scholarly journals Nonlinear water waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1501-1504 ◽  
Author(s):  
Adrian Constantin
Keyword(s):  
Water Waves ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Recent Developments

This introduction to the issue provides a review of some recent developments in the study of water waves. The content and contributions of the papers that make up this Theme Issue are also discussed.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals On the short-wavelength stabilities of some geophysical flows

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170090 ◽  
Author(s):  
Delia Ionescu-Kruse
Keyword(s):  
Water Waves ◽  
Short Wavelength ◽  
Travelling Wave ◽  
Geophysical Flows ◽  
Rotating Flows ◽  
Steady Flows ◽  
Azimuthal Direction ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Stability Method

This paper is a survey of the short-wavelength stability method for rotating flows. Additional complications such as stratification in the flow or the presence of non-conservative body forces are considered too. This method is applied to the specific study of some exact geophysical flows. For Gerstner-like geophysical flows one can identify perturbations in certain directions as a source of instabilities with an exponentially growing amplitude, the growth rate of the instabilities depending on the steepness of the travelling wave profile. On the other hand, for certain physically realistic velocity profiles, steady flows moving only in the azimuthal direction, with no variation in this direction, are locally stable to the short-wavelength perturbations. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Analytical approximation and numerical simulations for periodic travelling water waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170093 ◽  
Author(s):  
Konstantinos Kalimeris
Keyword(s):  
Water Waves ◽  
Optimization Problem ◽  
Analytical Approximation ◽  
Numerical Results ◽  
Numerical Continuation ◽  
Constrained Optimization Problem ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Constant Vorticity ◽  
Continuation Algorithm

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Shallow water equations for equatorial tsunami waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170100 ◽  
Author(s):  
Anna Geyer ◽  
Ronald Quirchmayr
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Water Model ◽  
Shallow Water Model ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Tsunami Waves ◽  
De Vries ◽  
Model Equations

We present derivations of shallow water model equations of Korteweg–de Vries and Boussinesq type for equatorial tsunami waves in the f -plane approximation and discuss their applicability. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Evolution of statistically inhomogeneous degenerate water wave quartets

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170101 ◽  
Author(s):  
R. Stuhlmeier ◽  
M. Stiassnie
Keyword(s):  
Water Waves ◽  
Wave Interaction ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Random Surface ◽  
Freak Wave ◽  
Long Time Behaviour ◽  
Long Time ◽  
Spatio Temporal ◽  
The Stability

A discretized equation for the evolution of random surface wave fields on deep water is derived from Zakharov's equation, allowing for a general treatment of the stability and long-time behaviour of broad-banded sea states. It is investigated for the simple case of degenerate four-wave interaction, and the instability of statistically homogeneous states to small inhomogeneous disturbances is demonstrated. Furthermore, the long-time evolution is studied for several cases and shown to lead to a complex spatio-temporal energy distribution. The possible impact of this evolution on the statistics of freak wave occurrence is explored. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Nonlinear water waves: introduction and overview

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170310 ◽  
Author(s):  
A. Constantin
Keyword(s):  
Water Waves ◽  
Essential Feature ◽  
Theme Issue ◽  
Research Activity ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Isaac Newton ◽  
Weakly Nonlinear Theory ◽  
Mathematical Sciences ◽  
To Come

For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170091 ◽  
Author(s):  
A. Compelli ◽  
R. Ivanov ◽  
M. Todorov
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Kdv Equation ◽  
Variable Coefficients ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Fluid Medium ◽  
Hamiltonian Models ◽  
Variable Bottom ◽  
The One

A single incompressible, inviscid, irrotational fluid medium bounded by a free surface and varying bottom is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet–Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected which leads to approximations of Boussinesq and Korteweg–de Vries (KdV) types, taking into account the effect of the slowly varying bottom. The arising KdV equation with variable coefficients is studied numerically when the initial condition is in the form of the one-soliton solution for the initial depth. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Solitary interfacial hydroelastic waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170099 ◽  
Author(s):  
Emilian I. Părău
Keyword(s):  
Dispersion Relation ◽  
Solitary Waves ◽  
Elastic Plate ◽  
Water Waves ◽  
Rigid Wall ◽  
Two Dimensional ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Two Fluids ◽  
Infinite Extent

Solitary waves travelling along an elastic plate present between two fluids with different densities are computed in this paper. Different two-dimensional configurations are considered: the upper fluid can be of infinite extent, bounded by a rigid wall or under a second elastic plate. The dispersion relation is obtained for each case and numerical codes based on integro-differential formulations for the full nonlinear problem are derived. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals On periodic geophysical water flows with discontinuous vorticity in the equatorial f -plane approximation

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170096 ◽  
Author(s):  
Calin Iulian Martin
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Wave Speed ◽  
Laminar Flows ◽  
Local Bifurcation ◽  
Two Dimensional ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Water Flows

We are concerned here with geophysical water waves arising as the free surface of water flows governed by the f -plane approximation. Allowing for an arbitrary bounded discontinuous vorticity, we prove the existence of steady periodic two-dimensional waves of small amplitude. We illustrate the local bifurcation result by means of an analysis of the dispersion relation for a two-layered fluid consisting of a layer of constant non-zero vorticity γ 1 adjacent to the surface situated above another layer of constant non-zero vorticity γ 2 ≠ γ 1 adjacent to the bed. For certain vorticities γ 1 , γ 2 , we also provide estimates for the wave speed c in terms of the speed at the surface of the bifurcation inducing laminar flows. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Capturing the flow beneath water waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170098 ◽  
Author(s):  
A. Nachbin ◽  
R. Ribeiro-Junior
Keyword(s):  
Flow Structure ◽  
Water Waves ◽  
Bottom Topography ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Stagnation Points ◽  
Link Type ◽  
Numerical Studies ◽  
Closed Orbits ◽  
Rotational Waves

Recently, the authors presented two numerical studies for capturing the flow structure beneath water waves (Nachbin and Ribeiro-Junior 2014 Disc. Cont. Dyn. Syst. A 34 , 3135–3153 ( doi:10.3934/dcds.2014.34.3135 ); Ribeiro-Junior et al. 2017 J. Fluid Mech. 812 , 792–814 ( doi:10.1017/jfm.2016.820 )). Closed orbits for irrotational waves with an opposing current and stagnation points for rotational waves were some of the issues addressed. This paper summarizes the numerical strategies adopted for capturing the flow beneath irrotational and rotational water waves. It also presents new preliminary results for particle trajectories, due to irrotational waves, in the presence of a bottom topography. This article is part of the theme issue ‘Nonlinear water waves’.

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