scholarly journals Approximation Models For Water Wave Equations

2019 ◽  
Vol 7 (08) ◽  
Author(s):  
Leonard Bezati ◽  
Shkelqim Hajrulla ◽  
Kristofor Lapa
Keyword(s):  
Phase Velocity ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Kdv Equation ◽  
Wave Theory ◽  
Finite Volume Methods ◽  
Linear Water Wave Theory ◽  
The Best Approximation ◽  
Non Linear

Abstract: In this work we are interested in developing approximate models for water waves equation. We present the derivation of the new equations uses approximation of the phase velocity that arises in the linear water wave theory. We treat the (KdV) equation and similarly the C-H equation. Both of them describe unidirectional shallow water waves equation. At the same time, together with the (BBM) equation we propose, we provide the best approximation of the phase velocity for small wave numbers that can be obtained with second and third-order equations. We can extend the results of [3, 4].  A comparison between the methods is mentioned in this article. Key words:  C-H equation, KdV equation, approximation, water wave equation, numerical methods. --------------------------------------------------------------------------------------------------------------------- [3]. D. J. Benney, “Long non-linear waves in fluid flows,” Journal of Mathematical           Physics, vol. 45, pp. 52–63, 1966. View at Google Scholar · View at Zentralblatt MATH  [4]. Bezati, L., Hajrulla, S., & Hoxha, F. (2018). Finite Volume Methods for Non-Linear          Eqs. International Journal of Scientific Research and Management, 6(02), M-  2018. 

scholarly journals Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations

2021 ◽  
Vol 67 (6 Nov-Dec) ◽  
Author(s):  
Gizel Bakicierler ◽  
Suliman Alfaqeih ◽  
Emine Misirli
Keyword(s):  
Traveling Wave ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Simple Equation ◽  
Wave Solutions ◽  
Non Linear ◽  
Package Program ◽  
Modified Simple Equation Method

Recently, non-linear fractional partial differential equations are used to model many phenomena in applied sciences and engineering. In this study, the modified simple equation scheme is implemented to obtain some new traveling wave solutions of the non-linear conformable time-fractional approximate long water wave equation and the non-linear conformable coupled time-fractional Boussinesq-Burger equation, which are used in the expression of shallow-water waves. The time- fractional derivatives are described in terms of conformable fractional derivative sense. Consequently, new exact traveling wave solutions of both equations are achieved. The graphics and correctness of the wave solutions are obtained with the Mathematica package program.

scholarly journals WATER WAVE SCATTERING BY A THIN VERTICAL SUBMERGED PERMEABLE PLATE

2021 ◽  
Vol 26 (2) ◽  
pp. 223-235
Author(s):  
Rupanwita Gayen ◽  
Sourav Gupta ◽  
Aloknath Chakrabarti
Keyword(s):  
Integral Equation ◽  
Water Waves ◽  
Water Wave ◽  
Wave Theory ◽  
Hypersingular Integral Equation ◽  
Cauchy Type ◽  
Hypersingular Integral ◽  
Permeable Plate ◽  
Linear Water Wave Theory ◽  
Surface Water Waves

An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock’s expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov’s book [7]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock’s theorems.

scholarly journals Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation

2018 ◽  
Vol 148 (6) ◽  
pp. 1201-1237
Author(s):  
Benjamin Melinand
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Kdv Equation ◽  
Boussinesq Equations ◽  
Spatial Dimension ◽  
Wave Approximation ◽  
Long Wave ◽  
Ostrovsky Equation ◽  
Long Wave Approximation

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

scholarly journals Introduction: some recent developments of nonlinear water wave theory

2007 ◽  
Vol 365 (1858) ◽  
pp. 2195-2201 ◽  
Author(s):  
Adrian Constantin
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Mathematical Theory ◽  
Wave Theory ◽  
Research Literature ◽  
Recent Developments

The opening article of this issue is intended to provide a review of some relevant topics of the mathematical theory of water waves that have recently received considerable attention in the research literature. We also provide a brief discussion about the content and contribution of the articles that make up this issue.

scholarly journals Three-dimensional water-wave scattering in two-layer fluids

2000 ◽  
Vol 423 ◽  
pp. 155-173 ◽  
Author(s):  
J. R. CADBY ◽  
C. M. LINTON
Keyword(s):  
Water Wave ◽  
Lower Layer ◽  
Three Dimensional ◽  
Single Layer ◽  
Wave Theory ◽  
Finite Depth ◽  
Wave Radiation ◽  
Infinite Layer ◽  
Linear Water Wave Theory ◽  
Submerged Sphere

We consider, using linear water-wave theory, three-dimensional problems concerning the interaction of waves with structures in a fluid which contains a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. In a single-layer fluid there are a number of reciprocity relations that exist connecting the various hydrodynamic quantities that arise, and these relations are systematically extended to the two-fluid case. The particular problems of wave radiation and scattering by a submerged sphere in either the upper or lower layer are then solved using multipole expansions.

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.

Mechanisms for the generation of edge waves over a sloping beach

1988 ◽  
Vol 186 ◽  
pp. 379-391 ◽  
Author(s):  
D. V. Evans
Keyword(s):  
Water Wave ◽  
Wave Theory ◽  
Dominant Mode ◽  
Mode Shapes ◽  
Edge Waves ◽  
Longshore Current ◽  
Linear Water Wave Theory ◽  
Pressure Distributions ◽  
Sloping Beach ◽  
Standing Edge Waves

Two mechanisms for the generation of standing edge waves over a sloping beach are described using classical linear water-wave theory. The first is an extension of the result of Yih (1984) to a class of localized bottom protrusions on a sloping beach in the presence of a longshore current. The second is a class of longshore surface-pressure distributions over a beach. In both cases it is shown that Ursell-type standing edge-wave modes can be generated in an appropriate frame of reference. Typical curves of the mode shapes are presented and it is shown how in certain circumstances the dominant mode is not the lowest.

scholarly journals Some symmetries, similarity solutions and various conservation laws of a type of dispersive water waves

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yufeng Zhang ◽  
Na Bai ◽  
Hongyang Guan
Keyword(s):  
Conservation Laws ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Similarity Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Lie Transformation ◽  
Group Invariant Solutions ◽  
Special Solutions

Abstract We investigate the point symmetries, Lie–Bäcklund symmetries for a type of dispersive water waves. We obtain some Lie transformation groups, various group-invariant solutions, and some similarity solutions. Besides, we produce different formats of conservation laws of the dispersive water waves by using different schemes. Finally, we consider some special solutions of the stationary dispersive water-wave equations.

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