Nonlinear Fourier Analysis with Sine Wave, Stokes Wave and Rogue Wave Basis Functions: A Paradigm Change in the Understanding of Nonlinear Waves

2020 ◽  
Author(s):  
Alfred Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Rogue Wave ◽  
Random Phase ◽  
Stokes Wave ◽  
Random Wave ◽  
Linear Dispersion ◽  
Wide Range ◽  
Stokes Waves ◽  
Wave Trains

<p>I give a new perspective for the description of nonlinear water wave trains using mathematical methods I refer to as nonlinear Fourier analysis (NLFA). I discuss how this approach holds for one-space and one time dimensions (1+1) and for two-space and one time dimensions (2+1) to all orders of approximation. I begin with the nonlinear Schroedinger (NLS) equation in 1+1 dimensions: Here the NLFA method is derived from the complete integrability of the equation by the periodic inverse scattering transform. I show how to compute the nonlinear Fourier series that exactly solve 1+1 NLS. I then show how to extend the order of 1+1 NLS to the Dysthe and the extended Dysthe equations. I also show how to include directional spreading in the formulation so that I can address the 2+1 NLS, the 2+1 Dysthe and the 2+1 Trulsen-Dysthe equations. This hierarchy of equations extends formally all the way to the Zakharov equations in the infinite order limit. Each order and extension from 1+1 to 2+1 dimensions is characterized by its own modulational dispersion relation that is required at each order of the NLFA formalism. NLFA is characterized by its own fundamental nonlinear Fourier series, which has particular nonlinear Fourier modes: sine waves, Stokes waves and breather trains. We are all familiar with sine waves (known for centuries) and Stokes waves (known since the Stokes paper in 1847). Breather trains have become known over the past three decades as a major source of rogue or freak waves in the ocean: Breather packets are known to pulse up and down during their evolution. At the moment of the maximum amplitude the largest wave in a breather packet is often referred to as a “rogue” or “freak” wave. Such extreme packets are known to be “coherent structures" so that pure linear dispersion does not occur as in a linear packet. Instead the breather packets have components that are phase locked with each other and hence remain coherent and are “long lived” just as vortices do in classical turbulence. Because the breathers live for a long time, the notion of risk based upon linear dispersion, as used in the oil and shipping industries, must be revised upwards. I discuss how to apply NLFA to (1) nonlinearly Fourier analyze time series, (2) to analyze wave fields from radar, lidar and synthetic aperture radar measurements, (3) how to treat NLFA to describe nonlinear, random wave trains using a kind of nonlinear random phase approximation and (4) how to compute the nonlinear power spectrum in terms of the parameters used to describe the rogue wave Fourier modes in a random wave train. Thus the emphasis here is to discuss a number of new tools for nonlinear Fourier analysis in a wide range of problems in the field of ocean surface waves.</p>

Theory of Nonlinear Fourier Analysis: The Construction of Quasiperiodic Fourier Series for Nonlinear Wave Motion

2020 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Kdv Equation ◽  
Wave Turbulence ◽  
Two Dimensional ◽  
Kp Equation ◽  
Wave Dynamics ◽  
Wide Range ◽  
Stokes Waves ◽  
Parallel Dynamics

Abstract I give a description of nonlinear water wave dynamics using a recently discovered tool of mathematical physics I call nonlinear Fourier analysis (NLFA). This method is based upon and is an application of a theorem due to Baker [1897, 1907] and Mumford [1984] in the field of algebraic geometry and from additional sources by the author [Osborne, 2010, 2018, 2019]. The theory begins with the Kadomtsev-Petviashvili (KP) equation, a two dimensional generalization of the Korteweg-deVries (KdV) equation: Here the NLFA method is derived from the complete integrability of the equation by finite gap theory or the inverse scattering transform for periodic/quasiperiodic boundary conditions. I first show, for a one-dimensional, plane wave solution, that the KP equation can be rotated to a solution of the KdV equation, where the coefficients of KdV are now functions of the rotation angle. I then show how the rotated KdV equation can be used to compute the spectral solutions of the KP equation itself. Finally, I write the spectral solutions of the KP equation as a finite gap solution in terms of Riemann theta functions. By virtue of the fact that I am able to write a theta function formulation of the KP equation, it is clear that the wave dynamics lie on tori and constitute parallel dynamics on the tori in the integrable cases and non-parallel dynamics on the tori for certain perturbed quasi-integrable cases. Therefore, we are dealing with a Kolmogorov-Arnold-Moser KAM theory for nonlinear partial differential wave equations. The nonlinear Fourier series have particular nonlinear Fourier modes, including: sine waves, Stokes waves and solitons. Indeed the theoretical formulation I have developed is a kind of exact two-dimensional “coherent wave turbulence” or “integrable wave turbulence” for the KP equation, for which the Stokes waves and solitons are the coherent structures. I discuss how NLFA provides a number of new tools that apply to a wide range of problems in offshore engineering and coastal dynamics: This includes nonlinear Fourier space and time series analysis, nonlinear Fourier wave field analysis, a nonlinear random phase approximation, the study of nonlinear coherent functions and nonlinear bi and tri spectral analysis.

