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.

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

Interaction of a Second-Order Solitary Wave With an Array of Four Vertical Cylinders

2006 ◽  
Author(s):  
A. Basmat ◽  
M. Markiewicz ◽  
S. Petersen
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Order Approximation ◽  
Boussinesq Equations ◽  
Second Order ◽  
Nonlinearity Parameter ◽  
Wave Forces ◽  
Weakly Nonlinear ◽  
First Order ◽  
Vertical Cylinders

In this paper the interaction of a plane second order solitary wave with an array of four vertical cylinders is investigated. The fluid is assumed to be incompressible and inviscid. The diffraction analysis assumes irrotationality, which allows for the use of Boussinesq equations. A simultaneous expansion in a small nonlinearity parameter (wave amplitude/depth) and small dispersion parameter (depth/horizontal scale) is performed. Boussinesq models, which describe weakly nonlinear and weakly dispersive long waves, are characterized by the assumption that the nonlinearity and dispersion are both small and of the same order. An incident plane second order solitary wave is the Laitone solution of Boussinesq equations. The representation of variables as the series of small nonlinearity parameters leads to the sequence of linear boundary value problems of increasing order. The first order approximation can be determined as a solution of homogeneous differential equations and the second order approximation follows as a solution of non-homogeneous differential equations, where the right hand sides may be computed from the first order solution. For the case of a single cylinder an analytical solution exists. However, when dealing with more complex cylinder configurations, one has to employ numerical techniques. In this contribution a finite element approach combined with an appropriate time stepping scheme is used to model the wave propagation around an array of four surface piercing vertical cylinders. The velocity potential, the free surface elevation and the subsequent evolution of the scattered field are computed. Furthermore, the total second order wave forces on each individual cylinder are determined. The effect of the incident wave angle is discussed.

An experimental study of the pressure variations in standing water waves

1951 ◽  
Vol 206 (1086) ◽  
pp. 424-435 ◽  
Keyword(s):  
Experimental Study ◽  
Surface Waves ◽  
Standing Wave ◽  
Water Waves ◽  
Wave Length ◽  
Second Order ◽  
First Order ◽  
Standing Water ◽  
Plane Barrier ◽  
Progressive Waves

Although the first-order pressure variations in surface waves on water are known to decrease exponentially downwards, it has recently been shown theoretically that in a standing wave there should be some second-order terms which are unattenuated with depth. The present paper describes experiments which verify the existence of pressure variations of this type in waves of period 0·45 to 0·50 sec. When the motion consists of two progressive waves of equal wave-length travelling in opposite directions, the amplitude of the unattenuated pressure variations is found to be proportional to the product of the wave amplitudes. This property is used to determine the coefficient of reflexion from a sloping plane barrier.

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.

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.

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.

Fundamentals concerning wave loading on offshore structures

1986 ◽  
Vol 173 ◽  
pp. 667-681 ◽  
Author(s):  
James Lighthill
Keyword(s):  
Fluid Mechanics ◽  
Natural Frequency ◽  
Potential Flow ◽  
Vortex Flow ◽  
Offshore Structures ◽  
Second Order ◽  
Mechanics Of Solids ◽  
Wave Loading ◽  
Velocity Fluctuations ◽  
Substantial Body

This article is aimed at relating a certain substantial body of established material concerning wave loading on offshore structures to fundamental principles of mechanics of solids and of fluids and to important results by G. I. Taylor (1928a,b). The object is to make some key parts within a rather specialised field accessible to the general fluid-mechanics reader.The article is concerned primarily to develop the ideas which validate a separation of hydrodynamic loadings into vortex-flow forces and potential-flow forces; and to clarify, as Taylor (1928b) first did, the major role played by components of the potential-flow forces which are of the second order in the amplitude of ambient velocity fluctuations. Recent methods for calculating these forces have proved increasingly important for modes of motion of structures (such as tension-leg platforms) of very low natural frequency.

Modeling the Interaction of Water Waves with Porous Coastal Structures

2016 ◽  
Vol 142 (6) ◽  
pp. 03116003 ◽  
Author(s):  
Inigo J. Losada ◽  
Javier L. Lara ◽  
Manuel del Jesus
Keyword(s):  
Water Waves ◽  
Coastal Structures
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