Stokes Wave

Simulation of more realistic ocean conditions in wave basins is becoming important for offshore industry. As spreading wave has become more desirable, the capability of reproducing oblique planar wave train is critical for a wave basin's performance. In this study, the oblique waves have been simulated in a numerical wave basin based on the volume of fluid (VOF) method and Navier-Stokes solver. Multi-element bottom hinged flap motion using the conventional snake principle has been simulated by moving boundary dynamic mesh in OpenFOAM. Finite length of the entire wave maker and finite width of each paddle caused considerable spatial variations in wave height and wave propagating direction. Beaches with slope have been implemented to minimize the sidewall reflection and improve the uniformity of the oblique wave field. The generated linear oblique waves have been compared with the analytical solutions for validation in terms of uniformity and wave height. The time history of the surface elevation at different locations have been computed to investigate the uniformities of general wave fields. In addition, the

failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of 120 • included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

In this paper a nonlinear response of a fixed offshore platform under the combined forces of waves, wind and sea currents is presented. Wave force acting on the elements is calculated using Morison equation. Hydrodynamic loads on horizontal and vertical tubular members and the dynamic response of offshore fixed platform coupled with distribution of displacement, axial force, and bending moment along the base of the platform for regular and severe cases have been investigated. The structure must be able maintain production in a one-year wave return period condition and also to be able to continue with one hundred-year storm return period. The results of this study show that bending moment values with a one-year wave return period condition for the base platform and junction of platform to deck are 70 percent and 59 percent, respectively more than bending moment with a one-year wave return period. The direction of wave and wind hit has significant effects on the shift platform response, also nonlinear response is important for the safe design and operation of offshore structures.

This paper describes an investigation of the generation of desired sea states in a numerical wave model. Bimodal sea states containing energetic swell components can be coastal hazards along coastlines exposed to large oceanic fetches. Investigating the effects of long-period bimodal seas requires large computational domains and increased running time to ensure the development of the desired sea state. Long computational runs can cause mass stability issues due to the Stokes drift and wave reflection, which in turn affect results through the variation of the water level. A numerical wave flume, NEWRANS, was used to investigate two wave generation methods: the wave paddle method, allowing for a smaller domain; and the internal mass source function method, providing an open boundary allowing reflected waves to leave the domain. The two wave generation methods were validated against experimental data by comparing the wave generation accuracy and the variance of mass in the model during simulations. Results show that the wave paddle method not only accurately generates the desired sea state but also provides a more stable simulation, in which mass fluctuation has less of an effect on the water depth during the long-duration simulations. As a result, it is suggested that the wave paddle method with active wave absorption is preferable to the internal wave maker option when investigating intermediate-depth long-period bimodal seas for long-duration simulations.

The two-scale approximation (hereafter, TSA) to the full Boltzman integral representation of quadruplet wave-wave interactions has recently been presented as a new method to estimate nonlinear transfer rates in wind waves, and has been tested for idealized spectral data, as well as for observed field measurements. TSA has been shown to perform well for wave spectra from field measurements, even for cases with directional energy shearing, compared to the Discrete Interaction Approximation (DIA), which is used in almost all operational wave forecast models. In this study, TSA is implemented in a modern operational wave model, WAVEWATCHIII Ò , hereafter WW3. Tests include idealized wave spectra based on field measurements, as well as additional tests for fetch-limited wave growth, and waves generated by hurricane Juan. Generally, TSA is shown to work well when its basic assumptions are met, when its first order, broad-scale term represents most of the spectrum, and its second order term is a perturbation-scale residual representing the rest of the spectrum. These conditions are easily met for test cases involving idealized JONSWAP-type spectra and in time-stepping cases when winds are spatially and temporally constant. To some extent, they also appear to be met in more demanding conditions, when storms move through their life cycles, with winds that change speed and direction, and with complex wave spectra, involving swell-windsea interactions, multiple peaks and directional shears.

