scholarly journals Oblique wave groups in deep water

1984 ◽  
Vol 146 ◽  
pp. 1-20 ◽  
Author(s):  
P. J. Bryant
Keyword(s):  
Deep Water ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Wave Pattern ◽  
Periodic Wave ◽  
Two Dimensions ◽  
Wave Groups ◽  
Oblique Wave ◽  
Parallel Lines ◽  
The Waves

Oblique wave groups consist of waves whose straight parallel lines of constant phase are oblique to the straight parallel lines of constant group phase. Numerical solutions for periodic oblique wave groups with envelopes of permanent shape are calculated from the equations for irrotational three-dimensional deep-water motion with nonlinear upper free-surface conditions. Two distinct families of periodic wave groups are found, one in which the waves in each group are in phase with those in all other groups, and the other in which there is a phase difference of π between the waves in consecutive groups. It is shown that some analytical solutions for oblique wave groups calculated from the nonlinear Schrödinger equation are in error because they ignore the resonant forcing of certain harmonics in two dimensions. Particular attention is given to oblique wave groups whose group-to-wave angle is in the neighbourhood of the critical angle tan−1√½, corresponding to waves on the boundary wedge of the Kelvin ship-wave pattern.

Drifting Rogue Packets

2018 ◽  
Author(s):  
Amin Chabchoub ◽  
Norbert Hoffmann ◽  
Nail Akhmediev ◽  
Takuji Waseda
Keyword(s):  
Modulation Instability ◽  
Wave Train ◽  
Modulation Frequency ◽  
Phase Speed ◽  
Periodic Wave ◽  
Unstable Wave ◽  
Extreme Waves ◽  
Wave Groups ◽  
The Waves ◽  
Good Agreement

Modulation instability (MI) is one possible mechanism to explain the formation of extreme waves in uni-directional and narrow-banded seas. It can be triggered, when side-bands around the main frequency are excited and subsequently follow an exponential growth. In physical domain this dynamics translates to periodic pulsations of wave groups that can reach heights up to three times the initial amplitude of the wave train. It is well-known that these periodic wave groups propagate with approximately half the waves phase speed in deep-water. We report an experimental study on modulationally unstable wave groups that propagate with a velocity that is higher than the group velocity since the modulation frequency is complex. It is shown that when this additional velocity to the wave groups is small a good agreement with exact nonlinear Schrödinger (NLS) models, that describe the nonlinear stage of MI, is reached. Otherwise a significant deviation is observed that could be compensated when increasing accuracy of the water wave modeling beyond NLS.

Progressive internal waves on slopes

1969 ◽  
Vol 35 (1) ◽  
pp. 131-144 ◽  
Author(s):  
Carl Wunsch
Keyword(s):  
Boundary Layers ◽  
Internal Waves ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Inhomogeneous Waves ◽  
Usual Manner ◽  
Propagating Waves ◽  
Viscous Boundary ◽  
The Waves ◽  
Interior Solutions

The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Väisälä frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.

scholarly journals REFLECTION OF IRREGULAR WAVES AT PARTIALLY REFLECTING STRUCTURES INCLUDING OBLIQUE WAVE APPROACH

1986 ◽  
Vol 1 (20) ◽  
pp. 162 ◽  
Author(s):  
Hans-Joachim Scheffer ◽  
Soren Kohlhase
Keyword(s):  
Reflection Coefficient ◽  
Three Dimensional ◽  
Diffraction Theory ◽  
Wave Pattern ◽  
Irregular Waves ◽  
Breaking Wave ◽  
Oblique Wave ◽  
Wave Flume ◽  
Wave Approach ◽  
Wave Basin

