# Waves and Fractals

AN ABSTRACT OF THE THESIS OF

Crystal L. Gilliland for the degree of Master of Science in Mathematics presented on December 5, 1991.

Title: Waves and Fractals

Abstract approved: Redacted for Privacy Dr. Ronald B.luenther

The goal of this research project is to determine the fractal nature, if any, which surface water waves exhibit when viewed on a microscopic scale.Due to the relatively recent development of this area of mathematics, a brief introduction to the study of fractal geometry, as well as several examples of fractals, are included in this paper. From that point, this paper addresses the specific situation of a surface wave as it nears the breaking point and attempts to detect the fractal structure of a wave at this given point when viewed on a microscopic scale. This is done from both a physical standpoint based on observations at the Hinsdale Wave Facility at Oregon State University and at Cape Perpetua, Oregon on the Pacific Coast, and from a theoretical standpoint based on a spring model. Waves and Fractals by

Crystal L. Gilliland

A THESIS

submitted to

Oregon State University

in partial fulfillment of the requirements for the degree of

Master of Science

Completed December 5, 1991

Commencement June 1992 APPROVED:

Redacted for Privacy Dr. Ronald B. Guenther, Professor1of Mathematicsin charge of major Redacted for Privacy Dr. Francis J. Flaherty, Chairtiff the Departmentof Mathematics A Redacted for Privacy Dr. Thomas J. esh, Dean duate School

Date thesis is presented December 5. 1991

Typed by Crystal L. Gilliland Acknowledgements

First and foremost, I would like to thankmy advisor, Dr. Ronald B. Guenther, for all of his guidance, patience andsupport.His expertise and encouragement have been invaluable to me.

I would also like to greatly acknowledge Dr. Kennan T. Smith for all of his knowledge and assistance with the digitization portion of this project. His kindness and expertise, as well as the use of his computer facilities andprograms, are greatly appreciated. I am also appreciative of Dr. Harold Parks for his advice concerning the curvature of a surface. In addition, I would liketo thank Dr. Gary Musser for the use of his nice computer set-up upon which this documentwas produced and for all of his interest, encouragement and support inme as a person. I am also extremely grateful for all of the photographic work done for this project by my cousin, Christopher F. Lamoureaux. Without all of his beautiful photographs, this project would not have been possible.I am especially appreciative to him for all of the special trips he made to Corvallis for the sakeof my project and for his wonderful friendship. Many of the photographswere taken at the Hinsdale Wave Facility at Oregon State University and I would like to thank the faculty and staff there for their cooperation. In particular, I would liketo thank

Dr. Chuck Sol lit and Terry Dibble for their help in coordinatingmy photographs at the wave facility.I am also appreciative of the support for this project been made available by the Office of Naval Research under ONR Grant 30-262-3352.

In addition to the above mentioned, I have receiveda tremendous amount of assistance with the computer aspects of my project from the followingpeople. To each of them, I extend my appreciation for theuse of their computer facilities, software and programming abilities, as well as for their patience in workingwith me on their various contributions. I extend my thanks to Russ Ruby for all of his help with the Sun Computer System and in particular, for his help in converting my digitized data into a form that could be graphed on the Sun System; to Peter White for the use of his Runge-Kutta program and his help therewith; to Eric Welch for the use of his computer and software, and for his enthusiasm about fractals and my project; and to Paul Palmer for his help with graphing on the Sun System. Finally, I would like to acknowledge my parents, Harlan and Shirley

Gilliland, for all of their love and support, not just while I have been workingon this project, but always. They are truly an inspiration to me and it is to them that this work is dedicated. Table of Contents

Chapter 1.Introduction to Fractals 1 Chapter 2.Further Examples and Subsequent Applications of

Fractals 24

Chapter 3.Fractal Dimension 36

Chapter 4.Classical Description of Water Waves 39 Chapter 5.A Physical Approach to the Fractal Nature of

Water Waves 44 Chapter 6.A Theoretical Approach to the Fractal Nature of

Water Waves 67

Chapter 7.Conclusion and Significant Findings 83

Bibliography 85

Appendix. Computer Software Utilized in this Project 88 List of Figures

Chapter 1

Figure 1.1 Weierstrass Function 4

Figure 1.2 Weierstrass Function 5

Figure 1.3 Van der Waerden Function 6

Figure 1.4 Cantor Set Construction. 8

Figure 1.5 Cantor Set 9

Figure 1.6 Koch Curve Construction. 10

Figure 1.7 Koch Curve 11

Figure 1.8 Peano Curve Construction 14

Figure 1.9 Peano Curve 16

Figure 1.10 Botanical Image One 19

Figure 1.11 Botanical Image Two 20

Figure 1.12 Botanical Image Three 21

Figure 1.13 Botanical Image Four 22

Chapter 2

Figure 2.1 Verhulst Bifurcation 26

Figure 2.2 Mandelbrot Set 27

Figure 2.3 Mandelbrot Set Close-ups 28

Figure 2.4 Julia Set for the Mandelbrot Set 31

Figure 2.5 Lorenz Attractor 33

Figure 2.6 ROssler Band Attractor 34

Chapter 4 Figure 4.1 Cross-sectional Cut of Water 40 Figure 4.2 Cross-sectional Cut of Water as a Function of a Particle of

Water and Time 41

Figure 4.3 Cross-sectional Cut of Water as Time Varies 42

Figure 4.4 Path of Motion of a Particular Particle as Time Varies 43

Chapter 5

Figure 5.1 Sequence of Photographs of a Breaking Wave 45

Figure 5.2 Photograph of Water1.135 50

Figure 5.3 Graph of Cross-sectional Cut of Water1.135 51 Figure 5.4 Graph of Box-counting Dimension Calculation for

Water1.135 52

Figure 5.5 Photograph of Water2.110 53

Figure 5.6 Graph of Cross-sectional Cut of Water2.110 54 Figure 5.7 Graph of Box-counting Dimension Calculation for

Water2.110 55

Figure 5.8 Photograph of Water3.150 56

Figure 5.9 Graph of Cross-sectional Cut of Water3.150 57 Figure 5.10 Graph of Box-counting Dimension Calculation for

Water3.150 58

Figure 5.11 Photograph of Beach1.149 59

Figure 5.12 Graph of Cross-sectional Cut of Beach1.149 60 Figure 5.13 Graph of Box-counting Dimension Calculation for

Beach1.149 61

Figure 5.14 Photograph of Smooth1.100 62

Figure 5.15 Graph of Cross-sectional Cut of Smooth1.100 63 Figure 5.16 Graph of Box-counting Dimension Calculation for

Smooth1.100 64

Figure 5.17 Table of Surface Area Ratios and Average Curvatures for Water1.135, Water2.110, Water3.150, Beach1.149,

Smooth1.10 and Mammogram 66

Chapter 6

Figure 6.1 Spring Model 67

Figure 6.2 Equilibrium Position of Spring Model 70

Figure 6.3 Spring Model for Nonbreaking Wave 72 Figure 6.4 Spring Model for Breaking Wave (Small Time Increments/ Small Time) 74

Figure 6.5 Spring Model for Breaking Wave (Larger Time Increments/

Longer Time) 75

Figure 6.6 Close-up of One Period of the Spring Model in Figure 6.5. 76

Figure 6.7 Close-up of Spring Model in Figure 6.6 for t = .20 78

Figure 6.8 Graph of Box-counting Dimension Calculation for

Figure 6.7 79

Figure 6.9 Close-up of Spring Model in Figure 6.6 for t = .25 80 Figure 6.10 Graph of Box-counting Dimension Calculation for

Figure 6.9 81 Waves and Fractals

Chapter 1 Introduction to Fractals

The study of fractals, although its roots extend back into the last century, is

a relatively new branch of mathematics. More generally, the study of chaos, or the

study of nonlinear dynamical systems, has only recently come into its own.

As has been the case in so many areas of mathematics, the study of fractals began as a small seed in the minds of a few mathematicians many years ago. This seed began to grow as a fascinating topic, but with no seemingly apparent application to real world phenomena.It is often said that there is a time for everything and it was only with the birth of the computer age and with great technological advances that the widespread application of fractal geometry began to show itself.

A fractal, as defined by Hans Lauwerier, is a geometrical figure in which an identical motif repeats itself on an ever diminishing scale. Although others may not exactly agree on the definition of a fractal, there are many properties that most would agree fractals possess.The self-similarity alluded to in Lauwerier's definition is one such property. The closely related concept of self-affinity is another such property. Self-affinity refers to the notion of self-similarity combined with a linear transformation such as a translation, rotation, dilation or contraction.