Fourier -Weak turbulence is a theoretical framework aimed at describing wave turbulence (in the weakly nonlinear limit) i.e. a statistical state involving a large number of nonlinearly coupled waves. For gravity waves at the surface of water, it provides a phenomenology that may describe the formation of the spectrum of the ocean surface. Analytical predictions of the spectra are made based on the fact that energy transfer occurs through 4-wave coupling. By using an advanced stereoscopic imaging technique, we measure in time the deformation of the water surface. We obtain a state of wave turbulence by using two small wedge wavemakers in a 13-m diameter wavetank. We then use high order correlator (bi-and tri-coherence) in order to get evidence of the active wave coupling present in our system as used successfully for gravity-capillary wave turbulence [1]. At odds with the weak turbulence theory we observe 3-wave interaction involving 2 quasi linear wave and a bound wave whose frequency lies on the first harmonics of the linear dispersion relation. We do not observe 4-wave coupling within the accuracy of our measurement.

turbulence is a statistical state made of a very large number of nonlinearly interacting waves. The Weak Turbulence Theory was developed to describe such a situation in the weakly nonlinear regime. Although, oceanic data tend to be compatible with the theory, laboratory data fail to fulfill the theoretical predictions. A space-time resolved measurement of the waves have proven to be especially fruitful to identify the mechanism at play in turbulence of gravity-capillary waves [1]. We developed an image processing algorithm to measure the motion of the surface of water with both space and time resolution. We first seed the surface with slightly buoyant polystyrene particles and use 3 cameras to reconstruct the surface. Our stereoscopic algorithm is coupled to PIV so that to obtain both the surface deformation and the velocity of the water surface. Such a coupling is shown to improve the sensitivity of the measurement by one order of magnitude. We use this technique to probe the existence of weakly nonlinear turbulence excited by two small wedge wavemakers in a 13-m diameter wave flume. We observe a truly weakly nonlinear regime of isotropic wave turbulence. [1] Aubourg & Mordant, Phys. Rev. Fluids 1, 2016

is a theory, in the limit of vanishing nonlinearity, that derive analytically statistical features of wave turbulence. The stationary spectrum for the surface elevation in the case of gravity waves, is predicted to E(k) ∝ k −5/2. This spectral exponent-5/2 remains elusive in all experiments. in which the measured exponent is systematically lower than the prediction. Furthermore in the experiments the weaker the nonlinearity the further the spectral exponent is from the prediction. In order to investigate the reason for this observation we developed an experiment in the CORIOLIS facility in Grenoble. It is a 13m-diameter circular pool filled with water with a 70 cm depth. We generate wave turbulence by using two wedge wavemakers. Surface elevation measurements are performed by a stereoscopic optical technique and by capacitive probes. The nonlinear coupling at work in this system are analyzed by computing 3-and 4-wave correlations of the Fourier wave amplitudes in frequency. Theory predicts that coupling should occur through 4-wave resonant interaction. In our data, strong 3-wave correlations are observed in addition to the 4-wave correlation. Most our observations are consistent with field observation in the Black Sea (Leckler et al 2015).

The diffraction of highly nonlinear Stokes waves by vertical cylinders of circular cross section is numerically simulated in the time domain. A finite-element method, based on Hamilton's principle, is used to discretize the fluid domain. The Stokes waves, input at the numerical wave maker, are obtained numerically from the two-dimensional steady solution of the finite element model. A new matching scheme is developed to match the two-dimensional wave at the far field and the three-dimensional diffracted wave in the near field. The method developed here can easily be extended to the diffraction of irregular, nonlinear waves. Numerical examples are presented for the diffraction of Stokes waves with various steepnesses by single and multiple circular cylinders. The wave elevation and runup on the cylinder are calculated and compared with the available theoretical and experimental results.

The diffraction of highly nonlinear Stokes waves by vertical cylinders of circular cross section is numerically simulated in the time domain. A finite-element method, based on Hamilton's principle, is used to discretize the fluid domain. The Stokes waves, input at the numerical wave maker, are obtained numerically from the two-dimensional steady solution of the finite element model. A new matching scheme is developed to match the two-dimensional wave at the far field and the three-dimensional diffracted wave in the near field. The method developed here can easily be extended to the diffraction of irregular, nonlinear waves. Numerical examples are presented for the diffraction of Stokes waves with various steepnesses by single and multiple circular cylinders. The wave elevation and runup on the cylinder are calculated and compared with the available theoretical and experimental results.

