Finite amplitude steady-state wave groups with multiple near resonances in deep water

2017 ◽  
Vol 835 ◽  
pp. 624-653 ◽  
Author(s):  
Z. Liu ◽  
D. L. Xu ◽  
S. J. Liao
Keyword(s):  
Steady State ◽  
Linear Operator ◽  
Deep Water ◽  
Wave Equations ◽  
Finite Amplitude ◽  
Wave Groups ◽  
Homogeneous Solutions ◽  
State Wave ◽  
Resonant Interactions ◽  
Auxiliary Linear Operator

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth

2019 ◽  
Vol 867 ◽  
pp. 348-373 ◽  
Author(s):  
Z. Liu ◽  
D. Xie
Keyword(s):  
Steady State ◽  
Water Depth ◽  
Deep Water ◽  
High Frequency ◽  
Finite Amplitude ◽  
Wave Groups ◽  
State Wave ◽  
Resonant Interactions ◽  
Frequency Components ◽  
Finite Water Depth

Finite-amplitude wave groups with multiple near-resonances are investigated to extend the existing results due to Liu et al. (J. Fluid Mech., vol. 835, 2018, pp. 624–653) from steady-state wave groups in deep water to steady-state wave groups in finite water depth. The slow convergence rate of the series solution in the homotopy analysis method and extra unpredictable high-frequency components in finite water depth make it hard to obtain finite-amplitude wave groups accurately. To overcome these difficulties, a solution procedure that combines the homotopy analysis method-based analytical approach and Galerkin method-based numerical approaches has been used. For weakly nonlinear wave groups, the continuum of steady-state resonance from deep water to finite water depth is established. As nonlinearity increases, the frequency bands broaden and more steady-state wave groups are obtained. Finite-amplitude wave groups with steepness no less than $0.20$ are obtained and the resonant sets configuration of steady-state wave groups are analysed in different water depths. For waves in deep water, the majority of non-trivial components appear around the primary ones due to four-wave, six-wave, eight-wave or even ten-wave resonant interactions. The dominant role of four-wave resonant interactions for steady-state wave groups in deep water is demonstrated. For waves in finite water depth, additional non-trivial high-frequency components appear in the spectra due to three-wave, four-wave, five-wave or even six-wave resonant interactions with the components around the primary ones. The amplitude of these high-frequency components increases further as the water depth decreases. Resonances composed by components only around the primary ones are suppressed while resonances composed by components around the primary ones and from the high-frequency domain are enhanced. The spectrum of steady-state resonant wave groups changes with the water depth and the significant role of three-wave resonant interactions in finite water depth is demonstrated.

scholarly journals Experimental study of particle trajectories below deep-water surface gravity wave groups

2019 ◽  
Vol 879 ◽  
pp. 168-186 ◽  
Author(s):  
T. S. van den Bremer ◽  
C. Whittaker ◽  
R. Calvert ◽  
A. Raby ◽  
P. H. Taylor
Keyword(s):  
Gravity Wave ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Stokes Drift ◽  
Surface Gravity ◽  
Return Flow ◽  
Surface Gravity Wave ◽  
Wave Groups ◽  
Wave Induced

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

scholarly journals On the steady-state nearly resonant waves

2016 ◽  
Vol 794 ◽  
pp. 175-199 ◽  
Author(s):  
Shijun Liao ◽  
Dali Xu ◽  
Michael Stiassnie
Keyword(s):  
Steady State ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Linear Part ◽  
Wave Frequency ◽  
Analytic Approximation ◽  
Approximation Approach ◽  
Full Wave ◽  
Resonant Waves

The steady-state nearly resonant water waves with time-independent spectrum in deep water are obtained from the full wave equations for inviscid, incompressible gravity waves in the absence of surface tension by means of a analytic approximation approach based on the homotopy analysis method (HAM). Our strategy is to mathematically transfer the steady-state nearly resonant wave problem into the steady-state exactly resonant ones. By means of choosing a generalized auxiliary linear operator that is a little different from the linear part of the original wave equations, the small divisor, which is unavoidable for nearly resonant waves in the frame of perturbation methods, is avoided, or moved far away from low wave frequency to rather high wave frequency with physically negligible wave energy. It is found that the steady-state nearly resonant waves have nothing fundamentally different from the steady-state exactly resonant ones, from physical and numerical viewpoints. In addition, the validity of this HAM-based analytic approximation approach for the full wave equations in deep water is numerically verified by means of the Zakharov’s equation. A thought experiment is discussed, which suggests that the essence of the so-called ‘wave resonance’ should be reconsidered carefully from both of physical and mathematical viewpoints.

On the near resonances of collinear steady-state wave groups in finite water depth

2019 ◽  
Vol 182 ◽  
pp. 584-593 ◽  
Author(s):  
Zeng Liu ◽  
Jianglong Sun ◽  
De Xie ◽  
Zhiliang Lin
Keyword(s):  
Steady State ◽  
Water Depth ◽  
Wave Groups ◽  
State Wave ◽  
Finite Water Depth

Steep gravity waves in water of arbitrary uniform depth

1977 ◽  
Vol 286 (1335) ◽  
pp. 183-230 ◽  
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Perturbation Parameter ◽  
Intermediate Energy ◽  
Finite Amplitude ◽  
Wave Profile ◽  
Accurate Calculation ◽  
Theoretical Developments ◽  
Deep Water Waves

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.

Trend to spatial homogeneity for solutions to semilinear damped wave equations

1987 ◽  
Vol 105 (1) ◽  
pp. 117-126 ◽  
Author(s):  
J. Solà-Morales ◽  
M. València
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Lipschitz Constant ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Spatial Homogeneity ◽  
Homogeneous Solutions ◽  
Spatially Homogeneous ◽  
Image Position

SynopsisThe semilinear damped wave equationssubject to homogeneous Neumann boundary conditions, admit spatially homogeneous solutions (i.e. u(x, t) = u(t)). In order that every solution tends to a spatially homogeneous one, we look for conditions on the coefficients a and d, and on the Lipschitz constant of f with respect to u.

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