Another property common to many fractals is that they may be defined as nonlinear dynamical systems. That is, a fractal may be defined as a nonlinear sequence in which each term is defined in terms of the preceding term or terms. The fractal is 2

formed by iterating this recursion formula.In a world full of well-behaved dynamical systems, it is the nonlinearity of some dynamicalsystems that

distinguish those systems as being fractal. Yet another propertycommon to fractals is that of nondifferentiability. From a mathematical point of view,many fractals may be viewed as nowhere differentiable functions (or curves) and often as

continuous nowhere differentiable functions (or curves).

In order to understand fully the rich history of fractalgeometry, one must venture back to the late 18th/early 19th centuries and look at the concept of function

as it was understood at that time. The history of the concept of function is in itself a

fascinating topic, but let us focus primarily on how thisconcept was understood by Leonhard Euler and Joseph Lagrange.

To Euler, the idea of function was connected with the analytical study of curves. A curve represented by one algebraic or transcendental equation was called a continuous curve.Curves requiring different equations for their different constituent parts were called discontinuous. Concerning "arbitrary"curves, the debate was whether or not it was legitimate toassume continuity. Lagrange's concept of a function is not much different from Euler's. Lagrange viewed functions as having power series representations. These notions of function were a continuing attempt to definea previously vague and inexact concept which appeared throughout the calculus that had been developed in the 17th century.

One of the primary focuses of this calculuswas the differentiation of functions. Geometrically, differentiation isa means of finding the slope of the tangent to a curve at any given point. It was understood that theremay be some singular points that caused problems. Specifically, points where thecurve was 3

discontinuous or points where the curve changedso drastically that the idea of a tangent lost its meaning were problematic.These points, however, were considered to be the exception, not the rule, and attentionwas focused on nice, well-behaved functions.

The general feeling around this time was that although theremay be a few

problems, functions were piecewise differentiable and, in particular,continuous functions were almost everywhere differentiable. A.M. Ampereeven purported to have proved this in 1806.

In the early 1870's, Karl Weierstrass quickly proved himwrong, though,

as he exhibitedacontinuousnowheredifferentiablefunction. Consider the function f(x) = Zia° ancos(2monx) where lal > 1 andco > 1 is real. This is co what is known today as a Weierstrass function.Figure 1.1(a) exhibits a 1 Weierstrass function with a= w=2, 2' n=20 on the interval [0,5].Figure 1.1(b) shows a close-up view of the same functionon the interval [0,1]. Figure 1.2(a) exhibits a graph of the function with a=-2-,1w =6, n=8 on the interval [0,5]. Figure

1.2(b) shows a close-up view of this functionon the interval [0,1]. As one looks at the graph of this function on very fine scales, the roughness of thecurve indicates the nondifferentiability. 4

-0.995849 I I ry 1

0 S

0 i 1

(b) Figure 1.1

Weierstrass Function S

0 x

(b) Figure 1.2

Weierstrass Function 6

Another example of a continuous nowhere differentiable functionis that of {1Onx B.L. van der Waerden. The van der Waerden function is f(x)= En; where 10n ( lOnx) denotes the distance from lOnx to thenearest integer. Figure 1.3(a) shows the van der Waerden function with n=300on the interval [0,1]. The self-similarity of this curve is very clear when one looks ata close-up view of the function. Figure 1.3(b) shows the same function with n=300on the interval [.05,.06].

fixi

0

0 x 1

(a)

Figure 1.3

Van der Waerden Function 7

0.06

(b)

Figure 1.3 (continued)

Georg Cantor also disturbed many mathematicians in 1870 when he showed

how a simple, repeated procedure could turn a solid line into a scattering of singular points.Cantor looked at what is known as the Cantor set.His procedure is relatively simple. To construct the Cantor set, begin with a solid line segment including its endpoints [Figure 1.4(a)]. From this line segment, remove the middle third, but not the endpoints, leaving two solid line segments both of which include their respective endpoints [Figure 1.4(b)]. From each of these two line segments, remove the middle third, again without the endpoints, leaving four solid line segments with their endpoints [Figure 1.4(c)]. Continue removing the middle thirds of each line segment in this manner. Figure 1.5 shows the first 6 steps in this construction. Continuing this process indefinitely leaves only an infinite 8 scattering of points which are today often referred to as Cantor "dust". In Chapter

3, we will see that this set is considered a fractal set.

(a)

(b)

(c)

Figure 1.4

Cantor Set Construction 9

IMO M. MIMI M. Mi NOM ..

M. MOM MOD Mel IBM111.

1110 41; .11 .IIIIM

Figure 1.5

Cantor Set

Helge von Koch, in one of the simplest yet most well-known fractalcurves today, showed in 1904 how a simple iterative process could leadto a curve of finite length into one of infinite length. In doing so she demonstrated how the indefinite repetition of a simple geometric figure can lead toa continuous nowhere differentiable curve. Her procedure is similar to that of constructinga Cantor set. To construct a Koch curve, begin with a line of finite length (Figure 1.6(a)). As with the Cantor set construction, consider the middle third of the linesegment. Construct an equilateral triangle with this middle segmentas the base. Now remove the base. This leaves a curve consisting of four line segments of finite length (Figure 1.6(b)). Notice that this is a continuouscurve of finite length, but of a 10 greater length than the original curve. Now consider the middle thirds of each of these four line segments. Construct equilateral triangles with each of these middle thirds as the bases and then remove the bases (Figure 1.6(c)). Again we are left with a continuous curve of finite length, but which has length greater than the previous curves. Figure 1.7 shows a Koch curve at a later step in the process. Each step preserves the continuity of the curve, but increases the length of the curve.Repeating this process indefinitely theoretically leads to a continuous nowhere differentiable curve of infinite length and yet zero area. This was quite a disturbing concept for mathematicians at the time.

(a)

(b)

(c)

Figure 1.6

Koch Curve Construction 11

Figure 1.7

Koch Curve Strange, unpredictable shapes began to showup in other areas of mathematics as well. In the 1880's, in his attemptto analyze the motion of celestial bodies by traditional methods, Poincare ran into problems andwas forced to develop a new approach to the problem. His approach pointedto the fact that while many initial motions amiably settled into the familiar curves, therewere also some strange "chaotic" orbits that never attained periodicity. These orbitswere, in fact, unpredictable.

Many of these shapes and functions seemed to be somewhere betweenthe familiar 0-dimensional points, 1-dimensional lines, 2-dimensionalplanes and 3- dimensional space. The concept of dimension developed by FelixHausdorff in 1919 seemed somehow remotely related, but itwas not until much later that the 12

concept of fractal dimension would be developed and the connection between Hausdorff dimension and fractal dimension would be explored. This will be

discussed in Chapter3.

The use of traditional methods in other areas of study suchas meteorology,

economics and hydrodynamics also began to reveal such chaotic behaviorand scientists were beginning to realize that they must deal with these problems.These problems simply could no longer be ignored.

The time for fractal geometry was here and itwas B.B. Mandelbrot who really drove that fact home and showed the world what fractalsare all about. Although many mathematicians had worked independentlyon the development of what is now called fractal geometry, it was Mandelbrot who really pulledit all together and, in fact, coined the term "fractal".

Mandelbrot was born in Poland in1924.However, he moved to France in

1935and thus received most of his education there. Mandelbrotwas not what his instructors at the Ecole Polytechnique would considera particularly promising mathematician because he rejected their strong analytic approachto mathematics in favor of a highly geometric approach. This approach in lateryears proved him indeed to be quite a successful mathematician in hisown right.

Mandelbrot continued his education at Caltech and in1958he joined IBM. In1974he was named an IBM Fellow and continues to workat the IBM Watson Research Center. In addition, he has also beenvery active as a visiting professor and guest lecturer at many universities.

Mandelbrot first stumbled onto what later led to his development offractal geometry while he was at Caltech. There he became involved with and intrigued by the study of fluid turbulence. After joining IBM, he continuedto be haunted by this 13

problem and encountered other similar problems that seemed to defy explanation by

traditional means. In his mind, he began to see a sort of structure- a structure that would grow and be developed over the yearsas he continued to ponder these problems.

The advances in technology coupled with his positionat IBM both allowed

and forced him to look at models of such baffling problems and he beganto be able to distinguish order in these chaotic systems. His "fractal" models began to take hold and in areas ranging from coastlines to cotton prices,experts in these respective fields confirmed the practicality and reliability of such models.

