Filtered Waves

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For more than 200 years, the Fourier Transform has been one of the most important mathematical tools for understanding the dynamics of linear wave trains. Nonlinear Ocean Waves and the Inverse Scattering Transform presents the development of the nonlinear Fourier analysis of measured space and time series, which can be found in a wide variety of physical settings including surface water waves, internal waves, and equatorial Rossby waves. This revolutionary development will allow hyperfast numerical modelling of nonlinear waves, greatly advancing our understanding of oceanic surface and internal waves. Nonlinear Fourier analysis is based upon a generalization of linear Fourier analysis referred to as the inverse scattering transform, the fundamental building block of which is a generalized Fourier series called the Riemann theta function. Elucidating the art and science of implementing these functions in the context of physical and time series analysis is the goal of this book. Presents techniques and methods of the inverse scattering transform for data analysis Geared toward both the introductory and advanced reader venturing further into mathematical and numerical analysis Suitable for classroom teaching as well as research

Wave breaking represents one of the most interesting and challenging problems for fluid mechanics and physical oceanography. Over the last fifteen years our understanding has undergone a dramatic leap forward, and wave breaking has emerged as a process whose physics is clarified and quantified. Ocean wave breaking plays the primary role in the air-sea exchange of momentum, mass and heat, and it is of significant importance for ocean remote sensing, coastal and ocean engineering, navigation and other practical applications. This book outlines the state of the art in our understanding of wave breaking and presents the main outstanding problems. It is a valuable resource for anyone interested in this topic, including researchers, modellers, forecasters, engineers and graduate students in physical oceanography, meteorology and ocean engineering.

Ten years ago, de Loor and co-workers at TNO, The Netherlands, were the first to report bottom topography patterns in real aperture radar (RAR) images of the southern North Sea. At that time, this was a real puzzle. The skin depth of microwaves for sea water is only of the order of centimeters while the sea bottom is about 20 meters below the surface. Electromagnetic radiation therefore cannot probe the bottom directly. Similar phenomena were found in radar imagery from SEASAT and SIR-AlB synthetic aperture radars (SAR's) of Nantucket Shoals, the English Channel and many other coastal areas. Since then theory and ocean field experiments (Le., Phelps Bank, Georgia Straits, SARSEX, TOWARD, FASINEX, etc.) have advanced our understanding considerably. We now know that these surface signatures are the results of surface currents, perturbed by the bottom topography, which refract the propagation and modulate the energy of (short) surface waves so as to cause microwave backscatter power variations. Hence, any large scale ocean features containing nonuniform surface currents (i.e. internal waves, eddies, fronts, etc.) will cause similar manifestations in the radar imagery by means of current-wave-microwave interactions. Observations confirm this.

The central theme of this book is the application of the linear filtering theory to the vibration of structures in a fluid. Emphasis is placed on the mathematical models which, in the theory of systems, characterize the state of a dynamic system. The mathematical models are in the form of linear Ito stochastic differential equations. Discretization of the models, which leads to straightforward computer applications, is also discussed. The book also presents an approach to nonlinear problems based on the expansion of random functions in a series. To elucidate the proposed approach, examples on the application of Kalman filters, which refer to the vibrations of cylinders in waves, are cited. This provides a practical orientation to complement the proposed theory and contributes to a clearer and deeper understanding of the subject matter.

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