Encyclopedia of Physical Science and Technology - Classical Physics

448 Pages·2001·14.76 MB·English

by  Robert A. Meyers (Editor-in-Chief)

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Acoustic Chaos
Werner Lauterborn
Universita¨t Go¨ttingen
I. The Problem of Acoustic Cavitation Noise
II. The Period-Doubling Noise Sequence
III. A Fractal Noise Attractor
IV. Lyapunov Analysis
V. Period-Doubling Bubble Oscillations
VI. Theory of Driven Bubbles
VII. Other Systems
VIII. Philosophical Implications
GLOSSARY a dynamic system. A point in phase space characterizes
a specific state of the system.
Bifurcation Qualitative change in the behavior of a sys- Strange attractor In dissipative systems, the motion
tem, when a parameter (temperature, pressure, etc.) is tends to certain limits forms (attractors). When the mo-
altered (e.g., period-doubling bifurcation); related to tion comes to rest, this attractor is called a fixed point.
phase change in thermodynamics. Chaotic motions run on a strange attractor, which has
Cavitation Rupture of liquids when subject to tension ei- involved properties (e.g., a fractal dimension).
ther in flow fields (hydraulic cavitation) or by an acous-
tic wave (acoustic cavitation).
Chaos Behavior (motion) with all signs of statistics de- THE PAST FEW years have seen a remarkable develop-
spite an underlying deterministic law (often, determin- ment in physics, which may be described as the upsurge
istic chaos). of “chaos.” Chaos is a term scientists have adapted from
Fractal Object (set of points) that does not have a smooth common language to describe the motion or behavior of
structure with an integer dimension (e.g., three dimen- a system (physical or biological) that, although governed
sional). Instead, a fractal (noninteger) dimension must by an underlying deterministic law, is irregular and, in the
be ascribed to them. long term, unpredictable.
Period doubling Special way of obtaining chaotic (irreg- Chaotic motion seems to appear in any sufficiently
ular) motion; the period of a periodic motion doubles complex dynamical system. Acoustics, that part of
repeatedly until in the limit of infinite doubling aperi- physics that descibes the vibration of usually larger en-
odic motion is obtained. sembles of molecules in gases, liquids, and solids, makes
Phase space Space spanned by the dependent variables of no exception. As a main necessary ingredient of chaotic
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118 Acoustic Chaos
dynamics is nonlinearity, acoustic chaos is closely related to ask what physical mechanisms are known to convert a
to nonlinear oscillations and waves in gases, liquids, single frequency to a broadband spectrum? This could not
and solids. It is the science of never-repeating sound be answered before chaos theory was developed. However,
waves. This property it shares with noise, a term having although chaos theory is now well established, a physical
its origin in acoustics and formerly attributed to every (intuitive) understanding is still lacking.
sound signal with a broadband Fourier spectrum. But
Fourier analysis is especially adapted to linear oscillatory
systems. The standard interpretation of the lines in a II. THE PERIOD-DOUBLING
Fourier spectrum is that each line corresponds to a (linear) NOISE SEQUENCE
mode of vibration and a degree of freedom of the system.
However, as examples from chaos physics show, a broad- To investigate the sound emission from acoustic cavita-
band spectrum can already be obtained with just three tion the experimental arrangement as depicted in Fig. 1
(nonlinear) degrees of freedom (that is, three dependent is used. To irradiate the liquid (water) a piezoceramic
variables). Chaos physics thus develops a totally new cylinder (PZT-4) of 76-mm length, 76-mm inner diameter,
view of the noise problem. It is a deterministic view, and 5-mm wall thickness is used. When driven at its main
but it is still an open question how far the new approach resonance, 23.56 kHz, a high-intensity acoustic field is
will reach in explaining still unsolved noise problems generated in the interior and cavitation is easily achieved.