Mandelbrot has been very effective in arguing thatmany of the traditional

models used in studying nature's forms andprocesses are inappropriate and that, in fact, many of the seemingly inapplicable "strange" shapes developed byearlier mathematicians were indeed appropriate and insome cases even vital to the study of such natural forms and processes.From human physiology to population dynamics to economics, fractals have offered new direction for previously vexing problems and, consequently, fractal geometry has developed intoan exciting new branch of mathematics with extremely widespread application.

There are many popular examples of fractals which exhibit the properties mentioned above. One commonly known andvery applicable example is what is known as a "space-filling" or "Peano" curve. A space-fillingcurve is a continuous one-to-one mapping of [0,1] into [0,1]n. There is a whole class of space-filling curves, but the first one was given in 1890 by G. Peano and, thus, theyare commonly known as Peano curves.Peano curves are continuous nowhere differentiable curves and can be shown to have fractal dimension. 14

Peano curves are classic examples of self-similar fractals because theyare constructed by repeating the same geometric figureover and over again on ever

diminishing scales. To construct the curve first given by Peano in 1890,we begin with a unit square (Figure 1.8(a)). Each side of thesquare is then replaced with a polygonal line consisting of nine segments each31 the unit length. The polygonal line looks like the original side with two squares sharing the middle third of the original side as a base. Figure 1.8(b) shows thecurve resulting from this step in the construction. This process is then repeated replacing each of the linesegments of the new curve with polygonal lines similar to those in the precedingstep (Figure 1.8(c)).Figure 1.9 shows the resulting figure after severalsteps in the construction.

(a)

Figure 1.8

Peano Curve Construction 15

(b)

IIIIISI

i

(c)

Figure 1.8 (continued) 16

Figure 1.9

Peano Curve

Historically, there have been two important applications of Peanocurves in computer science.These have been in the areas of data transmission and mathematical programming. In data transmission, the idea isto transmit a single piece of data instead of n pieces of dataso as to reduce the bandwidth of the waveforms. In 1969, T. Bially proposed usinga Peano curve to transform the n pieces of data into a single piece of data and then torestore the n pieces of data after transmission. The restored n pieces of data remain in the neighborhood of the initial data because of the continuity of the Peanocurve. These efforts led to the study of wavelets and wavelet transforms.

Peano curves have also found application in mathematical programming. In the late 1960's/early 1970's, A. Butz proposedan iterative method for solving n- 17

dimensional problems for which conventional means of programmingwere not

possible. In his method, a functional n-dimensional domain is mappedonto a one- dimensional domain by means of what is called a Peano scanning. A Peano

scanning is the reciprocal application of a Peano curve from [0,1]nto [0,1] and is almost everywhere continuous. Since the Peano scanning is almost everywhere

continuous, this allows one to use one-dimensional iterative methodsto solve the problem.

Another important class of fractals which clearly convey theconcepts of self-similarity and/or self-affinity are those used in encoding images. This is

similar to the notion of data transmission in the preceding example in that the goal is

to be able to reduce the amount of information needed to describe and/or analyzean

image. If an object has a highly intricate fractal structure that is self-similaror self- affine in nature, then that object can be encoded and stored bya small amount of

information. If such an object can be defined recursively, the iteration of sucha definition can produce surprisingly good pictures of the object.

Fractal models such as this seem to fit well with many naturally occurring objects such as trees, leaves, clouds, grasses, etc.Using intricate computer

graphics, some very realistic botanical images can be generated. Figures 1.10- 1.13 demonstrate some of the more basic botanical fractal images thatcan be produced. These images have all been drawn using the Lsystemcomputer program as incorporated into the Fractint software. They are all constructed from line segments using rules specified in the following drawing commands:

F Draw Forward nnIncrease Angle nn Degrees G Move Forward (without drawing) /nnDecrease Angle nn Degrees +Increase Angle (by specified angle measure)Cnn Select nn Color - Decrease Angle (by specified angle measure)

[ Push. Stores current angle and position ] Pop. Return to location of on a stack. last push.

I Try to turn 180° (if the angle is odd, the largest possible turn will be less than 180°) @nnn Multiply line segment by size nnn (number may be preceded by I for inverse or Q for square root) Starting with an initial string called the axiom, an angle is specified and transformation rules are applied a specified number of times (called the order) and the image is produced. 19

Figure 1.10 was constructed using the following parameters:

Angle: 22.5° Order: 4 Axiom: ++++F Iterations: 150 Transformation: F = FF-[-F+F-f-F]+[+F-F-F] Domain: -1 < x < 1 Range: -1 <_y51

41 . 0'A 24"30;1t01. 7 a 70...,7,0041.0fl.,, igPr.,44illi.liferA. flitdrAiliffr"' i IIPA 1/ .0 pt II0 l.0%. ifit ylye° i

44.1;11044,0. 1.,,,tali irti",_-..-..:

mli; ,1A w,'..;: O'FA Vrie;;;Illii 11.c;;47 -'4.-

110,

Figure 1.10

Botanical Image One 20

The parameters for Figure 1.11 were as follows:

Angle: 25.71° Order: 8 Axiom: X Iterations: 150 Transformation: X = F[+X][-X]FX, F = FF Domain: -1 < x < 1 Range: -1 1

Figure 1.11

Botanical Image Two 21

Figure 1.12 was based on the following parameters:

Angle: 25.71° Order: 8 Axiom: f Iterations: 150 Transformation: f = F[+F]F[-F]F Domain: -1 < x < 1 Range: -1 y 1

Al

Figure 1.12

Botanical Image Three 22

The following were the parameters for the construction of Figure 1.13:

Angle: 18° Order 6 Axiom: SLFFF Iterations: 150 Transformation: s = [+++Z][---Z]TS z = +H[-Z]L h = -Z[+H]L t = TL 1 = [-1-14F][+1-1-1-]F Domain: -1 < x < 1 Range: -1 5. y 1

Figure 1.13

Botanical Image Four 23

These are just two examples of classes of fractals and their subsequent applications, but they should instill the impression of the extremely applicable and still greatly unexplored realm of fractals. The upcoming chapters will lookmore deeply at some further examples of fractals and will address the issue of howto determine whether or not an object is indeed fractal. 24

Chapter 2 Further Examples and Subsequent Applications of Fractals

In Chapter 1 we constructed fractals based on the property of self-similarity or self-affinity.In this chapter, we will construct our fractals from recursive definitions. Then we will look for the self-similarityor self-affinity on ever diminishing scales.

Our first fractal developed out of the study of population dynamics.This is

often referred to as Verhulst dynamics and the population growthmodel used is often called the Malthusian or Verhulst model. This model is basedon a recursively

defined real-valued function. Let xo be the initial population size andlet xn be the population size after n years. Then the growthrate R is the relative increase per year,

(xn+1xn) xn

If this is a constant, then the dynamical law is

xn+1 = f(xn) = (1 + r)xn. After n years, this yields

xn = (1 + Onxo.

This implies, however, that populationsgrow without bound which we know is not true. Verhulst assumed, therefore, that R must vary with respect to the population size. He argued that any environmentcan only sustain a certain population size which can arbitrarily be set to one. Verhulst then postulatedthat R is proportional to 1xn and set R = r(1xn) where r>0 is called the growth parameter.

In this model, if xn<1, the populationmay keep growing until xn = 1 and then it will stop growing. The recursive definition then takes the form 25

xn+i = f(xn) = (1 + r)xn - rxg

Notice that if xo = 0 or xo = 1, the population size does not change. Itcan be shown, however, that both of these steady state pointsare unstable. That is, only a

slight deviation from xo = 0 or xo =1 will lead to unsteady behavior of thesystem. Figure 2.1 shows this model. One can see the self-affinity in the branching pattern of the graph of the model. This is often called the Verhuist bifurcation. The

term "bifurcation" is derived from the roots bi (= two) and furca (= forkor prong) and simply means splitting into two branches. In looking at the branching of the Verhuist process, analysis reveals an important law for the world of nonlinear

phenomena. This regards the lengths of the parameter intervals in whicha particular period is stable. If rn is the growth parameter at the nth bifurcation (that

is, the period where 2n becomes unstable and 2n+1 gains stability), then the ratioof the lengths of successive intervals is 8n rn - rn _1and was found by S. rn+1rn GroOmann and S. Thomae, as well as independently by M. Feigenbaum. They

found that as n--->00, sn)4.669201660910... (that is, the intervalsare shortened at each branching by a factor which approaches 8= 4.669201660910... as n>00). Feigenbaum established the universality of this number when he found it applied over and over again for many differentprocesses. Consequently, this number is often referred to as the "Feigenbaum" number. Thiswas extremely significant in that this bifurcation scenario actuallyoccurs in many natural systems. As a result, extensive studies have demonstrated thatmany of the features of complex systems can be reduced to or modeled by the Verhuist model thus shedding great light on the dynamics of these systems. 26

Figure 2.1

Verhulst Bifurcation

Our next example, the Mandelbrot set, is probably the most oftenseen and most familiar to anyone who has had any dealings with the study of chaosor fractals.In 1980, B. Mandelbrot discovered the set that isnow named in his honor. Mandelbrot's method is very similar to that of Verhulst in the mathematical sense. The difference was that whereas Verhulst looked at real-valued functions, Mandelbrot looked at complex-valued functions.

