The trajectories of particles in deep-water Stokes waves

Henry, David

doi:10.1155/imrn/2006/23405

Cited by 62 publications

(65 citation statements)

References 16 publications

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“…In this way we extend the propositions in [1,16], providing further understanding even for the irrotational case. For steady rotational waves, to our knowledge this is the first investigation of its kind apart from [13].…”

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confidence: 63%

On the streamlines and particle paths of gravitational water waves

Ehrnström¹

2008

Nonlinearity

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We investigate steady symmetric gravity water waves on finite depth. For nonpositive vorticity it is shown that the particles display a mean forward drift, and for a class of waves we prove that the size of this drift is strictly increasing from bottom to surface. We also provide detailed information concerning the streamlines and the particle trajectories. This includes the case of particles within irrotational waves.

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confidence: 63%

On the streamlines and particle paths of gravitational water waves

Ehrnström¹

2008

Nonlinearity

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“…This result was first obtained in [13] for gravity waves and [19] for capillary and capillary-gravity water waves. This results have been improved in [4,9,18] in the case of irrotational waves by describing the exact geometry of the actual particle paths. For gravity deep-water waves we reconfirm herein the results of [13], but our analysis is more precise and uses only elementary analysis methods.…”

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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“…For nonlinear water waves, the existence of the fully nonlinear equations was proven in [12,13,38,41], firstly for gravity water waves and then for waves incorporating surface tension. Further properties of the flow for these waves were enunciated in [4,7,8,17,21], such as the symmetry of the flow. Recent analytically rigorous results which provide a qualitative description of the particle trajectories for a wide variety of both linear and nonlinear flow are [3][4][5][6][7][8][9][10][11][15][16][17][18][19][20][21]26,27,39].…”

Section: Introductionmentioning

confidence: 99%

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

Hsu

Hwung

2010

J. Math. Fluid Mech.

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Abstract.A third-order Lagrangian asymptotic solution is derived for gravity-capillary waves in water of finite depth. The explicit parametric solution gives the trajectory of a water particle and the wave kinematics for Lagrangian points above the mean water level, and in a water column. The water particle orbits and mass transport velocity as functions of the surface tension are obtained. Some remarkable trajectories may contain one or multiple sub-loops for steep waves and large surface tension. Overall, an increase in surface tension tends to increase the motions of surface particles including the relative horizontal distance travelled by a particle as well as the time-averaged drift velocity.Mathematics Subject Classification (2010). Primary 76B15; Secondary 35Q35, 34C05, 34C25.

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The trajectories of particles in deep-water Stokes waves

Cited by 62 publications

References 16 publications

On the streamlines and particle paths of gravitational water waves

On the streamlines and particle paths of gravitational water waves

Particle trajectories in linear deep-water waves

A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth

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