On the steady-state fully resonant progressive waves in water of finite depth

2012 ◽  
Vol 710 ◽  
pp. 379-418 ◽  
Author(s):  
Dali Xu ◽  
Zhiliang Lin ◽  
Shijun Liao ◽  
Michael Stiassnie
Keyword(s):  
Steady State ◽  
Energy Spectrum ◽  
Wave Energy ◽  
Wave Equations ◽  
Finite Depth ◽  
Nonlinear Wave Equations ◽  
Wave System ◽  
Resonant Wave ◽  
Finite Water Depth ◽  
Progressive Waves

AbstractThe steady-state fully resonant wave system, consisting of two progressive primary waves in finite water depth and all components due to nonlinear interaction, is investigated in detail by means of analytically solving the fully nonlinear wave equations as a nonlinear boundary-value problem. It is found that multiple steady-state fully resonant waves exist in some cases which have no exchange of wave energy at all, so that the energy spectrum is time-independent. Further, the steady-state resonant wave component may contain only a small proportion of the wave energy. However, even in these cases, there usually exist time-dependent periodic exchanges of wave energy around the time-independent energy spectrum corresponding to such a steady-state fully resonant wave, since it is hard to be exactly in such a balanced state in practice. This view serves to deepen and enrich our understanding of the resonance of gravity waves.

Third-order approximation to short-crested waves

1979 ◽  
Vol 90 (1) ◽  
pp. 179-196 ◽  
Author(s):  
J. R. C. Hsu ◽  
Y. Tsuchiya ◽  
R. Silvester
Keyword(s):  
Order Approximation ◽  
Vertical Wall ◽  
Finite Depth ◽  
Wave System ◽  
Third Order ◽  
Inviscid Incompressible Fluid ◽  
Stokes Waves ◽  
Progressive Waves ◽  
New Formulation ◽  
Third Order Approximation

Short-crested wave systems, as produced by two progressive waves propagating at an oblique angle to each other, have an extremely important effect on a sedimentary bed. The complex water-particle motions are conducive to lifting material into suspension and sustaining it in motion. In order to study this phenomenon rigorously, the variables of this wave system are derived to a third-order approximation by a perturbation method. The case of waves reflecting obliquely from a vertical wall is examined under the assumptions of full reflexion, uniform finite depth and an inviscid incompressible fluid. The new formulation reduces to standing or Stokes waves at the limiting angles of approach. Expressions for kinematic quantities are also presented.

Steady-state resonance of multiple wave interactions in deep water

2014 ◽  
Vol 742 ◽  
pp. 664-700 ◽  
Author(s):  
Zeng Liu ◽  
Shi-Jun Liao
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Gravity Waves ◽  
Deep Water ◽  
Wave Energy ◽  
Two Dimensions ◽  
Physical Parameters ◽  
Wave System ◽  
State Resonance ◽  
Resonant Waves

AbstractThe steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.

Waves on a shear flow

1974 ◽  
Vol 11 (2) ◽  
pp. 263-277 ◽  
Author(s):  
K.K. Puri
Keyword(s):  
Free Surface ◽  
Steady State ◽  
Shear Flow ◽  
Initial Value Problem ◽  
Finite Depth ◽  
Steady State Solution ◽  
State Solution ◽  
Wave System ◽  
Initial Value ◽  
Oscillatory Pressure

The propogation of disturbance when a shear flow with a free surface, in a channel of infinite horizontal extent and finite depth, is disturbed by the application of time-oscillatory pressure, is studied. The initial value problem is solved by using transform techniques and the steady state solution is obtained therefrom in the limit t → ∞. The effect of the initial shear on the development of the wave system is investigated.

scholarly journals Quasiperiodic Solutions of Completely Resonant Wave Equations with Quasiperiodic Forced Terms

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Yixian Gao ◽  
Weipeng Zhang ◽  
Jing Chang
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Traveling Waves ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Linking Theorem ◽  
First Order ◽  
Quasiperiodic Solutions ◽  
Resonant Wave ◽  
Spatial Boundary

This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Tan ◽  
Zhao-Yang Yin
Keyword(s):  
Differential Equations ◽  
Wave Equations ◽  
Rogue Wave ◽  
Nonlinear Partial Differential Equations ◽  
Rogue Waves ◽  
Homoclinic Solutions ◽  
Nonlinear Wave Equations ◽  
Bilinear Method ◽  
Wave Solutions ◽  
Rogue Wave Solutions

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

scholarly journals A uniqueness result for the Sine-Gordon breather

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Rainer Mandel
Keyword(s):  
Nonlinear Wave ◽  
Wave Equations ◽  
Uniqueness Result ◽  
Nonlinear Wave Equations

AbstractIn this note we prove that the sine-Gordon breather is the only quasimonochromatic breather in the context of nonlinear wave equations in $$\mathbb {R}^N$$ R N .

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