Singularity and Breaking of the Third Order Stokes Waves

In this paper, the fact is revealed that the surface elevation of the third order Stokes waves in implicit form could have no solution or have simultaneously a trivial one and a singular one on certain conditions. Based on this fact, the relative breaking width, a more reasonable quantity in agreement with the definition of whitecapping coverage rate, is obtained directly from the assumption that no solution means breaking. The implications of the singular solution existing in the third order stokes waves are also discussed briefly.

A NEW WAY FOR DERIVING HIGH ORDER STOKES WAVES

Variational principle for non-vortex, non-linear wave theories is established in this paper. By using this variational principle and related functional minimum condition, the fifth and sixth order Stokes Vaves are given as an example and the results are compared with those in Reference (Skjel-breia, 1961).

ON THE RESONANT GENERATION OF WEAKLY NONLINEAR STOKES WAVES IN REGIONS WITH FAST VARYING TOPOGRAPHY AND FREE SURFACE CURRENT

The effect of nonlinearity on the free surface wave resonated by an incident flow over rippled beds, which consist of fast varying topography superimposed on an otherwise slowly varying mean depth, is studied using a WKBJ-type perturbation approach. Synchronous, superharmonic and in particular subharmonic resonance were selectively excited over the fast varying topography with corresponding wavelengths. For a steady current the dynamical system is autonomous and the possible nonlinear steady states and their stability were investigated. When the current has a small oscillatory component the dynamical system becomes non-autonomous, chaos is now possible.

FREE LONG WAVES INDUCED BY WEAKLY NONLINEAR STOKES WAVES PROPAGATING OVER STRONG AMBIENT CURRENT AND AN UNEVEN BOTTOM

This paper concerns the generation of forced and free long waves. The free long waves are due to uneven bottoms and ambient currents. The pure wave evolution equation of Liu & Dingemans is extended to include the effects of strong ambient currents, leading to more general third order governing equations for the evolution of the envelope of the short waves and for the generation and scattering of the long waves.

Numerical Simulation of Seabed Response and Liquefaction due to Nonlinear Waves

Based on Biot＇s consolidation theory, a two-dimensional model for computation of the seabed response to waves is presented with the finite element method. Numerical results for different wave conditions are obtained, and the effects of wave non-lineafity on the wave-induced seabed response are examined. Moreover, the wave-induced momentary liquefaction in uniform and inhomogeneous seabeds is investigated. It is shown that the wave non-linearity affects the distribution of the wave-induced pore pressure and effective stresses, while the influence of wave non-linearity on the seabed liquefaction potential is not so significant.

Third-Order Stokes Wave Solutions of the Free Surface Capillary- Gravity Wave and the Interfacial Internal Wave

Based on the Stokes wave theory, the capillary-gravity wave and the interfacial internal wave in two-layer constant depth＇s fluid system are investigated. The fluids are assumed to be incompressible, inviscid and irrotational. The third-order Stokes wave solutions are given by using a perturbation method. The results indicate that the third-order solutions depend on the surface tension, the density and the depth of each layer. As expected, the first-order solutions are the linear theoretical results （the small amplitude wave theoretical results）. The second-order and the third-order solutions describe the nonlinear modification and the nonlinear interactions. The nonlinear impact appears not only in the n （n〉~2） times＇ high frequency components, but also in the low frequency components. It is also noted that the wave velocity depends on the wave number, depth, wave amplitude and surface tension.

Experimental Study on Crescent Waves Diffracted by A Circular Cylinder

Crescent waves often observed on the sea surface are unusual wave pattern induced by the instability of Stokes wave.The paper presents the experimental results of the wave field around a circular cylinder generated by the diffraction of crescent wave in order to examine the difference of diffracted crescent waves from the commonly-used diffracted Stokes waves. The results show that with the existence of the cylinder, the crescent wave pattern can still get fully developed, and with the presence of this type of wave pattern, the symmetry breaking of the wave amplitude distribution occurs and there are extra wave components at the frequencies of 0.5 ω;, 1.5ω;and 2.5ω;（ω;is the frequency of Stokes waves） appearing in the wave amplitude spectrum.

