A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth: The Irrotational Green-Naghdi Model

2002 ◽  
Author(s):  
J. W. Kim ◽  
K. J. Bai ◽  
R. C. Ertekin ◽  
W. C. Webster
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Wave Train ◽  
Vertical Wall ◽  
Numerical Test ◽  
Boussinesq Equations ◽  
Approximate Model ◽  
Stokes Wave ◽  
Nonlinear Water Waves ◽  
True Solution

Recently, the authors have derived a new approximate model for the nonlinear water waves, the Irrotational Green-Naghdi (IGN) model. In this paper, we first derive the IGN equations applicable to variable water depth, then perform numerical tests to show whether and how fast the solution of the IGN model converges to the true solution as its level increases. The first example given is the steady solution of the progressive waves of permanent form, which includes the small amplitude sinusoidal wave, the solitary wave and the nonlinear Stokes wave. The second example is the run-up of a solitary wave on a vertical wall. The last example is the shoaling of a wave train over a sloping beach. In each numerical test, the self-convergence of the IGN model is shown first. Then the converged solution is compared to the known analytic solutions and/or solutions of other approximate models such as the KdV and the Boussinesq equations.

A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model

2003 ◽  
Vol 125 (1) ◽  
pp. 25-32 ◽  
Author(s):  
J. W. Kim ◽  
K. J. Bai ◽  
R. C. Ertekin ◽  
W. C. Webster
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Wave Train ◽  
Vertical Wall ◽  
Numerical Test ◽  
Boussinesq Equations ◽  
Approximate Model ◽  
Stokes Wave ◽  
Nonlinear Water Waves ◽  
True Solution

Recently, the authors have derived a new approximate model for the nonlinear water waves, the Irrotational Green-Naghdi (IGN) model. In this paper, we first derive the IGN equations applicable to variable water depth, and then perform numerical tests to show whether and how fast the solution of the IGN model converges to the true solution as its level increases. The first example given is the steady solution of progressive waves of permanent form, which includes the small-amplitude sinusoidal wave, the solitary wave and the nonlinear Stokes wave. The second example is the run-up of a solitary wave on a vertical wall. The last example is the shoaling of a wave train over a sloping beach. In each numerical test, the self-convergence of the IGN model is shown first. Then the converged solution is compared to the known analytic solutions and/or solutions of other approximate models such as the KdV and the Boussinesq equations.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

scholarly journals Dynamic-pressure distributions under Stokes waves with and without a current

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170103 ◽  
Author(s):  
Motohiko Umeyama
Keyword(s):  
Water Waves ◽  
Dynamic Pressure ◽  
Water Pressure ◽  
Travelling Waves ◽  
Stokes Wave ◽  
Front Face ◽  
Nonlinear Water Waves ◽  
Near Surface ◽  
Stokes Waves ◽  
A Current

To investigate changes in the instability of Stokes waves prior to wave breaking in shallow water, pressure data were recorded vertically over the entire water depth, except in the near-surface layer (from 0 cm to −3 cm), in a recirculating channel. In addition, we checked the pressure asymmetry under several conditions. The phase-averaged dynamic-pressure values for the wave–current motion appear to increase compared with those for the wave-alone motion; however, they scatter in the experimental range. The measured vertical distributions of the dynamic pressure were plotted over one wave cycle and compared to the corresponding predictions on the basis of third-order Stokes wave theory. The dynamic-pressure pattern was not the same during the acceleration and deceleration periods. Spatially, the dynamic pressure varies according to the faces of the wave, i.e. the pressure on the front face is lower than that on the rear face. The direction of wave propagation with respect to the current directly influences the essential features of the resulting dynamic pressure. The results demonstrate that interactions between travelling waves and a current lead more quickly to asymmetry. This article is part of the theme issue ‘Nonlinear water waves’.

