scholarly journals Global solution for the 3D gravity water waves system above a flat bottom

2019 ◽  
Vol 346 ◽  
pp. 805-886 ◽  
Author(s):  
Xuecheng Wang
Keyword(s):  
Global Solution ◽  
Water Waves ◽  
Flat Bottom ◽  
3D Gravity

An Efficient Numerical Model of Hyperbolic Mild-Slope Equation

2007 ◽  
Author(s):  
Zhiyao Song ◽  
Honggui Zhang ◽  
Jun Kong ◽  
Ruijie Li ◽  
Wei Zhang
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Water Waves ◽  
Computational Time ◽  
Transient Wave ◽  
Flat Bottom ◽  
The Real ◽  
Wave Elevation ◽  
Mild Slope Equation ◽  
Effective Wave

Introduction of an effective wave elevation function, the simplest time-dependent hyperbolic mild-slope equation has been presented and an effective numerical model for the water wave propagation has been established combined with different boundary conditions in this paper. Through computing the effective wave elevation and transforming into the real transient wave motion, then related wave heights are computed. Because the truncation errors of the presented model only induced by the dissipation terms, but those of Lin’s model (2004) contributed by the convection terms, dissipation terms and source terms, the error analysis shows that calculation stability of this model is enhanced obviously compared with Lin’s one. The tests show that this model succeeds to the merit in Lin’s one and the computer program simpler, computational time shorter because of calculation stability enhanced efficiently and computer memory decreased obviously. The presented model has the capability of simulating exactly the location of transient wave front by the speed of wave propagation in the first test, which is important for the real-time prediction of the arrival time of water waves generated in the deep sea. The model is validated against experimental data for combined wave refraction and diffraction over submerged circular shoal on a flat bottom in the second test. Good agreements are gained. The model can be applied to the theory research and engineering applications about the wave propagation in the coastal waters.

scholarly journals Adaptive Numerical Modeling of Tsunami Wave Generation and Propagation with FreeFem++

Geosciences ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 351
Author(s):  
Georges Sadaka ◽  
Denys Dutykh
Keyword(s):  
Water Waves ◽  
Tsunami Wave ◽  
New Technology ◽  
Boussinesq Approximation ◽  
Boussinesq System ◽  
The Finite Element Method ◽  
Flat Bottom ◽  
Source Codes ◽  
Variable Bottom ◽  
Surface Water Waves

A simplified nonlinear dispersive Boussinesq system of the Benjamin–Bona–Mahony (BBM)-type, initially derived by Mitsotakis (2009), is employed here in order to model the generation and propagation of surface water waves over variable bottom. The simplification consists in prolongating the so-called Boussinesq approximation to bathymetry terms, as well. Using the finite element method and the FreeFem++ software, we solve this system numerically for three different complexities for the bathymetry function: a flat bottom case, a variable bottom in space, and a variable bottom both in space and in time. The last case is illustrated with the Java 2006 tsunami event. This article is designed to be a pedagogical paper presenting to tsunami wave community a new technology and a novel adaptivity technique, along with all source codes necessary to implement it.

scholarly journals On the 3-dimensional water waves system above a flat bottom

2017 ◽  
Vol 10 (4) ◽  
pp. 893-928 ◽  
Author(s):  
Xuecheng Wang
Keyword(s):  
Water Waves ◽  
Flat Bottom ◽  
3 Dimensional

Two-Phase Problem for Two-Dimensional Water Waves of Finite Depth

1997 ◽  
Vol 07 (06) ◽  
pp. 791-821
Author(s):  
Tatsuo Iguchi
Keyword(s):  
Water Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Phase Problem ◽  
Flat Bottom ◽  
Two Phase ◽  
The Cauchy Problem ◽  
Lower Fluid ◽  
Quasi Linear System ◽  
Finite Smoothness

We consider the two-phase problem for two-dimensional and irrotational motion of incompressible ideal fluids in the case that the fluids are separated into the lower and the upper parts by an almost horizontal interface and that there is an almost flat bottom below the lower fluid. It is proved that the Cauchy problem is well-posed, locally in time, in a Sobolev space of finite smoothness, if the surface tension is taken into account and the initial data are suitably close to the equilibrium rest state. The main part of the proof is the reduction of the problem to a quasi-linear system of integro-differential equations for the function defining the interface and the horizontal component of the velocity of the lower fluid on the interface.

An approximation for the highest gravity waves on water of finite depth

1998 ◽  
Vol 372 ◽  
pp. 45-70 ◽  
Author(s):  
E. A. KARABUT
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Finite Depth ◽  
Simple Calculation ◽  
Ideal Incompressible Fluid ◽  
Finite Amplitude ◽  
Steady Flows ◽  
Heavy Fluid ◽  
Flat Bottom ◽  
New Approach

Planar steady gravity waves of finite amplitude at the surface of an ideal incompressible fluid above a flat bottom are studied theoretically. A new approach to the construction of some steady flows of heavy fluid with a partially free surface is proposed. The hypothesis is suggested and justified that these flows are close to gravity waves. For the case of the highest waves a one-parameter family of exact solutions describing free boundary flows above a flat bottom and under two uneven symmetrically located caps is derived. This family of solutions gives an approximation to the highest water waves in moderate to shallow water depths, enabling relatively simple calculation of their properties.

A pseudospectral approach for scattering of water waves

1994 ◽  
Vol 445 (1925) ◽  
pp. 619-636 ◽  
Keyword(s):  
Wave Field ◽  
Water Waves ◽  
Parabolic Model ◽  
Beach Profile ◽  
Flat Bottom ◽  
Propagation Models ◽  
Wave Refraction ◽  
Collocation Points ◽  
Mild Slope Equation ◽  
Backward Wave

Wave propagation models for scattering of water waves are developed based on the mild-slope equation. The pseudospectral Fourier approach is used to reduce the mild-slope equation to a set of ordinary differential equations for the mod­ified potential, ϕ√CC g , at collocation points in the alongshore direction. The wave field is then decoupled into a series of wave modes including all forward and backward propagating modes. Ignoring the backward wave field as a first approximation, a wide-angle parabolic model is derived. When the backward wave field is important, both forward and backward wave fields are obtained by construct­ing the Bremmer series solution. A small-angle parabolic model is also developed for comparison. Numerical results are presented for wave refraction over an equi­librium beach profile and wave focusing over a submerged circular shoal on a flat bottom. The importance of the backward scattering is illustrated by the latter example.

Solitary wave solutions for a model of the two-way propagation of water waves in a channel

1981 ◽  
Vol 90 (2) ◽  
pp. 343-360 ◽  
Author(s):  
J. F. Toland
Keyword(s):  
Initial Data ◽  
Water Waves ◽  
Coupled System ◽  
Finite Amplitude ◽  
Solitary Wave Solutions ◽  
Flat Bottom ◽  
One Dimensional ◽  
Image Position ◽  
System 1 ◽  
Relevant Class

Bona and Smith (6) have suggested that the coupled system of equationshas the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariantthey show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.

scholarly journals Traveling Quasi-periodic Water Waves with Constant Vorticity

2021 ◽  
Author(s):  
M. Berti ◽  
L. Franzoi ◽  
A. Maspero
Keyword(s):  
Periodic Solutions ◽  
Water Waves ◽  
Small Amplitude ◽  
Capillary Water ◽  
Flat Bottom ◽  
Borel Set ◽  
Free Interface ◽  
Constant Vorticity ◽  
Wave Solutions ◽  
Full Lebesgue Measure

AbstractWe prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.

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