Optimum fifth order Stokes wave theory

This Paper improves the existing fifth order Stokes wave theory by using least Square method, and givesthe optimum result in the meaning of minimum error Squares to satisfy the free surface boundary conditions, and thewave profile can be adjusted according to the measured data. This paper also gives a simplified method for derivingthe parameters of the existing fifth order Stokes wave.

Numerical Simulation of the Stokes Wave for the Flow around a Ship Hull Coupled with the VOF Model

The surface wave generated by flow around a ship hull moving near free surface of water is simulated numerically in this study. The three-dimensional implicit finite volume method （FVM） is applied to solve Reynolds averaged Navier-Stokes （RANS） equation. The realizable k-e turbulence model has been implemented to capture turbulent flow around the ship hull in the free surface zone. The volume of fluid （VOF） method coupled with the Stokes wave theory has been used to determine the free surface effect of water. By using is a six degrees of freedom model, the ship hull＇s movement is numerically solved with the Stokes wave together. Under the action of Stokes waves on the sea, the interface between the air and water waves at the same regular pattem and so does the pressure and the vertical velocity. The ship hull moves in the same way as the wave. The amplitude of the ship hull＇s heave is less than the wave height because of the viscosity damping. This method could provide an important reference for the study of ships＇ movement, wave and hydrodynamics.

Experimental study on the Stokes effect in disordered birefringent microstructure fibers

In this paper, a 120-fs pulse transmission experiment is carried out using disordered birefringent microstructure fibers with cladding ventages. Through this experiment, it is found for the first time that remarkable Stokes and anti-Stokes waves can also be produced when the central wavelength of the incident pulse is in the normal dispersion regime of the microstructure fiber. The generation of the two waves can be explained by the four-wave mixing phase matching theory. Properties of the two waves under the action of femtosecond laser pulses with different parameters are studied. The results show that the central wavelength of anti-Stokes waves and Stokes waves produced under the two orthogonal polarization states shift by 63 nm and 160 nm, respectively. The strengths and central positions of the two waves in birefringent fibers can be controlled by adjusting the phase match condition and the polarization directions of incident pulses.

Simplified Method and Improved Solution for Deriving Fifth Order Stokes Wave

As the solution of the two equations for determining the existing fifth order Stokes wave derived by Skjelbreia is complex and tedious, the two equations are simplified into one equation for determining d / L, i. e., f(H, T, d / L) = 0. According to this simplified method, three cases of the solution for the Skjelbreia equations have been found: one accurate solution; more than one accurate solution and no accurate solution (but there exists the optimum approximate solution in the area of satisfying Skjelbreia equations). As to the case of more than one accurate solution, the reasonable solution can be judged from the method of variational principle, by means elf which an optimum solution improved from the solution of Skjelbreia equations in the area of satisfying the original mathematical equations of non-vortex and nonlinear wave theory, i. e., the optimum fifth order Stokes wave, is given.

THE LIMITING STOKES WAVE WITH FINITE WATER DEPTH

In this paper we extend the method developed in[1] for limiting Stokes wave of infinite water depth to cover the case of finite depth. The method has high efficiency and the result is accurate.

A simplified model for extreme-wave kinematics in deep sea

Based on the fifth-order Stokes regular wave theory, a simplified model for extreme-wave kinematics in deep sea was developed. In this model, from the wave records the average of two neighboring wave periods for the extreme crest or trough was defined as the period of the Stokes wave by the up and down zero-crossing methods. Then the input wave amplitude was deduced by substituting the wave period and extreme crest or trough into the expression for the fifth-order Stokes wave elevation. Thus the corresponding formula for the wave velocity can be used to describe kinematics beneath the extreme wave. By comparison with the published numerical models and experimental data, the proposed model is validated to be able to calculate the extreme wave velocity rather easily and accurately.

An experimental and numerical study of the freak wave speed

The propagation speed is one of the most important characteristics for describing freak waves. The research of freak wave speed is not only helpful for understanding the generation mechanism and evolution process of freak waves, but also applicable to the prediction. A stable and accurate method is proposed for the cal- culation of the freak wave speed, in which physical model tests are carried out to measure the motion of the largest wave crest along the wave tank. The linear regression relationship between the spatial position of the largest wave crest and instantaneous moment is established to calculate the speed of totally 248 cases of experimental freak waves and 312 supplementary cases of numerical freak waves. Based on the calculate results, a semitheoretical and semiempirical formula is proposed by using a regression analysis method to predict the speed of the freak wave, and the nonlinear characteristic of the freak wave speed is also investi- gated.

