On dispersion of directional surface gravity waves

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.

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Measurement of the dispersion relation for random surface gravity waves

Journal of Fluid Mechanics ◽

10.1017/jfm.2015.25 ◽

2015 ◽

Vol 766 ◽

pp. 326-336 ◽

Cited By ~ 12

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.

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Hamiltonian structure for two-dimensional extended Green–Naghdi equations

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0127 ◽

2016 ◽

Vol 472 (2190) ◽

pp. 20160127 ◽

Cited By ~ 5

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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