Three-wave solitons backward propagating with respect to a pump wave are generated in nonlinear optical media through stimulated Brillouin scattering (SBS) in optical fibers or through the non-degenerate three-wave interaction in quadratic (χ (2)) nonlinear media. In an optical fiber-ring cavity, nanosecond solitary wave morphogenesis takes place when it is pumped with a continuous wave (c.w.). A backward dissipative Stokes soliton is generated from the hypersound waves stimulated by electrostriction between the forward pump wave and the counterpropagating Stokes wave. Superluminous and subluminous dissipative solitons are controlled via a single parameter: the feedback or reinjection for a given pump intensity or the pump intensity for a given feedback. In a c.w. pumped optical parametric oscillator (OPO), backward picosecond soliton generation takes place for non-degenerate three-wave interaction in the quadratic medium. The resonant condition is automatically satisfied in stimulated Brillouin backscattering when the fiber-ring laser contains a large number of longitudinal modes beneath the Brillouin gain curve. However, in order to achieve quasiphase matching between the three optical waves (the forward pump wave and the backward signal and idler waves) in the χ (2) medium, the nonlinear susceptibility should be periodically structurated by an inversion grating of sub-µm period in an optical parametric oscillator (OPO). The stability analysis of the inhomogeneous stationary solutions presents a Hopf bifurcation with a single control parameter which gives rise to temporal modulation and then to backward three-wave solitons. Above OPO threshold, the nonlinear dynamics yields selfstructuration of a backward symbiotic solitary wave, which is stable for a finite temporal walk-off, i.e. different group velocities, between the backward propagating signal and idler waves. We also study the dynamics of singly backward mirrorless OPO (BMOPO) pumped by a broad bandwidth field and also with a highly incoherent pump, in line with the recent experimental demonstration of this BMOPO configuration in a KTP crystal. We show that this system is characterized, as a general rule, by the generation of a highly coherent backward field, despite the high degree of incoherence of the pump field. This remarkable property finds its origin in the convection-induced phase-locking mechanism that originates in the counterstreaming configuration: the incoherence of the pump is transferred to the co-moving field, which thus allows the backward field to evolve towards a highly coherent state. We propose other realistic experimental conditions that may be implemented with currently available technology and in which backward coherent wave generation from incoherent excitation may be observed and studied.

The physical processes associated with propagation of a high-power (power > critical power for self-focusing) laser beam in water include nonlinear focusing, stimulated Raman scattering (SRS), optical breakdown, and plasma formation. The interplay between nonlinear focusing and SRS is analyzed for cases where a significant portion of the pump power is channeled into the Stokes wave. Propagation simulations and an analytical model demonstrate that the Stokes wave can re-focus the pump wave after the power in the latter falls below the critical power. It is shown that this novel focusing mechanism is distinct from cross-phase focusing. The phenomenon of gain-focusing discussed here for propagation in water is expected to be of general occurrence applicable to any medium supporting nonlinear focusing and stimulated Raman scattering.

Expansions have been given in the past for steady Stokes waves at or near a largest wave with a 120° corner. It is shown here that the solution is more complicated than has been assumed: that the corner is not a regular singular point, and that waves of less than maximum amplitude have singularities of a different order.

A new formalism of spectral filtering for the description of the modulation processes is proposed. The method allows one to study the classical problem of multi-phase modulations in dispersive systems. In the present paper, deepwater waves are considered. Spectral filtering results in a system of coupled equations that describe the modulations of the carrier wave and its harmonics. The formalism may find applications in a broad range of physical situations with multi-phase dynamics.

