The runup of solitary waves

1987 ◽  
Vol 185 ◽  
pp. 523-545 ◽  
Author(s):  
Costas Emmanuel Synolakis
Keyword(s):  
Solitary Wave ◽  
Linear Theory ◽  
Solitary Waves ◽  
Nonlinear Theory ◽  
Laboratory Experiments ◽  
Breaking Waves ◽  
Asymptotic Result ◽  
Approximate Theory ◽  
Sloping Beach

This is a study of the runup of solitary waves on plane beaches. An approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves. A series of laboratory experiments is described to support the theory. It is shown that the linear theory predicts the maximum runup satisfactorily, and that the nonlinear theory describes the climb of solitary waves equally well. Different runup regimes are found to exist for the runup of breaking and non-breaking waves. A breaking criterion is derived for determining whether a solitary wave will break as it climbs up a sloping beach, and a different criterion is shown to apply for determining whether a wave will break during rundown. These results are used to explain some of the existing empirical runup relationships.

scholarly journals ON THE MAXIMUM RUNUP OF CNOIDAL WAVES

1988 ◽  
Vol 1 (21) ◽  
pp. 39
Author(s):  
Costas Emmanuel Synolakis ◽  
Manas Kumar
Keyword(s):  
Solitary Waves ◽  
Nonlinear Theory ◽  
Laboratory Experiments ◽  
Laboratory Data ◽  
Numerical Results ◽  
Asymptotic Result ◽  
Approximate Theory ◽  
Cnoidal Waves ◽  
Linear And Nonlinear ◽  
Monochromatic Waves

This is a study of the maximum runup of cnoidal waves on plane beaches. An approximate theory is described for determining the maximum runup of nonbreaking cnoidal waves. It is shown that the linear and nonlinear theory predict mathematically identical maximum runup heights. An asymptotic result is derived for the maximum runup of solitary waves, which are one limiting form of cnoidal waves. A series of laboratory experiments is described to support the theory. Other numerical results are presented that suggest that the runup of cnoidal waves is significantly higher than the runup of monochromatic waves with the same waveheight and wavelength. Preliminary laboratory data are also presented which suggest that, for certain cnoidal waves, the maximum runup is not a monotonically varying function of the normalized wavelength.

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.

On the vertical structure of internal solitary waves in the northeastern South China Sea

2021 ◽  
Author(s):  
Gong Yi ◽  
Song Haibin ◽  
Zhao Zhongxiang ◽  
Guan Yongxian ◽  
Kuang Yunyan
Keyword(s):  
South China Sea ◽  
Structure Function ◽  
Linear Theory ◽  
Solitary Waves ◽  
South China ◽  
Nonlinear Theory ◽  
Seismic Data ◽  
Vertical Structure ◽  
Internal Solitary Waves ◽  
First Order

<p>Internal solitary waves (ISWs) make important contributions to energy cascade, ocean mixing and material transport in the ocean. However, there are few observational studies on the vertical structure of ISWs. The high-spatial resolution of seismic data enables us to obtain clear internal structure image of ISWs, so we can conduct a detailed research on their vertical structure. In this article, we report 11 ISWs near Dongsha Atoll in the South China Sea using two-dimensional seismic data.</p><p>We first extracted the amplitudes of ISW from seismic section, and obtained a series of discrete amplitude points. Then, the least-squares spline fitting was used to fit these amplitude points into a vertical structure curve. We calculated vertical structures by linear theory and first-order nonlinear theory, respectively, and compared the observed vertical structure with the two theories. We found that three ISWs conform to the linear vertical structure function, four ISWs conform to the first-order nonlinear vertical structure function, and four ISWs do not conform to the two theories. In order to figure out the reason why the observation did not conform to the theories, we decomposed the fitted vertical structures of these four ISWs by the empirical mode decomposition (EMD) algorithm, and compare the residuals of decomposition with the two theories. The results showed that the residuals of two ISWs are in agreement with the linear vertical structure function, the residual of one ISW conforms to the first-order nonlinear vertical structure function, and one residual of ISW still cannot conform to the two theories. We calculated key parameters of these ISWs to analyze the reasons for difference between observation and theory.</p><p>In summary, we found that the shape of vertical structure is mainly determined by nonlinearity. The vertical structure with low degree nonlinearity can be described by linear theory, while ISW with high degree nonlinearity conform to the first-order nonlinear theory. Besides, for an ISW with large amplitude propagating in shallow water, its vertical structure is more susceptible to be affected by the topography. Moreover, the background flow can also affect the vertical structure. We found an ISW was passing through an eddy which was trapped near seafloor, and resulted in the bottom of vertical structure decayed rapidly.</p>

Comment on ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’

2014 ◽  
Vol 80 (3) ◽  
pp. 513-516
Author(s):  
Frank Verheest
Keyword(s):  
Shock Wave ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Plasma Phys ◽  
Theoretical Underpinning ◽  
Pair Plasma ◽  
Electrons And Positrons ◽  
Small Dissipation ◽  
Pseudopotential Approach ◽  
Wave Properties