(e.g., the 1/ f -noise spectrum encountered so often). The The noise is picked up by a broadband hydrophone and
detailed relationship between chaos and noise is still an digitized at rates up to 60 MHz after suitable lowpass
area of active research. An example, where the properties filtering (for correct analog-to-digital conversion for later
of acoustic noise could be related to chaotic dynamics, is processing) and strong filtering of the driving frequency,
given below for the case of acoustic cavitation noise. which would otherwise dominate the noise output. The
Acoustic chaos appears in an experiment when a liq- experiment is fully computer controlled. The amplitude of
uid is irradiated with sound of high intensity. The liquid the driving sound field can be made an arbitrary function
then ruptures to form bubbles or cavities (almost empty of time via a programmable synthesizer. In most cases,
bubbles). The phenomenon is known as acoustic cavita- linear ramp functions are applied to study the buildup of
tion and is accompanied by intense noise emission—the noise when the driving pressure amplitude in the liquid is
acoustic cavitation noise. It has its origin in the bubbles set increased.
into oscillation in the sound field. Bubbles are nonlinear From the data stored in the memory of the transient
oscillators, and it can be shown both experimentally and recorder, power spectra are calculated via the fast-Fourier-
theoretically that they exhibit chaotic oscillations after a transform algorithm from usually 4096 samples out of the
series of period doublings. The acoustic emission from 128 × 1024 samples stored. This yields about 1000 short-
these bubbles is then a chaotic sound wave (i.e., irregular time spectra when the 4096 samples are shifted by 128
and never repeats). This is acoustic chaos. samples from one spectrum to the next.
Figure 2 shows four power spectra from one such
experiment. Each diagram gives the excitation level at
I. THE PROBLEM OF ACOUSTIC
CAVITATION NOISE
The projection of high-intensity sound into liquids has
been investigated since the application of sound to locate
objects under water became used. It was soon noticed that
at too high an intensity the liquid may rupture, giving rise
to acoustic cavitation. This phenomenon is accompanied
by broadband noise emission, which is detrimental to the
useful operation of, for instance, a sonar device.
The noise emission presents an interesting physical
problem that may be formulated in the following way.
A sound wave of a single frequency (a pure tone) is trans-
formed into a broadband sound spectrum, consisting of
an (almost) infinite number of neighboring frequencies.
What is the physical mechanism that causes this transfor- FIGURE 1 Experimental arrangement for measurements on
mation? The question may even be shifted in its emphasis acoustic cavitation noise (chaotic sound).
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Acoustic Chaos 119
FIGURE 2 Power spectra of acoustic cavitation noise at different excitation levels (related to the pressure amplitudes
of the driving sound wave). (From Lauterborn, W. (1986). Phys. Today 39, S-4.)
the transducer in volts, the time since the experiment systems has been found to show period doubling, among
(irradiating the liquid with a linear ramp of increasing them driven nonlinear oscillators. A peculiar feature
excitation) has started in milliseconds, and the power of the period-doubling bifurcation is that it occurs in
spectrum at this time. At the beginning of the experiment, sequences; that is, when one period-doubling bifurcation
at low sound intensity, only the driving frequency f0 shows has occurred, it is likely that further period doubling will
up. In the upper left diagram of Fig. 2 the third harmonic, occur upon altering a parameter of the system, and so on,
3 f0, is present. When comparing both lines it should often in an infinite series. Acoustic cavitation has been one
be remembered that the driving frequency is strongly of the first experimental examples known to exhibit this
damped by filtering. In the lower left-hand diagram, many series. In Fig. 2, the upper right-hand diagram shows the
1
more lines are present. Of special interest is the spectral noise spectrum after further period doubling to f0. The
4
1 1 1
line at
2 f0 (and their harmonics). A well-known feature of doubling sequence can be observed via 8 f0 and 16 f0 up
1
nonlinear systems is that they produce higher harmonics. to f
0 (not shown here). It is obvious that the spectrum is
32
Not yet widely known is that subharmonics can also be rapidly “filled” with lines and gets more and more dense.
produced by some nonlinear systems. These then seem The limit of the infinite series yields an aperiodic motion,
to spontaneously divide the applied frequency f0 to a densely packed power spectrum (not homogeneously),
yield, for example, exactly half that frequency (or exactly that is, broadband noise (but characteristically colored by
one-third). This phenomenon has become known as a lines). One such noise spectrum is shown in Fig. 2 (lower
period-doubling (-tripling) bifurcation. A large class of right-hand diagram). Thus, at least one way of turning
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120 Acoustic Chaos
a pure tone into broadband noise has been found—via
successive period doubling.
