Stokes Wave Research Articles

Undular Bore Due to a Low Pressure or Bottom Trough Moving at the Critical Speed

AbstractA low pressure or bottom obstacle of negative displacement moves at the critical speed along a fluid layer of constant depth. The disturbance causes a depression of the surface at its forward position. The nonlinear dynamics generates a group of short waves attached to the disturbance. The number of two to six crests have a wavelength of 5.7–12 times the water depth. The wave height increases with the decreasing wavelength. The waves of the group do not follow the dispersion properties of cnoidal waves or Stokes waves. The group is rather characterised as an undular bore. The bore develops during a travel distance of 7–15 times the length of the disturbance. Its front eventually moves ahead of the driving disturbance where the leading crest develops into a solitary wave. A short disturbance is more powerful, but generates fewer crests compared to a long one. Comparison to the generation phase of upstream waves due to a high pressure (ship) or a bottom elevation shows that the wavelength in the two cases is approximately equal.

Peaked Stokes waves as solutions of Babenko’s equation

Babenko’s equation describes traveling water waves in holomorphic coordinates. It has been used in the past to obtain properties of Stokes waves with smooth profiles analytically and numerically. We show in the deep-water limit that properties of Stokes waves with peaked profiles can also be recovered from the same Babenko’s equation. In order to develop the local analysis of singularities, we rewrite Babenko’s equation as a fixed-point problem near the maximal elevation level. As a by-product, our results rule out a corner point singularity in the holomorphic coordinates, which has been obtained in a local version of Babenko’s equation.

An adaptive internal mass source wave-maker for short wave generation

The mass source wave-maker is commonly employed for generating water waves in numerical simulations, during which a correct amount of mass is introduced or subtracted from the internal flow region to produce target waves. The method has proven to be effective in producing waves in shallow and intermediate water depths, while its efficiency is declined for short wave generation. The main reason for this efficiency declination is that the internal mass source in deeper water region is not effective to generate short waves with their motions primarily on water surface. In order to overcome this shortcoming, many of the previous numerical treatments have introduced various enhancement factors into the source functions, which are empirically obtained and also violate the law of mass conservation. In this study, we develop a new adaptive internal wave-maker model that can be self-adjusted to suit different wave conditions. The line source starts from the bottom and extends to the computational cell right beneath free surface at each time step. The depth dependent weighting coefficient is introduced to the source function based on the linear wave theory for each wave component. No empirical coefficients are necessary, and the mass conservation is strictly and explicitly enforced. In principle, the method can be applied to all types of linear waves in the entire range of kh. The numerical experiments show that the present method can produce very good results for linear waves with kh up to 16.11, adequate for most of wave conditions in coastal engineering. For generation of fifth-order Stokes waves, the method can be extended straightforwardly for each of five wave components. For irregular waves composed of many linear wave components, different weighting coefficients can be readily calculated for each of them, respectively. As a result, the new model can generate irregular waves with overall better performance of reproducing wave spectrum, whose high-frequency part has been underestimated by previous methods. The numerical experiments also show that the new model can produce better results for focused waves where many linear waves of different frequencies start from the same point with specific phase angles, due to its capability of generating shorter wave components.

Theory for plunger-type wavemakers to generate second-order Stokes waves and Smoothed Particle Hydrodynamics verification

Plunger-type wavemakers are used in ocean engineering laboratories globally. However, the wave-making theories remain incomplete due to the complex geometries of the plungers. This work derives the motion equations for an arbitrary-geometry plunger to generate second-order Stokes waves, along with the generic constraints on wave parameters. Taking wedge-shaped and cylinder-shaped plungers as examples, the motion equations and constraints are detailed. Subsequently, a smoothed particle hydrodynamics (SPH) model for the plunger-induced waves is established and validated by accurately reproducing published experimental results. Finally, the SPH model is applied to simulate the wave generation by the two plungers. The computed wave profiles, water pressures, and flow velocities are compared with their analytical solutions in terms of spatial distribution and time history. Additionally, the wave field stability and the wave generation process are evaluated. The results demonstrate that second-order Stokes waves with target waveforms can be precisely generated using an arbitrary-geometry plunger-type wavemaker within a short transition distance and time, thereby verifying the reliability of the proposed wavemaker theory.

