Nonlinear Fourier Analysis for Shallow Water Waves

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Shallow Water ◽  
Nonlinear Wave ◽  
Water Waves ◽  
Wave Equations ◽  
Superposition Principle ◽  
Broad Band ◽  
Kp Equation ◽  
Linear Superposition

Abstract I consider nonlinear wave motion in shallow water as governed by the KP equation plus perturbations. I have previously shown that broad band, multiply periodic solutions of the KP equation are governed by quasiperiodic Fourier series [Osborne, OMAE 2020]. In the present paper I give a new procedure for extending this analysis to the KP equation plus shallow water Hamiltonian perturbations. We therefore have the remarkable result that a complex class of nonlinear shallow water wave equations has solutions governed by quasiperiodic Fourier series that are a linear superposition of sine waves. Such a formulation is important because it was previously thought that solving nonlinear wave equations by a linear superposition principle was impossible. The construction of these linear superpositions in shallow water in an engineering context is the goal of this paper. Furthermore, I address the nonlinear Fourier analysis of experimental data described by shallow water physics. The wave fields dealt with here are fully two-dimensional and essentially consist of the linear superposition of generalized cnoidal waves, which nonlinearly interact with one another. This includes the class of soliton solutions and their associated Mach stems, both of which are important for engineering applications. The newly discovered phenomenon of “fossil breathers” is also characterized in the formulation. I also discuss the exact construction of Morison equation forces on cylindrical piles in terms of quasiperiodic Fourier series.

scholarly journals Nonlinear Fourier Analysis: Rogue Waves in Numerical Modeling and Data Analysis

2020 ◽  
Vol 8 (12) ◽  
pp. 1005
Author(s):  
Alfred R. Osborne
Keyword(s):  
Nonlinear Dynamics ◽  
Fourier Series ◽  
Fourier Analysis ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Phase Locking ◽  
Rogue Waves ◽  
Nonlinear Wave Equations ◽  
Nls Equation ◽  
Linear Superposition

Nonlinear Fourier Analysis (NLFA) as developed herein begins with the nonlinear Schrödinger equation in two-space and one-time dimensions (the 2+1 NLS equation). The integrability of the simpler nonlinear Schrödinger equation in one-space and one-time dimensions (1+1 NLS) is an important tool in this analysis. We demonstrate that small-time asymptotic spectral solutions of the 2+1 NLS equation can be constructed as the nonlinear superposition of many 1+1 NLS equations, each corresponding to a particular radial direction in the directional spectrum of the waves. The radial 1+1 NLS equations interact nonlinearly with one another. We determine practical asymptotic spectral solutions of the 2+1 NLS equation that are formed from the ratio of two phase-lagged Riemann theta functions: Surprisingly this construction can be written in terms of generalizations of periodic Fourier series called (1) quasiperiodic Fourier (QPF) series and (2) almost periodic Fourier (APF) series (with appropriate limits in space and time). To simplify the discourse with regard to QPF and APF Fourier series, we call them NLF series herein. The NLF series are the solutions or approximate solutions of the nonlinear dynamics of water waves. These series are indistinguishable in many ways from the linear superposition of sine waves introduced theoretically by Paley and Weiner, and exploited experimentally and theoretically by Barber and Longuet-Higgins assuming random phases. Generally speaking NLF series do not have random phases, but instead employ phase locking. We construct the asymptotic NLF series spectral solutions of 2+1 NLS as a linear superposition of sine waves, with particular amplitudes, frequencies and phases. Because of the phase locking the NLF basis functions consist not only of sine waves, but also of Stokes waves, breather trains, and superbreathers, all of which undergo complex pair-wise nonlinear interactions. Breather trains are known to be associated with rogue waves in solutions of nonlinear wave equations. It is remarkable that complex nonlinear dynamics can be represented as a generalized, linear superposition of sine waves. NLF series that solve nonlinear wave equations offer a significant advantage over traditional periodic Fourier series. We show how NLFA can be applied to numerically model nonlinear wave motions and to analyze experimentally measured wave data. Applications to the analysis of SINTEF wave tank data, measurements from Currituck Sound, North Carolina and to shipboard radar data taken by the U. S. Navy are discussed. The ubiquitous presence of coherent breather packets in many data sets, as analyzed by NLFA methods, has recently led to the discovery of breather turbulence in the ocean: In this case, nonlinear Fourier components occur as strongly interacting, phase locked, densely packed breather modes, in contrast to the previously held incorrect belief that ocean waves are weakly interacting sine waves.

