Water waves in a deep square basin

1995 ◽  
Vol 302 ◽  
pp. 65-90 ◽  
Author(s):  
Peter J. Bryant ◽  
Michael Stiassnie
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Three Dimensional ◽  
Standing Waves ◽  
Two Dimensional ◽  
Initial State ◽  
Dimensional Properties ◽  
Periodic Standing Waves ◽  
The Waves ◽  
Small Disturbances

The form and evolution of three-dimensional standing waves in deep water are calculated analytically from Zakharov's equation and computationally from the full nonlinear bounddary value problem. The water is contained in a basin with a square cross-cection, when three-dimensional properties to pairs of sides are the same. It is found that non-periodic standing waves commonly follow forms of cyclic recurrence over times. The two-dimensional Stokes type of periodic standing waves (dominated by the fundamental harmonic) are shown to be unstable to three dimensional disturbances, but over long times the waves return cyclically close to their initial state. In contrast, the three-dimensional Stokes type of periodic standing waves are found to be stabel to small disturbances. New two-dimensional periodic standing waves with amplitude maxima at other than the fundamental harmonic have been investigated recently (Bryant & Stiassnie 1994). The equivalent three-dimensional standing waves are described here. The new two-dimensional periodic standing waves, like the two-dimensional Stokes standing waves, are found to be unstable to three-dimensional disturbances, and to exhibit cyclic recurrence over long times. Only some of the new three-dimensional periodic standing waves are found to be stable to small disturbances.

On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

Effects of plane progressive irrotational waves on thermal boundary layers

1971 ◽  
Vol 50 (2) ◽  
pp. 321-334 ◽  
Author(s):  
James Witting
Keyword(s):  
Boundary Layers ◽  
Deep Water ◽  
Maximum Amplitude ◽  
Capillary Waves ◽  
Temperature Gradients ◽  
Two Dimensional ◽  
Inviscid Incompressible Fluid ◽  
Mean Temperature ◽  
The Waves ◽  
Thermal Boundary Layers

The average changes in the structure of thermal boundary layers at the surface of bodies of water produced by various types of surface waves are computed. the waves are two-dimensional plane progressive irrotational waves of unchanging shape. they include deep-water linear waves, deep-water capillary waves of arbitrary amplitude, stokes waves, and the deep-water gravity wave of maximum amplitude.The results indicate that capillary waves can decrease mean temperature gradients by factors of as much as 9·0, if the average heat flux at the air-water interface is independent of the presence of the waves. Irrotational gravity waves can decrease the mean temperature gradients by factors no more than 1·381.Of possible pedagogical interest is the simplicity of the heat conduction equation for two-dimensional steady irrotational flows in an inviscid incompressible fluid if the velocity potential and the stream function are taken to be the independent variables.

Wave Kinematics Computed With the Nonlinear Schro¨dinger Method for Deep Water

1999 ◽  
Vol 121 (2) ◽  
pp. 126-130 ◽  
Author(s):  
K. Trulsen
Keyword(s):  
Time Series ◽  
Gravity Waves ◽  
Deep Water ◽  
Water Wave ◽  
Surface Displacement ◽  
Horizontal Position ◽  
Two Dimensional ◽  
Limited Bandwidth ◽  
Wave Kinematics ◽  
The Waves

The nonlinear Schro¨dinger method for water wave kinematics under two-dimensional irregular deepwater gravity waves is developed. Its application is illustrated for computation of the velocity and acceleration fields from the time-series of the surface displacement measured at a fixed horizontal position. The method is based on the assumption that the waves have small steepness and limited bandwidth.

scholarly journals Reflection of nonlinear deep-water waves incident onto a wedge of arbitrary angle

1990 ◽  
Vol 32 (1) ◽  
pp. 61-96 ◽  
Author(s):  
T. R. Marchant ◽  
A. J. Roberts
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Wave Reflection ◽  
Mach Reflection ◽  
Wedge Angle ◽  
Arbitrary Angle ◽  
Weakly Nonlinear ◽  
Progressive Waves ◽  
The Waves ◽  
Wave Properties