A venting resonator emitting monochromatic periodic traveling waves in dispersive media

Dynamically stable traveling periodic waves are notoriously difficult to realize experimentally. Manifestations of such waves inherently become unstable in any dispersive medium, whether it be at continuum length scales in fluid mechanics, in electrodynamics, or in nonlinear optics. Here, a simple experimental system is proposed where Stokes waves are emitted from a resonator whose cavity accommodates standing waves of same wavelength that is emitted out. A single characteristic wavelength is found for each driving frequency despite the coexistence of standing and traveling waves, external noise, and potentially parasitic reflections. The system's response is shown to be reliable at very small values of quality factor. A unique utility of this model system is suggested in that such a venting resonator can be used as an inhibitor of modulation instability in dispersive medium where an engineering application demands stable periodic traveling waves. Some parallels are drawn with other unbounded resonators found in society, whereby their leakiness is central to their application, such as the role of air-filled cavity to achieve impedance matching in stringed musical instruments, or energy harvesting from oscillating fluid columns in nature.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Self-similarity and recurrence in stability spectra of near-extreme Stokes waves

We consider steady surface waves in an infinitely deep two-dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 $^{\circ }$ corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.

Comparison between linear and quadratic power take off for a single chamber land-fixed oscillating water column (OWC)

This study examines the influence of turbine response type on the performance of a land-fixed Oscillating Water Column (OWC) under second-order Stokes waves of varying heights. Numerical modelling techniques, including the Finite Element Method, incompressible Reynolds-Averaged Navier–Stokes (RANS) equations with the Shear-Stress Transport (SST) k-ω turbulence model, and an Arbitrary Lagrangian–Eulerian (ALE) scheme, were employed to gain insights into OWC behaviour. The linear turbine consistently displayed reduced efficiency with increasing wave height, maintaining a stable peak non-dimensional wave number (kh). Conversely, the quadratic turbine exhibited peak efficiency within a narrower wave height range, strongly influenced by wave height. Non-dimensional volume flow rate, non-dimensional chamber pressure, chamber volume, and maximum force on the front wall remained unaffected by changes in wave height for the linear turbine. In contrast, the quadratic turbine was notably impacted by variations in wave height. Moreover, the quadratic turbine revealed stronger odd-numbered higher-order harmonic components in both force and pressure due to its non-linear response. Notably, both linear and quadratic response types exhibited sloshing at similar wave numbers, suggesting minimal influence of turbine response on sloshing behaviour. Finally, the derived non-dimensional turbine coefficients proved both turbines could scale OWC models with negligible scaling effects from the turbine.

The steady-state solution of wave–current interaction based on the third-order Stokes wave theory

This manuscript reports on the interaction of a current-free monochromatic surface wave field with a wave-free uniform current field. The existing reasonable theories of wave–current interactions are primarily based on weak current assumptions and derived from linear theory, resulting in calculation bias in the analysis of nonlinear wave–current interactions. Moreover, experimental data on high-order wave–current interactions still need to be collected. Thus, steady-state solutions named the third-order wave–current theory based on the third-order wave dispersion relationship and the principle of wave–current energy conservation were derived. The wave–current interaction experiment was set up to cover 164 sets of experimental conditions, including 33 types of periodic waves from the second to the fifth order and six different current velocities. The effects of water depth, current velocity, wave period, and height on the wave height and wavelength in the wave–current interaction field were investigated. A comparison of the mean relative error (MRE) and the determination coefficient (R2) of the wavelength with the experimental data revealed that the third-order wave–current theory outperformed the traditional linear theory, with an optimal reduction of 75% and an enhancement of 25%, respectively. Additionally, the third-order wave–current theory reduces the MRE by 25%–40% in the wave height calculation, with R2 consistently outperforming the linear theory. The third-order wave–current theory can significantly improve the calculation accuracy of the theoretical method in solving nonlinear wave–current interactions.

