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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

WATER

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

WATER

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

WATER

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

WATER

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

WATER

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

WATER

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

WATER

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

WATER

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

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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

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WATER 9, 1-27, OCTOBER 25, 2017 15

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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

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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

Web ref. 1 – Telfer, 2010: http://cymaticmusic.

co.uk/water-experiments.htm [07-07-17]

Web ref. 2 – Reid, 2015: http://www.cymascope.

com/cyma_research/biology.html [07-07-17]

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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