The Mandelbrot set is generated by iterating the complex-valued function

4+17--- f(zn) = A 27

where zk are complex numbers for all integers lc. Figure 2.2 shows the Mandelbrot

set for 150 iterations. That is, for each point zk on the complex plane, theprogram calculates 150 iterations and looks at the sequence of pointszi, z2, ... z150If this sequence tends to zero or infinity as n increases, the point is not in the Mandelbrot set. If the sequence does not tend to zero or infinity, it is in the set. The coloring of the complex plane is dependent upon the rates at which thesequences corresponding to each point tend to zero or infinity. The blue portion in thecenter of the picture represents the actual points in the Mandelbrot set. Variationson the Mandelbrot set may be constructed by looking at the functionzn+i = f(zn) = 4 + c where zk, c are both complex numbers for all integers k.

Figure 2.2

Mandelbrot Set 28

The self-similarity of the boundary of the Mandelbrot set becomes clear as one looks at the boundary on increasingly fine scales. Figure 2.3 (a)- (d) show a progression of magnifications of the Mandelbrot set with n=150. The particular area magnified in any particular picture is shown in the white box in the preceding picture.

(a)

Figure 2.3

Mandelbrot Set Close-ups 29

(b)

(c)

Figure 2.3 (continued) 30

(d) Figure 2.3 (continued)

In constructing the Mandelbrot set, one sees that allzk such that lzkl<1 tend to zero, all zk such that lzkl>1 tend to infinity and all zk such that lzkl=1 either get caught in some loop or wander around infinitely withno set pattern. Thus, the Mandelbrot set divides the complex plane into two parts. One part being the stable points which when recursively defined by the given function tendto zero and the other part being the stable points which tend to infinity. The boundary between these two parts is the unit circle and corresponds to those points which donot tend to either zero or infmity. The desire to know more about the transition froma stable state to a chaotic state has led to the study of the boundary between these two parts. 31

The points on such a boundary form what is commonly knownas a Julia

set.These points are represented by the blue portion in the middle of thepicture. Julia sets are named for G. Julia who, along with fellowFrench mathematician P. Fatou, studied their properties for themore general case of rational mappings in the complex plane. Julia and Fatou did theirwork during

World War I, but it remained largely unknown because without theaid of computer

graphics, these ideas were extremely difficult to comprehend. Figure2.4 shows the Julia set for the Mandelbrot set with n=150.

Figure 2.4

Julia Set for the Mandelbrot Set

Our last two examples of fractalsare what are known as strange attractors and are examples of how a fractalcan occur in a continuous nonlinear dynamical 32

system. A continuous dynamical system may be thought of as a time-independent recursively defined function.In looking at such an iterated scheme,we are primarily interested in the behavior of thesequences of iterates, or orbits, for various initial values. Sometimes thissequence may converge to a fixed point.

Other times it may settle into a set orbit abouta fixed point. Still other times it may

appear to move about at random, but always remain close to a certain set. Thisset to which all nearby orbits converge is what is known as an attractor. If theset is a fractal set, this is referred to as a "strange"or "fractal" attractor. The first of these two examples is the classic Lorenzattractor. E. Lorenz

was trained as a mathematician and worked as a meteorologist. Hewas particularly interested in weather forecasting.The problem that he saw with weather forecasting was that the slightest change in initial conditionscan make the best

forecast meaningless. In his study of weather systems and his contemplationof the

possibility of weather forecasting, Lorenz became particularly involved instudying the thermal convection of fluid heated from below. As the fluidis heated from below, it rises. As it reaches the surface, it cools and again headsdownward. This results in parallel rotating cylindrical rolls in the fluid. Lorenz usedthe continuity equation and the Navier-Stokes equations from fluid dynamics andthe heat conduction equation to come up with a description ofone of these rotating cylindrical rolls. After much experimentation, hecame up with what are now known as the Lorenz equations:

dx a(yx)

EIX_ dt4x Y dz --di = xy - Cz 33 where x is the rate of rotation of the cylinder, y is the difference in temperature at opposite ends of the cylinder, z is the deviation from a linear vertical temperature gradient, a is the Prandtl number of the air which involves both viscosity and thermal conductivity, C depends on the width to height ratio of the horizontal layer of fluid, and t is a control parameter representing the fixed temperature difference between the bottom and top of the system.

Figure 2.5 shows the Lorenz attractor with a= 10, t = 20 and C = 3.

Figure 2.5

Lorenz Attractor 34

Our final example is that of the ROssler bandattractor. Like the Lorenz

attractor, this is also a system of differential equations that hasa fractal attractor. Rassler's equationsare:

dx c= -yz dy dt =x+4Y dz v + z(x co)

Figure 2.6 shows the Riissler bandattractor with 4 = .2, D = .2 and co 5.7.

Figure 2.6

ROssler Band Attractor 35

These are just a few of the many widespread examples of fractals that have already been explored. There are surely many areas where fractals are lurking that are yet to be explored. This vast appearance of fractals in nature and the subsequent application thereof may lead one to look at one of natures most perplexing wonders - the ocean. One may suspect that fractals are present here as well. Specifically, one may ask whether ocean waves or water waves have a fractal structure and, if so, what this means for the future of oceanic research.Chapters 4 - 7 are devoted to the exploration of this issue. 36

Chapter 3 Fractal Dimension

Having seen the basic definitive properties of fractals inChapter 1 along with several examples of fractals,one may wonder how it is that fractals are determined in practice. The answer lies in theconcept of fractal dimension. In

order to determine whether or not an object hasa fractal structure, it is necessary to calculate the fractal dimension of the object. If the fractaldimension of an object with integer dimension n is notn, the object is said to be fractal. If the object has fractal dimension n, it is not fractal innature. The problem lies in the actual calculation of this fractal dimension. This isnot always an easy task. Before one

can deal directly with this problem, though, one must first define what ismeant by fractal dimension and this alonecan stir up much debate.

It is difficult to nail down a precise definition offractal dimension. Definitions differ among different people often times accordingto the specific fractal, the ease of calculation and the intendedpurpose of such a calculation. Although definitions may differ slightly, theyare all basically the same, some

simply being more restrictive than others. For thisreason, we will not refer to any one definition as being "the" definition of fractal dimension. Two of themost popular dimensions used are Hausdorff dimension and box-counting dimension. Probably the most prominent dimension used in determiningthe fractal nature of an object is Hausdorff dimension.This particular dimension is advantageous in that it is defined forany set.Also, since it is based on the mathematically well established concept of Hausdorffmeasure, it is widely accepted and theoretically relatively easy to work with. Thedisadvantage of Hausdorff dimension is that in most cases it isvery difficult to calculate or estimate and, in that 37

sense, is quite inconvenient. For this reason, Hausdorff dimension is really only

used in "very nice" situations. For example, in thecase of self-similar fractals where a recursive definition is possible. Fora further discussion and definition of Hausdorff dimension, see Falconer (1990), Smith (1983). The definition of dimension thatwe are going to look at, namely box-

counting dimension, is very popular primarily because of itsease of calculation and estimation. The following definitiongoes back to the 1930's and is also sometimes called Kolmogorov entropy, entropy dimension, capacitydimension, metric dimension, logarithmic density and information dimension.To avoid confusion, we will just call it box-counting dimension.

Definition :(Falconer, 1990) Let F be any non-empty bounded subsetof in and let NS(F) be the smallest number of sets of diameterat most 8 which can cover F. The lower and upper box-counting dimensions of F loglo NF) respectively are defined as di_rBn F 8-30-g8 logNog8(F) lim . dim BF8_90 -1 8 If these are equal, their common value, logNs(F), dimBF = 8-40-log5 is referred to as the box counting dimension of

F. In this definition, NS(F) can be takento be any one of the following:

(i) the smallest number of closed balls of radius 8 thatcover F;

(ii) the smallest number of cubes of side 8 thatcover F; (iii) the number of 8-mesh cubes that intersect F;

(iv) the smallest number of sets of diameterat most 8 that cover F;

(v) the largest number of disjoint balls of radius 8 withcenters in F. In practice, to find the dimension ofa plane set F, one draws a mesh of squares or boxes of side 8 and counts the number 1\15(F) that overlap theset for 38

various small 5. The dimension is the logarithmic rateat which No(F) increases as 5)0. This gives a good intuitive idea of the meaning ofbox-counting dimension.

