scholarly journals Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves

2012 ◽  
Vol 370 (1964) ◽  
pp. 1572-1586 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Gravity Waves ◽  
Finite Amplitude ◽  
Numerical Results ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Surface Gravity Waves ◽  
Stokes Waves ◽  
Irrotational Motion ◽  
Dimensional Surface

The velocity and other fields of steady two-dimensional surface gravity waves in irrotational motion are investigated numerically. Only symmetric waves with one crest per wavelength are considered, i.e. Stokes waves of finite amplitude, but not the highest waves, nor subharmonic and superharmonic bifurcations of Stokes waves. The numerical results are analysed, and several conjectures are made about the velocity and acceleration fields.

scholarly journals Accurate fast computation of steady two-dimensional surface gravity waves in arbitrary depth

2018 ◽  
Vol 844 ◽  
pp. 491-518 ◽  
Author(s):  
Didier Clamond ◽  
Denys Dutykh
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Fast Computation ◽  
Surface Gravity Waves ◽  
Irrotational Motion ◽  
Arbitrary Precision ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral ◽  
Dimensional Surface

This paper describes an efficient algorithm for computing steady two-dimensional surface gravity waves in irrotational motion. The algorithm complexity is $O(N\log N)$, $N$ being the number of Fourier modes. This feature allows the arbitrary precision computation of waves in arbitrary depth, i.e. it works efficiently for Stokes, cnoidal and solitary waves, even for quite large steepnesses, up to approximately 99 % of the maximum steepness for all wavelengths. In particular, the possibility to compute very long (cnoidal) waves accurately is a feature not shared by other algorithms and asymptotic expansions. The method is based on conformal mapping, the Babenko equation rewritten in a suitable way, the pseudo-spectral method and Petviashvili iterations. The efficiency of the algorithm is illustrated via some relevant numerical examples. The code is open source, so interested readers can easily check the claims, use and modify the algorithm.

Surface gravity waves over a two-dimensional random seabed

2002 ◽  
Vol 66 (1) ◽  
Author(s):  
Jørgen H. Pihl ◽  
Chiang C. Mei ◽  
Matthew J. Hancock
Keyword(s):  
Gravity Waves ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Surface Gravity Waves

The Intermediate Water Depth Limit of the Zakharov Equation and Consequences for Wave Prediction

2007 ◽  
Vol 37 (10) ◽  
pp. 2389-2400 ◽  
Author(s):  
Peter A. E. M. Janssen ◽  
Miguel Onorato
Keyword(s):  
Gravity Waves ◽  
Modulational Instability ◽  
Finite Amplitude ◽  
Surface Gravity ◽  
Intermediate Water ◽  
Zakharov Equation ◽  
Depth Limit ◽  
Surface Gravity Waves ◽  
Wave Prediction ◽  
Starting Point

Abstract Finite-amplitude deep-water waves are subject to modulational instability, which eventually can lead to the formation of extreme waves. In shallow water, finite-amplitude surface gravity waves generate a current and deviations from the mean surface elevation. This stabilizes the modulational instability, and as a consequence the process of nonlinear focusing ceases to exist when kh < 1.363. This is a well-known property of surface gravity waves. Here it is shown for the first time that the usual starting point, namely the Zakharov equation, for deriving the nonlinear source term in the energy balance equation in wave forecasting models, shares this property as well. Consequences for wave prediction are pointed out.

A note on the uniqueness of small-amplitude water waves travelling in a region of varying depth

1979 ◽  
Vol 86 (3) ◽  
pp. 511-519 ◽  
Author(s):  
G. F. Fitz-Gerald ◽  
R. H. J. Grimshaw
Keyword(s):  
Linear Theory ◽  
Gravity Waves ◽  
Water Waves ◽  
Small Amplitude ◽  
Velocity Potential ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Physical Interest ◽  
Surface Gravity Waves ◽  
Convexity Conditions

The two-dimensional, irrotational, linear theory used in the investigation of the propagation of monochromatic surface gravity waves in a region of varying depth is considered. Uniqueness of the velocity potential is established for bottom profiles satisfying certain convexity conditions. These include the majority of profiles of physical interest.

scholarly journals Hamiltonian structure for two-dimensional extended Green–Naghdi equations

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160127 ◽  
Author(s):  
Yoshimasa Matsuno
Keyword(s):  
Dispersion Relation ◽  
Gravity Waves ◽  
Hamiltonian Structure ◽  
Bottom Topography ◽  
Higher Order ◽  
Surface Gravity ◽  
Two Dimensional ◽  
Linear Dispersion ◽  
Surface Gravity Waves ◽  
Linear Dispersion Relation

The two-dimensional Green–Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original Green–Naghdi (GN) model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

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