The reflection of irregular seas is increasingly considered in coastal engineering and harbour design as well with respect to wave pattern at the structure and energy dissipation as regarding the dimensioning of structures exposed to waves. It becomes evident that the three-dimensional sea state (oblique wave approach, irregularity of the waves) at partially-reflecting structures of a complex design cannot be described by means of a constant reflection coefficient alone, as is common practice. This is due to the fact that the coefficient is largely frequency-dependent and the physically effective reflection point of the structure cannot be clearly specified. In the light of this, basic investigations on wave reflection have been performed with different partially-reflecting structures, wave spectra and wave approach angles. In addition to laboratory experiments using both a wave flume and a wave basin, a theoretical solution based on diffraction theory was determined to describe the wave field in the reflection area of various structures. The investigations were restricted to non-breaking wave conditions. The reflection behaviour of structures is expressed by a complex reflection coefficient, containing two parameters, which have to be determined by model tests.

Three-dimensional wave patterns generated by moving disturbances at transcritical speeds

1988 ◽  
Vol 196 ◽  
pp. 39-63 ◽  
Author(s):  
Geir Pedersen
Keyword(s):  
Numerical Solutions ◽  
Three Dimensional ◽  
Stationary Wave ◽  
Wave Pattern ◽  
Implicit Difference Scheme ◽  
Pressure Fields ◽  
Time Simulation ◽  
Wave Patterns ◽  
Long Time ◽  
Radiation Conditions

Disturbances in the form of pressure fields, source distributions and time-dependent bottom topographies are discussed and found to produce similar wave patterns. Results obtained for wide channels are discussed in the light of the features of soliton reflection at a wall. Comparison with experiments shows excellent agreement. The introduction of radiation conditions enables long-time simulation of the development of wave patterns in infinite and semi-infinite fluids. A stationary wave pattern is also found to emerge for slightly supercritical Froude numbers, but contrary to linear results the leading divergent waves may originate ahead of the disturbance. This behaviour is due to nonlinear interactions similar to those governing collisions between solitons. This study on wave generation by a moving disturbance is based on numerical solutions of Boussinesq-type equations. The equations in their most general form are integrated by an implicit difference method. Strongly supercritical cases are described by a simplified set of equations which is solved by a semi-implicit difference scheme.

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 Electromagnetic Waves in Annular Regions

2020 ◽  
Vol 10 (5) ◽  
pp. 1780
Author(s):  
Daniele Funaro
Keyword(s):  
Electromagnetic Waves ◽  
Three Dimensional ◽  
Periodic Wave ◽  
Vortex Rings ◽  
Periodic Wave Solutions ◽  
Special Cases ◽  
Time Periodic Solutions ◽  
The Waves ◽  
Analytical Expressions ◽  
Time Periodic

In suitable bounded regions immersed in vacuum, time periodic wave solutions solving a full set of electrodynamics equations can be explicitly computed. Analytical expressions are available in special cases, whereas numerical simulations are necessary in more complex situations. The attention here is given to selected three-dimensional geometries, which are topologically equivalent to a toroid, where the behavior of the waves is similar to that of fluid-dynamics vortex rings. The results show that the shape of the sections of these rings depends on the behavior of the eigenvalues of a certain elliptic differential operator. Time-periodic solutions are obtained when at least two of such eigenvalues attain the same value. The solutions obtained are discussed in view of possible applications in electromagnetic whispering galleries or plasma physics.

scholarly journals A new type of three-dimensional deep-water wave of permanent form

1980 ◽  
Vol 101 (4) ◽  
pp. 797-808 ◽  
Author(s):  
Philip G. Saffman ◽  
Henry C. Yuen
Keyword(s):  
Deep Water ◽  
Wave Train ◽  
Three Dimensional ◽  
Wave Pattern ◽  
Weakly Nonlinear ◽  
New Class ◽  
Wave Heights ◽  
New Type ◽  
Deep Water Wave ◽  
Permanent Form

A new class of three-dimensional, deep-water gravity waves of permanent form has been found using an equation valid for weakly nonlinear waves due to Zakharov (1968). These solutions appear as bifurcations from the uniform two-dimensional wave train. The critical wave heights are given as functions of the modulation wave vector. The three-dimensional patterns may be skewed or symmetrical. An example of the skewed wave pattern is given and shown to be stable. The results become exact in the limit of very oblique modulations.