This finding has a deeper implication. If a system be-
comes aperiodic through the phenomenon of repeated pe-
riod doubling, then this is a strong indication that the ir-
regularity attained in this way is of simple deterministic
origin. This implies that acoustic cavitation noise is not a
basically statistical phenomenon but a deterministic one. It
also implies that a description of the system with usual sta-
tistical means may not be appropriate and that a successful
description by some deterministic theory may be feasible.
III. A FRACTAL NOISE ATTRACTOR
In Section II the sound signal has been treated by Fourier
analysis. Fourier analysis is a decomposition of a signal
into a sum of simple waves (normal modes) and is said to
give the degrees of freedom of the described system. Chaos
theory shows that this interpretation must be abandoned.
Broadband noise, for instance, is usually thought to be due
to a high (nearly infinite) number of degrees of freedom
that superposed yield noise. Chaotic systems, however,
have the ability to produce noise with only a few (nonlin-
ear) degrees of freedom, that is, with only a few dependent
variables. Also, it has been found that continuous systems FIGURE 3 Strange attractor of acoustic cavitation noise obtained
with only three dependent variables are capable of chaotic by phase–space analysis of experimental data (a time series of
pressure values sampled at 1 MHz). The attractor is rotated to
motions and thus, producing noise. Chaos theory has de-
visualize its three-dimensional structure. (Courtesy of J. Holzfuss.
veloped new methods to cope with this problem. One of
From Lauterborn, W. (1986). In “Frontiers in Physical Acoustics”
these is phase-space analysis, which in conjunction with
(D. Sette, ed.), pp. 124–144, North Holland, Amsterdam.)
fractal dimension estimation is capable of yielding the in-
trinsic degrees of freedom of the system. This method has
been applied to inspect acoustic cavitation noise. The an-
object. This suggests that the dynamical system produc-
swer it may give is the dimension of the dynamical system
ing the noise has only a few nonlinear degrees of freedom.
producing acoustic cavitation noise. See SERIES.
The flat appearance of the attractor in a three-dimensional
The sampled noise data are first used to construct a
phase space (Fig. 3) suggests that only three essential de-
noise attractor in a suitable phase space. Then the (frac-
grees are needed for the system. This is confirmed by a
tal) dimension of the attractor is determined. The pro-
fractal dimension analysis, which yields a dimension of
cedure to construct an attractor in a space of chosen di-
d = 2.5 for this attractor. Unfortunately, a method has not
mension n simply consists in combining n samples (not
yet been conceived of how to construct the equations of
necessarily consecutive ones) to an n-tuple, whose en-
motion from the data.