In a recent paper ‘Propagation of solitary waves and shock wavelength in the pair plasma (J. Plasma Phys. 78, 525–529, 2012)’, Malekolkalami and Mohammadi investigate nonlinear electrostatic solitary waves in a plasma comprising adiabatic electrons and positrons, and a stationary ion background. The paper contains two parts: First, the solitary wave properties are discussed through a pseudopotential approach, and then the influence of a small dissipation is intuitively sketched without theoretical underpinning. Small dissipation is claimed to lead to a shock wave whose wavelength is determined by linear oscillator analysis. Unfortunately, there are errors and inconsistencies in both the parts, and their combination is incoherent.

scholarly journals Existence and linearized stability of solitary waves for a quasilinear Benney system

2016 ◽  
Vol 146 (3) ◽  
pp. 547-564 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira ◽  
Filipe Oliveira
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Compact Support ◽  
Travelling Waves ◽  
Standing Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Linearized Stability ◽  
Stability Of Solitary Waves ◽  
Image Position

We prove the existence of solitary wave solutions to the quasilinear Benney systemwhere , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

SOLITARY WAVES OF THE NONLINEAR KLEIN-GORDON EQUATION COUPLED WITH THE MAXWELL EQUATIONS

2002 ◽  
Vol 14 (04) ◽  
pp. 409-420 ◽  
Author(s):  
VIERI BENCI ◽  
DONATO FORTUNATO FORTUNATO
Keyword(s):  
Electromagnetic Field ◽  
Solitary Wave ◽  
Electric Field ◽  
Solitary Waves ◽  
Maxwell Equations ◽  
Gordon Equation ◽  
Klein Gordon ◽  
Klein Gordon Equation

This paper is divided in two parts. In the first part we construct a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field. In the second part we study the electrostatic case. We prove the existence of infinitely many pairs (ψ, E), where ψ is a solitary wave for the nonlinear Klein-Gordon equation and E is the electric field related to ψ.

Stability of ion–acoustic solitary waves in a multi-species magnetized plasma consisting of non-thermal and isothermal electrons

2009 ◽  
Vol 75 (5) ◽  
pp. 593-607 ◽  
Author(s):  
SK. ANARUL ISLAM ◽  
A. BANDYOPADHYAY ◽  
K. P. DAS
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Perturbation Expansion ◽  
Magnetized Plasma ◽  
Wave Number ◽  
First Order ◽  
Zk Equation ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

AbstractA theoretical study of the first-order stability analysis of an ion–acoustic solitary wave, propagating obliquely to an external uniform static magnetic field, has been made in a plasma consisting of warm adiabatic ions and a superposition of two distinct populations of electrons, one due to Cairns et al. and the other being the well-known Maxwell–Boltzmann distributed electrons. The weakly nonlinear and the weakly dispersive ion–acoustic wave in this plasma system can be described by the Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation and different modified KdV-ZK equations depending on the values of different parameters of the system. The nonlinear term of the KdV-ZK equation and the different modified KdV-ZK equations is of the form [φ(1)]ν(∂φ(1)/∂ζ), where ν = 1, 2, 3, 4; φ(1) is the first-order perturbed quantity of the electrostatic potential φ. For ν = 1, we have the usual KdV-ZK equation. Three-dimensional stability analysis of the solitary wave solutions of the KdV-ZK and different modified KdV-ZK equations has been investigated by the small-k perturbation expansion method of Rowlands and Infeld. For ν = 1, 2, 3, the instability conditions and the growth rate of instabilities have been obtained correct to order k, where k is the wave number of a long-wavelength plane-wave perturbation. It is found that ion–acoustic solitary waves are stable at least at the lowest order of the wave number for ν = 4.

A regularized model for strongly nonlinear internal solitary waves

2009 ◽  
Vol 629 ◽  
pp. 73-85 ◽  
Author(s):  
WOOYOUNG CHOI ◽  
RICARDO BARROS ◽  
TAE-CHANG JO
Keyword(s):  
Stability Analysis ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Wave Solution ◽  
Wave Amplitude ◽  
Solitary Wave Solution ◽  
Shear Instability ◽  
Internal Solitary Waves ◽  
Strongly Nonlinear ◽  
Regularized Model

The strongly nonlinear long-wave model for large amplitude internal waves in a two-layer system is regularized to eliminate shear instability due to the wave-induced velocity jump across the interface. The model is written in terms of the horizontal velocities evaluated at the top and bottom boundaries instead of the depth-averaged velocities, and it is shown through local stability analysis that internal solitary waves are locally stable to perturbations of arbitrary wavelengths if the wave amplitudes are smaller than a critical value. For a wide range of depth and density ratios pertinent to oceanic conditions, the critical wave amplitude is close to the maximum wave amplitude and the regularized model is therefore expected to be applicable to the strongly nonlinear regime. The regularized model is solved numerically using a finite-difference method and its numerical solutions support the results of our linear stability analysis. It is also shown that the solitary wave solution of the regularized model, found numerically using a time-dependent numerical model, is close to the solitary wave solution of the original model, confirming that the two models are asymptotically equivalent.

scholarly journals Numerical modeling of flow and morphology induced by a solitary wave on a sloping beach

2019 ◽  
Vol 82 ◽  
pp. 259-273 ◽  
Author(s):  
Jinzhao Li ◽  
Meilan Qi ◽  
David R. Fuhrman
Keyword(s):  
Numerical Modeling ◽  
Solitary Wave ◽  
Sloping Beach
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