Nonlinear Decomposition of Transmitted Wave Trains Behind Submerged Reef Structures Using “Nonlinear Fourier Transform”: The Nonlinear Spectral Basic Components

2012 ◽  
Author(s):  
Markus Bruehl ◽  
Hocine Oumeraci
Keyword(s):  
Fourier Transform ◽  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Ocean Engineering ◽  
Alternative Analysis ◽  
Wave Interactions ◽  
Nonlinear Decomposition ◽  
Nonlinear Fourier Transform ◽  
Soliton Fission

The nonlinear Fourier transform (NLFT) is introduced as an alternative analysis method for nonlinear waves in shallow water. In physics the NLFT is the application of the inverse scattering transform (IST) for the solution of the Korteweg-deVries (KdV) equation that gouverns the evolution of waves in shallow water. In coastal and ocean engineering the NLFT can be regarded as an extension of the conventional Fourier transform (FT) as it uses nonlinear shallow water waves (cnoidal waves) as basic components for the spectral decomposition and explicitly considers the nonlinear wave-wave interactions during the analysis. A first description of the numerical implementation and its application for the analysis of soliton fission over and behind submerged reefs is given in a former paper [1]. This paper presents a closer view on the interpretation of both types of spectral basic components of the nonlinear decomposition: solitons and nonlinear oscillatory waves.

Application of “Nonlinear Fourier Transform (NLFT)” for the Analysis of Soliton Fission Behind Submerged Reefs With Finite Width

2011 ◽  
Author(s):  
Markus Bru¨hl ◽  
Hocine Oumeraci
Keyword(s):  
Fourier Transform ◽  
Water Waves ◽  
Numerical Test ◽  
Measured Signal ◽  
Finite Width ◽  
Wave Interactions ◽  
Nonlinear Approach ◽  
Nonlinear Fourier Transform ◽  
Soliton Fission ◽  
Submerged Reefs

The conventional Fourier transform (FT) is the most common analysis method in the frequency domain. The measured signal is decomposed into sine and cosine waves as linear basic components. Every basic component is represented in the Fourier spectrum by its wave amplitude, frequency and phase. As an extension to the conventional FT, this paper presents the advanced “nonlinear Fourier transform (NLFT)” as an alternative nonlinear approach for the analysis of nonlinear shallow water waves. The NLFT uses nonlinear cnoidal waves as basic components and considers their nonlinear wave-wave interactions in the analysis. The application of the NLFT to the analysis of numerical test data is shown using examples of soliton fission over and behind submerged reefs. The determined nonlinear Fourier spectra illustrate the capability of this method and its advantages as compared to the conventional FT.

Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques

2020 ◽  
Vol 34 (07) ◽  
pp. 2050045 ◽  
Author(s):  
Naila Nasreen ◽  
Aly R. Seadawy ◽  
Dianchen Lu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Nonlinear Problems ◽  
Applied Physics ◽  
Kawahara Equation ◽  
Generalized Riccati Equation ◽  
All Solutions ◽  
Fifth Order

The modified Kawahara equation also called Korteweg-de Vries (KdV) equation of fifth-order arises in shallow water wave and capillary gravity water waves. This study is based on the generalized Riccati equation mapping and modified the F-expansion methods. Several types of solitons such as Bright soliton, Dark-lump soliton, combined bright dark solitary waves, have been derived for the modified Kawahara equation. The obtained solutions have significant applications in applied physics and engineering. Moreover, stability of the problem is presented after being examined through linear stability analysis that justify that all solutions are stable. We also present some solution graphically in 3D and 2D that gives easy understanding about physical explanation of the modified Kawahara equation. The calculated work and achieved outcomes depict the power of the present methods. Furthermore, we can solve various other nonlinear problems with the help of simple and effective techniques.

Phenomena on Rogue Waves and Rational Solution of Korteweg-de Vries Equation

2013 ◽  
Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Nls Equation ◽  
Engineering Structures ◽  
Weakly Nonlinear ◽  
Nonlinear Mechanism ◽  
De Vries

An appropriate nonlinear mechanism may create the rogue waves. Perhaps the simplest mechanism, which is able to create considerate changes in the wave amplitude, is the nonlinear interaction of shallow-water solitons. The most well-known examples of such structure are Korteweg-de Vries (KdV) solitons. The Korteweg-de Vries (KdV) equation, which describes the shallow water waves, is a basic weakly dispersive and weakly nonlinear model. Basing on the homogeneous balanced method, we achieve the general rational solution of a classical KdV equation. Numerical simulations of the solution allow us to explain rare and unexpected appearance of the rogue waves. We compare the rogue waves with the ones generated by the nonlinear Schrödinger (NLS) equation which can describe deep water wave trains. The numerical results illustrate that the amplitude of the KdV equation is higher than the one of the NLS equation, which may causes more serious damage of engineering structures in the ocean. This nonlinear mechanism will provide a theoretical guidance in the ocean and physics.