The number of mesh squares of side 5 that intersecta set signifies how irregular the set is when examined at scale 8.The dimension reflects how rapidly the irregularities develop as 5>0.

This is one of the classic definitions of dimension thatcould be considered

"fractal" dimension. We will find this useful in the remainderof the paper as we try to determine whether or not objects do indeed havea fractal structure.

To get an intuitive idea of fractal dimension, letus reflect back to two of our

examples in Chapter 1.If one calculated the fractal dimension by either ofthe log2 above means, we would see that the Cantor Set hasa fractal dimension of log3 10E4 0.631 and the von Koch curve hasa fractal dimension of log3 1.262. Since the

fractal dimension of each of these sets is fractional,we would conclude that both of these sets are fractal. Notice that the Cantorset has dimension that is between 0 and

1 which we might suspect because it hasan infinite number of points (greater than

0-dimensional) and yet has zero length (less than 1-dimensional).The von Koch curve has dimension between 1 and 2 which again we mightsuspect because although it has infinite length (greater than 1-dimensional),it has zero area (less than 2-dimensional). Further, dimensions between 2 and3 would signify infinite

area (greater than 2-dimensional) and zero volume (less than 3-dimensional) andso on.It should also be noted that if acurve is continuously differentiable, the

dimension is one. Since fractal curves have, by definition,fractional dimensions,

we could conclude that fractal curves are not continuously differentiable.Thisis

consistentwith the propertyof nondifferentiabilityof fractalcurves mentioned in Chapter 1. 39

Chapter 4 Classical Description of Water Waves

Traditionally, there have been two fundamentalways of looking at a particle in motion. These two approachesare known as the Eulerian and Lagrangian

approaches. Both approaches are actually due to Leonhard Euler. However,Euler dealt primarily with fluids and so preferredone approach, while Joseph Lagrange

dealt mainly with solids and so preferred the other approach.It was this difference in their preferences that dictated the respectivenames of the two approaches. In order to get a simplified overview of thesetwo views and to get a glimpse of the differentiating factors, consider the following scenario. Suppose you are watching a circus parade andyou want to study the parade's motion. That is, you want to study the motion of theparticipants in the parade. One way to do this would be to stand inone fixed place and observe the phenomenon as the parade goes by. Thus,you are seeing the overall parade

without respect to any one particular participant. This is the Eulerianapproach.

A second approach would be to hop onto the back ofsome random elephant and join in the parade yourself. Nowyou are moving with respect to time so your position is no longer constant. Youare no longer seeing the entire parade, but rather you are seeing the motion of one random participant in theparade. This is the Lagrangian approach.

Both of these approaches provide a legitimate and usefulmeans of studying the motion of a particle, but if we consider eacha little closer, the distinction between the two will suggest which approachwe will want to take in our study of water waves. 40

Suppose we have a vertical cross-sectional cut through a finite body of water. Let a be any particle on the surface of our cut of water (Figure 4.1).

Figure 4.1

Cross-sectional Cut of Water

Then the surface of our cut of water may be described as a function ofour particle a and time t. Denote this function c(a,t). For any given t, thismay be thought of as a single-variable function and may be graphed two dimensionally (Figure 4.2). 41

(p(a,t)

0- a 0 Figure 4.2

Cross-sectional Cut of Wateras a Function of a Particle of Water and Time

Let ai be any particular value ofa. Then as t varies, each ai will randomly assume a new position and (p(a,t) will change accordingly (Figure 4.3). 42

9(a,t)

a j+1

0 a Figure 4.3

Cross-sectional Cut of Water as Time Varies

Although we have no way of knowing where each individualai will go, we can consider the path between their initial position and their finalposition to be continuously differentiable.

If we vary t and look at the sequence of 9(a,tj)ts,we are taking the Eulerian approach. We are, in essence, looking ata series of snapshots of our overall

surface from a constant viewpoint as time varies. It should be noted,however, that we do not know what any particular (p(a,tj) will look like and we are not guaranteed that (p(a,tj) will necessarily be continuously differentiable.

If, on the other hand, we want to trackone individual particle, we would

need to hold our specific particle a constant and lookat our surface as a function of

t. Since, as mentioned above, the path ofa distinct particle from tj to tj+1 may be

considered to be continuously differentiable, (p(a,t)may considered a continuously differentiable curve (Figure 4.4). This is the Lagrangian approach. 43

(p(a,t)

11(1)(a, ti+i))

(tim(a, ti ))

(ti_1,(P(a, ti-1))

0- t 0 Figure 4.4

Path of Motion of a Particular Particle as Time Varies

In our particular study of waves, we are going to want to considera cross- sectional cut across the surface of a wave at any particular time t. Thus,we will want to take the Eulerian approach. This will give us a series of "snapshots" of the cross-sectional cut across the surface of the wave. This will allowus to obtain a "snapshot" of the cut as the wave nears the breaking point. By observing where each particle is at the breaking point, we will be able to calculate the dimension of the waves surface and consequently detect the fractal structure of thewave at this point.

Chapters 5 and 6 will demonstrate this in two ways. Chapter 5 takesa physical approach to the problem and Chapter 6 takes a theoretical approachto the problem. In both cases, we will show that a cross-sectional cut of the surface ofa wave near the breaking point exhibits a fractal structure. 44

Chapter 5 A Physical Approach to the Fractal Nature of Water Waves

To convince ourselves of the fractal nature ofwaves, we decided to look at

photographs of water waves. The turbulence occurringas a wave breaks led us to suspect that the breaking point of a wave would be where we would most clearly see the fractal structure of a wave, so our basic plan was to try to get pictures of

waves as close to the breaking point as possible. This was not an easy task as it is

highly improbable that anyone would be able capture the instantaneousbreaking of a wave with a camera. Nevertheless, we set out to do this. The photos were taken at a speed of 5 framesper second. Thus, the pictures are grouped in sequences and the wave is capturedover a period of time ranging from shortly before breaking to shortly after breaking. The photographs

were taken at two locations. For a more controlled environment, photos were taken

of waves in the 100 meter wave tank at the O.H. Hinsdale Wave Facilityat Oregon State University. The waves being run at the timewe took the photographs were randomly generated. There was a sand bar constructed in thewave tank which served as a catalyst in the breaking of waves resulting in big, breakingwaves. This was desirable because the larger waves would be easierexamine in our search for

a fractal structure.In addition to the wave tank, we wanted to takesome

photographs in a naturally occurring setting. For this settingwe ventured out to the Oregon coast to obtain some pictures of Pacific Oceanwaves. The photographs were taken at the Devil's Churn which is located at Cape Perpetua, Oregon. All of

the photographs were taken with black and white film of the varying speeds of 100,

400 and 3200 according to the conditions of each respective location. 45

Once the photographs were obtained, we noticed a phenomena which seemed to be repeating itself to varying degrees in the differentsequences of waves. It appeared that as the wave approached the breaking point and the surface tension consequently was mounting, the face of the wavewas becoming increasingly

"crumpled" or irregular. The moment the surface tensionwas broken and the wave began to topple over, the face of the wave becamevery smooth. To use the analogy of a sheet for the face of the wave, it was as if the sheetwas becoming increasingly wrinkled as the surface tension approached the breaking point andas soon as the surface tension reached the breaking point, the sheetwas pulled taut and was very smooth. Figure 5.1 (a)- (f) shows one such sequence of pictures.

(a)

Figure 5.1

Sequence of Photographs of a Breaking Wave 46

(c)

Figure 5.1 (continued) 47

(d)

(e)

Figure 5.1 (continued) 48

(f) Figure 5.1 (continued)

This phenomena led us to the idea of looking ata cross-sectional cut of the face of the wave in an attempt tomeasure this irregularity. To do this, we proceeded to digitize several of the photographs.The digitization process numerically calculated the varying gray levels appearing ina given photograph. We then employed a computer program which would allowus to graph a cross- sectional cut of the wave based on these numerical calculations. As expected,these graphs appeared to be very rough and to the eye certainly looked nondifferentiable.

