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Communication Dans Un Congrès Année : 2005

Three-dimensional instabilities of periodic gravity waves in shallow water

Marc Francius

Résumé

Water-wave motion in coastal areas is a subject of prime importance to wave forecastingfor meteorological and oceanographic purposes and ship navigation, etc....Yet, thedynamics of water waves of large amplitude with typical wavelengths greater than thewater depth are not well understood, although major practical problems are associatedwith surface coastal waves.Theoretical models for shallow water waves often assume that nonlinear three-wavenear-resonant and four-wave resonant interactions are the main nonlinear processesacting to transfer energy among the components of the wave field. It is well knownthat these interactions influence the growth rates of wind-generated gravity waves aswell as other intrinsic properties of coherent surface gravity waves which can affecttheir own dynamics. Nevertheless, which wave-wave interactions dominate in shallowwater remains unclear. Besides, it is well known that waves are usually more nonlinearin finite depth than the waves on deep water. Despite this, there appears that the higherorderresonant interactions problem has received little attention in finite depth andshallow water in comparison with the three-wave near resonant interactions problem.In the present paper, we consider the linear stability analysis of finite-amplitude periodicsurface waves propagating steadily on water of finite depth. This is a first step, inshallow water, towards the characterization of the instabilities resulting from higherorderresonances, i.e between more than five waves. This study has extended existingresults to steeper waves and shallower water. This also completes the previous work ofBryant (1974, 1978) and McLean (1982). Some new types of instability are found forshallow water. When the water depth decreases, higher-order resonances lead to thedominant instabilities. In contrast with the deep water case, we have found that in shallowwater the dominant instabilities are usually associated with resonant interactionsbetween five, six, seven and eight waves. For small steepness, dominant instabilitiesare quasi two-dimensional. For moderate and large steepness, the dominant instabilitiesare three-dimensional and phased-locked with the unperturbed nonlinear wave.These results suggest, at the margin of instability diagram, the existence of bifurcatedthree-dimensional steady waves. These new waves possess few symmetry propertiesthat are discussed in this paper.
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Dates et versions

hal-00090455 , version 1 (31-08-2006)

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  • HAL Id : hal-00090455 , version 1

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Marc Francius, Christian Kharif. Three-dimensional instabilities of periodic gravity waves in shallow water. European Geosciences Union - General Assembly 2005, 2005, Vienne, Austria. ⟨hal-00090455⟩
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