tries are interpreted as the coordinate values of a point in
n-dimensional Euclidian space. An example of a noise at-
tractor constructed in this way is given in Fig. 3. The attrac- IV. LYAPUNOV ANALYSIS
tor has been obtained from a time series of pressure values
{p(kts); t = 1, . . . , 4096; ts = 1 µsec} taken at a sampling Chaotic systems exhibit what is called sensitive depen-
frequency of fs = 1/ts = 1 MHz by forming the three- dence on initial conditions. This expression has been intro-
tuples [p(kts), p(kts + T ), p(kts + 2T )], k = 1, . . . , 4086, duced to denote the property of a chaotic system that small
with T = 5 µsec. The frequency of the driving sound field differences in the initial conditions, however small, are
has been 23.56 kHz. The attractor in Fig. 3 is shown from persistently magnified because of the dynamics of the sys-
different views to demonstrate its nearly flat structure. It is tem. This property is captured mathematically by the no-
most remarkable that not an unstructured cluster of points tion of Lyapunov exponents and Lyapunov spectra. Their
is obtained as is expected for noise, but a quite well-defined definition can be illustrated by the deformation of a small
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Acoustic Chaos 121
In dissipative systems, the final motion takes place on
attractors. Besides the fractal dimension, as discussed in
the previous section, the Lyapunov spectrum may serve to
characterize these attractors. When at least one Lyapunov
exponent is greater than zero, the attractor is said to be
chaotic. Progress in the field of nonlinear dynamics has
made possible the calculation of the Lyapunov spectrum
from a time series. It could be shown that acoustic cavita-
tion in the region of broadband noise emission is charac-
FIGURE 4 Idea for defining Lyapunov exponents. A small sphere
terized by one positive Lyapunov exponent.
in phase space is deformed to an ellipsoid, indicating expansion
or contraction of neighboring trajectories.
V. PERIOD-DOUBLING BUBBLE
sphere of initial conditions along a fiducial trajectory (see
OSCILLATIONS
Fig. 4). The expansion or contraction is used to define the
Lyapunov exponents λi , i = 1, 2, . . . , m, where m is the
Thus far, only the acoustic signal has been investigated.
dimension of the phase space of the system. When, on the
An optic inspection of the liquid inside the piezoelectric
average, for example, r1(t) is larger than r1(0), then λ1 > 0
cylinder (see Fig. 1) reveals that a highly structured cloud
and there is a persistent magnification in the system. The
of bubbles or cavities is present (Fig. 5) oscillating and
set {λi , i = 1, . . . , m}, whereby the λi usually are ordered
moving in the sound field. It is obviously these bubbles
λ1 ≥ λ2 ≥ · · · ≥ λm, is called the Lyapunov spectrum.
that produce the noise. If this is the case, the bubbles must
FIGURE 5 Acoustic cavitation bubble field in water inside a cylin- FIGURE 6 Reconstructed images from (a) a holographic series
drical piezoelectric transducer of about 7 cm in diameter. Two taken at 23.100 holograms per second of bubbles inside a piezo-
planes in depth are shown about 5 mm apart. The pictures are ob- electric cylinder driven at 23.100 Hz and (b) the corresponding
tained by photographs from the reconstructed three-dimensional power spectrum of the noise emitted. Two period-doublings have
image of a hologram taken with a ruby laser. taken place.
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122 Acoustic Chaos
FIGURE 7 Period-doubling route to chaos for a driven bubble oscillator. Left column: radius-time solution curves;
middle left column: trajectories in phase space; middle right column: Poincare´ section plots: right column: power
spectra. Rn is the radius of the bubble at rest, Ps and v are the pressure amplitude and frequency of the driving sound
field, respectively. (From Lauterborn, W., and Parlitz, U. (1988). J. Acoust. Soc. Am. 84, 1975.)
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Acoustic Chaos 123
FIGURE 7 (Continued )
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124 Acoustic Chaos
move chaotically and should show the period-doubling The third column shows so-called Poincare´ section plots.
sequence encountered in the noise output. This has been Here, only the dots after the lapse of one full period of
confirmed by holographic investigations where once per the driving sound field are plotted in the radius–velocity
period of the driving sound field a hologram of the bub- plane of the bubble motion. Period doubling is seen most
ble field has been taken. Holograms have been taken be- easily here and also the evolution of a strange (or chaotic)
cause the bubbles move in three dimensions, and it is attractor. The rightmost column gives the power spectra
difficult to photograph them at high resolution when an of the radial bubble motion. The filling of the spectrum
extended depth of view is needed. In one experiment the with successive lines in between the old lines is evident,
driving frequency was 23,100 Hz, which means 23,100 as is the ultimate filling when the chaotic motion is
holograms per second have been taken. The total num- reached.
