Extreme Events in Nonlinear Random Seas

2005 ◽  
Vol 128 (1) ◽  
pp. 11-16 ◽  
Author(s):  
Francesco Fedele
Keyword(s):  
Extreme Events ◽  
Water Waves ◽  
Surface Displacement ◽  
Wave Crest ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Random Seas ◽  
Wave Resonance ◽  
Starting Point ◽  
Spectral Components

In this paper, the occurrence of extreme events due to the four-wave resonance interaction in weakly nonlinear water waves is investigated. The starting point is the Zakharov equation, which governs the dynamics of the spectral components of the surface displacement. It is proven that the optimal spectral components giving an extreme crest are solutions of a well-defined constrained optimization problem. A new analytical expression for the probability of exceedance of the wave crest is then proposed for the prediction of freak wave events.

scholarly journals Study on the Interaction of Nonlinear Water Waves considering Random Seas

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Marten Hollm ◽  
Leo Dostal ◽  
Hendrik Fischer ◽  
Robert Seifried
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Random Seas

A new Boussinesq method for fully nonlinear waves from shallow to deep water

2002 ◽  
Vol 462 ◽  
pp. 1-30 ◽  
Author(s):  
P. A. MADSEN ◽  
H. B. BINGHAM ◽  
HUA LIU
Keyword(s):  
Laplace Equation ◽  
Infinite Series ◽  
Deep Water ◽  
Water Waves ◽  
Modulational Instability ◽  
Surface Boundary ◽  
Nonlinear Water Waves ◽  
Finite Series ◽  
Starting Point ◽  
Highly Nonlinear

A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant

1991 ◽  
Vol 229 (-1) ◽  
pp. 135 ◽  
Author(s):  
S. W. Joo ◽  
A. F. Messiter ◽  
W. W. Schultz
Keyword(s):  
Water Waves ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear

scholarly journals Water waves, nonlinear Schrödinger equations and their solutions

1983 ◽  
Vol 25 (1) ◽  
pp. 16-43 ◽  
Author(s):  
D. H. Peregrine
Keyword(s):  
Water Waves ◽  
Rational Solution ◽  
Three Dimensional ◽  
Length Scale ◽  
Nls Equation ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Schrödinger ◽  
Wave Induced ◽  
Modulation Length

AbstractEquations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrödinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

scholarly journals On the existence of positive solutions for a nonlinear elliptic class of equations in R2 and R3

2021 ◽  
Vol 20 ◽  
pp. 188-210
Author(s):  
Jose Quintero
Keyword(s):  
Positive Solutions ◽  
Water Waves ◽  
Nonlinear Term ◽  
Homogeneous Function ◽  
Concentration Compactness ◽  
Nonlinear Water Waves ◽  
Weakly Nonlinear ◽  
Nonlinear Elliptic ◽  
Existence Of Positive Solutions ◽  
Localized Waves

We study the existence of positive solutions for an elliptic equation in RN for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H1(RN) with values in R.

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