scholarly journals Theoretical and experimental study of particle trajectories for nonlinear water waves propagating on a sloping bottom

2012 ◽  
Vol 370 (1964) ◽  
pp. 1543-1571 ◽  
Author(s):  
Yang-Yih Chen ◽  
Meng-Syue Li ◽  
Hung-Chu Hsu ◽  
Chiu-On Ng
Keyword(s):  
Asymptotic Solution ◽  
Water Waves ◽  
Surface Profile ◽  
Lagrangian Description ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
Particle Trajectories ◽  
Free Surface Profile ◽  
Sloping Beach ◽  
Sloping Bottom

A third-order asymptotic solution in Lagrangian description for nonlinear water waves propagating over a sloping beach is derived. The particle trajectories are obtained as a function of the nonlinear ordering parameter ε and the bottom slope α to the third order of perturbation. A new relationship between the wave velocity and the motions of particles at the free surface profile in the waves propagating on the sloping bottom is also determined directly in the complete Lagrangian framework. This solution enables the description of wave shoaling in the direction of wave propagation from deep to shallow water, as well as the successive deformation of wave profiles and water particle trajectories prior to breaking. A series of experiments are conducted to investigate the particle trajectories of nonlinear water waves propagating over a sloping bottom. It is shown that the present third-order asymptotic solution agrees very well with the experiments.

scholarly journals Prediction of the free-surface elevation for rotational water waves using the recovery of pressure at the bed

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170102 ◽  
Author(s):  
D. Henry ◽  
G. P. Thomas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Surface Profile ◽  
Two Dimensions ◽  
Arbitrary Distribution ◽  
Nonlinear Water Waves ◽  
Steady Current ◽  
Linear Waves ◽  
Free Surface Profile ◽  
Near Shore

This paper considers the pressure–streamfunction relationship for a train of regular water waves propagating on a steady current, which may possess an arbitrary distribution of vorticity, in two dimensions. The application of such work is to both near shore and offshore environments, and in particular, for linear waves we provide a description of the role which the pressure function on the seabed plays in determining the free-surface profile elevation. Our approach is shown to provide a good approximation for a range of current conditions. This article is part of the theme issue ‘Nonlinear water waves’.

A Nonlinear Coupled-Mode Model for Water Waves Over a General Bathymetry

2002 ◽  
Author(s):  
Gerassimos A. Athanassoulis ◽  
Konstandinos A. Belibassakis
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Vertical Structure ◽  
Local Mode ◽  
Surface Mode ◽  
Nonlinear Water Waves ◽  
Intermediate Depth ◽  
Coupled Mode ◽  
End Conditions ◽  
Sloping Bottom

A non-linear coupled-mode system of horizontal equations is derived with the aid of Luke’s (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bathymetry. The vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential, Athanassoulis & Belibassakis (2000). This series contains the usual propagating and evanescent modes, plus two additional modes, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The system fully accounts for the effects of non-linearity and dispersion.

scholarly journals Modified Babenko’s Equation For Periodic Gravity Waves On Water Of Finite Depth

2019 ◽  
Vol 72 (4) ◽  
pp. 415-428
Author(s):  
E Dinvay ◽  
N Kuznetsov
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Free Surface Profile ◽  
The Mean ◽  
New Equation

Summary A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko’s equation considered in Kuznetsov and Dinvay (Water Waves, 1, 2019). Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a parametric representation for a symmetric free surface profile. The latter is a component of a solution of the two-dimensional, nonlinear problem describing steady waves propagating in the absence of surface tension. Bifurcation curves (including a branching one) are obtained numerically for solutions of the new equation; they are compared with known results.

A ninth-order solution for the solitary wave

1972 ◽  
Vol 53 (2) ◽  
pp. 257-271 ◽  
Author(s):  
John Fenton
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Wave Amplitude ◽  
Surface Profile ◽  
Approximate Solutions ◽  
Series Expansions ◽  
Third Order ◽  
The Third ◽  
Order Solution ◽  
Physical Quantities

Several solutions for the solitary wave have been attempted since the work of Boussinesq in 1871. Of the approximate solutions, most have obtained series expansions in terms of wave amplitude, these being taken as far as the third order by Grimshaw (1971). Exact integral equations for the surface profile have been obtained by Milne-Thomson (1964,1968) and Byatt-Smith (1970), and these have been solved numerically. In the present work an exact operator equation is developed for the surface profile of steady water waves. For the case of a solitary wave, a form of solution is assumed and coefficients are obtained numerically by computer to give a ninth-order solution. This gives results which agree closely with exact numerical results for the surface profile, where these are available. The ninth-order solution, together with convergence improvement techniques, is used to obtain an amplitude of 0.85for the solitary wave of greatest height and to obtain refined approximations to physical quantities associated with the solitary wave, including the surface profile, speed of the wave and the drift of fluid particles.

A Unified Coupled-Mode Approach to Nonlinear Waves in Finite Depth: Potential Flow

2008 ◽  
Author(s):  
G. A. Athanassoulis ◽  
K. A. Belibassakis
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Water Wave ◽  
Bottom Topography ◽  
Local Mode ◽  
Surface Mode ◽  
Nonlinear Water Waves ◽  
Coupled Mode ◽  
Wide Range ◽  
Sloping Bottom

A non-linear coupled-mode system of horizontal equations is presented, as derived from Luke’s (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bottom topography. The vertical structure of the wave field is represented by means of a complete local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and has the following main features: (i) various standard models of water-wave propagation are recovered by appropriate simplifications, and (ii) it exhibits fast convergenge, and thus, a small number of modes (up to 5) are usually enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady traveling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate to shallow-water wave conditions and its results are compared vs. Stokes and cnoidal wave theories, respectively. Also, numerical results are presented for waves propagating over variable bathymetry regions and compared with second-order Stokes theory and experimental data.

Experimental and Lagrangian modeling of nonlinear water waves propagation on a sloping bottom

2013 ◽  
Vol 64 ◽  
pp. 36-48 ◽  
Author(s):  
Meng-Syue Li ◽  
Yang-Yih Chen ◽  
Hung-Chu Hsu ◽  
A. Torres-Freyermuth
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Lagrangian Modeling ◽  
Waves Propagation ◽  
Sloping Bottom

scholarly journals Nonlinear water waves (KdV) equation and Painleve technique

2015 ◽  
Vol 4 (2) ◽  
pp. 216
Author(s):  
Attia Mostafa
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Nonlinear Pde ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
The Third ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

<p>The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.</p>

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