Hydrodynamic Performance of Multiple-Row Slotted Breakwaters

This study examines the hydrodynamic performance of multiple-row vertical slotted breakwaters. We developed a mathematical model based on an eigenfunction expansion method and a least squares technique for Stokes second-order waves. The numerical results obtained for limiting cases of double-row and triple-row walls are in good agreement with results of previous studies and experimental results. Comparisons with experimental measurements of the reflection, transmission, and dissipation coefficients （CR, Cr, and CE） for double-row walls show that the proposed mathematical model adequately reproduces most of the important features. We found that for double-row walls, the CR increases with increasing wave number, kd, and with a decreasing permeable wall part, din. The Cr follows the opposite trend. The CE slowly increases with an increasing kd for lower kd values, reaches a maximum, and then decreases again. In addition, an increasing porosity of dm would significantly decrease the CR while increasing the Cr. At lower values of kd, a decreasing porosity increases the CE, but for high values of kd, a decreasing porosity reduces the Ce. The numerical results indicate that, for triple-row walls, the effect of the arrangement of the chamber widths on hydrodynamic characteristics is not significant, except when kd〈0.5 Double-row slotted breakwaters may exhibit a good wave-absorbing performance at kd〉0.5, where by the horizontal wave force may be smaller than that of a single wall. On the other hand, the difference between double-row and triple-row vertical slotted breakwaters is marginal.

Spectral Analysis of Nonlinear Drag Forces

The spectral properties of nonlinear drag forces of random waves on vertical circular cylinders are analyzed in this paper by means of nonlinear spectral analysis. The analysis provides basic parameters for estimation of the characteristic drag forces. Numerical computation is also performed for the investigation of the effects of nonlinearity of the drag forces. The results indicate that the wave drag forces calculated by linear wave theory are larger than those calculated by the third order Stokes wave theory for given waves. The difference between them increases with wave height. The wave drag forces calculated by use of linear approximation are about 5% smaller than their actual values when measured in the peak values of spectral densities. This will result in a safety problem for the design of offshore structures. Therefore, the nonlinear effect of wave drag forces should be taken into consideration in design and application of important offshore structures.

Nonlinear effect on inertia component of wave forces on a cylinder

A series of experiments on wave forces on a cylinder have been carried out when inertia component isdominant for a small she cylinder. The influence of nonlinear effect on the inertia component of wave forces on a cylinder is analyzed. The applicable range of nonlinear wave theories, such as Stokes and cnoidal wave theories, in calculating wave forces on a cylinder is discussed. A correction method is suggested for linear wave theory in calculated waveforces on a cylinder under the nonlinear condition.

A NUMERICAL METHOD FOR NONLINEAR WATER WAVES

This article presents a numerical method for modeling nonlinear water waves based on the High Order Spectral （HOS） method proposed by Dommermuth and Yue and West et al., involving Taylor expansion of the Dirichlet problem and the Fast Fourier Transform （FFT） algorithm. The validation and efficiency of the numerical scheme is illustrated by a number of case studies on wave and wave train configuration including the evolution of fifth-order Stokes waves, wave dispersive focusing and the instability of Stokes wave with finite slope. The results agree well with those obtained by other studies.

On the universal third order Stokes wave solution

This paper presents a universal third-order Stokes solution with uniform current. This solution is derived on the basis of potential theory by expanding the free surface and potential function in Fourier series and determining the Fourier coefficients by solving a set of nonlinear algebraic equations through the Taylor expansion and perturbation method. The universal solution is expressed upon the still water depth with the still water level as datum and retains a global perturbation parameter. The wave set-up term generated by the self-interaction of oscillatory waves is explicitly included in the free surface function. With the use of different definitions for the wave celerity, different water levels as the datum, different non-dimensional variables as the perturbation parameter, and different treatments for the total head, the universal solution can be reduced to the existing various Stokes solutions, thus explaining the reasons and the physical significance of different non-periodic terms in them, such as the positive or negative constant term in the free surface expression and the time-or space-proportional term in the potential function.