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1 Brief History and Overview of Nonlinear Water Waves 1 Brief History and Overview of Nonlinear Water Waves 1.1 Linear and Nonlinear Fourier Analysis Man has long been intrigued by the study of water waves, one of the most ubiquitous of all known natural phenomena. Who has not been fascinated by the rolling and churning of the surf on a beach or the often-imposing presence of large waves at sea? How many countless times have ship captains logged the treacherous encounters with high waves in the deep ocean or later reported (if they were lucky) the damage to their ships? Man’s often strained friendship with the world’s oceans, and its waves and natural resources, has endured at least since the beginning of recorded history and perhaps even to the invention of ocean going vessels thousands of years ago. But it is only in the last 200 years that the study of water waves has been placed on a firm foundation, not only from the point of view of the physics and mathematics, but also from the perspective of experimental science and engineering. While water waves are one of the most common of all natural phenomena, they possess an extremely rich mathematical structure. Water waves belong to one of the most difficult areas of fluid dynamics (Batchelor, 1967; Lighthill, 1986) and wave mechanics (Whitham, 1974; Stoker, 1957; LeBlond and Mysak, 1978; Lighthill, 1978; Mei, 1983; Drazin and Johnson, 1989; Johnson, 1997); Craik, 2005, namely the study of nonlinear, dispersive waves in two-space and one-time dimensions. The governing equations of motion are coupled nonlinear partial dif- ferential equations in two fields: the surface elevation, Z(x, t), and the velocity potential, f(x, t). Analytically, these equations are difficult to solve because of the nonlinear boundary conditions that are imposed on an unknown free surface. This set of equations is known as the Euler equations, which are based upon sev- eral physical assumptions: (1) the waves are irrotational, (2) the motion is inviscid, (3) the fluid is incompressible, (4) surface tension effects are negligible, and (5) the pressure over the free surface is a constant. While one may question a number of these assumptions, it is safe to say that they allow us to study a wide variety of wave phenomena to an excellent order of approximation. Generally speaking, the Euler equations of motion (Chapter 2) which govern the behavior of water waves are highly nonlinear and nonintegrable. The term “non- linear” implies that the larger the waves are, the more their shapes deviate from simple sinusoidal behavior. The term “integrable” means that the equations of # 2010 Elsevier Inc. All rights reserved. Doi: 10.1016/S0074-6142(10)97001-0 4 Nonlinear Ocean Waves & Inverse Scattering Transform motion can be exactly solved for particular boundary conditions. It is often fash- ionable in modern times to discuss higher-order “nonintegrability” in terms of such exotic phenomena as bifurcations, singular perturbation theory, and chaos. Clearly, the special case of linear wave motion, for a well-defined dispersion relation, can be solved exactly by the method of the Fourier transform (Chapter 2). The Fourier method allows one to project the free surface elevation (and other dynamical properties such as the velocity potential) onto linear modes that are simple sinusoidal waves. Linear superposition of the sine waves gives the exact solution for the wave dynamics for all space and time. Modern research devel- opments have led to the development of the discrete Fourier transform and its much celebrated and accelerated algorithm, the fast Fourier transform (FFT). These developments of course emphasize the importance of periodic boundary conditions in the analysis of time series data and in numerical modeling situa- tions, because the discrete Fourier transform is a periodic function. A large number of scientific fields have embraced the Fourier approach. These include the study of laboratory water waves, oceanic surface and internal waves, light waves in fiber optics, acoustic waves, mechanical vibrations, etc. Both scientists and engineers in such diverse fields as optics, ocean engineering, communications engineering, spectroscopy, image analysis, remotely sensed satellite data acquisition, plasma physics, etc., have all benefited from the use of Fourier methods. Tens of thousands of scientific papers have contributed to the various fields and a number of books have provided a clear pathway through the difficulties and pitfalls of linear (space and) time series analysis, not only from the point of view of data analysis procedures, but also from the point of view of numerical algorithms. Clearly, linear Fourier analysis is one of the most important tools ever developed for the scientific and engineer- ing study of wave-like phenomena. The power of the Fourier method for determining the exact solution of linear wave equations is often cast in terms of the Cauchy problem for one-space and