Equations for bi-directional waves over an uneven bottom

Chen, M.

doi:10.1016/s0378-4754(02)00193-3

Cited by 30 publications

(47 citation statements)

References 9 publications

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“…The derivation of the Boussinesq system is only briefly sketched. For a full derivation, the interested reader may consult [8] and [31]. The variables are non-dimensionalized using the following scaling:…”

Section: Derivation Of the Systemmentioning

confidence: 99%

“…For comparison, the work of Chen [8] is considered. Chen presented equations for bi-directional waves over an uneven bottom, which may be written in non-dimensional, unscaled variables and disregarding terms of order O(αβ , β 2 ) as…”

Section: Comparison To Mild Slope Model Systemsmentioning

confidence: 99%

“…Wei et al [42] used Nwogu's approach to derive a fully nonlinear extension of Boussinesq equations, which further extended the range of validity of Boussinesq models without the weak nonlinearity restriction. It is worth mentioning that in [8,31] the Boussinesq model (1) is extended to the moving bottom topography, where the bottom topography depends on x, y, and t. In [31], a Benjamin-Bora-Mahony (BBM-BBM) type system (see [3]) is derived and solved numerically using a finite element method. One aspect in which the BBM system differs from Peregrine's Boussinesq' system is that it is amenable to numerical integration.…”

Section: Introductionmentioning

confidence: 99%

“…Shoaling and wave breaking are studied numerically. This paper compares two models: the coupled BBM-BBM type system derived by Chen [8] and the one in Mitsotakis [31] with respect to the evolution of solitary waves. This comparison is concerned with initial wave profiles and wave shoaling on slopes that correspond to unidirectional propagation.…”

Section: Introductionmentioning

confidence: 99%

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On the influence of wave reflection on shoaling and breaking solitary Waves

Senthilkumar¹

2016

Proc. Estonian Acad. Sci.

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Abstract. A coupled BBM system of equations is studied in the situation of water waves propagating over a decreasing fluid depth. A conservation equation for mass and also a wave breaking criterion, both valid in the Boussinesq approximation, are found. A Fourier collocation method coupled with a 4-stage Runge-Kutta time integration scheme is employed to approximate solutions of the BBM system. The mass conservation equation is used to quantify the role of reflection in the shoaling of solitary waves on a sloping bottom. Shoaling results based on an adiabatic approximation are analysed. Wave shoaling and the criterion of the breaking of solitary waves on a sloping bottom are studied. To validate the numerical model the simulation results are compared with reference results and a good agreement between them can be observed. Shoaling of solitary waves is calculated for two different types of mild slope model systems. Comparison with reference solutions shows that both of these models work well in their respective regimes of applicability.

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Section: Derivation Of the Systemmentioning

confidence: 99%

Section: Comparison To Mild Slope Model Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the influence of wave reflection on shoaling and breaking solitary Waves

Senthilkumar¹

2016

Proc. Estonian Acad. Sci.

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“…The case of uneven bottoms has been less investigated; some of the significant references are Peregrine [26], Madsen et al [21], Nwogu [25], and Chen [10]. Peregrine was the first one to formulate the classical Boussinesq equations for waves in shallow water with variable depth on a three-dimensionnal domain.…”

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confidence: 99%

Influence of bottom topography on long water waves

Chazel¹

2007

M2AN

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Abstract. We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by Bona et al. [Arch. Rational Mech. Anal. 178 (2005) 373-410], we derive new symetric asymptotic models for each regime. The solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.Mathematics Subject Classification. 76B15, 35L55, 35C20, 35Q35.

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Large time existence for 3D water-waves and asymptotics

2007

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We rigorously justify in 3D the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili (KP) approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a ``variable'' nondimensionalized version of the water-waves equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep-water, from small to large surface and bottom variations, and from fully to weakly transverse waves. The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and the energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.Comment: Revised version of arXiv:math.AP/0702015 (notations simplified and remarks added) To appear in Inventione

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Equations for bi-directional waves over an uneven bottom

Cited by 30 publications

References 9 publications

On the influence of wave reflection on shoaling and breaking solitary Waves

On the influence of wave reflection on shoaling and breaking solitary Waves

Influence of bottom topography on long water waves

Large time existence for 3D water-waves and asymptotics

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