Numerical Study of Skipping Motion of Blended-Wing–Body Aircraft Ditching on Calm/Wavy Water

The blended-wing–body (BWB) is hoped to be the main configuration of next-generation transport aircraft. However, its large flat belly may cause a violent skipping motion during the ditching on water. In this paper, the skipping motions of a BWB aircraft ditching on calm and wavy water are numerically investigated. The finite volume method with the volume-of-fluid approach is employed to solve the unsteady Reynolds-averaged Navier–Stokes equations. A coupled method between global moving mesh and overset mesh is proposed to avoid a large background domain and improve the prediction accuracy of the free surface, and its accuracy is well validated by predicting the motions of a skipping stone, the hydrodynamic load and pressure on a flat plate in high-speed ditching, and the fifth-order Stokes wave in a dynamic wave tank. At the beginning of every surfing stage, the flat belly of BWB aircraft impacts heavily on the water surface at a smaller pitch angle and larger speed, resulting in the local vertical overload peak, which is much larger than the peak in the porpoising motion. The wavy water surface intensifies the heave and pitch motions and increases the risk of the cockpit diving into the water.

Modeling Ocean Swell and Overtopping Waves: Understanding Wave Shoaling with Varying Seafloor Topographies

One risk posed by hurricanes and typhoons is local inundation as ocean swell and storm surge bring a tremendous amount of energy and water flux to the shore. Numerical wave tanks are developed to understand the dynamics computationally. The three-dimensional equations of motion are solved by the software ‘Open Field Operation And Manipulation’ v2206. The ‘Large Eddy Simulation’ scheme is adopted as the turbulence model. A fifth-order Stokes wave is taken as the inlet condition. Breaking, ‘run-up’, and overtopping waves are studied for concave, convex, and straight-line seafloors for a fixed ocean depth. For small angles of inclination (<10°), a convex seafloor displays wave breaking sooner than a straight-line one and thus actually delivers a smaller volume flux to the shore. Physically, a convex floor exhibits a greater rate of depth reduction (on first encounter with the sloping seafloor) than a straight-line one. Long waves with a speed proportional to the square root of the depth thus experience a larger deceleration. Nonlinear (or ‘piling up’) effects occur earlier than in the straight-line case. All these scenarios and reasoning are reversed for a concave seafloor. For large angles of inclination (>30°), impingement, reflection, and deflection are the relevant processes. Empirical dependence for the setup and swash values for a convex seafloor is established. The reflection coefficient for waves reflected from the seafloor is explored through Fourier analysis, and a set of empirical formulas is developed for various seafloor topographies. Understanding these dynamical factors will help facilitate the more efficient designing and construction of coastal defense mechanisms against severe weather.

Dual-wavelength solid-state Raman laser at 555 and 579.5 nm for spectrophotometric determination of carboxyhemoglobin

For accurate measurement of carboxyhemoglobin level in the blood, a compact dual-wavelength laser at 555 and 579.5 nm with conversion efficiency up to 27.5% is originally developed by using Nd:YVO4/KGW/LBO laser with intracavity stimulated Raman scattering (SRS), second harmonic generation (SHG), and sum frequency generation (SFG). The SRS material is an Np-cut KGW crystal to produce the Stokes wave at 1159 nm from the fundamental wave at 1064 nm. The SHG of the Stokes wave and the SFG of the Stokes and fundamental waves are efficiently achieved by employing two LBO crystals. The temperature of the first LBO crystal is steadily fixed at the optimal phase-matching for generating the output power at 579.5 nm, whereas the temperature of the second one is varied to manipulate the power ratio between the yellow and green emissions. At the balanced temperature of the second LBO crystal, the output powers of the yellow and green emissions can simultaneously reach 5.5 W at a pump power of 40 W.