Quantification of shape by use of Fourier analysis: the Mississippian blastoid genus Pentremites

Paleobiology ◽  
1977 ◽  
Vol 3 (3) ◽  
pp. 288-299 ◽  
Author(s):  
Johnny Arlton Waters
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
External Morphology ◽  
Taxonomic Character ◽  
Fossil Species ◽  
Growth Series ◽  
Wide Range ◽  
Lawrence County ◽  
Psychological Evidence ◽  
Important Character

Psychological evidence suggests that the visual outline of an object is the most important character for discriminating differences in external morphology. External morphology is an important taxonomic character in describing living and fossil species, for example, the blastoid Pentremites. Comparison of different views of the same specimen utilizing Fourier series suggests that the skeleton of Pentremites commonly is not rotationally symmetrical and that the asymmetry is not associated with any specific ray. Analysis of a growth series indicates that the amplitudes of the second and third harmonics are significantly correlated with growth, which is demonstrated to be anisometric. Definable changes in the lateral outline can be attributed to changes in harmonic amplitudes and a wide range of morphological forms comparable to known taxa can be generated by systematically varying the amplitudes of the first four harmonics. The population used in this study probably represents Pentremites robustus Lyon and is from Bangor Limestone in the abandoned Moulton Quarry, Lawrence County, Alabama.

Nonlinear Fourier Analysis for Two-Dimensional Ocean Surface Waves Described by the Zakharov Equation

2021 ◽  
Author(s):  
Alfred R. Osborne
Keyword(s):  
Fourier Series ◽  
Fourier Analysis ◽  
Theta Functions ◽  
Soliton Solutions ◽  
Ocean Waves ◽  
Two Dimensional ◽  
Zakharov Equation ◽  
Stokes Waves ◽  
Arbitrary Potential ◽  
Dysthe Equation

<p>The physical hierarchy of two-dimensional ocean waves studied here consists of the 2+1 nonlinear Schrödinger equation (NLS), the Dysthe equation, the Trulsen-Dysthe equation, etc. on to the Zakharov equation. I call this the SDTDZ hierarchy. I demonstrate that the nonlinear Schrödinger equation with arbitrary potential is the natural way to treat this hierarchy, for any member of the hierarchy can be determined by an appropriate choice of the potential. Furthermore, the NLS equation with arbitrary potential can be written in terms of two bilinear forms and thereby has one and two-soliton solutions. To access the inverse scattering approach, I find a nearby equation which has N-soliton solutions: Such an equation is completely integrable by the IST on the infinite plane and by finite gap theory for periodic boundary conditions. In this way the entire SDTDZ hierarchy is closely related to a nearby integrable hierarchy which I refer to as the iSDTDZ hierarchy. Every member of this hierarchy has solutions in terms of ratios of Riemann theta functions and therefore every member has general spectral solutions in terms of quasiperiodic Fourier series. This last step occurs because ratios of theta functions are single valued, multiply periodic meromorphic functions. Once the quasiperiodic Fourier series are found, one can then invert these to determine the Riemann spectrum, namely, the Riemann matrix, wavenumbers, frequencies and phases. This means that the solutions of the nonlinear wave equations of the iSDTDZ hierarchy are generalized Fourier series indistinguishable from those of Paley and Weiner [1935] and therefore allows one to classify nonlinear wave motion in terms of a linear superposition of sine waves. How do the generalized quasiperiodic Fourier series differ from ordinary, standard periodic Fourier series? This can be seen by recognizing that the frequencies are incommensurable, and the phases can be phase locked. The nonlinear Fourier modes are Stokes waves and the coherent structure solutions are nonlinearly interacting, phase-locked Stokes waves, including breathers and superbreathers. Other types of coherent packets include fossil breathers and dromions. Techniques are developed for (1) numerical modeling of ocean waves (a fast algorithm for the Zakharov equation) and for (2) the nonlinear Fourier analysis of two-dimensional measured wave fields and space/time series (a 2D nonlinear Fourier analysis, implemented as a fast algorithm called the 2D NFFT). Examples of both applications are discussed.</p>

scholarly journals 2- AND 3-DIMENSIONAL DETERMINISTIC FREAK WAVES

1982 ◽  
Vol 1 (18) ◽  
pp. 43
Author(s):  
Soren Peter Kjeldsen
Keyword(s):  
Dimensional Case ◽  
Linear Wave ◽  
Linear Dispersion ◽  
New Techniques ◽  
Freak Waves ◽  
3 Dimensional ◽  
Non Linear ◽  
Stokes Waves ◽  
First Results ◽  
Wave Trains

The present paper deals with a new non-linear technique for generation of violent breaking freak waves (plunging breakers) at specified positions and times in wave basins. First, results concerning generation of non-linear wave trains, Stokes-waves and wave solitons in deep water are given. Then the technique for generation of non-linear wave transients are given with specified non-linear dispersion properties. Finally, the new techniques are used to obtain collisions between non-linear solitons coming both from the same direction (2-dimensional case), and from different directions (3-dimensional case) leading to generation of steep and violent plunging breakers.

