Surface Water Waves Research Articles

Forecasting a water-surface wave train with artificial intelligence (Part 2) – Can the occurrence of freak waves be predicted with AI?

In the author's previous paper published under the same title, the author showed that in a water-surface wave train, the peak height of a next-coming wave can be forecasted fairly well with recurrent neural networks (RNN). It has been also shown that the motion peak heights of a floating body in a wave train can be forecasted even better than the forecasting of wave peak heights. As the Part-2 paper, freak events such as extremely large waves or motions that are being reported taking place more often than previously thought are dealt with to investigate if such extreme events could be forecasted in advance with artificial intelligence. Two kinds of wave time series were submitted to the present work. One is that numerically produced in a computer while linearly superposing a certain number of regular waves of different frequencies and amplitudes, in which freak-like waves appeared. The other one is that measured in a real sea collected in a 4-year project conducted by Ship Research Institute (currently National Maritime Research Institute), Japan. By courtesy of the institute, the author could access quite a large amount of digitized raw data of measured wave time series and analyze them in the present study. Quite a few freak-like waves appeared in the data. As the result of the analyses conducted in the present work using the numerically produced wave time series, it has been found that the occurrence timing of freak-like waves as well as their heights could be forecasted well with RNN. On the other hand, the results of the analyses obtained using the wave data actually measured in a real sea suggest the occurrence timing of freak-like waves could be forecasted fairly well with RNN, while the peak heights of freak-like waves predicted with RNN are noticeably lower than their actual heights.

Coupling characteristics of bubbles with a free surface initially disturbed by water waves

The interactions between bubbles and water waves have important applications in ocean engineering, and their coupling characteristics are strongly associated with the wave phase angle, wavelength, and wave amplitude. Based on the assumption that the liquid is inviscid and incompressible, the coupling characteristics between bubbles and water waves are solved by the Euler equations with the finite volume method, and the bubble surface and water wave surface are tracked by the front tracking method. The accuracy of the numerical method is verified by comparison with a spark-generated bubble experiment. Compared with the bubble near the initially plane free surface, the rising height of the water spike is reduced by water waves in the crest state, where a concave shape forms on the falling water wave during bubble contraction when the wavelength λ≤ 4.00 and the wave amplitude h≥ 0.364. The rising height of the water spike is significantly strengthened by water waves in the trough state with smaller wavelengths and larger wave amplitudes, which produce a thinner and higher water spike. The bubble cycle is shortened by water waves in the crest state with smaller wavelengths and prolonged by water waves in the trough state with smaller wavelengths and larger wave amplitudes. The results presented in this paper provide guidance for the study of underwater explosions in complex water wave environments.

Numerical Simulation of Micro-Bubbles Dispersion by Surface Waves

This paper presents an algorithm for numerical modeling of bubble dispersion occurring in the near-surface layer of the upper ocean under the action of non-breaking two-dimensional (2D) surface waves. The algorithm is based on a Eulerian-Lagrangian approach where full, 3D Navier-Stokes equations for the carrier flow induced by a waved water surface are solved in a Eulerian frame, and the trajectories of individual bubbles are simultaneously tracked in a Lagrangian frame, taking into account the impact of the bubbles on the carrier flow. The bubbles diameters are considered in the range from 200 to 400 microns (thus, micro-bubbles), and the effects related to the bubbles deformation and dissolution in water are neglected. The algorithm allows evaluation of the instantaneous as well as statistically stationary, phase-averaged profiles of the carrier-flow turbulence, bubble concentration (void fraction) and void-fraction fluxes for different flow regimes, both with and without wind-induced surface drift. The simulations results show that bubbles are capable of enhancing the carrier-flow turbulence, as compared to the bubble-free flow, and that the vertical water velocity fluctuations are mostly augmented, and increasingly so by larger bubbles. The results also show that the bubbles dynamics are governed by buoyancy, the surrounding fluid acceleration force and the drag force whereas the impact of the lift force remains negligible.

