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.

scholarly journals MACH-REFLECTION AS A DIFFRACTION PROBLEM

1976 ◽  
Vol 1 (15) ◽  
pp. 45 ◽  
Author(s):  
Udo Berger ◽  
Soren Kohlhase
Keyword(s):  
Wave Height ◽  
Incident Wave ◽  
Water Waves ◽  
Gas Dynamics ◽  
Mach Reflection ◽  
Diffraction Problem ◽  
Vertical Wall ◽  
Oblique Wave ◽  
Wave Heights ◽  
The Waves

As under oblique wave approach water waves are reflected by a vertical wall, a wave branching effect (stem) develops normal to the reflecting wall. The waves progressing along the wall will steep up. The wave heights increase up to more than twice the incident wave height. The £jtudy has pointed out that this effect, which is usually called MACH-REFLECTION, is not to be taken as an analogy to gas dynamics, but should be interpreted as a diffraction problem.

Fully nonlinear internal waves in a two-fluid system

1999 ◽  
Vol 396 ◽  
pp. 1-36 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
ROBERTO CAMASSA
Keyword(s):  
Solitary Wave ◽  
Euler Equations ◽  
Deep Water ◽  
Large Amplitude ◽  
Nonlinear Models ◽  
Weak Nonlinearity ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Waves ◽  
Two Fluid

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

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.

Vorticity effects on nonlinear wave–current interactions in deep water

2015 ◽  
Vol 778 ◽  
pp. 314-334 ◽  
Author(s):  
R. M. Moreira ◽  
J. T. A. Chacaltana
Keyword(s):  
Shear Flow ◽  
Nonlinear Wave ◽  
Deep Water ◽  
Water Waves ◽  
Wave Breaking ◽  
Surface Profile ◽  
Boundary Integral ◽  
Progressive Waves ◽  
Wave Blocking ◽  
Deep Water Waves

The effects of uniform vorticity on a train of ‘gentle’ and ‘steep’ deep-water waves interacting with underlying flows are investigated through a fully nonlinear boundary integral method. It is shown that wave blocking and breaking can be more prominent depending on the magnitude and direction of the shear flow. Reflection continues to occur when sufficiently strong adverse currents are imposed on ‘gentle’ deep-water waves, though now affected by vorticity. For increasingly positive values of vorticity, the induced shear flow reduces the speed of right-going progressive waves, introducing significant changes to the free-surface profile until waves are completely blocked by the underlying current. A plunging breaker is formed at the blocking point when ‘steep’ deep-water waves interact with strong adverse currents. Conversely negative vorticities augment the speed of right-going progressive waves, with wave breaking being detected for strong opposing currents. The time of breaking is sensitive to the vorticity’s sign and magnitude, with wave breaking occurring later for negative values of vorticity. Stopping velocities according to nonlinear wave theory proved to be sufficient to cause wave blocking and breaking.

scholarly journals Theoretical and experimental investigations of wave field around vessel in shallow water

2020 ◽  
pp. 72-89
Author(s):  
Н.В. Єфремова ◽  
A.Є. Нильва ◽  
Н.Н. Котовська ◽  
М.В. Дрига
Keyword(s):  
Shallow Water ◽  
Deep Water ◽  
Theoretical Data ◽  
Experimental Investigations ◽  
Arbitrary Angle ◽  
Actual Problem ◽  
Transformation Zone ◽  
Non Linear ◽  
The Waves ◽  
3D Waves

Un-running vessel at the shallow-water road anchorage is under exposure to waves that come at arbitrary angle from the high sea. 3D waves from deep-sea area become practically 2D when entering shallow water. While mean periods are kept, waves become shorter and their crests become higher and sharpener than for deep-water ones. As a result of diffraction of waves that come from the deep-water sea at the vessel, a transformation zone appears where waves become 3D again. Dimensions of the waves’ transformation zone, character and height of waves in this zone specify safety of auxiliary crafts, e.g. tugboats, bunker vessels, pilot and road crafts, oil garbage collectors and boom crafts. In the complex 3D waves the trajectory of auxiliary vessel’s movement has to be safe, vessel’s motions have to be moderate. Besides waves’ height is one of the parameters that are used for forecast of movement of spilled oil. Last years the biggest part examination of waves’ problems was devoted to estimation of waves’ impact onto stationary or floating shelf facilities. For validity estimation, waves’ characteristics defined due to different theories, are compared with experimental ones. But characteristics of the waves around shelf facilities are hardly able to be compared to same ones of waves around bodies with vessel-type shape.  At the experiments with vessels’ models, waves’ impact onto vessel was examined, but not the transformation of the waves themselves. So, comparing of waves area’s characteristics defined by both theoretical experimental ways is an actual problem.  Aim of the paper is verification of results of wave area investigation; wave area is located around a vessel that is exposed of arbitrary angle waves at shallow water conditions. Description of experimental investigations of transformed waves in the towing tank is done; transformation zone appears around vessel’s model while running waves diffract on it. Distribution of waves’ amplitudes at the designated points was fixed by the special designed and manufactured unit. Experimental data is compared with computation results both of linear and non-linear theories. It was assumed that experimental results and theoretical data satisfactory meet each other; also that non-linear computations define the maximal values of waves’ amplitudes at all cases.

