scholarly journals Four-wave resonant interactions in the classical quadratic Boussinesq equations

2009 ◽  
Vol 618 ◽  
pp. 263-277 ◽  
Author(s):  
M. ONORATO ◽  
A. R. OSBORNE ◽  
P. A. E. M. JANSSEN ◽  
D. RESIO
Keyword(s):  
Energy Transfer ◽  
Kinetic Equation ◽  
Shallow Water ◽  
Water Waves ◽  
Wave Spectrum ◽  
Boussinesq Equations ◽  
Wave Model ◽  
Flat Bottom ◽  
Weakly Nonlinear ◽  
Resonant Interactions

We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

scholarly journals COMBINED REFRACTION-DIFFRACTION OF NONLINEAR WAVES IN SHALLOW WATER

1984 ◽  
Vol 1 (19) ◽  
pp. 68
Author(s):  
James T. Kirby ◽  
Philip L.F. Liu ◽  
Sung B. Yoon ◽  
Robert A. Dalrymple
Keyword(s):  
Shallow Water ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Evolution Equations ◽  
Boussinesq Equations ◽  
Two Dimensions ◽  
Parabolic Approximation ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Dimensional Domain

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow water waves. Two methods of approach are taken. In the first method Boussinesq equations are used to derive evolution equations for spectral wave components in a slowly varying two-dimensional domain. The second method modifies the equation of Kadomtsev s Petviashvili to include varying depth in two dimensions. Comparisons are made between present numerical results, experimental data and previous numerical calculations.

Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions

2018 ◽  
Vol 15 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Aly R. Seadawy
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

scholarly journals Far-Field Characteristics of Linear Water Waves Generated by a Submerged Landslide over a Flat Seabed

2020 ◽  
Vol 8 (3) ◽  
pp. 196
Author(s):  
Haixiao Jing ◽  
Yanyan Gao ◽  
Changgen Liu ◽  
Jingming Hou
Keyword(s):  
Shallow Water ◽  
Water Depth ◽  
Water Waves ◽  
Wave Model ◽  
Linear Wave ◽  
Far Field ◽  
Constant Depth ◽  
Low Froude Number ◽  
Dispersive Model ◽  
Wave Models

Understanding the propagation of landslide-generated water waves is of great help against tsunami hazards. In order to investigate the effects of landslide shapes on the far-field leading wave generated by a submerged landslide at a constant depth, three linear wave models with different degrees of dispersive properties are employed in this study. The linear fully dispersive model is then validated by comparing the results against the experimental data available for landslides with a low Froude number. Three simplified shapes of landslides with the same volume, which are unnatural for a body of incoherent material, are used to investigate the effects of landslide shapes on the far-field properties of the generated leading wave over a flat seabed. The results show that the far-field leading crest over a constant depth is independent of the exact landslide shape and is invalid at a shallow water depth. Therefore, the most popular non-dispersive model (also called the shallow water wave model) cannot be used to reproduce the phenomenon. The weakly dispersive wave model can predict this phenomenon well. If only the leading wave is considered, this model is accurate up to at least μ = h0/Lc = 0.6, where h0 is the water depth and Lc denotes the characteristic length of the landslide.

Implementing New Nonlinear Term in Third Generation Wave Models

2014 ◽  
Author(s):  
Odin Gramstad ◽  
Alexander Babanin
Keyword(s):  
Kinetic Equation ◽  
Spread Spectrum ◽  
Time Scale ◽  
Source Term ◽  
Wave Spectrum ◽  
Wave Model ◽  
Third Generation ◽  
Spectral Evolution ◽  
Linear Interaction ◽  
Wavewatch Iii

The non-linear interaction term is one of the three key source functions in every third-generation spectral wave model. An update of physics of this term is discussed. The standard statistical/phase-averaged description of the nonlinear transfer of energy in the wave spectrum (wave-turbulence) is based on Hasselmann’s kinetic equation [1]. In the derivation of the kinetic equation (KE) it is assumed that the evolution takes place on the slow O(ε−4) time scale, where ε is the wave steepness. This excludes the effects of near-resonant quartet interactions that may lead to spectral evolution on the ‘fast’ O(ε−2) time scale. Generalizations of the KE (GKE) that enable description of spectral evolution on the O(ε−2) time scale [2–4] are discussed. The GKE, first solved numerically in [4], is implemented as a source term in the third generation wave model WAVEWATCH-III. The new source term (GKE) is tested and compared to the other nonlinear-interaction source terms in WAVEWATCH-III; the full KE (WRT method) and the approximate DIA method. It is shown that the GKE gives similar results to the KE in the case of a relatively broad banded and directional spread spectrum, while it shows somewhat larger difference in the case of a more narrow banded spectrum with narrower directional distribution. We suggest that the GKE may be a suitable replacement to the KE in situations where ‘fast’ spectral evolution takes place.

Characteristic of the algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations

2019 ◽  
Vol 35 (07) ◽  
pp. 2050028 ◽  
Author(s):  
Jian-Gen Liu ◽  
Xiao-Jun Yang ◽  
Yi-Ying Feng
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Traveling Wave Solutions ◽  
Rational Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Restoring Forces ◽  
First Time

With the aid of the planar dynamical systems and invariant algebraic cure, all algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations, which can be used to model shallow water waves with weakly nonlinear restoring forces and to describe waves in ferromagnetic media, were obtained. Meanwhile, some new rational solutions are also yielded through an invariant algebraic cure with two different traveling wave transformations for the first time. These results are an effective complement to existing knowledge. It can help us understand the mechanism of shallow water waves more deeply.

Diverse wave propagation in shallow water waves with the Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and Benney–Luke integrable models

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 808-818
Author(s):  
Usman Younas ◽  
Aly R. Seadawy ◽  
Muhammad Younis ◽  
Syed T. R. Rizvi ◽  
Saad Althobaiti
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Three Dimensional ◽  
Wave Model ◽  
Algebraic Method ◽  
Integrable Models ◽  
Computational Technique ◽  
Shallow Water Waves ◽  
Physical Problems ◽  
Contour Plots

Abstract The shallow water wave model is one of the completely integrable models illustrating many physical problems. In this article, we investigate new exact wave structures to Kadomtsev–Petviashvili–Benjamin–Bona–Mahony and the Benney–Luke equations which explain the behavior of waves in shallow water. The exact structures are expressed in the shapes of hyperbolic, singular periodic, rational as well as solitary, singular, shock, shock-singular solutions. An efficient computational strategy namely modified direct algebraic method is employed to construct the different shapes of wave structures. Moreover, by fixing parameters, the graphical representations of some solutions are plotted in terms of three-dimensional, two-dimensional and contour plots, which explain the physical movement of the attained results. The accomplished results show that the applied computational technique is valid, proficient, concise and can be applied in more complicated phenomena.

Energy transfer between external and internal gravity waves

1964 ◽  
Vol 19 (3) ◽  
pp. 465-478 ◽  
Author(s):  
F. K. Ball
Keyword(s):  
Energy Transfer ◽  
Internal Wave ◽  
Surface Waves ◽  
Shallow Water ◽  
Gravity Waves ◽  
External Surface ◽  
Internal Gravity Waves ◽  
Liquid System ◽  
Resonant Interactions ◽  
Non Linear

In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail.

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