ber of holograms, however, was limited to a few hundred. A compact way to show the period-doubling route to
Figure 6a gives an example of a series of photographs chaos is by plotting the radius of the bubble as a func-
taken from a holographic series. In this case, two period- tion of a parameter of the system that can be varied, e.g.,
doubling bifurcations have already taken place since the the frequency of the driving sound field. Figure 8a gives
oscillations only repeat after four cycles of the driving an example for a bubble of radius at rest of Rn = 10 µm,
sound wave. The first period doubling is strongly visible; driven by a sound field of frequency ν between 390 kHz
the second one can only be seen by careful inspection. and 510 kHz at a pressure amplitude of Ps = 290 kPa.
Figure 6b gives the noise power spectrum taken simulta- The period-doubling cascade to chaos is clearly visible.
neously with the holograms. The acoustic measurements In the chaotic region, “windows” of periodicity show
show both period doublings more clearly than the optical
1
measurement (documented in Fig. 6a) as the
4 f0 ( f0 =
23.1 kHz) spectral line is strongly present together with its
harmonics.
VI. THEORY OF DRIVEN BUBBLES
A theory has not yet been developed that can account for
the dynamics of a bubble field as shown in Fig. 5. The most
advanced theory is only able to describe the motion of a
single spherical bubble in a sound field. Even with suitable
neglections the model is a highly nonlinear ordinary
differential equation of second order for the radius R of
the bubble as a function of time. With a sinusoidal driving
term (sound wave) the phase space is three dimensional,
just sufficient for a dynamical system to show irregular
(chaotic) motion. The model is an example of a driven
nonlinear oscillator for which chaotic solutions in certain
parameter regions are by now standard. However, period
doubling and irregular motion were found in the late 1960s
in numerical calculations when chaos theory was not yet
available and thus the interpretation of the results difficult.
The surprising fact is that already this simple model of a
purely spherically oscillating bubble set into oscillation
by a sound wave yields successive period doubling up
to chaotic oscillations. Figure 7 demonstrates the period-
doubling route to chaos in four ways. The leftmost column
gives the radius of the bubble in the sound field as a func-
tion of time, where the dot on the curve indicates the lapse
of a full period of the driving sound field. The next column
FIGURE 8 (a) A period-doubling cascade as seen in the bifurca-
shows the corresponding trajectories in the plane spanned
tion diagram. (b) The corresponding largest Lyapunov exponent
by the radius of the bubble and its velocity. The dots again λ
max. (c) The winding number w. (From Parlitz, U. et al. (1990).
mark the lapse of a full period of the driving sound field. J. Acoust. Soc. Am. 88, 1061.)
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Acoustic Chaos 125
up as regularly experienced with other chaotic systems. VII. OTHER SYSTEMS
In Fig. 8b the largest Lyapunov exponent λmax is plot-
ted. It is seen that λmax > 0 when the chaotic region is Are there other systems in acoustics with chaotic dynam-
reached. Figure 8c gives a further characterization of the ics? The answer is surely yes, although the subtleties of
system by the winding number w. The winding number chaotic dynamics make it difficult to easily locate them.
describes the winding of a neighboring trajectory around When looking for chaotic acoustic systems, the ques-
the given one per period of the bubble oscillation. It can tion arises as to what ingredients an oscillatory system, as
be seen that this quantity changes quite regularly in the an acoustic one, must possess to be susceptible to chaos.