Stability of weakly nonlinear deep-water waves in two and three dimensions

1981 ◽  
Vol 105 (-1) ◽  
pp. 177 ◽  
Author(s):  
Donald R. Crawford ◽  
Bruce M. Lake ◽  
Philip G. Saffman ◽  
Henry C. Yuen
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Three Dimensions ◽  
Weakly Nonlinear ◽  
Deep Water Waves

Some effects of surface tension on steep water waves. Part 2

1980 ◽  
Vol 96 (3) ◽  
pp. 417-445 ◽  
Author(s):  
S. J. Hogan
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Expansion Method ◽  
Capillary Waves ◽  
Maximum Wave Height ◽  
Pure Gravity ◽  
Wave Profiles ◽  
The Waves ◽  
Wave Properties

This paper continues an investigation of the effects of surface tension on steep water waves in deep water begun in Hogan (1979a). A Stokes-type expansion method is given which can be applied to most wavelengths. For capillary waves (2 cm or less) it is found that the surface of the highest wave encloses a bubble of air, as was found for pure capillary waves by Crapper (1957). For intermediate waves (20 cm) the wave profiles are similar to those of pure gravity waves and the wave properties increase monotonically. For gravity waves (200 cm) the wave properties all exhibit a maximum just short of the maximum wave height obtained by the method. The integral properties for all the waves are drawn and given in numerical form in the appendix.

scholarly journals Equations for Deep Water Counter Streaming Waves and New Integrals of Motion

Fluids ◽  
2019 ◽  
Vol 4 (1) ◽  
pp. 47 ◽  
Author(s):  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Water Waves ◽  
Integrals Of Motion ◽  
Wave Interactions ◽  
The Right ◽  
The Waves

The waves on a free surface of 2D deep water can be split into two groups: the waves moving to the right, and the waves moving to the left. A specific feature of the four-wave interactions of water waves allows to describe the evolution of the two groups as a system of two equations. The fundamental consequence of this decomposition is the conservation of the “number of waves” in each particular group. The envelope approximation for the waves in each group of counter streaming waves is obtained.

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.

scholarly journals On the formation of water waves by wind (second paper)

1926 ◽  
Vol 110 (754) ◽  
pp. 241-247 ◽  
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Deep Water ◽  
Water Waves ◽  
Velocity Potential ◽  
Systematic Difference ◽  
Small Extent ◽  
Short Waves ◽  
First Approximation ◽  
The Waves

In a previous paper I investigated the problem of the formation of waves on deep water by wind, and found that the available data were consistent with the hypothesis that the growth of the waves is due principally to a systematic difference between the pressures of the air on the front and rear slopes. Lamb had already discussed the maintenance of waves against viscosity by an approximate method, but without obtaining numerical results. Being under the incorrect impression that Lamb’s approximation would not hold for the short waves I was chiefly considering, I proceeded on more elaborate lines. It now appears, however, that Lamb’s method is not only applicable to the problem of waves on deep water, but is readily extended to cover the case when the water is comparatively shallow, and to allow for surface tension. The fundamental approximations are first, the usual one that squares of the displacements from the steady state can be neglected, and second, that viscosity modifies the motion of the water to only a small extent. The motion of the water can then, to a first approximation, be considered as irrotational. With the previous notation, let ζ be the elevation of the free surface x, y, z the position co-ordinates, t the time, U the undisturbed velocity of the water, h the depth, and φ the velocity potential. Also let σ, p, q , and ϑ denote respectively ∂/∂ t , ∂/∂ x , ∂/∂ y , and ∂/∂ z , and write p 2 + q 2 = - r 2 .

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