AbstractWave reflection by a wedge in deep water is examined, where the wedge can represent a breakwater of finite length or the bow of a ship heading directly into the waves. In addition, the form of the solution allows the results to apply to ships heading at an angle into the waves. We consider a deep-water wavetrain approaching the wedge head on from infinity and being reflected. Far from the wedge there is a field of progressive waves (the incident wavetrain) while close to the wedge there is a short-crested wavefield (the incident and reflected wavetrains). A weakly-nonlinear slowly-varying averaged Lagrangian theory is used to describe the problem (see Whitham [16]) as the theory includes the nonlinear interaction between the incident and reflected wavetrains. This modelling of a short-crested wavefield allows the nonlinear wavefield to be found for broad wedges, as opposed to previous theories which are applicable to thin wedges only.It is shown that the governing partial differential equations are hyperbolic and that the solution comprises two regions, within which the wave properties are constant separated by a wave jump. Given the wedge angle and the incident wavefield, the jump angle and the wave steepness and wavenumber of the short-crested wave-field behind the wave jump can be determined. Two solution branches are found to exist: one corresponds to regular reflection, while for small amplitudes the other is similar to Mach-reflection and so it is called near Mach-reflection. Results are presented describing both solution branches and the transition between them.

On the observability of finite-depth short-crested water waves

1996 ◽  
Vol 322 ◽  
pp. 1-19 ◽  
Author(s):  
M. Ioualalen ◽  
A. J. Roberts ◽  
C. Kharif
Keyword(s):  
Standing Wave ◽  
Water Waves ◽  
Numerical Study ◽  
Standing Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Wave Fields ◽  
Progressive Waves ◽  
Wave Limit ◽  
Narrow Bands

A numerical study of the superharmonic instabilities of short-crested waves on water of finite depth is performed in order to measure their time scales. It is shown that these superharmonic instabilities can be significant-unlike the deep-water case-in parts of the parameter regime. New resonances associated with the standing wave limit are studied closely. These instabilities ‘contaminate’ most of the parameter space, excluding that near two-dimensional progressive waves; however, they are significant only near the standing wave limit. The main result is that very narrow bands of both short-crested waves ‘close’ to two-dimensional standing waves, and of well developed short-crested waves, perturbed by superharmonic instabilities, are unstable for depths shallower than approximately a non-dimensional depth d= 1; the study is performed down to depth d= 0.5 beyond which the computations do not converge sufficiently. As a corollary, the present study predicts that these very narrow sub-domains of short-crested wave fields will not be observable, although most of the short-crested wave fields will be.

scholarly journals Three-dimensional surface gravity waves of a broad bandwidth on deep water

2021 ◽  
Vol 926 ◽  
Author(s):  
Yan Li
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Order Approximation ◽  
Three Dimensional ◽  
Finite Depth ◽  
Second Order ◽  
Leading Order ◽  
Broad Bandwidth ◽  
Stokes Waves ◽  
Deep Water Waves

A new nonlinear Schrödinger equation (NLSE) is presented for ocean surface waves. Earlier derivations of NLSEs that describe the evolution of deep-water waves have been limited to a narrow bandwidth, for which the bound waves at second order in wave steepness are described in leading-order approximations. This work generalizes these earlier works to allow for deep-water waves of a broad bandwidth with large directional spreading. The new NLSE permits simple numerical implementations and can be extended in a straightforward manner in order to account for waves on water of finite depth. For the description of second-order waves, this paper proposes a semianalytical approach that can provide accurate and computationally efficient predictions. With a leading-order approximation to the new NLSE, the instability region and energy growth rate of Stokes waves are investigated. Compared with the exact results based on McLean (J. Fluid Mech., vol. 511, 1982, p. 135), predictions by the new NLSE show better agreement than by Trulsen et al. (Phys. Fluids, vol. 12, 2000, pp. 2432–2437). With numerical implementations of the new NLSE, the effects of wave directionality are investigated by examining the evolution of a directionally spread focused wave group. A downward shift of the spectral peak is observed, owing to the asymmetry in the change rate of energy in a more complex manner than that for uniform Stokes waves. Rapid oblique energy transfers near the group at linear focus are observed, likely arising from the instability of uniform Stokes waves appearing in a narrow spectrum subject to oblique sideband disturbances.

