observed competition
between alternative patterns (Figure 13).
A similar phenomenon was reported by
Ciliberto and Gollub (1984), who describe
competition between alternate patterns in
regions close to the point at which neigh-
boring stability curves intersect, giving
rise either to slow oscillations between al-
ternative patterns, or chaotic behavior. An
analogous phenomenon was described by
Thom (1975) in situations where dynamical
systems must “choose from several possible
resonances.” leading to the “competition
of resonances.” The variation that we ob-
serve between replications (Figure 11 and
15) may be due to the stochastic outcome of
competition between adjacent patterns, or
the sensitivity of the unstable cusp points
to tiny perturbations, described by Thom
as “innitesimal vibrations” which “play
a controlling part in the choice” (Thom,
1975). For example, we observed a mild ef-
fect of atmospheric pressure on the pattern
that formed at two transition sequences,
115 Hz and 184 Hz. At higher pressures
there was a greater tendency for the alter-
native patterns to form: at 115 Hz the usual
10-fold pattern was increasingly replaced
by a 16-fold pattern at higher atmospheric
pressure, and likewise at 184 Hz, the usual
14-fold pattern was replaced more often by
a 26-fold pattern (Figure 15).
The framework of basins of attraction and
catastrophes also helps to explain our ob-
servations that at 111 Hz and 180 Hz very
high amplitudes caused a sudden shift in
the Faraday wave pattern (Figure 4). At 111
Hz the 10-fold pattern shifted to a 16-fold
pattern, and at 180 Hz the 14-fold pattern
shifted to an unstable 20-fold pattern; Fig-
ure 4), suggesting that high amplitudes can
shift the stability boundary of the patterns,
WATER 9, 1-27, OCTOBER 25, 2017 24
WATER
in effect “bouncing” the pattern into an ad-
jacent basin of attraction. Abraham (1975)
describes an analogous phenomenon
whereby pattern-shifting catastrophes in-
crease at higher amplitudes. In accordance
with this interpretation, the 56 Hz pattern,
which lacked adjacent alternative patterns
(Figure 11), and thus lacked adjacent basins
of attraction (Figure 13), did not exhibit this
behavior at high amplitudes. Conversely,
we found that at transition frequencies at
the boundaries between basins, a reduction
in the amplitude could also shift the pattern
to an alternate form: at 115 Hz the 10-fold
pattern shifted in some cases to a 16-fold
pattern, and at 184 Hz the 14-fold pattern
sometimes changed to a 26-fold pattern.
Broader Applicability of
This Experimental System
The experimental system we describe per-
mits the reconstruction of the dynamics
and stability boundaries of a dynamical
system, including basins of attraction and
attractors (Simonelli and Gollub, 1988). By
extension, it may be treated as an analogue
model system for the study of vibrational
modal phenomena in dynamical systems
in general. Indeed, Abraham describes an
experimental Faraday wave system compa-
rable to the one we present as an “analogue
computer for dynamic catastrophes” (Abra-
ham, 1972).
There are many situations where such ana-
logue processors may provide more ef-
cient and predictive simulations than clas-
sical mathematical approaches based on
equations arising from physical theories
(Buluta and Nori, 2009), an idea origi-
nally suggested by Richard Feynman with
regard to quantum systems. According to
Feynman, an analogue processor would not
compute numerical algorithms for differ-
ential equations to approximate some nat-
ural phenomenon (as classical mathemati-
cal simulation approaches do), but rather
exactly simulate the natural phenomenon
(Feynman, 1982). In classical mathemati-
cal simulations, the computational power
required to solve the equations grows ex-
ponentially with the number of elements in
the system, meaning that even moderately
complex systems prove intractable (Roos,
2012). Moreover, as we point out in the In-
troduction, even if the equations for a given
system can be adequately solved by com-
putational brute force, they do not account
for the sort of perturbation that the system
is likely to encounter in the real world, se-
verely limiting the predictive power of the
approach (Ball, 2009).
Various natural phenomena behave ac-
cording to the resonant harmonic modes
represented in the experimental system de-
scribed here, and have been modelled using
an analogous system. These include: the
laser-induced vibrations of ions in a crystal
lattice (which can be used as an analogue
simulator of quantum magnetism; Britton
et al., 2012); the wave function of a free
particle provided by the time-independent
Schrödinger equation (Schrödinger, 1926;
Moon et al., 2008); pattern formation in
animal coats, shells and body plans (Xu et
al., 1983; Lauterwasser, 2015; Reid, Web
ref. 2); plant phyllotaxis (Lauterwasser,
2015); and cortical activity in the human
brain (Atasoy et al., 2016). Moreover, nu-
merous aspects of biological morphogen-
esis have been shown to operate according
to oscillatory mechanisms. These include
communication amongst and between bac-
terial colonies (Matsuhashi et al., 1998; Liu
et al., 2017); discrimination between self
and non-self in plant root growth (Grunt-
man and Novoplansky, 2004); and che-
motaxis and self-organisation in the slime
mould Dictyostelium discoideum (Gholami
et al., 2015).
We plan to extend these studies to investi-
gate Faraday wave patterns formed by dif-
ferent uids with a range of viscosities, and
also to investigate the inuence of different
shapes of container. Faraday wave patterns
may not only provide analogue models for
WATER 9, 1-27, OCTOBER 25, 2017 25
WATER
morphogenesis, but point to underlying vi-
bratory processes in developmental biology
and in the functioning of nervous systems.
Acknowledgements
We thank John Stuart Reid for his help
and advice on the technical details of our
research and Ruben Dechamps for assis-
tance with photography. We are grateful
to the following for their nancial support:
the Peter Hesse Foundation, Dusseldorf;
Sabine Uhlen, Dusseldorf; the Planet Heri-
tage Foundation, Naples, Florida; the Gaia
Foundation, London, and the Watson Fam-
ily Foundation and the Institute of Noetic
Sciences, Petaluma, California. The manu-
script was improved by comments from two
anonymous reviewers.
References
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Web ref. 2 – Reid, 2015: http://www.cymascope.
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Web ref. 3 – http://www.ralph-abraham.org/
articles/Blurbs/blurb012.shtml [07-07-17]
Web ref. 4 – http://www.ralph-abraham.org/
articles/Blurbs/blurb016.shtml [07-07-17]
Web ref. 5 – http://www.ralph-abraham.org/
articles/Blurbs/blurb015.shtml [07-07-17]
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Quatrième Racontre entre Mathematiciens et
Physiciens, Lyon, Département de Mathematiques
de l ‘Université de Lyon-1, Volume 4 — Fasicule 1,
pp. 38-114 and F1-23. (Web ref. 3)
Abraham R. 1975. Macroscopy of resonance. In:
Lecture Notes in Mathematics: Structural Stability,
the Theory of Catastrophes, and Applications in
the Sciences. A Dold, B. Eckmann (eds.). Springer-
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