Analysis of Subaerial Landslide Data Using Nonlinear Fourier Transform Based on Korteweg-deVries Equation (KdV-NLFT)

2018 ◽  
Vol 12 (02) ◽  
pp. 1840002 ◽  
Author(s):  
Markus Brühl ◽  
Matthias Becker
Keyword(s):  
Fourier Transform ◽  
Shallow Water ◽  
Frequency Domain Analysis ◽  
Spectral Structure ◽  
Rock Falls ◽  
Surface Data ◽  
Subaerial Landslide ◽  
Impulse Waves ◽  
Nonlinear Fourier Transform ◽  
The Impact

Subaerial and underwater landslides, rock falls and glacier calvings can generate impulse waves in lakes, fjords and the open sea. Experiments with subaerial landslides have shown that, depending on the slide characteristics, different wave types (Stokes, cnoidal or bore-like waves) are generated. Each of these wave types shows different wave height decay with increasing distance from the impact position. Furthermore, in very shallow water, the first impulse wave shows characteristic properties of a solitary wave. The nonlinear Fourier transform based on the Korteweg–deVries equation (KdV-NLFT) is a frequency-domain analysis method that decomposes shallow-water free-surface data into nonlinear cnoidal waves instead of linear sinusoidal waves. This method explicitly identifies solitons as spectral components within the given data. In this study, we apply the KdV-NLFT for the very first time to available 2D and 3D landslide-test data. The objective of the nonlinear decomposition is to identify the hidden nonlinear spectral structure of the impulse waves, including solitons. Furthermore, we analyze the determined solitons at different downstream positions from the impact point with respect to soliton propagation and modification. Finally, we draw conclusions for the prediction of the expected landslide-generated downstream solitons in the far-field.

The Inverse KdV-Based Nonlinear Fourier Transform (KdV-NLFT): Nonlinear Superposition of Cnoidal Waves and Reconstruction of the Original Data

2014 ◽  
Author(s):  
Markus Brühl ◽  
Hocine Oumeraci
Keyword(s):  
Fourier Transform ◽  
Nonlinear Wave ◽  
Original Data ◽  
Wave Interactions ◽  
Nonlinear Superposition ◽  
Cnoidal Waves ◽  
Scattering Transform ◽  
Wave Heights ◽  
Nonlinear Fourier Transform ◽  
Oscillatory Waves

Since 2008, at Leichtweiß-Institute for Hydraulic Engineering and Water Resources at TU Braunschweig a KdV-based nonlinear Fourier transform is implemented and successfully applied to numerical and hydraulic model test data of solitary wave fission behind submerged reefs [1]. The KdV-NLFT is the application of the direct and inverse scattering transform for the solution of the Korteweg-deVries equation. This approach explicitly considers both solitons and oscillatory waves (cnoidal waves) as spectral basic components for the decomposition of the original data. Furthermore, the nonlinear wave-wave interactions between the nonlinear spectral basic components are explicitly considered in the analysis. The direct KdV-NLFT decomposes the original data into cnoidal waves and provides wave heights, wave numbers or frequencies, phases and the moduli which are a measure of the nonlinearity of cnoidal waves. Details of this procedure are given in Brühl & Oumeraci [2]. The interpretation of the nonlinear spectral basic components is described in Brühl & Oumeraci [3]. The inverse KdV-NLFT which is addressed here calculates the nonlinear wave-wave interactions between cnoidal waves and provides the original data by superposition of cnoidal waves and their nonlinear interactions. The practical application of the KdV-NLFT for the analysis of long-wave propagation in shallow water is presented in Brühl & Oumeraci [4].

Shallow-water waves, the Korteweg-deVries equation and solitons

1971 ◽  
Vol 47 (4) ◽  
pp. 811-824 ◽  
Author(s):  
N. J. Zabusky ◽  
C. J. Galvin
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Water Tank ◽  
Shallow Water Waves ◽  
Downstream Location ◽  
Low Dissipation ◽  
The Kdv Equation ◽  
Upstream Station

A comparison of laboratory experiments in a shallow-water tank driven by an oscillating piston and numerical solutions of the Korteweg-de Vries (KdV) equation show that the latter can accurately describe slightly dissipative wavepropagation for Ursell numbers (h1L2/h03) up to 800. This is an input-output experiment, where the initial condition for the KdV equation is obtained from upstream (station 1) data. At a downstream location, the number of crests and troughs and their phases (or relative locations within a period) agree quantitatively with numerical solutions. The crest-to-trough amplitudes disagree somewhat, as they are more sensitive to dissipative forces. This work firmly establishes the soliton concept as necessary for treating the propagation of shallow-water waves of moderate amplitude in a low-dissipation environment.

Nonlinear interaction of ice cover with shallow water waves in channels

2002 ◽  
Vol 467 ◽  
pp. 259-268 ◽  
Author(s):  
XUN XIA ◽  
HUNG TAO SHEN
Keyword(s):  
Nonlinear Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Amplitude ◽  
Kdv Equation ◽  
Ice Cover ◽  
River Channels ◽  
Weakly Nonlinear ◽  
Wave Celerity

A nonlinear analysis of the interaction between a water wave and a floating ice cover in river channels is presented. The one-dimensional weakly nonlinear equation for shallow water wave propagation in a uniform channel with a floating ice cover is derived. The ice cover is assumed to be a thin uniform elastic plate. The weakly nonlinear equation is a fifth-order KdV equation. Analytical solutions of the nonlinear periodic wave equation are obtained. These solutions show that the shape, wavelength and celerity of the nonlinear waves depend on the wave amplitude. The wave celerity is slightly smaller than the open water wave celerity. The wavelength decreases as the wave amplitude increases. Based on these solutions the fracture of the ice cover is analysed. The spacing between transverse cracks varies form 50 m to a few hundred metres with the corresponding wave amplitude varying from 0.2 to 0.8 m, depending on the thickness and strength of the cover. These results agree well with limited field observations.

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