Session: 02-04-01 Extreme and Freak Waves (Co-hosted with the Ian Young Honoring Symposium)

Paper Number: 101978

101978 - Nonlinear Fourier Analysis Using Quasiperiodic Fourier Series 

            “Linear Fourier analysis” is based on the application of “periodic Fourier series” to a wide range of wave simulation and data analysis problems. A key feature of the method is the direct computation of the “correlation function” and its Fourier transform the “power spectrum.” A key idea in these applications is the assumption that the “phases” of the periodic Fourier transform are “random numbers on zero to two pi.” The method leads to simple ways to compute the “coherence functions” and “transfer functions” between two signals. Many books have been written about the “linear model” and its myriad applications. In the present talk I address the infinite number of “nonlinear wave equations” that are solvable by “quasiperiodic inverse scattering transform” and, amazingly, I show how such equations are solvable with “quasiperiodic Fourier series.” This means that all operations of the nonlinear problems of water waves are isomorphic to those of the linear problem! All operations that one can conduct with periodic Fourier series can also be conducted with quasiperiodic Fourier series: This means the correlation function, power spectrum, coherence functions, transfer function for the nonlinear problem can be conducted in analogy with the linear problem. The nonlinear problem, is just as easy as the linear problem! I give a fast Fourier transform for quasiperiodic Fourier series and show how to conduct numerical simulations of nonlinear waves using the procedure. The methods discussed here provide for a new future in which nonlinear problems are just as easy to treat as linear ones.

Presenting Author: Alfred R. Osborne Nonlinear Waves Research Corporation

Presenting Author Biography: Alfred R. Osborne has worked in the field of ocean waves and their application to offshore structure design for over 50 years. He is an expert in the study of extreme ocean surface and internal waves. He began his employment with Exxon Production Research Company in Houston in 1967, where we worked for 15 years. He has since consulted in the offshore for over half a dozen different oil companies. He is presently CEO of the company Nonlinear Waves Research Corporation in Alexandria, Virginia.

Authors:

Alfred R. Osborne Nonlinear Waves Research Corporation