Shallow Water Nonlinear Waves: Their Wave Equations, Quasiperiodic Fourier Series Solutions and Statistical Distributions

Abstract

Abstract The goal of this paper is to bring new mathematical and physical methods to the problem of shallow water wave motions, for which there are many important engineering applications. There are four issues we address in this paper: (1) Nonlinear integrable wave equations have (quasiperiodic) Fourier series solutions. This means that, even though the wave dynamics are nonlinear, they can nevertheless be described as a linear superposition of sine waves, a surprising result. Nonlinearity arises from particular phase locking of the Fourier modes. (2) We discuss how these Fourier series solutions can always be computed to high accuracy for a particular nonlinear wave equation. While quasiperiodic Fourier series are more complicated than ordinary periodic Fourier series (a standard tool of ocean engineering as the Fast Fourier transform, or FFT), the advantage of dealing with a linear superposition law for nonlinear equations is quite useful. (3) We apply these Fourier series to the modelling of water waves. Amazingly, we use the linearity property of the quasiperiodic Fourier series, and we find a new wave model for solving nonlinear wave motion. Our model, which we call quasilinear, has properties similar to the well-known linear model already applied to ocean engineering problems for over 90 years [Paley and Weiner, 1935] [Longuet-Higgins, 1957]. An additional particular transformation (due to Baker [1907], Its & Matveev [1976], Mumford [1982]) to the quasilinear model allows us to simply simulate typical oceanic nonlinear wave motions and to analyze data. (4) We discuss determination of the important probability distributions of water waves for amplitudes, heights and crests from knowledge of the quasiperiodic Fourier series solutions. The results given herein are based upon a simply stated Theorem: Given an integrable wave equation and/or its Hamiltonian perturbations we can write the full spectral solutions of the equations as quasiperiodic Fourier series and we can also analytically determine the probability distributions of various properties of the wave field (including, amplitudes, crest heights and wave heights). We are interested in shallow water wave motion in the present paper and our nonlinear wave equation of choice is the Korteweg-deVries equation (KdV). Our simple discussion has therefore introduced quasiperiodic Fourier methods and probability distributions to the solutions of the KdV equation. Furthermore, the Kadomtsev-Petviashvili equation is a two-dimensional shallow water wave equation (a generalization of the KdV equation, also with Hamiltonian perturbations) and it too can be treated in a similar manner. In this context this paper is dedicated to a discussion of important physical and engineering tools for the shallow water domain.