Extreme Response Prediction for Fixed Offshore Structures by Monte Carlo Time Simulation Technique

2016 ◽  
Author(s):  
M. K. Abu Husain ◽  
N. I. Mohd Zaki ◽  
M. B. Johari ◽  
G. Najafian
Keyword(s):  
Monte Carlo ◽  
Statistical Properties ◽  
Simulation Technique ◽  
Random Wave ◽  
Wave Loading ◽  
Offshore Structure ◽  
Sampling Variability ◽  
Time Simulation ◽  
Wide Range ◽  
Extreme Response

For an offshore structure, wind, wave, current, tide, ice and gravitational forces are all important sources of loading which exhibit a high degree of statistical uncertainty. The capability to predict the probability distribution of the response extreme values during the service life of the structure is essential for safe and economical design of these structures. Many different techniques have been introduced for evaluation of statistical properties of response. In each case, sea-states are characterised by an appropriate water surface elevation spectrum, covering a wide range of frequencies. In reality, the most versatile and reliable technique for predicting the statistical properties of the response of an offshore structure to random wave loading is the time domain simulation technique. To this end, conventional time simulation (CTS) procedure or commonly called Monte Carlo time simulation method is the best known technique for predicting the short-term and long-term statistical properties of the response of an offshore structure to random wave loading due to its capability of accounting for various nonlinearities. However, this technique requires very long simulations in order to reduce the sampling variability to acceptable levels. In this paper, the effect of sampling variability of a Monte Carlo technique is investigated.

Stability-Optimized Clearance Configuration of Fluid-Film Bearings

2006 ◽  
Vol 129 (1) ◽  
pp. 106-111 ◽  
Author(s):  
Koichi Matsuda ◽  
Shinya Kijimoto ◽  
Yoichi Kanemitsu
Keyword(s):  
Fourier Series ◽  
Load Capacity ◽  
Fluid Film ◽  
Eccentricity Ratio ◽  
Rotating Speed ◽  
Fluid Film Bearings ◽  
Wide Range ◽  
Jeffcott Rotor ◽  
The Stability ◽  
Frequency Ratios

The whirl instability occurs at higher rotating speeds for a full circular fluid-film journal bearing, and many types of clearance configuration have been proposed to solve this instability problem. A clearance configuration of fluid-film journal bearings is optimized in a sense of enhancing the stability of the full circular bearing at high rotational speeds. A performance index is chosen as the sum of the squared whirl-frequency ratios over a wide range of eccentricity ratios, and a Fourier series is used to represent an arbitrary clearance configuration of fluid-film bearings. An optimization problem is then formulated to find the Fourier coefficients to minimize the index. The designed bearing has a clearance configuration similar to that of an offset two-lobe bearing for smaller length-to-diameter ratios. It is shown that the designed bearing cannot destabilize the Jeffcott rotor at any high rotating speed for a wide range of eccentricity ratio. The load capacity of the designed bearings is nearly in the same magnitude as that of the full circular bearing for smaller length-to-diameter ratios. The whirl-frequency ratios of the designed bearing are very sensitive to truncating higher terms of the Fourier series for some eccentricity ratio. The designed bearings successfully enhance the stability of a full circular bearing and are free from the whirl instability.

Stability of periodic waves in shallow water

1974 ◽  
Vol 66 (1) ◽  
pp. 81-96 ◽  
Author(s):  
P. J. Bryant
Keyword(s):  
Shallow Water ◽  
Wave Train ◽  
Periodic Wave ◽  
Fundamental Wave ◽  
Cnoidal Waves ◽  
Margin Of Stability ◽  
Fundamental Wavelength ◽  
Stokes Waves ◽  
Wave Trains ◽  
Permanent Shape

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

ANALYTICAL SOLUTIONS OF COUPLED NONLINEAR SCHRODINGER EQUATIONS FOR TWO NON LINEARLY INTERACTING STOKES WAVE TRAINS OVER INFINITE DEPTH

2021 ◽  
Vol 10 (3) ◽  
pp. 1137-1144
Author(s):  
S. Manna ◽  
A.K. Dhar
Keyword(s):  
Analytical Solutions ◽  
Nonlinear Schrödinger Equations ◽  
Stokes Wave ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Wave Trains ◽  
Coupled Nonlinear Schrodinger Equations

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.

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