scholarly journals Measurement of the dispersion relation for random surface gravity waves

2015 ◽  
Vol 766 ◽  
pp. 326-336 ◽  
Author(s):  
Tore Magnus A. Taklo ◽  
Karsten Trulsen ◽  
Odin Gramstad ◽  
Harald E. Krogstad ◽  
Atle Jensen
Keyword(s):  
Dispersion Relation ◽  
Gravity Waves ◽  
Laboratory Experiments ◽  
Surface Gravity ◽  
Linear Dispersion ◽  
Random Surface ◽  
Zakharov Equation ◽  
Surface Gravity Waves ◽  
Linear Dispersion Relation ◽  
Jonswap Spectrum

AbstractWe report laboratory experiments and numerical simulations of the Zakharov equation, designed to have sufficient resolution in space and time to measure the dispersion relation for random surface gravity waves. The experiments and simulations are carried out for a JONSWAP spectrum and Gaussian spectra of various bandwidths on deep water. It is found that the measured dispersion relation deviates from the linear dispersion relation above the spectral peak when the bandwidth is sufficiently narrow.

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.

scholarly journals On dispersion of directional surface gravity waves

2017 ◽  
Vol 812 ◽  
pp. 681-697 ◽  
Author(s):  
Tore Magnus A. Taklo ◽  
Karsten Trulsen ◽  
Harald E. Krogstad ◽  
José Carlos Nieto Borge
Keyword(s):  
Gravity Waves ◽  
Nonlinear Evolution ◽  
Laboratory Data ◽  
Surface Gravity ◽  
Linear Dispersion ◽  
Marine Research ◽  
Surface Gravity Waves ◽  
Linear Dispersion Relation ◽  
Directional Waves ◽  
Crest Length

Using a nonlinear evolution equation we examine the dependence of the dispersion of directional surface gravity waves on the Benjamin–Feir index (BFI) and crest length. A parameter for describing the deviation between the dispersion of simulated waves and the theoretical linear dispersion relation is proposed. We find that for short crests the magnitude of the deviation parameter is low while for long crests the magnitude is high and depends on the BFI. In the present paper we also consider laboratory data of directional waves from the Marine Research Institute of the Netherlands (MARIN). The MARIN data confirm the simulations for three cases of BFI and crest length.

On modifications of the Zakharov equation for surface gravity waves

1984 ◽  
Vol 143 ◽  
pp. 47-67 ◽  
Author(s):  
Michael Stiassnie ◽  
Lev Shemer
Keyword(s):  
Integral Equation ◽  
Gravity Waves ◽  
Higher Order ◽  
Class I ◽  
Surface Gravity ◽  
Stokes Wave ◽  
Zakharov Equation ◽  
Infinite Depth ◽  
Surface Gravity Waves ◽  
New Equation

The Zakharov integral equation for surface gravity waves is modified to include higher-order (quintet) interactions, for water of constant (finite or infinite) depth. This new equation is used to study some aspects of class I (4-wave) and class II (5-wave) instabilities of a Stokes wave.

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.

Internal Gravity Waves

2013 ◽  
Author(s):  
Gary A. Glatzmaier
Keyword(s):  
Dispersion Relation ◽  
Phase Velocity ◽  
Gravity Waves ◽  
Thermal Convection ◽  
Fluid Velocity ◽  
Computer Code ◽  
Internal Gravity Waves ◽  
Linear Analysis ◽  
Linear Dispersion ◽  
Linear Dispersion Relation

This chapter focuses on internal gravity waves in a stable thermal stratification. When the amplitude of the fluid velocity is small relative to the amplitude of the phase velocity, a linear analysis, which neglects advection, provides insight to the relation between the wavelength and frequency of internal gravity waves. Furthermore, when thermal and viscous diffusion play relatively minor roles the system can be further simplified by neglecting diffusion. The chapter first describes the linear dispersion relation before discussing the computer code modifications and simulations. In particular, it explains what modifications would be needed to convert one's thermal convection code to a code that simulates internal gravity waves, including the nonlinear and diffusive terms. Finally, it considers the computer analysis of wave energy.

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