Two-dimensional steady edge waves. Part II: Solitary waves

Wave Motion ◽  
2009 ◽  
Vol 46 (6) ◽  
pp. 372-378 ◽  
Author(s):  
Mats Ehrnström ◽  
Joachim Escher ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Solitary Waves ◽  
Two Dimensional ◽  
Edge Waves

Comparison principles for free-surface flows with gravity

1991 ◽  
Vol 230 ◽  
pp. 231-243 ◽  
Author(s):  
Walter Craig ◽  
Peter Sternberg
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Immiscible Fluids ◽  
Steady Flows ◽  
Two Dimensional ◽  
Surface Flows ◽  
Ship Hulls ◽  
Geometrical Information ◽  
Supercritical Flows

This article considers certain two-dimensional, irrotational, steady flows in fluid regions of finite depth and infinite horizontal extent. Geometrical information about these flows and their singularities is obtained, using a variant of a classical comparison principle. The results are applied to three types of problems: (i) supercritical solitary waves carrying planing surfaces or surfboards, (ii) supercritical flows past ship hulls and (iii) supercritical interfacial solitary waves in systems consisting of two immiscible fluids.

scholarly journals The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1586
Author(s):  
Yury Stepanyants
Keyword(s):  
Solitary Waves ◽  
Asymptotic Theory ◽  
Deep Ocean ◽  
Two Dimensional ◽  
Asymptotic Approach ◽  
Suggested Approach ◽  
Symmetric Wave ◽  
Petviashvili Equation ◽  
Wave Patterns ◽  
Symmetric Soliton

The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

scholarly journals On the Dynamics of Two-Dimensional Capillary-Gravity Solitary Waves with a Linear Shear Current

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Dali Guo ◽  
Bo Tao ◽  
Xiaohui Zeng
Keyword(s):  
Solitary Waves ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Study ◽  
Two Dimensional ◽  
Shear Current ◽  
Linear Shear ◽  
The Stability ◽  
Depression Wave ◽  
Elevation Wave

The numerical study of the dynamics of two-dimensional capillary-gravity solitary waves on a linear shear current is presented in this paper. The numerical method is based on the time-dependent conformal mapping. The stability of different kinds of solitary waves is considered. Both depression wave and large amplitude elevation wave are found to be stable, while small amplitude elevation wave is unstable to the small perturbation, and it finally evolves to be a depression wave with tails, which is similar to the irrotational capillary-gravity waves.

scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

2009 ◽  
Vol 465 (2109) ◽  
pp. 2725-2749 ◽  
Author(s):  
M. J. Ablowitz ◽  
T. S. Haut
Keyword(s):  
Surface Tension ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Asymptotic Series ◽  
High Order ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Small Surface ◽  
Petviashvili Equation ◽  
Dimensional Series

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

A note on tuning in Roseau's alternative edge waves

1999 ◽  
Vol 382 ◽  
pp. 245-262 ◽  
Author(s):  
ULF TORSTEN EHRENMARK
Keyword(s):  
Continuous Spectrum ◽  
Two Dimensional ◽  
Edge Waves ◽  
Modal Number ◽  
Stagnation Points ◽  
Bounded State ◽  
Wave Guides ◽  
Recent Method ◽  
Wave Profiles ◽  
Standing Edge Waves

Ursell's edge waves are derived systematically using a new method. Computed profiles are then compared with the lesser known shoreline singular waves first constructed by Roseau (1958). A recent method of writing the continuous spectrum solutions on a plane beach is thereby extended to the discrete spectrum to enable the reconstruction of both types of edge waves so that, in particular, the unbounded wave profiles are more easily computed. The existence of stagnation points on the bed for standing edge waves is considered and demonstrated for the first few modes. A ramification of this is the existence of (two-dimensional-cross-shore) dividing ‘streamlines’ from the bed to the surface also, the number of which appears to equate to the modal number of the edge wave. These dividing streamlines, along with other streamlines, are computed for the first few modes of both the Ursell and the (alternative) singular waves constructed by Roseau.It follows that these waves can also exist in the presence of solid cylinders bounded by fixed streamlines and, in particular therefore, that the hitherto unbounded Roseau waves can exist in a bounded state since a region including the origin can be removed from the flow by exploiting a dividing streamline. It is confirmed that the wavenumbers of the Roseau waves interlace those of the Ursell waves. An examination of available evidence leaves open to further research the question of whether the alternative Roseau waves have been ‘inadvertently’ observed either in the laboratory or, by means of contamination of data, in the field. Further laboratory simulations using longshore solid cylinders as ‘wave guides’ are proposed.

scholarly journals Dynamics of steep two-dimensional gravity–capillary solitary waves

2010 ◽  
Vol 664 ◽  
pp. 466-477 ◽  
Author(s):  
PAUL A. MILEWSKI ◽  
J.-M. VANDEN-BROECK ◽  
ZHAN WANG
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Infinite Depth ◽  
Linear Dispersion Relation ◽  
Dimensional Gravity ◽  
Nonlinear Free Surface ◽  
The Waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

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