scholarly journals THE DEVELOPMENT OF A SAND BEACH BY DEEP-WATER WAVES

2000 ◽  
Vol 1 (4) ◽  
pp. 12
Author(s):  
Harold Flinsch
Keyword(s):  
Grain Size ◽  
Deep Water ◽  
Water Waves ◽  
Equilibrium Theory ◽  
Sand Beach ◽  
Shore Line ◽  
Natural Development ◽  
Accurate Forecast ◽  
Sand And Gravel ◽  
The Waves

In a previous paper**, it was shown that the mechanism of the trochoidal waves can be used to determine the equilibrium slope of a sand beach under any wave conditions. As a start it was assumed that the beach material was of uniform grain size, and that the waves approached the beach directly with all motion in planes at right angles to the shore line. In the present paper, the application of the theory is shown in the development of various sand and gravel beaches. The equilibrium theory is studied in the light of the fact that there is usually considerable transportation of material along the shore. In particular, attention is called to the characteristics of beaches with rounded or pointed contours, of beaches whose ends are closed off by rocks or cliffs, or whose ends are open and extend into deep water without barriers of any kind. A method of study and analysis is demonstrated which can be applied to all beaches. Finally, it is shown that an accurate forecast of the natural development of a beach can be made on the basis of the equilibrium slope equation, as well as a forecast of the effect of any structure placed in a naturally developing beach.

Quasi-three-dimensional simulation of crescent-shaped waves

2020 ◽  
Author(s):  
Alexander Dosaev ◽  
Yuliya Troitskaya
Keyword(s):  
Water Waves ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Boundary Integral ◽  
Main Direction ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Wave Interactions

<p>Many features of nonlinear water wave dynamics can be explained within the assumption that the motion of fluid is strictly potential. At the same time, numerically solving exact equations of motion for a three-dimensional potential flow with a free surface (by means of, for example, boundary integral method) is still often considered too computationally expensive, and further simplifications are made, usually implying limitations on wave steepness. A quasi-three-dimensional model, put forward by V. P. Ruban [1], represents another approach at reducing computational cost. It is, in its essence, a two-dimensional model, formulated using conformal mapping of the flow domain, augmented by three-dimensional corrections. The model assumes narrow directional distribution of the wave field and is exact for two-dimensional waves. It was successfully applied by its author to study a nonlinear stage of of Benjamin-Feir instability and rogue waves formation.</p><p>The main aim of the present work is to explore the behaviour of the quasi-three-dimensional model outside the formal limits of its applicability. From the practical point of view, it is important that the model operates robustly even in the presence of waves propagating at large angles to the main direction (although we do not attempt to accurately describe their dynamics). We investigate linear stability of Stokes waves to three-dimensional perturbations and suggest a modification to the original model to eliminate a spurious zone of instability in the vicinity of the perpendicular direction on the perturbation wavenumber plane. We show that the quasi-three-dimensional model yields a qualitatively correct description of the instability zone generated by resonant 5-wave interactions. The values of the increment are reasonably close to those obtained from the exact equations of motion [2], despite the fact that the corresponding modes of instability consist of harmonics that are relatively far from the main direction. Resonant 5-wave interactions are known to manifest themselves in the formation of the so-called “horse-shoe” or “crescent-shaped” wave patterns, and the quasi-three-dimensional model exhibits a plausible dynamics leading to formation of crescent-shaped waves.</p><p>This research was supported by RFBR (grant No. 20-05-00322).</p><p>[1] Ruban, V. P. (2010). Conformal variables in the numerical simulations of long-crested rogue waves. <em>The European Physical Journal Special Topics</em>, <em>185</em>(1), 17-33.</p><p>[2] McLean, J. W. (1982). Instabilities of finite-amplitude water waves. <em>Journal of Fluid Mechanics</em>, <em>114</em>, 315-330.</p>

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