A Coupled-Mode Technique for the Run-Up of Non-Breaking Dispersive Waves on Plane Beaches

2006 ◽  
Author(s):  
K. A. Belibassakis ◽  
G. A. Athanassoulis
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Wave Equations ◽  
Neumann Boundary ◽  
Breaking Waves ◽  
Numerical Results ◽  
Coupled Mode Theory ◽  
Mode Theory ◽  
Coupled Mode ◽  
Run Up

A coupled-mode model is developed and applied to the transformation and run-up of dispersive water waves on plane beaches. The present work is based on the consistent coupled-mode theory for the propagation of water waves in variable bathymetry regions, developed by Athanassoulis & Belibassakis (1999) and extended to 3D by Belibassakis et al (2001), which is suitably modified to apply to a uniform plane beach. The key feature of the coupled-mode theory is a complete modal-type expansion of the wave potential, containing both propagating and evanescent modes, being able to consistently satisfy the Neumann boundary condition on the sloping bottom. Thus, the present approach extends previous works based on the modified mild-slope equation in conjunction with analytical solution of the linearised shallow water equations, see, e.g., Massel & Pelinovsky (2001). Numerical results concerning non-breaking waves on plane beaches are presented and compared with exact analytical solutions; see, e.g., Wehausen & Laitone (1960, Sec. 18). Also, numerical results are presented concerning the run-up of non-breaking solitary waves on plane beaches and compared with the ones obtained by the solution of the shallow-water wave equations, Synolakis (1987), Li & Raichlen (2002), and experimental data, Synolakis (1987).

scholarly journals A conservative fully-discrete numerical method for the regularised shallow water wave equations

2021 ◽  
Author(s):  
Dimitrios Mitsotakis ◽  
Hendrik Ranocha ◽  
David I Ketcheson ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Numerical Method ◽  
Solitary Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Mixed Finite Element ◽  
Boussinesq System ◽  
Fully Discrete ◽  
Long Time

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

scholarly journals New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 137-144 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Bo Xu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Shallow Water Waves ◽  
Bilinear Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

scholarly journals Breather Turbulence: Exact Spectral and Stochastic Solutions of the Nonlinear Schrödinger Equation

Fluids ◽  
2019 ◽  
Vol 4 (2) ◽  
pp. 72 ◽  
Author(s):  
Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Water Waves ◽  
Ocean Waves ◽  
Airborne Lidar ◽  
Ocean Dynamics ◽  
Theoretical Interpretation ◽  
Maximum Height ◽  
Nonlinear Phenomena ◽  
Nonlinear Schrödinger

I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also referred to as finite gap theory (FGT) in the Russian literature) and (2) quasiperiodic Fourier series, both of which enhance the physical and mathematical understanding of complicated nonlinear phenomena in water waves. The basic approach I refer to is nonlinear Fourier analysis (NLFA). The formulation describes wave motion with spectral components consisting of sine waves, Stokes waves and breather packets that nonlinearly interact pair-wise with one another. This contrasts to the simpler picture of standard Fourier analysis in which one linearly superposes sine waves. Breather trains are coherent wave packets that “breath” up and down during their lifetime “cycle” as they propagate, a phenomenon related to Fermi-Pasta-Ulam (FPU) recurrence. The central wave of a breather, when the packet is at its maximum height of the FPU cycle, is often treated as a kind of rogue wave. Breather turbulence occurs when the number of breathers in a measured time series is large, typically several hundred per hour. Because of the prevalence of rogue waves in breather turbulence, I call this exceptional type of sea state a breather sea or rogue sea. Here I provide theoretical tools for a physical and dynamical understanding of the recent results of Osborne et al. (Ocean Dynamics, 2019, 69, pp. 187–219) in which dense breather turbulence was found in experimental surface wave data in Currituck Sound, North Carolina. Quasiperiodic Fourier series are important in the study of ocean waves because they provide a simpler theoretical interpretation and faster numerical implementation of the NLFA, with respect to the IST, particularly with regard to determination of the breather spectrum and their associated phases that are here treated in the so-called nonlinear random phase approximation. The actual material developed here focuses on results necessary for the analysis and interpretation of shipboard/offshore platform radar scans and for airborne lidar and synthetic aperture radar (SAR) measurements.

scholarly journals Observation of broad-band water waveguiding in shallow water: a revival

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Fabián Sepúlveda-Soto ◽  
Diego Guzmán-Silva ◽  
Edgardo Rosas ◽  
Rodrigo A. Vicencio ◽  
Claudio Falcón
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Open Channel ◽  
Fluid Layer ◽  
Broad Band ◽  
Shallow Water Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Phenomena

Abstract We report on the observation and characterization of broad-band waveguiding of surface gravity waves in an open channel, in the shallow water limit. The waveguide is constructed by changing locally the depth of the fluid layer, which creates conditions for surface waves to propagate along the generated guide. We present experimental and numerical results of this shallow water waveguiding, which can be straightforwardly matched to the one-dimensional water wave equation of shallow water waves. Our work revitalizes water waveguiding research as a relevant and controllable experimental setup to study complex phenomena using waveguide geometries.

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