Asymptotic theory for the almost-highest solitary wave

1996 ◽  
Vol 317 ◽  
pp. 1-19 ◽  
Author(s):  
M. S. Longuet-Higgins ◽  
M. J. H. Fox
Keyword(s):  
Solitary Wave ◽  
Wave Height ◽  
Solitary Waves ◽  
Deep Water ◽  
Asymptotic Theory ◽  
Finite Depth ◽  
Periodic Waves ◽  
Wave Parameters ◽  
Asymptotic Formulae ◽  
Energy And Momentum

The behaviour of the energy in a steep solitary wave as a function of the wave height has a direct bearing on the breaking of solitary waves on a gently shoaling beach. Here it is shown that the speed, energy and momentum of a steep solitary wave in water of finite depth all behave in an oscillatory manner as functions of the wave height and as the limiting height is approached. Asymptotic formulae for these and other wave parameters are derived by means of a theory for the ‘almost-highest wave’ similar to that formulated previously for periodic waves in deep water (Longuet-Higgins & Fox 1977, 1978). It is demonstrated that the theory fits very precisely some recent calculations of solitary waves by Tanaka (1995).

Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

2004 ◽  
Vol 34 (12) ◽  
pp. 2774-2791 ◽  
Author(s):  
Roger Grimshaw ◽  
Efim Pelinovsky ◽  
Tatiana Talipova ◽  
Audrey Kurkin
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Asymptotic Theory ◽  
Vries Equation ◽  
Internal Solitary Waves ◽  
Variable Depth ◽  
North West ◽  
Wave Parameters ◽  
De Vries

Abstract Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg–de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg–de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg–de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

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.

LARGE SCALE EXPERIMENTS ON EVOLUTION AND RUN-UP OF BREAKING SOLITARY WAVES

2007 ◽  
Vol 01 (03) ◽  
pp. 257-272 ◽  
Author(s):  
KAO-SHU HWANG ◽  
YU-HSUAN CHANG ◽  
HWUNG-HWENG HWUNG ◽  
YI-SYUAN LI
Keyword(s):  
Solitary Wave ◽  
Wave Height ◽  
Solitary Waves ◽  
Water Depth ◽  
Large Scale ◽  
Scale Effects ◽  
Wave Heights ◽  
Run Up ◽  
Hydraulics Laboratory ◽  
The Waves

The evolution and run-up of breaking solitary waves on plane beaches are investigated in this paper. A series of large-scale experiments were conducted in the SUPER TANK of Tainan Hydraulics Laboratory with three plane beaches of slope 0.05, 0.025 and 0.017 (1:20, 1:40 and 1:60). Solitary waves of which relative wave heights, H/h0, ranged from 0.03 to 0.31 were generated by two types of wave-board displacement trajectory: the ramp-trajectory and the solitary-wave trajectory proposed by Goring (1979). Experimental results show that under the same relative wave height, the waveforms produced by the two generation procedures becomes noticeably different as the waves propagate prior to the breaking point. Meanwhile, under the same relative wave height, the larger the constant water depth is, the larger the dimensionless run-up heights would be. Scale effects associated with the breaking process are discussed.

Mass transport in gravity waves

1953 ◽  
Vol 49 (1) ◽  
pp. 145-150 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Solitary Wave ◽  
Mass Transport ◽  
Gravity Waves ◽  
Deep Water ◽  
Wave Motion ◽  
Finite Depth ◽  
Particle Displacement ◽  
Slow Drift

ABSTRACTA proof is given of the existence of a slow drift associated with the passage of gravity waves over the surface of a frictionless fluid when the wave motion is irrotational. The drift is in the direction of propagation and is greatest near the surface, decreasing steadily towards the bottom, where it may be negative if the fluid is of finite depth. The particle displacement due to a solitary wave has similar properties. The proof confirms the approximate calculations of Stokes (6) and extends the proof of Rayleigh applicable only to deep water.

Oblique runup of non-breaking solitary waves on an inclined plane

2011 ◽  
Vol 668 ◽  
pp. 582-606 ◽  
Author(s):  
GEIR K. PEDERSEN
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Evolution Equations ◽  
Boussinesq Equations ◽  
Normal Incidence ◽  
Finite Depth ◽  
Breaking Waves ◽  
Wave Pattern ◽  
Angle Of Incidence ◽  
Inclined Plane

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.