We found the photographs taken in the wave tank to be much easierto deal with because of the uniformity in their breaking points. Itwas difficult to predict, much less capture, the breaking point of the waves at the beach. Thiswas largely 49 due to the extraneous factors affecting the waves such as the windy condition and our inability to be at as close of a range as in the wave facility.

The task then remained to measure the irregularity of these graphs. To do this, we employed the box-counting dimension discussed in Chapter 3. Recall, if we let F be our graph, the box-counting dimension is defined as dimBF= logN8(F) 8>0 -log8 where N8(F) is in our case taken to be the smallest number of squares of side 8 that cover F. To calculate the limit, we manually looked at several decreasing values of 8 and then employed a statistical curve-fittingprogram to extrapolate the limit. Figures 5.2, 5.5, 5.8, 5.11 and 5.14 show the photographs of

5 of the waves. The arrow in each figure indicates where the horizontalcross- sectional cut across the face of the wave was taken. The last three digits in the title of each photograph indicate the line number measured from thetop out of a total of 450 lines where the cross-sectional cut was taken. Figures 5.3, 5.6, 5.9, 5.12 and

5.15 show the resulting graphs of the respective photographs where Xrepresents the horizontal position of a given particle of wateron the cut and Y represents the position of the particle perpendicular to the face of the picture. The unitsof measurement are based on the digitization grid of 700 points horizontally and 450 points vertically.Figures 5.4, 5.7, 5.10, 5.13 and 5.16 givea graphic representation of the computation of their respective box-counting dimensions. For the sake of comparison, we have included three waves at the breaking point from the wave facility, one wave at the breaking point from the beach andone wave slightly past the breaking point from the wave facility which demonstrates the smoothness that appears to follow the breaking of the surface tension. 50

Water1.135

Figure 5.2 Photograph of Water1.135 Y

210.000 200.000 - 190.000 180.000 170.000 160.000 150.000 140.000 130.000 120.000 110.000 100.000 90.000 80.000 70.000 60.000 50.000 40.000 30.000 20.000 10.000 0.000 -10.000 X 0.000 50.000 100.000 150.000 200.0043 250.000 300.000 350.000 400 000 450.000 500 000 Figure 5.3

Graph of Cross-sectional Cut ofWater1.135 52

Water1.135 dimBF 2.25 2.20 2.15 2.10 2.05 2.00 1.95 1.90 1.85 1.80

8 (decimeters) .01 .02.03 .04.05 .06.07 .08.09.10 Figure 5.4

Graph of Box-counting Dimension Calculation for Water1.135

8 dimBF log 163 .10 dm dimBF - - 2.212 log 197 .09 dm dimBF-log.09 -2.194 log 249 .08 dm dimB F = . log.08- 2 184 .07 dm rlog 300- 2.145 dim B' -log.07 log 334 .06 dm log.06 -2.066 log 432 .05 dm log.05- 2.026 .04 dm log 552 log.04- 1.961 883 .03 dm dimBF-l_oglog.03 -1.934

The curve-fitting program employed indicated the best exponentialcurve to fit our logN8(F) data was dimBF = 1.821e2-18. Thus, dimBF = lim 1.821. 8->0-log8 53

Water2.110 Figure 5.5 Photograph of Water2.110 Y

190.000

180.000

170.000

160.000

150.000

140.000

130.000

120.000

110.000

100.000

90.000

80.000

70.000

60.000

50.000

X 50.000 100.000 150.000 200 000 250.000 300.0(X) 350.000 400.000 450.0(X) 500.000 550 000 Figure 5.6 tft Graph of Cross-sectional Cutof Water2.110 4=, 55

Water2.110 dimBF

2.3 2.25 2.20

2.15 2.10 2.0 2.0 1.95 1.90 1.85

8 (decimeters) .01.02 .03.04.05 .06.07.08.09.10 Figure 5.7

Graph of Box-counting Dimension Calculation for Water2.110

dimBF 170 .10 dm 2.230 B -log.10 log 201 .09 dm dimBF --log.09- 2.202 .08 dm dimBF -log.08 2.200 l g 317 .07 dm dimBF -log.07- 2.166 log 344 .06 dm dimBF -log.06- 2.076 log 445 .05 dm dimBF --log.05 -2.036 log 553 .04 dm dimBF - - 1.962 -log.04 log 881 .03 dm dimBF - -log.03- 1.934

The curve-fitting program employed indicated the best exponentialcurve to fit our logN&(F) data was dimBF = 1.818e2.2018. Thus, dimBF=run 1.818. 8--)0-log8 56

Water3.150 Figure 5.8 Photograph of Water3.150 Y

160.000

150.000

140.000

1130.000

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100.000

90.000

80.000

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50.000

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10.000

0.000

0.000 50.000 10(1 (WM 150.000 200.0(x) 250.000 300.0(X) 350.000 400.600 450.000 500.000 Figure 5.9

Graph of Cross-sectionalCut of Water3.150 58

Water3.150 dimBF

2.15 2.10

2.05 2.00 1.95 1.90

1.85 1.80

1.75 1.70

5 (decimeters) .01.02 .03.04.05 .06.07.08.09.10 Figure 5.10

Graph of Box-counting Dimension Calculation for Water3.150

_a- dimBF log116 .10 dm - 2.064 B -log.10 log147 .09 dm dim F - 2.072 B -10g.09 icaLY12 .08 dm - 2.049 B -log.08 log 203 .07 dm dimBF - -log.07- 1.998 log 224- 1.924 B -log.06 _log_3_04 .05 dm dim -log - 1.908 log 383 .04 dm dimBF - -log.04- 1.848 loog.03g570 .03 dm dimBF - - 1.810 -1

The curve-fitting program employed indicated the best exponentialcurve to fit our logN6(F) data was dimBF = 1.710e2.0756. Thus, dimBF=lirn 1.710. 8-40-log6 59

Beach1.149 Figure 5.11 Photograph of Beach1.149 Y

150.000

140.000

130.000

120.000

110.000

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90.000

80.000

70.000 60.000 -

50.000

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0.000

0.000 100.000 200.0(X) 300.000 4(0.000 500.000 600.0(X) 700.000 Figure 5.12

Graph of Cross-sectional Cut ofBeach1.149 61

Beach1.149 dimBF

2.15 2.10

2.05 2.00 1.95 1.90

1.85 1.80

1.75 1.70

5 (decimeters) .01.02 .03.04.05 .06.07.08.09.10 Figure 5.13

Graph of Box-counting Dimension Calculation for Beach1.149

dimBF .10 dm dimBF - 2.158 log 167 .09 dm dimBF - 2.125 -log.09 .08 dm dimBFlog 195- 2.088 imB -log.08 log 209 .07 dm dimBF-log.07- 2.009 log 252 .06 dm dim BF --log.06- 1.965 .05 dm dimBFlog 353- 1.958 .04 dm log 423 dimBF-log.04- 1.879 log 573 .03 dm dimBF -log.03- 1.811

The curve-fitting program employed indicated the best exponentialcurve to fit our logNs(F) data was dimBF = 1.702e2.4498. Thus, dimBF=inn 1.702. 8->0-log8 62

Smooth1.100 Figure 5.14 Photograph of Smooth1.100 Y

210.000 200.000 190.000 180.000 170.000 160.000 150.000 140.000 130.000 120.000 110.000 100.000 90.000 80.000 70.000 60.000 50.000 40.000 30.000 20.000

10.000 0.000 -10.000 0.000 100.000 200.000 300.000 400.000 500.000 Figure 5.15

Graph of Cross-sectional Cutof Smooth1.100 64

Smooth1.100 dimBF

1.90 1.85

1.80 1.75 1.70 1.65

1.6

1.55 1.50 1.45

8 (decimeters) .01.02.03.04.05 .06.07.08.09.10 Figure 5.16

Graph of Box-counting Dimension Calculation for Smooth1.100

dimBF .10 dm '11-1 -log.10 tog 72 .09 dm dimBF-log.09 -1.776 log 87 .08 dm dimBF 1.768 log 95 .07 dm dimBF-log.07 1.712 .06 dm dimBF og1/ -log 1.644 log 129 .05 dm dimBF--log.05 -1.622 log 167 .04 dm dimBF --log.04- 1.590 log239 .03 dm dimBF - -log.03- 1.562

The curve-fitting program employed indicated the best exponential curve to fit our lim logN3(F) data was dimBF = 1.459e2.1928. Thus, dimBF = 1.459. 8-30-log8 65

The fractal dimensions calculated on these five waves range from 1.459 to 1.821. They are all clearly not equal to one (our graphs all have integer dimension of one) and thus we may conclude that the graphs resulting from the digitization of

these photographs are indeed indicative of a fractal structure to thewaves near the

breaking point. Even the smooth wave has a fractional dimension, although,as would be expected due to its smooth appearance, it is substantially lower than the other waves.

