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.