We describe a means for efficient, energy-scalable generation of first-order anti-Stokes light by four-wave mixing, using collimated pump and first-order Stokes seed beams. We performed a one-dimensional, plane-wave analysis of the process, assuming steady-state conditions and monochromatic beams. A prescription for designing the optimum anti-Stokes converter with this technique is derived from this analysis, and we show that energy conversion efficiencies from pump light to anti-Stokes light of-30% might be possible. We also report an unoptimized, proof-of-concept experiment that produced 40 mJ of anti-Stokes-shifted light at 432 nm from 1.2 J of 527-nm pump light in a 60-cm-long hydrogen cell.

The present paper describes a model for the distribution of maximum wave and crest heights in a given sea state. The standard Rayleigh and Weibull distributions for wave and crest heights are modified using a third order Stokes expansion of the wave envelope. The maximum wave or crest height in a record is calculated using the asymptotic Gumbel distribution. It was validated by fitting the model distributions to North Sea measurement data and provided much better predictions of maximum crest and wave heights than standard models. To make up for the fact that directionality and spectral shape are not taken into account in the model, correction terms (specific to the studied site) are introduced and mainly one of them will give the optimized mean wave number to consider in the model. The proposed model is relatively easy to apply and could be an effective tool to determine extreme wave and crest heights for offshore structure design purposes.

The transformation of an internal second mode solitary wave over a bottom step in a computational tank filled with a three-layer stratified fluid was studied. The convex waveforms were generated by a collapse mechanism for stratification with a thin mid-layer. The wave transformation depends on the blocking parameter B which is a ratio of the amplitude of the incident wave to the thickness of the lower water layer over the step. Three regimes of second mode wave transformation over the step are identified. In regime I (2 < B < 6) this wave generates a breather-like internal wave (BLIW) packet behind the wave of mode-2 due to the impulse-like effect of the step and a long wave of mode-1 ahead of the wave. The BLIW degenerates into a dispersive wave packet at large B > 6. In regime II (0.5 < B < 2) the mode-2 wave is permanently disintegrated, generating a chain of waves of mode-1 ahead of wave of mode-2 and tail of short waves of mode-1 behind the wave. In regime III (B < 0.5) only waves of elevation of mode-1 penetrate into the fluid layer over the step.

We present general analytical expressions of Stokes and anti-Stokes spectral photon-flux densities that are spontaneously generated by a single monochromatic pump wave propagating in a single-mode optical fiber. We validate our results by comparing them with experimental data. Limiting cases of the general expressions corresponding to interesting physical situations are discussed.

A statistical model of random wave is developed using Stokes wave theory of water wave dynamics. A new nonlinear probability distribution function of wave height is presented. The results indicate that wave steepness not only could be a parameter of the distribution function of wave height but also could reflect the degree of wave height distribution deviation from the Rayleigh distribution. The new wave height distribution overcomes the problem of Rayleigh distribution that the prediction of big wave is overestimated and the general wave is underestimated. The prediction of small probability wave height value of new distribution is also smaller than that of Rayleigh distribution. Wave height data taken from East China Normal University are used to verify the new distribution. The results indicate that the new distribution fits the measurements much better than the Rayleigh distribution.

Evolution of Stokes wave side-band instability along a super tank was studied experimentally and theoretically. The initial exponential growth of the resonant side-bands is followed by asymmetrical growth rates for the side-bands: lower side-band growth is much faster, and finally the main energy is concentrated in it and the primary wave. An active breaking process increases the frequency downshift during the latter stages of the wave propagation. Some type of wave stabilization takes place during the final post-breaking stages of the process and can be characterized by a sharp decreasing of wave breaking activity and stabilization of wave modulations with essentially high characteristic steepness. The adjusted dissipative model based on the Nonlinear Schrödinger Equation is suggested for adequate description of the obtained experimental data. Sink/Source terms due to wave breaking processes in its right side correspond to the well known Tulin (1996) [20] model. The wave dissipation function includes the wave steepness threshold function and is applied only in the regions with active wave breaking. Permanent frequency downshift as a result of wave breaking process and post-breaking wave modulations described by the model have the satisfactory quantitative correspondence to results of experiments conducted along a super tank.