Numerical simulation of the influence of wave parameters on the horizontal cylinder in the nonlinear coupling of the wave–current

Considering that marine structures are frequently subjected to the combined effects of waves and currents, the wave–current coupling environment largely determines the structural load and the surrounding water–air mixed flow, which is a typical feature of offshore structures, such as ships and offshore platforms. This study focuses on the interaction between a horizontal cylinder and a free surface in a wave–current nonlinear coupled environment using numerical simulations. The numerical method is based on the incompressible Navier–Stokes equation, with the volume-of-fluid method used to capture the free surface. Based on the second-order Stokes wave theory, we studied the impact of wave height and steepness on the cylinder force, vorticity field, free-surface deformation, and spatial–temporal distribution characteristics of entrainment bubbles. The results showed that the wave height and wave steepness have opposite effects on the root mean square (RMS) value of the force and influence the amplitude and period of the force curve. The stretching of the negative vortex led to varying degrees of double-frequency oscillation modes in the force curves. The main sources of bubbles in the wake are the breaking of the free surface and the entrainment caused by the cylinder vortex, and the bubbles caused by the former account for the majority. Compared with the wave height, an increase in wave steepness can cause a more severe interface breaking, resulting in more entrainment bubbles.

Raman Lasing and Transverse Mode Selection in a Multimode Graded-Index Fiber with a Thin-Film Mirror on Its End Face.

Multimode fibers are attractive for high-power lasers if transverse modes are efficiently controlled. Here, a dielectric thin-film mirror (R~20%) is micro-fabricated on the central area of the end face of a 1 km multimode 100/140 µm graded-index fiber and tested as the output mirror of a Raman laser with highly multimode (M2~34) 940 nm diode pumping. In the cavity with highly reflective input FBG, Raman lasing of the Stokes wave at 976 nm starts at the threshold pump power of ~80 W. Mode-selective properties of mirrors with various diameters were tested experimentally and compared with calculations in COMSOL, with the optimum diameter found to be around 12 µm. The measured Raman laser output beam at 976 nm has a quality factor of M2~2 near the threshold, which confirms a rather good selection of the fundamental transverse mode. The power scaling capabilities, together with a more detailed characterization of the output beam's spatial profile, spectrum, and their stability, are performed. An approximately 35 W output power with an approximately 60% slope efficiency and a narrow spectrum has been demonstrated at the expense of a slight worsening of beam quality to M2~3 without any sign of mirror degradation at the achieved intensity of >30 MW/cm2. Further power scaling of such lasers as well as the application of the proposed technique in high-power fiber lasers are discussed.

Power optimization of high-power random Raman laser with a full-open cavity.

Since the concept of distributed feedback fiber random laser was put forward, random Raman fiber laser (RRFL) has made great progress in high power operation. For RRFL with a full-open cavity, the simplest cavity structure, further power scaling was restricted by the rapid increase of high-order Stokes wave. In this paper, we demonstrate that the output power of the RRFL can be further improved by optimizing the fiber length. The relationship between the RRFL output power and fiber length is researched theoretically and experimentally. Results show that to optimize the RRFL with a full-open cavity in output power to the best, the fiber length should be as short as possible, under the premise of avoiding causing strong four-wave mixing (FWM) and ensuring the sufficient absorption of signal light.

Numerical simulation of near-surface turn maneuver of a submarine in waves

Aiming toward a better understanding of the limitations of submarine maneuverability in complex marine environments, systematic investigations of near-surface maneuvers of a generic submarine in waves were conducted. By using the sliding grid for propeller rotation and the overset grid for control plane deflection, numerical simulations of the free running submarine model under different near-surface maneuver conditions were performed. The fifth-order Stokes wave was utilized to model the regular wave. Simulated results of near-surface self-propulsion and turn maneuvers with different wavelengths, wave heights, and submergence depths were analyzed. The results demonstrated that in near-surface turn maneuvers in waves, to balance the dramatic fluctuation of the submarine's attitude, a significant dynamic deflection of rudders with the maximum angle probably reaching the upper limit was observed. The resultant effectiveness of rudders varied greatly, which directly led to a significant difference in the trajectories. The research can provide more comprehensive guidance for the prediction and evaluation of submarine maneuverability and seaworthiness.

Compact actively Q-switched Nd:YVO4/YVO4 Raman laser at 1525 nm with frequency up to 150 kHz.