Two other tests were run on the graphs of these waves to giveus a more

general idea of the roughness of the surface. The first was the calculation ofa surface area ratio. In this test, a box was prescribed for thearea to be measured.

The program used to perform this calculation then calculated the ratio of thearea of the surface of the wave within this box to the area of the base of the box. By this definition, a flat surface would have a surface area ratio of 1 andvery rough surfaces would have large surface area ratios. The second testwas the calculation of the average curvature of the surface. The program usedwas written based on a definition of average curvature of a surface as conjectured by Dr. Harold Parks.

This definition was based on the definition of curvature fora closed polygonal curve. The surface was approximated by a polygonal surface by means of building up triangles from each of the vertices of the digitization grid. Then, whereas with a polygonal curve you look at the change in the directions of the tangentsat each vertex, here we looked at the edges on our polygonal surface and calculated the change in the direction of the tangents. To compensate for the extra dimension (that is, an edge versus a vertex), the program then multiplied the angle representing the change in direction of the tangent vectors by the length of the edge. Thecurvature was then averaged over the area of the prescribed box on which you were taking the 66

curvature. According to this definition, the average curvature of a flat surface would be zero and a very rough surface would have a large average curvature. Figure 5.17 gives a table of the surface area ratios and average curvatures

obtained for our five photos. As a reference for comparison purposes,we were advised by Dr. Kennan Smith that the rough spots on amammogram yielded curvature readings ranging from 2.49 to 16.54 and surface area ratios ranging from 2.36 to 13.62 depending upon the sharpening of the image. This data is also in

Figure 5.17. Although we do not have a precise frame of reference, basedon this frame of reference, our surfaces do indeed appear to be extremely rough.

Box Size Average PhotographBox Origin(Width x Height)Surface Area RatioCurvature Waterl (40,125) 300 x 65 19.04 23.92 Water2 (55,45) 450 x 125 13.70 15.12

Water3 (10,60) 435 x 90 9.11 11.77

Beachl (85,85) 650 x 100 7.98 9.54

Smoothl (110,80) 465 x 60 4.24 5.85

Mammogram (200,325) 100 x 70 2.35 to 13.62 2.49 to 16.52

Figure 5.17 Table of Surface Area Ratios and Average Curvatures for Water1.135, Water2.110,

Water3.150, Beach1.149, Smooth1.100 and Mammogram The significance of these findings will be discussed later in thispaper. The following chapter looks at the fractal structure of a wave froma different standpoint. Whereas the photographic approach offered usan actual physical look at waves, the following chapter will look at the waves from a more theoretical standpoint. 67

Chapter 6 A Theoretical Approach to the Fractal Nature of Water Waves

To approach the problem described in Chapter 4 ina more theoretical manner, we decided to create a spring model of a wave. In our model, each particle

of water in a cross-sectional cut of the surface of a wave, would be viewedas an individual mass which would be attached by springs to the particleson each side of it as well as to a base point representing a point on the floor of the modeled body of water. One end of the spring configuration is fixed while the other end is allowed to vary. Figure 6.1 shows this model.

y

0 11 1 1 1 1 1 1x 0 Figure 6.1

Spring Model 68

Representing the cut of the surface of the wave in this manner, we would then set the system in motion and observe the behavior of the particles. Again, we are taking the Eulerian approach, so we will want to look at the positions of the

particles at any particular time t. The Lagrangian of the system is defined to be L= T - V where T is the kinetic energy of the system and V is the potential energy of

the system. Let T and V be functions of the variables x and y. Also, let ,Yi and

zi,Yi denote the first and second derivatives of xi, yi with respect to t.Then the

Lagrangian equations of motion are:

d aL dt axi d aL dt a);i ayi

where (xi, yi) is the position of the ith particle at time t.

We findthat in our system:

T = =n 1Mii2 y.i2)]

V = n1 xi-1)2 (YiYi-1)2 -1)2+]

[2(4(xi-ti)2 + (yi-vi)2_ 4_Mg212]

f rmi'2 L =n1 LT(xi Yi2)1 r2(4(XiXi-1)2 + (yiyi-1)2

(xi - ti)2 + (yi - ivi)2 +M gY 12] 214-4 2 and the subsequent equations of motion are: 69

rYi1 k(4(xixi-1)2 + (yiYi-1)21) rxixi -11 i J mq(xi- xi -1)2 + (yi - yi-1)2 `Yi kol(xi- - + 402 + (yi,i)2 - q) [xi - ti mq(xiti )2 + (yiVi)2 Y Wi

k(4(xi+i-xi)2 + (yi+i-yi)2- 1) rxi+- xil mq(xi+ixi)2 + (yi+iyi)2 LYi+1Yij

where k = spring constant, 1 = horizontal spring length,q = vertical spring length,

m = mass, (ti,xifi) = fixed base point, (x0,y0) = fixed point on left, and (xi,yi)= position of the ith particle. Note that we are assuming all of the particleshave the same mass and all of the springs have the same constant. This gives us a system of 2n second order linear differential equations

where n is the number of particles in the model. To solve thissystem of differential

equations, we employed a computer program which used the 4th orderRunge- Kutta method, the 4th order Adams-Bashforth Predictor method and the5th order Adams-Moulton method. The results were then graphed using theXgraph program.

In performing these calculations, we wanted to be consistent with those in

Chapter 5, so we set up our model on a decimeter scale. This would bethe logical scale to use since that is the scale upon which the fractalstructure was observed in Chapter 5.

Initially we set up our model so that in the equilibrium position each particle is one decimeter apart and 10 decimeters from the base ofour model. We let each particle have a mass of 1 gram. The spring constantswe employed were 100 Newtons per decimeter. Figure 6.1 shows the model in the equilibrium position. 70

Y

10u=2:03=110==ez=n10==.= = = me=110

0 x 1 2 3 0 4 5 49 50 Figure 6.2

Equilibrium Position of Spring Model

In contrast to Chapter 6 where we looked at a horizontal cut of the face of the wave, we are now going to look at a vertical cut of surface of thewave. That is, we are going to take a side view of the wave andsee what happens as a wave is set in motion.

First of all, we observed the behavior of a nice, smoothwave which is not near the breaking point. To model such a wave with our system, we began with the system in the equilibrium position. We then grouped the particles in groups of five and we alternated giving each particle in one groupno initial velocities and giving each particle in the next group horizontal initial velocities of 2 decimetersper second and no vertical initial velocities. Figure 6.3 shows the graph of thesystems as time varies from 0 to .55 with time increments of .05 seconds. The behavior of 71 the system is as expected and we get a nice sinusoidal type of wave developing for small periods of time. It should be noted that since this isa nonlinear system, we only expect to get nice results for small periods of time. For longer periods of time the nonlinearities set in and we can no longer predict the system. =0 10.000 - .05 9.000 ,...... = .10 8.000 =.15 =72ir 7.000 % t= 6.000 1 - u i. _ _ - . . 5.000 .36S- 4.000 3.000 11 1= .45 t= .50 2.000 t= .55 1.000 ..- - 1 .i,4------. - - . 0.000 t

1 -1.000 -2.000 1 P. 1. -3.000 -4.000

-5.000 -6.000 -7.000 Iltommillen11111111- - - -8.000 \ \ -9.000 V___ -10.000

0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 45.000 50.000 Figure 6.3

Spring Model for Nonbreaking Wave 73

As mentioned in Chapter 5, our expectation of seeing a fractal lies with waves near the breaking point. As the surface tension mounts, we expect to see a very crumpled surface. This is the point at which we wish to model the wave. To obtain such a model, we again began in the equilibrium position and applied horizontal initial velocities ranging from -33 decimeters per second to 9 decimeters per second and vertical initial velocities ranging from 0 decimeters per second to 9 decimeters per second. In order to see how the wave initially develops Figure 6.4 shows the system as t varies from 0 to .15 with time increments of .01 seconds. Since it is really the breaking point we are interested in, we then took larger time increments and observed the behavior of the system over a longer period of time.Figure 6.5 shows the system as t varies from 0 to .30 with time increments of .05 seconds. Figure 6.6 shows a close-up of one period of this wave. ,Y

t =0

10.000 - t.. --..,,n-r, sn.:&.t -,..-ute t= .01 ...... ' - - - - - .- -...... ------...... -- t ... ., - - ..... t =.02 9.500 -- - .... -- A...- ...... - ...... - ..- s. " -- ... i..... , .--.053 9.000 , .- .." -...... r=:.111- V .... .--, 8.500 / ._. 1;15.5- t -. . -,. , .\ - , .-- . .-,. _ / ...... t= .06 8.000 ./ -,--.. / / - / , t= .07 AIM ANL 7.500 t= .08 7.000 IirV t= .09 "". t= .10 6.500 ...... t=.11 - 4 6.000 -t 5.500 - -, e - , / r I .. , t, t= .14 5.000 I. 1__= f 4.500 L / . / l / 1 1 1 / i 1 / ...... - / .. 4.000 / ... --.... / 3.500 / -, I / i 3.000 '-` / / .-- i 2.500 / 1, ---.. 7 ---7 .1 . --- % ...... / r --- / --. /_____ 2.000 ------,--- , -- k (-. I 1.500 / 7 / 7 / / / ./... 7 / .... I, .-- . --. r 1.000 "--- -.__ / ---_--_/ --l