This paper describes an investigation of the generation of desired sea states in a numerical wave model. Bimodal sea states containing energetic swell components can be coastal hazards along coastlines exposed to large oceanic fetches. Investigating the effects of long-period bimodal seas requires large computational domains and increased running time to ensure the development of the desired sea state. Long computational runs can cause mass stability issues due to the Stokes drift and wave reflection, which in turn affect results through the variation of the water level. A numerical wave flume, NEWRANS, was used to investigate two wave generation methods: the wave paddle method, allowing for a smaller domain; and the internal mass source function method, providing an open boundary allowing reflected waves to leave the domain. The two wave generation methods were validated against experimental data by comparing the wave generation accuracy and the variance of mass in the model during simulations. Results show that the wave paddle method not only accurately generates the desired sea state but also provides a more stable simulation, in which mass fluctuation has less of an effect on the water depth during the long-duration simulations. As a result, it is suggested that the wave paddle method with active wave absorption is preferable to the internal wave maker option when investigating intermediate-depth long-period bimodal seas for long-duration simulations.

The present paper and its companion (Higuera et al., 2012) introduce OpenFOAM® as a tool to consider for coastal engineering applications as it solves 3D domains and considers two-phase flow. In this first paper, OpenFOAM® utilities are presented and the free surface flow solvers are analysed. The lack of specific boundary conditions for realistic wave generation is overcome with their implementation combined with active wave absorption. Wave generation includes all the widely used theories plus specific piston-type wavemaker replication. Also standalone active wave absorption implementation is explained for several formulations, all of which are applicable to 3D cases. Active wave absorption is found to enhance stability by decreasing the energy of the system and to correct the increasing water level on long simulations. Furthermore, it is advantageous with respect to dissipation zones such as sponge layers, as it does not increase the computational domain. The results vary depending on the theory (2D, Quasi-3D and 3D) but overall performance of the implemented methods is very good. The simulations and results of the present paper are purely theoretical. Comparisons with laboratory data are presented in the second paper (Higuera et al., 2012).

An experimental setup and a method to obtain the Brillouin scattering spectrum (BSS) out of optical fibers are proposed. The setup is described and experimentally validated by developing the measurement of the Brillouin spectral distribution of a birefringent microstructuted optical fiber. The setup here proposed is based on a Brillouin ring cavity that uses the fiber under test as the active medium. The measurements are obtained in base band by beating the Stokes wave with a reference wave that is taken from the optical pump. The data can be obtained with high resolution frequency.

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.

An attempt to find the exact analytical solutions of the two coupled nonlinear Schrodinger equations of 3rd order occurring from the oblique interaction of two capillary gravity wave trains in the case of crossing sea states in deep water is the main premise of the present paper. The solutions obtained here are due to the nonlinear interaction of two Stokes wave trains in one spatial dimension. Graphs have been plotted to investigate the influence of capillarity on the amplitudes of such wave trains. From 3D figures it has been observed that the capillarity has diminishing influence on the amplitudes of the either wave packet.

The nonlinear evolution equations of fourth-order have been established for two surface gravity waves in infinite depth of water including the effect of air flowing over water. In the present paper, we have applied a general approach depending upon the integral equation due to Zakharov. Based on these equations, the stability analysis has been studied in the appearance of a uniform gravity wave packet with another wave packet of the identical group velocity. Graphs have been drawn for the instability growth rate of the uniform wave packet with shorter wave number versus the perturbed wave number for several values of the amplitude of the wave packet of larger wave-number and several values of wind velocity. We have observed from the figures that the instability growth rate of the second wave packet enhances with the enhancement of nondimensional wind velocity for the settled value of the amplitude of the first wave packet.

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A laboratory experiment was conducted inside a wind wave tank to investigate the wave induced turbulence. In this experiment, the wave surface elevation and velocity beneath the water surface were measured simultaneously to investigate the relation between the wave status and wave induced turbulence. The profile of the turbulent dissipation rate and Reynolds stress were calculated using experimental data. The effect of the wave status on turbulence is investigated with regard to the wind wave, swell, and mixed wave conditions. It was depicted that the turbulence decreased with increasing depth from the water surface and that the turbulence that was induced by a wave with larger wavelength and wave height is much stronger for the same wave status. Finally, we observed that the wind wave is more effective in activating the wave induced turbulence.