We develop a compact high-frequency actively Q-switched Nd:YVO4/YVO4 Raman laser at 1525 nm. The mode size stability and the mode overlapping are numerically analyzed to craft the resonator. Experimental results reveal that the compact cavity and the cavity dumping effect lead to the considerable narrowing of the pulse width. In addition, the quality factor of the cavity is significantly strengthened by using the YVO4 Raman crystal with a dichroic coating to minimize the scattering and absorption losses for the Stokes wave. Furthermore, we investigate the influence of the gate-on time of the Q-switcher on the output performance. Under the optimal condition, the average output power can be generally greater than 4.2 W at the pump power of 26 W for the repetition rate within 50-150 kHz, and the corresponding optical efficiency higher than 16.1%. The maximum peak powers can reach 53 kW and 25 kW for the repetition rates of 50 kHz and 100 kHz, respectively.

SBS suppression in fiber amplifiers with a broadband seed.

We present a scalar, time-dependent, plane-wave model for stimulated Brillouin scattering (SBS) within a fiber amplifier having a seed linewidth comparable to, or greater than, the Brillouin frequency shift. The broadband model introduces the existence of a backward anti-Stokes wave, in addition to the Stokes wave present in the narrowband model. The model also incorporates a Fresnel reflection within the fiber or at the exit face. In the absence of Fresnel reflections, the SBS threshold increases linearly with seed linewidth, up to linewidths of at least three times the Brillouin frequency shift. In the presence of Fresnel reflections that seed the Stokes wave, the threshold increases sub-linearly, which agrees with previously reported experimental results. For non-zero reflections and modulation formats with very compact spectra, the threshold can decrease with bandwidth in the region 16-24 GHz due to seeding of the Stokes wave.

A model ODE for the exponential asymptotics of nonlinear parasitic capillary ripples

Abstract In this work, we develop a linear model ordinary differential equation (ODE) to study the parasitic capillary ripples present on steep Stokes waves when a small amount of surface tension is included in the formulation. Our methodology builds upon the exponential asymptotic theory of Shelton & Trinh (J. Fluid Mech., vol. 939, 2022, A17), who demonstrated that these ripples occur beyond-all-orders of a small-surface-tension expansion. Our model equation, a linear ODE forced by solutions of the Stokes wave equation, forms a convenient tool to calculate numerical and asymptotic solutions. We show analytically that the parasitic capillary ripples that emerge in solutions to this linear model have the same asymptotic scaling and functional behaviour as those in the fully nonlinear problem. It is expected that this work will lead to the study of parasitic capillary ripples that occur in more general formulations involving viscosity or time-dependence.

Demonstration of standing cavity Brillouin random fiber lasers using double fiber Bragg grating arrays.

Bidirectional feedback by fiber Bragg grating arrays (FBGAs) reduced the loss of the cavity and increased stimulated Brillouin scattering (SBS) gain by bi-directional Stokes wave through FBGA associated Rayleigh feedback of the pump wave. As a result, the Q value of the Brillouin random fiber laser (BRFL) increased significantly, which leads to narrow linewidth. This is different from the ring configuration with unidirectional SBS gain versus dual SBS gain of the same fiber length. Highly efficient use of the SBS gain fiber for coherent SBS amplification suppressed thermal noise associated Stokes wave. Such an efficient SBS laser is realized by a standing cavity BRFL based on double FBGAs. Multiple scattering of light traveling in strong scattering FBGAs enables light localization and the generation of high-Q reflection peaks. Coherent SBS amplification with high Q help to reduce laser relative intensity noise (RIN) and laser linewidth. Experimental results demonstrate that the BRFL supports localized modes by increasing the scattering strength of the FBGA random feedback, resulting in long lifetime and single-frequency emission with 20 dB noise floor reduction. The BRFL with a 1 km Brillouin gain fiber exhibits lower RIN and narrower linewidth than that with a 10 km Brillouin gain fiber due to the stronger gain competition of more modes in the longer cavity length. The optimized standing caivty BRFL with 1 km gain fiber leads to 3.5 kHz linewidth versus 40 kHz from the pump laser. These findings provide experimental evidence that double FBGAs offer a unique setting to control mode dynamics, realizing low-noise single-frequency lasing.