period-doubling sequence, and rules can be given for this The full answer is not yet known, but some understanding
change. is emerging. A necessary, but unfortunately not sufficient,
The driven bubble system shows resonances at vari- ingredient is nonlinearity. Next, period doubling is known
ous frequencies that can be labeled by the ratio of the to be a precursor of chaos. It is a peculiar fact that, when
linear resonance frequency of the bubble to the driving one period doubling has occurred, another one is likely to
frequency of the sound wave. Figure 9 gives an example appear, and indeed a whole series with slight alterations of
of the complicated response characteristic of a driven bub- parameters. Further, the appearance of oscillations when
ble. At somewhat higher driving than given in the figure a parameter is altered points to an intrinsic instability of a
the oscillations start to become chaotic. A chaotic bubble system and thus to the possibility of becoming a chaotic
attractor is shown in Fig. 10. To better reveal its structure, one. After all, two distinct classes can be formulated: (1)
it is not the total trajectory that is plotted but only the periodically driven passive nonlinear systems (oscillators)
points in the velocity–radius plane of the bubble wall at a and (2) self-excited systems (oscillators). Passive means
fixed phase of the driving. These points hop around on the that in the absence of any external driving the system
attractor in an irregular fashion. These chaotic bubble os- stays at rest as, for instance, a pendulum does. But a
cillations must be considered as the source of the chaotic pendulum has the potential to oscillate chaotically when
sound output observed in acoustic cavitation. being driven periodically, for instance by a sinusoidally
FIGURE 9 Frequency response curves (resonance curves) for a bubble in water with a radius at rest of Rn = 10 µm
for different sound pressure amplitudes pA of 0.4, 0.5, 0.6, 0.7, and 0.8 bar. (From Lauterborn, W. (1976). J. Acoust.
Soc. Am. 59, 283.)
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126 Acoustic Chaos
as it is a three-dimensional (space), nonlinear, dynamical
(time) system; that is, it requires three space coordinates
and one time coordinate to be followed. This is at the
border of present-day technology both numerically and
experimentally. The latest measurements have singled out
mode competition as the mechanism underlying the com-
plex dynamics. Figure 11 gives two examples of oscilla-
tory patterns: a periodic hexagonal structure (Fig. 11a) and
a
FIGURE 10 A numerically calculated strange bubble attractor
(Ps = 300 kPa, v = 600 kHz). (Courtesy of U. Parlitz.)
varying torque. This is easily shown experimentally by
the repeated period doubling that soon appears at higher
periodic driving. Self-excited systems develop sustained
oscillations from seemingly constant exterior conditions.
One example is the Rayleigh-Be´nard convection, where a
liquid layer is heated from below in a gravitational field.
The system goes chaotic at a high enough temperature
difference between the bottom and surface of the liquid
layer. Self-excited systems may also be driven, giving
an important subclass of this type. The simplest model
in this class is the driven van der Pol oscillator. A real
physical system of this category is the weather (the
atmosphere). It is periodically driven by solar radiation
with the low period of 24 hr, and it is a self-excited
system, as already constant heating by the sun may lead to b
Rayleigh-Be´nard convection as observed on a faster time
scale.
The first reported period-doubled oscillation from a pe-
riodically driven passive system dates back to Faraday in
1831. Starting with the investigation of sound-emitting, vi-
brating surfaces with the help of Chladni figures, Faraday
used water instead of sand, resulting in vibrating a layer
of liquid vertically. He was very astonished about the re-
sult: regular spatial patterns of a different kinds appeared
and, above all, these patterns were oscillating at half the
frequency of the vertical motion of the plate. Photography
was not yet invented to catch the motion, but Faraday may
well have seen chaotic motion without knowing it. It is in-
teresting to note that there is a connection to the oscillation
of bubbles as considered before. Besides purely spherical
oscillations, bubbles are susceptible to surface oscillations
as are drops of liquid. The Faraday case of a vibrating flat
surface of a liquid may be considered as the limiting case
of either a bubble of larger and larger size or a drop of
FIGURE 11 Two patterns appearing on the surface of a liq-
larger and larger size, when the surface is bent around up
uid layer vibrated vertically in a cylindrical container: (a) regular
or down. Today, the Faraday patterns and Faraday oscil- hexagonal pattern at low amplitude, and (b) pattern when ap-
lations can be observed better, albeit still with difficulties proaching chaotic vibration. (Courtesy of Ch. Merkwirth.)