The effect of the induced mean flow on solitary waves in deep water

1998 ◽  
Vol 355 ◽  
pp. 317-328 ◽  
Author(s):  
T. R. AKYLAS ◽  
F. DIAS ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Phase Speed ◽  
Mean Flow ◽  
Nls Equation ◽  
Envelope Soliton

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

scholarly journals Energetics of internal solitary waves in a background sheared current

2010 ◽  
Vol 17 (5) ◽  
pp. 553-568 ◽  
Author(s):  
K. G. Lamb
Keyword(s):  
Solitary Wave ◽  
Surface Waves ◽  
Solitary Waves ◽  
Deep Water ◽  
Pressure Change ◽  
Boussinesq Approximation ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Deep Basin ◽  
Internal Seiche

Abstract. The energetics of internal waves in the presence of a background sheared current is explored via numerical simulations for four different situations based on oceanographic conditions: the nonlinear interaction of two internal solitary waves; an internal solitary wave shoaling through a turning point; internal solitary wave reflection from a sloping boundary and a deep-water internal seiche trapped in a deep basin. In the simulations with variable water depth using the Boussinesq approximation the combination of a background sheared current, bathymetry and a rigid lid results in a change in the total energy of the system due to the work done by a pressure change that is established across the domain. A final simulation of the deep-water internal seiche in which the Boussinesq approximation is not invoked and a diffuse air-water interface is added to the system results in the energy remaining constant because the generation of surface waves prevents the establishment of a net pressure increase across the domain. The difference in the perturbation energy in the Boussinesq and non-Boussinesq simulations is accounted for by the surface waves.

scholarly journals A Preliminary Laboratory Study of Motion of Floating Debris Generated by Solitary Waves Running up a Beach

2014 ◽  
Vol 08 (03) ◽  
pp. 1440006 ◽  
Author(s):  
Yao Yao ◽  
Zhenhua Huang ◽  
Edmond Y. M. Lo ◽  
Hung-Tao Shen
Keyword(s):  
Solitary Wave ◽  
Wave Height ◽  
Solitary Waves ◽  
Water Depth ◽  
Laboratory Study ◽  
Current Understanding ◽  
Final Position ◽  
Coastal Structures ◽  
The Difference

Destructive tsunamis can destroy coastal structures and move huge amounts of tsunami debris. Our current understanding of motion of tsunami debris in tsunami flows is limited. In this paper, we present a preliminary laboratory study of motion of model debris under the action of solitary waves running up a beach. The difference between the waterline of maximum inundation and the final position of debris was examined under various conditions. Effects of solitary wave height, water depth, and the distance of debris source to the shoreline on the maximum inundation, the debris limit, and the final position of debris were examined. In general, the final positions of the debris are different from the waterline at maximum inundation and there is a low possibility that a large amount of debris can be carried by retreating water offshore into the sea.

scholarly journals THE EXACT SOLUTION OF THE HIGHEST WAVE DERIVED FROM A UNIVERSAL WAVE MODEL

1984 ◽  
Vol 1 (19) ◽  
pp. 70
Author(s):  
Yang Yih Chen ◽  
Frederick L.W. Tang
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Gravity Wave ◽  
Wave Height ◽  
Water Depth ◽  
Series Solution ◽  
Wave Model ◽  
Finite Depth

The solitary wave is first established in this paper by extending the series solution of periodic gravity wave as the wavelength approaches to infinite. Then, the highest gravity wave of permanent type in finite depth of water is immediately analyzed. The maximum ratio of wave height to water depth is obtained as 0.85465')..., and the angle at the crest for the considered highest wave is estimated to be 90°.

Wave Instability in Finite Depths

2014 ◽  
Author(s):  
Alexander V. Babanin ◽  
Kevin Ewans
Keyword(s):  
Deep Water ◽  
Modulational Instability ◽  
Finite Depth ◽  
Rogue Waves ◽  
Design Criterion ◽  
Extreme Wave ◽  
Research Attention ◽  
Practical Applications ◽  
Wave Heights ◽  
Energy And Momentum

Waves in finite-depth and shallow water represent an important and challenging research topic, both from practical and academic perspective. Most of practical applications related to the ocean are conducted in coastal areas, but most of research attention has been dedicated to the surface waves in deep water. Shallow areas are effectively a different physical environment, where wave kinematics change, waves become steeper and consequently more nonlinear but less or non-dispersive, and as a result their nonlinear behaviors change dramatically, waves directly interact with the bottom, in a number of different ways, and most importantly release their energy and momentum through intensive and extensive breaking. In the paper, modulational instability in finite depths will be discussed. This mechanism is regarded responsible for rogue waves, and hence extreme wave heights in deep water. Extreme wave heights are a key design criterion for the ocean industry, including shallow environments too. It is believed, however, that wave breaking rather than wave instabilities is the primary mechanism that controls the maximum possible wave height in depth-limited environments. Here, data of the LoWish JIP will be used in order to identify the criteria for transition from modulational instability being active in the deep-water environments to being suppressed in the shallow waters.

Sign in / Sign up

Export Citation Format

Share Document