one-time dimensions: Given the wave profile as a function of space, x,atsome initial value of time, t ¼ 0, determine the solution of the surface wave dynamics for all values of x for all future (and past) times, t, that is, given the initial surface elevation Z(x,0)computeZ(x, t) for all t. In two-space dimensions (x, y, t), this perspective has the obvious generalization. Of course, the major goal of the field of nonlinear wave mechanics is to fully describe the surface elevation, Z(x, y, t), and the velocity potential, f(x, y, z, t), for all space and time. Within this theoretical context, an important aspect of the Fourier transform is the extension of the approach to the analysis of experimental data. Typically, (1) the wave amplitude is measured as a function of the spatial variable, x,at some fixed time, t ¼ 0 (this approach is often discussed in terms of remote sens- ing methods) or (2) the amplitude is measured as a function of time, t, at some fixed spatial location, x ¼ 0 (for which one obtains a time series). Clearly, one may also consider an array of fixed locations at which the wave amplitude is measured as a function of time. From a mathematical point of view, the first of these approaches is naturally associated with the Cauchy problem 1 Brief History & Overview of Nonlinear Water Waves 5 (one measures space series and Fourier analysis is defined over the spatial variable in terms of wavenumber) while the second method is associated with a boundary value problem (one measures time series and Fourier analysis is defined over the time variable and the associated frequency). Extension of the Fourier method to other aspects of the data analysis problem, such as the filter- ing of data and the analysis of random data, are also well known and are used often by researchers whose goal is to better understand wave-like phenomena. For those familiar with the analysis of measured space or time series the most often used numerical tool is the FFT, a discrete algorithm that obeys periodic boundary conditions. The Fourier transform for infinite-line or infinite-space boundary conditions has also been an important mathematical development; it solves the famous “rock-in-a-pond” problem. For most data analysis purposes, the discrete, periodic Fourier transform is most often preferred. As simple as the picture is for linear, dispersive wave motion, the extension of the Fourier approach to nonlinear wave dynamics has followed a long and difficult road. Analytical approaches for solving nonlinear wave equations have been slow to evolve and it is only in the last 50 years that general methods have become available. This theoretical work was a natural evolution that began, at least in modern terms, with the work of Fermi et al. (1955) who discovered a marvelous temporal recurrence property for a chain of nonlinearly connected oscillators. A few years later, Zabusky and Kruskal (1965) discovered the soliton in numerical solutions of the Korteweg-deVries (KdV) equation (small-but-finite amplitude, long waves in shallow water). Then the exact solution of the Cauchy problem for the KdV equation was found for infinite-line boundary conditions (Gardner, Green, Kruskal, and Miura, GGKM, 1967) using a new mathematical method now known as the inverse scattering transform (IST). This work was only the beginning of many new approaches for integrating nonlinear wave equations and for discovering their physical properties (Leibovich and Seebass, 1974; Lonngren and Scott, 1978; Lamb, 1980; Ablowitz and Segur, 1981; Eilenberger, 1981; Calogero and Degasperis, 1982; Newell, 1983; Matsuno, 1984; Novikov et al., 1984; Tracy, 1984; Faddeev and Takhtajan, 1987; Drazin and Johnson, 1989; Fordy, 1990; Infeld and Rowlands, 1990; Makhankov, 1990; Ablowitz and Clarkson, 1991; Dickey, 1991; Gaponov-Grekhov and Rabinovich, 1992; Newell and Moloney, 1992; Belokolos et al., 1994; Ablowitz and Fokas, 1997; Johnson, 1997; Remoissenet, 1999; Polishchuk, 2003; Ablowitz et al., 2004; Hirota, 2004). From data analysis and numerical modeling points of view, the IST plays a role in the study of nonlinear wave dynamics similar to the linear, periodic Fourier transform provided that the IST exists for periodic boundary condi- tions for a physically suitable nonlinear wave equation. One motivation for periodic boundary conditions for nonlinear equations rests with the fact that most applications of linear Fourier analysis are based upon the FFT, a periodic algorithm. The periodic formulation for the IST was discovered for the KdV equation in the mid-1970s (see Belokolos et al., 1994 and cited references) and subsequently applied to a number of other physically important wave 6 Nonlinear Ocean Waves & Inverse Scattering Transform equations. In this chapter, the Riemann theta function plays the central theoret- ical and experimental roles. Of course, one can see that the nonlinear Fourier analysis of time series data must contain a number of pit falls.
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