Analysis of singular inertial modes in a spherical shell: the slender toroidal shell model

2002 ◽  
Vol 463 ◽  
pp. 345-360 ◽  
Author(s):  
M. RIEUTORD ◽  
L. VALDETTARO ◽  
B. GEORGEOT
Keyword(s):  
Spherical Shell ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Toroidal Shell ◽  
Perturbative Approach ◽  
Quadratic Potential ◽  
Ekman Number ◽  
Damping Rate ◽  
Thin Spherical Shell

We derive the asymptotic spectrum (as the Ekman number E → 0) of axisymmetric inertial modes when the problem is restricted to two dimensions. We show that the damping rate of such modes scales with the square root of the Ekman number and that the width of the shear layers of the eigenfunctions scales with E1/4. The eigenfunctions obey a Schrödinger equation with a quadratic potential; we provide the analytical expression for eigenvalues (frequency and damping rate). These results validate the picture that attractors act like a potential well, trapping inertial waves which resist confinement owing to viscosity. Using three-dimensional numerical solutions, we show that the results can be applied to equatorially trapped modes in a thin spherical shell; in fact, these two-dimensional solutions give the first step (the zeroth order) of a perturbative approach to three-dimensional solutions in a spherical shell. Our method is applicable in a straightforward way to any other container where bi-dimensionality dominates.

Frequency downshift in three-dimensional wave trains in a deep basin

1997 ◽  
Vol 352 ◽  
pp. 359-373 ◽  
Author(s):  
KARSTEN TRULSEN ◽  
KRISTIAN B. DYSTHE
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Wave Breaking ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Deep Basin ◽  
Weakly Nonlinear ◽  
Wave Trains ◽  
Dimensional Wave

The conservative evolution of weakly nonlinear narrow-banded gravity waves in deep water is investigated numerically with a modified nonlinear Schrödinger equation, for application to wide wave tanks. When the evolution is constrained to two dimensions, no permanent shift of the peak of the spectrum is observed. In three dimensions, allowing for oblique sideband perturbations, the peak of the spectrum is permanently downshifted. Dissipation or wave breaking may therefore not be necessary to produce a permanent downshift. The emergence of a standing wave across the tank is also predicted.

scholarly journals Interaction of near-inertial waves with an anticyclonic vortex

2021 ◽  
Author(s):  
Hossein A. Kafiabad ◽  
Jacques Vanneste ◽  
William R. Young
Keyword(s):  
Numerical Solutions ◽  
Energy Levels ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Kinetic Energy Density ◽  
Nonlinear Feedback ◽  
Inertial Waves ◽  
Energy Signature ◽  
Averaged Model ◽  
The Waves

AbstractAnticyclonic vortices focus and trap near-inertial waves so that near-inertial energy levels are elevated within the vortex core. Some aspects of this process, including the nonlinear modification of the vortex by the wave, are explained by the existence of trapped near-inertial eigenmodes. These vortex eigenmodes are easily excited by an initialwave with horizontal scale much larger than that of the vortex radius. We study this process using a wave-averaged model of near-inertial dynamics and compare its theoretical predictions with numerical solutions of the three-dimensional Boussinesq equations. In the linear approximation, the model predicts the eigenmode frequencies and spatial structures, and a near-inertial wave energy signature that is characterized by an approximately time-periodic, azimuthally invariant pattern. The wave-averaged model represents the nonlinear feedback of the waves on the vortex via a wave-induced contribution to the potential vorticity that is proportional to the Laplacian of the kinetic energy density of the waves. When this is taken into account, the modal frequency is predicted to increase linearly with the energy of the initial excitation. Both linear and nonlinear predictions agree convincingly with the Boussinesq results.

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