Simulation of more realistic ocean conditions in wave basins is becoming important for offshore industry. As spreading wave has become more desirable, the capability of reproducing oblique planar wave train is critical for a wave basin's performance. In this study, the oblique waves have been simulated in a numerical wave basin based on the volume of fluid (VOF) method and Navier-Stokes solver. Multi-element bottom hinged flap motion using the conventional snake principle has been simulated by moving boundary dynamic mesh in OpenFOAM. Finite length of the entire wave maker and finite width of each paddle caused considerable spatial variations in wave height and wave propagating direction. Beaches with slope have been implemented to minimize the sidewall reflection and improve the uniformity of the oblique wave field. The generated linear oblique waves have been compared with the analytical solutions for validation in terms of uniformity and wave height. The time history of the surface elevation at different locations have been computed to investigate the uniformities of general wave fields. In addition, the

Theoretical modeling of stimulated Raman scattering (SRS) in fibers is presented for the near-infrared band around 2 m, where pump and Stokes wave have different absorption. This model takes into account amplified spontaneous emission (ASE), SRS towards Stokes and anti-Stokes waves, absorption of the Raman medium and Rayleigh backscattering in fibers. Depending on the fiber configuration, this model includes the cavity parameters of either external or internal mirrors at the fiber ends. Input parameters are, among others, temporal profiles of the pump radiation, absorption, and gain curve of the Raman medium. The model agrees well with experimental results obtained with a GeO 2 doped core fiber pumped by a pulsed and tunable Tm:silica fiber laser emitting around 2 m.

The second-order Stokes wave expansion is obtained for #ow over a beach of uniform slope with beach angle α = (2r − 1)π/(2m), m, r ∈ N. The "rst-order solution is written as an inverse Mellin transform and the Van Dyke principle of minimum singularity is invoked at the shoreline, whereby higher-order terms are no more singular than their predecessors. This condition establishes uniqueness at higher orders and in particular, for the "rst-order bounded standing wave, we obtain bounded second-order solutions with the presence of a quanti"able radiating second harmonic. This scattered wave is obtained as a "nite sum of inverse Mellin transforms for r = 1 and is computed for this case. For r > 1 the Mellin transform of the velocity potential may be written as a (multiple) integral.

By using a Stokes-type expansion method, a fourth-order solution has been derived for nonlinear interaction among multiple directional wave trains. Since an arbitrary number of free-wave modes with arbitrary propagation directions can be imposed as first-order inputs, this solution may demonstrate more realistic wave fields, as compared to those obtained from periodic wave theories. For the cases of one and two free-wave modes, the present theory is found to coincide with conventional theories such as the Stokes and standing wave theories, indicating the validity and general applicability of the solution. As an application, two particular wave fields are calcutidted, involving 17 free wave modes. The results indicate that the third-and fourth-order components may produce isolated large crests in random wave fields, and that the fourth-order nonlinearity significantly contributes to lowfrequency modulation bound by wave trains.

Evolution of Stokes wave side-band instability along a super tank was studied experimentally and theoretically. The initial exponential growth of the resonant side-bands is followed by asymmetrical growth rates for the side-bands: lower side-band growth is much faster, and finally the main energy is concentrated in it and the primary wave. An active breaking process increases the frequency downshift during the latter stages of the wave propagation. Some type of wave stabilization takes place during the final post-breaking stages of the process and can be characterized by a sharp decreasing of wave breaking activity and stabilization of wave modulations with essentially high characteristic steepness. The adjusted dissipative model based on the Nonlinear Schrödinger Equation is suggested for adequate description of the obtained experimental data. Sink/Source terms due to wave breaking processes in its right side correspond to the well known [20] model. The wave dissipation function includes the wave steepness threshold function and is applied only in the regions with active wave breaking. Permanent frequency downshift as a result of wave breaking process and post-breaking wave modulations described by the model have the satisfactory quantitative correspondence to results of experiments conducted along a super tank.