Interaction of surface water waves with a finite dock over two-stepped bottom profile

Interaction of surface water waves with a finite dock over two-stepped bottom profile

Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre–Green–Naghdi equations

We consider strongly-nonlinear and weakly-dispersive surface water waves governed by equations of Boussinesq type, known as the Serre–Green–Naghdi system; it describes future states of the free water surface and depth averaged horizontal velocity, given their initial state. The lack of knowledge of the velocity field as well as the initial states provided by measurements lead to an ill-posed problem that cannot be solved by traditional techniques. To this end, we employ physics-informed neural networks (PINNs) to generate solutions to such ill-posed problems using only data of the free surface elevation and depth of the water. PINNs can readily incorporate the physical laws and the observational data, thereby enabling inference of the physical quantities of interest. In the present study, both experimental and synthetic (generated by numerical methods) training data are used to train PINNs. Furthermore, multi-fidelity data are used to solve the inverse water wave problem by leveraging both high- and low-fidelity data sets. The applicability of the PINN methodology for the estimation of the impact of water waves onto solid obstacles is demonstrated after deriving the corresponding equations. The present methodology can be employed to efficiently design offshore structures such as oil platforms, wind turbines, etc. by solving the corresponding ill-posed inverse water waves problem.

Novel localized waves and interaction solutions for a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions

The Boussinesq equation has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave overtopping, inundation, and near-shore wave process. Consequently, the study deals with the dynamics of localized waves and interaction solutions to a dimensionally reduced (2 + 1)-dimensional Boussinesq equation from N-soliton solutions by Hirota’s bilinear method. The integrability of the equation of interest was confirmed via the Painlevé test. Taking the long-wave limit approach in coordination with some constraint parameters available in the N-soliton solutions, the localized waves and interaction solutions are constructed. The interaction solutions can be obtained among the localized waves, such as (1) one breather or one lump from the two solitons, (2) one stripe and one breather, and one stripe and one lump from the three solitons, and (3) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. The interactions among the solitons are found to be elastic. The energy, phase shift, shape, period, amplitude, and propagation direction of each of the localized waves and interaction solutions are found to be influenced and controlled by the parameters involved. The dynamical characteristics of these localized waves and interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.

Field theory spin and momentum in water waves.

Spin is a fundamental yet nontrivial intrinsic angular momentum property of quantum particles or fields, which appears within relativistic field theory. The spin density in wave fields is described by the theoretical Belinfante-Rosenfeld construction based on the difference between the canonical and kinetic momentum densities. These quantities are usually considered as abstract and non-observable per se. Here, we demonstrate, both theoretically and experimentally, that the Belinfante-Rosenfeld construction naturally arises in gravity (water surface) waves. There, the canonical momentum is associated with the generalized Stokes drift phenomenon, while the spin is generated by subwavelength circular motion of water particles. Thus, we directly observe these fundamental field theory properties as microscopic mechanical properties of a classical wave system. Our findings shed light onto the nature of spin and momentum in wave fields, demonstrate the universality of relativistic field theory concepts, and offer a new platform for their studies.

On the scattering and radiation of water waves by a finite dock floating over a rectangular trench

In this study, the scattering and radiation of surface water waves by a finite dock floating over an asymmetric rectangular trench-type bottom topography are analyzed. The boundary value problems are handled for solutions analytically using a matched eigenfunction expansion method and numerically using the boundary element method. Various physical quantities associated with the scattering and radiation problems are studied for various values of wave and structural parameters. The results demonstrate that the scattering coefficients are significantly influenced due to the variation in dock length and depth ratios of the bottom trench. In the case of radiation problems, the dock length plays a crucial role in analyzing the radiation coefficients and wave excitation forces associated with each mode of motion. On the other hand, the effect of depth ratios of the bottom trench is negligible on the radiation coefficients. The present study reveals that the scattering of water waves by a floating dock over an abrupt change in bottom topography plays an important role in the construction of floating structures having more durability to resist harsh wave environments and reduce the wave impact near the seashore.

