On Gerstner's Water Wave

Henry, David

doi:10.2991/jnmp.2008.15.s2.7

Cited by 138 publications

(99 citation statements)

References 29 publications

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“…More recently, a three-dimensional version describing explicitly edge waves propagating along a sloping beach was provided in Constantin [40]. Finally, for a modern exposition of Gerstner's wave, the papers by Constantin [41] and Henry [42] are recommended. (k) Compared with deep-water waves, the case of solitary waves of finite depth seems to be easier.…”

Section: Remark 42mentioning

confidence: 99%

Regularity of rotational travelling water waves

Escher¹

2012

Phil. Trans. R. Soc. A.

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Section: Remark 42mentioning

confidence: 99%

Regularity of rotational travelling water waves

Escher¹

2012

Phil. Trans. R. Soc. A.

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“…The theory consists of the exact solutions required to express a possible form of gravity wave motion in the Lagrangian system, despite the fact that the motion is rotational. Modern approaches towards Gerstner's wave with a particular distribution of vorticity can be found in Constantin [2] and Henry [3]. In the theory, water particles are described as circular orbits whose radii diminish with water depth.…”

Section: Introductionmentioning

confidence: 99%

Eulerian–Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry

Umeyama

2012

Phil. Trans. R. Soc. A.

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This paper investigates the velocity and the trajectory of water particles under surface waves, which propagate at a constant water depth, using particle image velocimetry (PIV). The vector fields and vertical distributions of velocities are presented at several phases in one wave cycle. The third-order Stokes wave theory was employed to express the physical quantities. The PIV technique's ability to measure both temporal and spatial variations of the velocity was proved after a series of attempts. This technique was applied to the prediction of particle trajectory in an Eulerian scheme. Furthermore, the measured particle path was compared with the positions found theoretically by integrating the Eulerian velocity to the higher order of a Taylor series expansion. The profile of average travelling distance is also presented with a solution of zero net mass flux in a closed wave flume.

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“…Due to the mathematical intractability of the governing equations for water waves, for irrotational water waves (Gerstner's wave has a peculiar nonvanishing vorticity) the classical approach [15,20,23,24,28,29,30] relies on analyzing the particle motion after linearization of the governing equations. However, even within the linear water wave theory, the ordinary differential equations system describing the motion of the particles is nevertheless nonlinear and explicit solutions of this system are not available, but qualitative features of the underlying flow are known [6,9,10,11,21,26,27,33] and could possibly lead to a confirmation of the features we prove here, within the confines of linear water waves (see [4]).…”

Section: Introductionmentioning

confidence: 99%

Particle trajectories in linear deep-water waves

Constantin

Ehrnström

Villari

2008

Nonlinear Analysis: Real World Applications

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Abstract. We determine the phase portrait of a Hamiltonian system of equations describing the motion of the particles in linear deep-water waves. The particles experience in each period a forward drift which decreases with greater depth. IntroductionThe motion of water particles under the waves which advance across the water is a classical problem in this field. Watching the sea it is oft possible to trace a wave as it propagates on the water's surface, but what one observes traveling across the sea is not the water but a wave pattern. Although the wave travels from one place to another, the substance through which it travels moves very little. As the wave advances across the water and can be followed for a long way, a typical water particle moves slightly up and down, forward and backward as the wave passes it. If an object hovered in the water, like a water particle, its motion would be synchronized with that of a floating object lying on the water's surface, with its orbit diminishing with the distance from the surface. As waves generated by wind in an area move towards a region where the wind has ceased, we observe swell-long crested two-dimensional waves approaching a smooth sinusoidal shape and moving over long distances. Deep-water waves are modelled mathematically as periodic two-dimensional waves in water of infinite depth. The motion of the water particle in the fluid below swell is of great interest. The classical description of these particle paths is obtained within the framework of linear water wave theory [1,15,16,23,24,25,29,30]: all water particles trace a circular orbit, the diameter of which decreases with depth so that the orbital motion practically ceases at depth equal to one-half the wavelength. These features have important practical consequences. For example, a submarine at a depth below half a wavelength would hardly notice the motion of the surface wave, for this reason submarines dive during storms in the open sea.The only known explicit solution with a non-flat free surface of the governing equations for gravity water waves is Gerstner's wave [17]: a deep-water wave solution for which all particle paths are circles of diameters decreasing with the distance from the free surface (see the discussion in [2,3]). Due to the mathematical intractability of the governing equations for water waves, for irrotational water waves (Gerstner's wave has a peculiar nonvanishing vorticity) the classical approach [15,20,23,24,28,29,30] relies on analyzing the particle motion after linearization of the governing equations. However, even within the linear water wave theory, the ordinary differential equations system describing the motion of the particles is nevertheless nonlinear and explicit solutions of this system are not available, but qualitative features of the underlying flow 2000 Mathematics Subject Classification. 76B15, 34C25, 35Q35.

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On Gerstner's Water Wave

Cited by 138 publications

References 29 publications

Regularity of rotational travelling water waves

Regularity of rotational travelling water waves

Eulerian–Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry

Particle trajectories in linear deep-water waves

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