By using a Stokes-type expansion method, a fourth-order solution has been derived for nonlinear interaction among multiple directional wave trains. Since an arbitrary number of free-wave modes with arbitrary propagation directions can be imposed as first-order inputs, this solution may demonstrate more realistic wave fields, as compared to those obtained from periodic wave theories. For the cases of one and two free-wave modes, the present theory is found to coincide with conventional theories such as the Stokes and standing wave theories, indicating the validity and general applicability of the solution. As an application, two particular wave fields are calcutidted, involving 17 free wave modes. The results indicate that the third-and fourth-order components may produce isolated large crests in random wave fields, and that the fourth-order nonlinearity significantly contributes to lowfrequency modulation bound by wave trains.

A-coordinate non-hydrostatic model, combined with the embedded Boussinesq-type-like equations, a reference velocity, and an adapted top-layer control, is developed to study the evolution of deep-water waves. The advantage of using the Boussinesq-type-like equations with the reference velocity is to provide an analytical-based non-hydrostatic pressure distribution at the top-layer and to optimize wave dispersion property. The-based non-hydrostatic model naturally tackles the so-called overshooting issue in the case of non-linear steep waves. Efficiency and accuracy of this non-hydrostatic model in terms of wave dispersion and nonlinearity are critically examined. Overall results show that the newly developed model using a few layers is capable of resolving the evolution of non-linear deep-water wave groups.

A new formalism of spectral filtering for the description of the modulation processes is proposed. The method allows one to study the classical problem of multi-phase modulations in dispersive systems. In the present paper, deep- water waves are considered. Spectral filtering results in a system of coupled equations that describe the modulations of the carrier wave and its harmonics. The formalism may find applications in a broad range of physical situations with multi-phase dynamics.

The first two orders of stimulated Raman scattering have been observed by pumping a single-mode fiber with a cw Nd:YAG laser. Broadband first Stokes with intensity of up to a few watts has been observed as a function of the input power. The critical powers have been measured.

We present a new algorithm to numerically simulate two-dimensional viscous incompressible flows with moving interfaces. The motion is updated in time by using the backward difference formula through an iterative procedure. At each iteration, the pseudo-spectral technique is applied in the horizontal direction. The resulting semi-discretized equations constitute a boundary value problem in the vertical coordinate which is solved by decoupling growing and decaying solutions. Numerical tests justify that this method achieves fully second-order accuracy in both the temporal variable and vertical coordinate. As an application of this algorithm, we study the motion of Stokes waves in the presence of viscosity. Our numerical results are consistent with the recently published asymptotic solution for Stokes waves in slightly viscous fluids.

The variational principle of Hamilton is applied to derive the volume viscosity coefficients of a reacting fluid with multiple dissipative processes. The procedure, as in the case of a single dissipative process, yields two dissipative terms in the Navier-Stokes equation: The first is the traditional volume viscosity term, proportional to the dilatational component of the velocity; the second term is proportional to the material time derivative of the pressure gradient. Each dissipative process is assumed to be independent of the others. In a fluid comprising a single constituent with multiple relaxation processes, the relaxation times of the multiple processes are additive in the respective volume viscosity terms. If the fluid comprises several relaxing constituents ͑each with a single relaxation process͒, the relaxation times are again additive but weighted by the mole fractions of the fluid constituents. A generalized equation of state is derived, for which two special cases are considered: The case of "low-entropy production," where entropy variation is neglected, and that of "high entropy production," where the progress variables of the internal molecular processes are neglected. Applications include acoustical wave propagation, Stokes flow around a sphere, and the structure and thickness of a normal shock. Finally, it is shown that the analysis presented here resolves several misconceptions concerning the volume viscosity of fluids.