Controllability and stabilization of gravity-capillary surface water waves in a basin

<p style='text-indent:20px;'>The paper concerns the controllability and stabilization of surface water waves in a two-dimensional rectangular basin under the forces of gravity and surface tension. The surface waves are generated by a wave-maker placed at the left side-boundary and it is physical relevant to see whether the surface waves are controllable or can be stabilized using appropriate motion of the wave-maker. Due to the surface tension, an edge condition must be imposed at the contact point between the free surface and a solid boundary. Two types of wave-makers are considered: "flexible" or "rigid". It is shown that the surface waves are approximately controllable, but not exactly controllable, for both "flexible" and "rigid" wave-makers. In addition, under a static feedback to control a "rigid" wave-maker, the strong stability of feedback control system is obtained.</p><p style='text-indent:20px;'> </p><p style='text-indent:20px;'>Correction: The page numbers on each page of the PDF file have been corrected. We apologize for any inconvenience this may cause.</p>

Distribution of energy in propagation for ocean extreme wave generation in hydrodynamics laboratory

BACKGROUND AND OBJECTIVES: The hydrodynamic uncertainty of the ocean is the reason for testing marine structures as an initial consideration. This uncertainty has an impact on the natural structure of the topography as well as marine habitats. In the hydrodynamics laboratory, ships and offshore structures are tested using mathematical models as input to the wave marker. For large wavenumbers, Benjamin Bona Mahony's equation has a stable direction and position in the wave tank. During their propagation, the generated waves exhibit modulation instability and phase singularity phenomena. These two factors refer to Benjamin Bona Mahony as a promising candidate for generating extreme waves in the laboratory. The aim of this research is to investigate the distribution of energy in each modulation frequency change. The Hamiltonian formula that describes the phenomenon of phase singularity is used to observe energy. This data is critical in determining the parameters used to generate extreme waves.METHODS: The envelope of the Benjamin Bona Mahony wave group can be used to study the Benjamin Bona Mahony wave. The Benjamin Bona Mahony wave group is known to evolve according to the Nonlinear Schrodinger equation. The Hamiltonian governs the dynamics of the phase amplitude and proves the Nonlinear Schrodinger equation's singularity for finite time. The Hamiltonian is derived from the appropriate Lagrangian for Nonlinear Schrodinger and then transformed into the Hamiltonian with the displaced phase-amplitude variable.FINDINGS: Potential energy is related to wave amplitude and kinetic energy is related to wave steepness in the study of surface water waves. When , the maximum wave amplitude and steepness are obtained. When , extreme waves cannot be formed due to steepness. This is due to the possibility of breaking waves into smaller waves on the shore. In terms of position, the energy curve is symmetrical.CONCLUSION: According to Hamiltonian's description of the energy distribution, the smaller the modulation frequency, the greater the potential and kinetic energy involved in wave propagation, and vice versa. While the wave's amplitude and steepness will be greatest for a low modulation frequency, and vice versa. The modulation frequency considered as an extreme wave generator is , because the resulting amplitude is quite high and the energy in the envelope is also quite large.

A Simulation Study on the Focusing of Water Surface Waves Induced by Gradient-Index Lens

A Simulation Study on the Focusing of Water Surface Waves Induced by Gradient-Index Lens

Development of teaching materials for simulating interference of water surface waves and its practical use in classroom

Development of teaching materials for simulating interference of water surface waves and its practical use in classroom

Finite element modeling of extreme wave analysis with dissipation in arbitrary shaped domain

A finite element numerical model is presented considering the effect of refraction, diffraction, and energy dissipation by breakwater. As wave move from deep to shallow water coastal regions, wave parameter changes as a result of depth variations, presence of irregular shore boundaries, and wave reflections from the coastline. Therefore, to understand the wave transformation phenomenon near the coastal region, mild slope equation is solved using the finite element method. The numerical model is validated against the experimental and numerical results available for the rectangular domain and is applied to a real coastal Paradip port, situated in Odissa (India). The wave height is computed at the selected key stations in the port. From the simulated results, it is observed that the present numerical model with dissipation parameter is an efficient and useful tool for the analysis of water surface wave problem in realistic domains.

Hyperbolic (3+1)‐Dimensional Nonlinear Schrödinger Equation: Lie Symmetry Analysis and Modulation Instability

The hyperbolic nonlinear Schrödinger equation in the (3 + 1)‐dimension depicts the evolution of the elevation of the water wave surface for slowly modulated wave trains in deep water. Many researchers have studied the applicability and practicality of this model, but the analytical approach has been virtually absent from the literature. We adapted the lie symmetry analysis method to obtain a new complex solution in this work. The obtained complex solution contains bright and dark solitons. Furthermore, modulation instability is applied to this model to explain the interplay between nonlinear and dispersive effects. As a result, the modulation instability condition and the explosive rate are also discussed.