Interaction of a Second-Order Solitary Wave With a Vertical Permeable Plane Breakwater

2004 ◽  
Author(s):  
A. Basmat
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Boussinesq Equations ◽  
Second Order ◽  
Wave Loading ◽  
Coastal Structures ◽  
Transformation Techniques ◽  
Incident Plane ◽  
Impermeable Wall ◽  
Plane Barrier

The purpose of this paper is to develop mathematical models to investigate the interaction between long non-linear water waves and dissipative/absorbing coastal structures. The diffraction of a plane second-order solitary wave at a vertical permeable plane barrier standing in front of an impermeable wall, with calculation of the second-order wave loading is investigated. An incident plane second-order solitary wave is the Laitone solution of Boussinesq equations. The analytical solution is obtained by means of a small parameter development and Fourier transformation techniques. Computational results were performed using the software MATHEMATICA version 4.0.1.0.

scholarly journals Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

Bifurcations of Traveling Wave Solutions for Fully Nonlinear Water Waves with Surface Tension in the Generalized Serre–Green–Naghdi Equations

2020 ◽  
Vol 30 (01) ◽  
pp. 2050019
Author(s):  
Jibin Li ◽  
Guanrong Chen ◽  
Jie Song
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Water Waves ◽  
Traveling Wave Solutions ◽  
Wave Theory ◽  
Solitary Wave Solutions ◽  
Nonlinear Water Waves ◽  
Fully Nonlinear ◽  
Wave Solutions

For the generalized Serre–Green–Naghdi equations with surface tension, using the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] for their traveling wave systems, in different parameter conditions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) are obtained. More than 26 explicit exact parametric representations are given. It is interesting to find that this fully nonlinear water waves equation coexists with uncountably infinitely many smooth solitary wave solutions or infinitely many pseudo-peakon solutions with periodic solutions or compacton solutions. Differing from the well-known peakon solution of the Camassa–Holm equation, the generalized Serre–Green–Naghdi equations have four new forms of peakon solutions.

Interaction of a Solitary Wave With a Vertical Permeable Elliptical Breakwater

2005 ◽  
Author(s):  
A. Basmat
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Numerical Technique ◽  
Wave Length ◽  
Boussinesq Equations ◽  
Perturbation Parameter ◽  
Elliptical Cylinder ◽  
Total Force ◽  
Surface Geometry ◽  
First Order

In this paper the diffraction of a plane first-order solitary wave by a vertical permeable breakwater with calculation of the wave loading is investigated. The interaction between long non-linear water waves and dissipative/absorbing coastal structures that are commonly used in coastal engineering is investigated using both analytical and numerical technique. The breakwater consists of a vertical permeable surface-piercing elliptical cylinder fixed in the ground. The analytical model herein is based on the application of Boussinesq equations to describe the diffraction of the first-order solitary wave by an elliptical breakwater. The method of solution is based on perturbation theory, using a perturbation parameter defined in terms of surface geometry of the cylinder. The analysis includes terms up to the first-order in this parameter, where the zeroth-order solution corresponds to a circular cylinder. The velocity potentials at the zeroth and first orders are expressed as eigenfunction expansion involving unknown coefficients that are determined through the boundary conditions on unperturbed cylinder. The flow through the porous breakwater is assumed to obey Darcy’s law. The total force onto the elliptical cylinder is obtained by integration of the pressure over permeable cylindrical wall. The analytical solution is obtained by means of a Fourier transformation technique. The effects of porosity, relative wave length and the incident wave angle are discussed. Numerical results compare well with previous predictions for the limiting cases of a permeable cylindrical breakwater.

scholarly journals The dynamic pressure in deep-water extreme Stokes waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170095 ◽  
Author(s):  
Tony Lyons
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Dynamic Pressure ◽  
Stokes Wave ◽  
Wave Crest ◽  
Maximum Principles ◽  
Nonlinear Water Waves ◽  
Stagnation Points ◽  
Wave Trough ◽  
Crest Line

In this paper, we consider the dynamic pressure in a deep-water extreme Stokes wave. While the presence of stagnation points introduces a number of mathematical complications, maximum principles are applied to analyse the dynamic pressure in the fluid body by means of an excision process. It is shown that the dynamic pressure attains its maximum value beneath the wave crest and its minimum beneath the wave trough, while it decreases in moving away from the crest line along any streamline. This article is part of the theme issue ‘Nonlinear water waves’.

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