We theoretically analyse the impact of subsurface currents induced by internal waves on nonlinear Stokes surface waves. We present analytical and numerical solutions of the modulation equations under conditions that are close to group velocity resonance. Our results show that smoothing of the downcurrent surface waves is accompanied by a relatively high-frequency modulation, while the profile of the opposing current is reproduced by the surface wave&#39;s envelope. We confirm the possibility of generating an internal wave forerunner that is a modulated surface wave packet. Long surface waves can create such a wave modulation forerunner ahead of the internal wave, while other relatively short surface waves comprise the trace of the internal wave itself. Modulation of surface waves by a periodic internal wavetrain may exhibit a characteristic period that is less than the internal wave period. This period can be non-uniform while the wave crosses the current zone. Our results confirm t...

The effect of subsurface currents induced by internal waves on nonlinear surface waves is theoretically analyzed. An analytical and numerical solution of the modulation equations are found under the conditions close to the group velocity resonance. It is shown that smoothing of the down current surface waves is accompanied by a relatively highfrequency modulation while the profile of the opposing current is reproduced by the surface wave's envelope. Long surface waves can form the wave modulation forerunner ahead of the internal wave, while the relatively short surface waves create the trace of the internal wave.

Evolution of Stokes wave side-band instability along a super tank was studied experimentally and theoretically. The initial exponential growth of the resonant side-bands is followed by asymmetrical growth rates for the side-bands: lower side-band growth is much faster, and finally the main energy is concentrated in it and the primary wave. An active breaking process increases the frequency downshift during the latter stages of the wave propagation. Some type of wave stabilization takes place during the final post-breaking stages of the process and can be characterized by a sharp decreasing of wave breaking activity and stabilization of wave modulations with essentially high characteristic steepness. The adjusted dissipative model based on the Nonlinear Schrödinger Equation is suggested for adequate description of the obtained experimental data. Sink/Source terms due to wave breaking processes in its right side correspond to the well known Tulin (1996) [20] model. The wave dissipation function includes the wave steepness threshold function and is applied only in the regions with active wave breaking. Permanent frequency downshift as a result of wave breaking process and post-breaking wave modulations described by the model have the satisfactory quantitative correspondence to results of experiments conducted along a super tank.

The evaluation of the complex wave regime due to wave interaction with a large group of cylinders placed in proximity requires an efficient and accurate numerical model. This paper presents the application of a two-phase Computational Fluid Dynamics (CFD) model to carry out a detailed investigation of wave forces and flow around vertical circular cylinders placed in groups of different configurations at low Keulegan-Carpenter (KC) numbers. The 3D numerical wave tank is validated by comparing the numerical results with experimental data. Further, the hydrodynamic effects associated with three cylinders placed in tandem, side by side and in a 3 × 3 square array of nine cylinders are investigated. Wave forces are seen to reduce along the row in a tandem array. In a side-by-side arrangement, the central cylinder experiences the highest force. A combination of these effects is seen in the 3 × 3 square array. The variation of the wave forces on the cylinders in the array for different center-to-center distances and incident wavelengths is evaluated and the results show that the wave forces are the highest on the cylinders when the center-to-center distance is slightly less than half the incident wavelength.

The diffraction of highly nonlinear gravity waves by vertical cylinders is numerically simulated in the time domain. A finite-element method, based on Hamilton's principle, is used to discretize the fluid domain. The incoming waves are generated from two-dimensional numerical solution to provide a far-field closure condition on three-dimensional solution. Waves of extreme wave height are generated either by self-modulation of Stokes wave or focusing multi-component waves at a specific location.

A vertically integrated fully dispersive nonlinear wave model is expressed in curvilinear coordinates with non-orthogonal grids for the simulation of broad-banded nonlinear random water waves in regions of arbitrary geometry. The transformation is performed for both dependent and independent variables, hence an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Use of contravariant velocity components as dependent variables ensures easy and accurate satisfaction of the wall condition for lateral enclosures surrounding a physical domain, such as a coastal area, channel, or harbor. The numerical scheme is based on finitedifference approximations with staggered grids which results in implicit formulations for the momentum equations and a semi-explicit formulation for the continuity equation. Linear long wave propagation in a channel of varying crosssection and linear random wave propagation in a circular channel are presented as test cases for comparisons with the corresponding analytical solutions. Cnoidal and Stokes waves in a circular channel are also simulated as examples to nonlinear wave propagation within curved walls.