New bright soliton solutions for Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equations and bidirectional propagation of water wave surface

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for [Formula: see text] and [Formula: see text]-dimensional nonlinear KP-BBM equations. The simplified version of Hirota’s technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions.

The Close Relationship between Internal Wave and Ocean Free Surface Wave

A numerical model was used to simulate the propagation of internal waves (IW) along the surface layer. The results show that strong water exchange during IW propagation results in strong free surface flow and produces small but distinct free surface waves. We found a close relationship between the internal and ocean surface waves. Our intuitive reaction is that by training the relationship between the water surface wave height and the internal wave waveform, the internal wave waveform can be reversed from the water surface wave height value. This paper intends to validate our intuition. The artificial neural network (ANN) method was used to train the Fluent simulated results, and then the trained ANN model was used to predict the inner waves below by the free surface wave signal. In addition, two linear internal wave equations (I and II) were derived, one based on the Archimedes principle and the other based on the long wave and Boussinesq approximation. The prediction by equation (II) was superior to the prediction of equation (I), which is independent of depth. The predicted IW of the proposed ANN method was in good agreement with the simulated results, and the predicted quality was much better than the two linear wave formulas. The proposed simple method can help researchers infer the magnitude of IW from the free surface wave signal. In the future, the spatial distribution of IW below the sea surface might be obtained by the proposed method without costly field investigation.

Higher-dimensional extended shallow water equations and resonant soliton radiation

Higher order equations governing long surface waves in shallow water beyond the standard Korteweg-de Vries and Kadomtsev-Petviashvili equations are derived from the full water wave equations. The higher order dispersive and nonlinear terms in these equations lead to resonance between nonlinear wave structures and dispersive radiation. This resonance between solitary waves and dispersive shock waves (undular bores) is studied numerically and analytically using exponential asymptotic theory. A key result is that there is a near node in the resonant wave amplitude for the higher order equations with the water wave coefficients.

Transformation Metamaterials.

Based on the form-invariance of Maxwell's equations under coordinate transformations, mathematically smooth deformation of space can be physically equivalent to inhomogeneous and anisotropic electromagnetic (EM) medium (called a transformation medium). It provides a geometric recipe to control EM waves at will. A series of examples of achieving transformation media by artificially structured units from conventional materials is summarized here. Such concepts are firstly implemented for EM waves, and then extended to other wave dynamics, such as elastic waves, acoustic waves, surface water waves, and even stationary fields. These shall be cataloged as transformation metamaterials. In addition, it might be conceptually attractive and practically useful to control diverse waves for multi-physics designs.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

Rogue wave generation in wind-driven water wave turbulence through multiscale phase-amplitude coupling, phase synchronization, and self-focusing by curved crests

Rogue wave events (RWEs), localized high amplitude extreme events, uncertainly emerge in various nonlinear waves. For RWE generation, modulation instability leading to amplitude soliton formation for one-dimensional (1D) systems; and the additional wave directional property and the ratio of nonlinearity to spectrum bandwidth on the modulation instability for two-dimensional (2D) systems, are the accepted mechanisms. However, those studies have mainly focused on RWEs in weakly disordered wave states dominated by a single scale, but to a much lesser extent on wave turbulence with multiscale excitations. Wind-driven water surface wave turbulence widely occurs in nature. Unraveling RWE generation in wind-driven water surface wave turbulence is an important issue. Here, using multidimensional empirical mode decomposition, we experimentally investigate the dynamics of decomposed multiscale spatiotemporal waveforms of wind-driven water wave turbulence in the 2 + 1D space. We demonstrate how the cascaded amplitude modulation of the faster (higher frequency) modes by the phases of the slower modes, the phase synchronization of the largest peaks in the bursts of fast modes emerging in the crest regions of the medium modes, and self-focusing by the curved crests of the three fastest modes lead to RWE generation.