0.000 5.000 10.000 15.000 20.000 25.0(X) 30.000 35.0(X) 40 000 45.000 50.000 Figure 6.4

Spring Model for Breaking Wave (Small Time Increments/Stnall Time) -F Y

1=0 10.000

1= .05 9.000 11...... '...... -.-.5' ...... -----...... _...... 1= .10 8.000 .15 7.000 6.000 ..--, .25 (0.....- . -- - ...... 30 5.000 ...... 4.000 'I 3.000 2.000

1.000 II.. 0.000 -1.000 i -2.000 ti - -3.000 ;-c--,--, ---- - _ 4' 4" F.- -4.000 il -5.000

-6.000 ti I \ -7.000 I,...------8.000 -9.000 _...... C.2-- mow.. ...a. _-_...... =.-...a...... 1r2r.....,.,.--. ---,.....too.....-... -10.000 --...... X 0.000 5.0(X) 10.0(X) 15.(X)() 20.000 25.(X)0 30.000 35.000 40.000 45.000 50.000 Figure 6.5

Spring Model for Breaking Wave (Larger TimeIncrements/Longer Time) 76

Y 12.000 t= t= .05 11.000 t=.10 10.000 t=.15 ...... t= .20 ...... t=.25 8.000 t= .30

7.000 ...... 6.000...... 5.000

4.000

3.000

2.000 _ - _- r7-- 1.000

0.000

-1.000

-2.000

-3.000 a./

-4.000

-5.000

-6.000 ..... 41111.

-8.000

-9.000

-10.000

-11.000

-12.000 15.000 20.000 25.000

Figure 6.6

Close-up of One Period of the Spring Model in Figure 6.5 77

We see that at t = .30, the graph intersects itself indicating thatour modeled wave has broken. The steps right before that are where we would expect to see the fractal. Since the fractal is expected to be observed on the face of thewave, we looked at a close-up of that area.Figure 6.7 shows the face of the wave at t= .20 and Figure 6.9 shows the face of the wave at t= .25. Using the same method to calculate the box-counting dimensionas in Chapter 5, we obtained the results shown on the followingpages. A graphic representation of these results is seen in Figure 6.8 and Figure 6.10 for t= .20 and t = .25 respectively. Y

-2.100 -2.200 -2.300 -2.400 -2.500

-2.600

-2.700

-2.800 -2.900 -3.000 -3.100

-3.200 -3.300 -3.400 -3.500

-3.600 -3.700

-3.800

-3.900

-4.000 -4.100

14.000 15.0(10 16.000 17.0(X) 18. )00 19.000 20.000 21.000 22.000 Figure 6.7

Close-up of Spring Model in Figure 6.6 for t= .20 00 79

Spring Problem 2(t = .20) dimBF

1.70 1.65

1.60 1.55 1.50 1.45

1.40 1.35 1.30 1.25

0 8 (decimeters) 0 .01.02.03.04.05.06.07.08.09.10 Figure 6.8

Graph of Box-counting Dimension Calculation for Figure 6.7

5 dimRF .10 dm dimBF'-log.10 1.544 log 42 .09 dm 1.552 dimBF-log.09 .08 dm log 47 dimBF --log.08= 1.524 .07 dm log 53 dinIBF --log.07= 1.493 tog 54 .06 dm dimBF 1.418 -log.06 - .05 dm log 70 dimBF-log.05 1.418 .04 dm log85 dimBF-log.04 -1.380 tog114 .03 dm dimBF- 1'351

The curve-fitting program employed indicated the best exponentialcurve to fit our iim logN5(F) data was dimBF = 1.269e2.1378. Thus, dimBF= 1.269. -log8 Y

-5.600 t= .25

-5.800 -6.000 111111111111111111111 -6.200 111111111111.1.11111 -6.400

-6.600

-6.800

-7.000 -7.200 1111.5111111111Will11111MMEN -7.400

-7.600

-7.800 arig141.0111 -8.000 Will WilimAWITITione -8.200

-8.400

14.000111111111111111116MINTI115.000 16.000 17.000 1RANX1 19.000 20.000 21.000 22.000 23.000 Figure 6.9 Close-up of Spring Model in Figure 6.6 for t = .25 81

Spring Problem 2(t = .25) dimBF A 1.70 1.65- 1.60- 1.55- 1.50- 1.45- 1.40- 1.35- I 1.30- 1.25- 0 8 (decimeters) 0 .01.02.03.04.05 .06.07.08.09.10 Figure 6.10

Graph of Box-counting Dimension Calculation for Figure 6.9

dimBF .10 dm log 36 dimBF-log.10 1.556 .09 dm =1g45 -log.09 1.581 .08 dm log50 dimBF-log.08- 1.549 log52 .07 dm dimBF - log. 1.486 .06 dm log57 dimBF-log.06 -1.437 .05 dm log 70 dimBF --log.05 1.418 .04 dm log 82 dimBF-log.04 1.369 _log.120 .03 dm dim -log 1'365

The curve-fitting program employed indicated the best exponentialcurve to fit our limlogIsTa(F) data was dimBF = 1.269e2-1378. Thus, dimBF= 1.269. 6-30 82

In both of the close-up views of the latter spring model,we see that the wave has a fractional fractal dimension of 1.269. Thus, the wave does indeed appear to exhibit a fractal structure at this point. The significance of our results will be discussed in the next chapter. 83

Chapter 7 Conclusion and Significant Findings

The significance of this research lies primarily in the fact that it pointsto a very problematic situation that occurs at the surface of a wave as it nears the

breaking point. Although no rigorous mathematical proof has been madeto this effect, the results of this research are highly indicative ofa fractal structure of waves as they approach the breaking point.

All indications, from this research, are that as awave nears the breaking point, one obtains a surface that is highly nondifferentiableor at least a surface that would be very difficult to model with a system of differential equations. Suchan approach to modeling waves has long been the tradition. At thevery least, this research should suggest the difficulty one would encounter in tryingto model waves with differential equations. If indeed, as this research seems to indicate, the surface is fractal at any given point in time, and consequently nondifferentiableat that point in time, the attempts to model waves accuratelyover a substantial period of time with differential equations would prove impossible. It isour hope that this research will encourage and promote new approaches to the modeling ofwaves that do not rely upon the differentiability of the waves surface.

The study of fractals is a very young discipline. It is as of yeta relatively vaguely defined area of mathematics. Fractals are beginningto be observed in all areas of study and the rumblings of fractal geometry are being heard with increased frequency. As with any new phenomenon, one has first to observe it. Only then can one begin to disect and study it, and, ultimately, learn to respond to and deal with it. The discipline of fractals is at this stage.They are being observed with increasing regularity and people in many varied fieldsare beginning to look at and 84

study them carefully. Many questions are arising, but as of now the solutions are not clear. How to deal with or respond to fractals is not necessarily evident. It will take time, but it is our belief that, as we learn more about fractals, this will

happen. It is also our belief that with that will come a new way of modeling waves that is not dependent upon the differentiability of the surface and that is accurate over substantial periods of time. It is our hope to contribute to making that happen. 85

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Computer Software Utilized in this Project

Fractint by Bert Tyler, Tim Wegner, Mark Peterson,Pieter Branderhorst,et al Xgraph by David Harrison, University of California

Math Cad by Soft Math

HP48SX Statistical Curve Fitting Program

Curvature Program by Dr. Kennan T. Smithas advised by Dr. Harold Parks Surface Area Ratio Program by Dr. Kennan T. Smith

Digitization Graphing Programs by Dr. Kennan T. Smith Conversion Program by Russ Ruby

Runge-Kutta/Adams-Bashforth/Adams-Moulton Program by Peter White