Surface Gravity Waves In Water Research Articles

General solution for linear 2D anisotropic surface gravity water waves on uniform current

Measuring and modeling ocean waves is a routine task for oceanographers and marine engineers. Accounting for current effects on waves is less common; however, such effects can be non-trivial, particularly for strong currents. Recent research has also found that currents are increasing with global temperatures. Thus, accounting for currents is of potential importance to a growing subset of real-world environments. One significant impact of currents is their modification of dispersion. It is known that, in contrast to dispersion without current, wave–current dispersion is multivalued and anisotropic, meaning it varies with direction and can yield multiple solutions for the same frequency. Strictly speaking, a complete analysis ought to account for all dispersion solutions. This study presents a general wave field solution for linear steady-state two-dimensional surface gravity waves on current. In contrast to existing formulations, it accounts for the complete set of dispersion solutions. The multiple solutions correspond to linearly independent (same frequency) spatial modes with no analogy in models without current. To fully determine the complete temporal and spatial features of a wave field, one must determine the amplitudes of all spatial modes. This can be achieved through the solution of an inverse problem, provided one has sufficient initial conditions. The significance of accounting for all spatial modes is demonstrated through examples. It is shown that a single time-series measurement is insufficient for determining the spatial features of a monochromatic (single frequency) wave. Rather, additional spatial information must be introduced to render the problem well-posed. The theory and methodology presented can be used to improve wave models and measurements.

Correction: Lin, G.; Huang, H. The Dynamical and Kinetic Equations of Four-Five-Six-Wave Resonance for Ocean Surface Gravity Waves in Water with a Finite Depth. Symmetry 2024, 16, 618

Addition of an author and his email [...]

Observation of a phase space horizon with surface gravity water waves

In 1974, Stephen Hawking predicted that quantum effects in the proximity of a black hole lead to the emission of particles and black hole evaporation. At the very heart of this process lies a logarithmic phase singularity which leads to the Bose-Einstein statistics of Hawking radiation. An identical singularity appears in the elementary quantum system of the inverted harmonic oscillator. In this Letter we report the observation of the onset of this logarithmic phase singularity emerging at a horizon in phase space and giving rise to a Fermi-Dirac distribution. For this purpose, we utilize surface gravity water waves and freely propagate an appropriately tailored energy wave function of the inverted harmonic oscillator to reveal the phase space horizon and the intrinsic singularities. Due to the presence of an amplitude singularity in this system, the analogous quantities display a Fermi-Dirac rather than a Bose-Einstein distribution.

The Dynamical and Kinetic Equations of Four-Five-Six-Wave Resonance for Ocean Surface Gravity Waves in Water with a Finite Depth

Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions, the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii.

Bohmian mechanics of the three-slit experiment in the linear potential

We report on a three-slit experiment in the presence of a linear potential with surface gravity water waves. For these classical waves, we reconstruct the Bohm trajectories as well as the corresponding quantum potentials.

Machine learning simulation of one-dimensional deterministic water wave propagation

Deterministic phase-resolved prediction of the evolution of surface gravity waves in water is challenging due to their complex spatio-temporal dynamics. Physics-based methods of varying complexity are available, but the conflicting objectives of numerical efficiency and accuracy impede real-time wave prediction. Data-driven methods may be able to overcome this challenge by using training data generated by complex numerical methods. This work explores the potential of a machine learning (ML) approach based on a fully convolutional encoder–decoder architecture for the efficient and accurate prediction of water waves. The high-order spectral (HOS) method forms the foundation for the generation of the training data. The HOS method is applied for different, consecutive orders of nonlinearity starting from first order up to fourth order. The JONSWAP wave energy spectrum serves as the basis for modeling the one-dimensional irregular sea states. The overall objective of this work is to evaluate whether the complex non-linear physical processes can be identified and learned by the ML approach. The trained ML flow mapper is used to perform time integration of an initial sea state. The results indicate that the proposed ML approach is able to reproduce the distinctive physical processes of the different orders of nonlinearities. It is shown that the ML approach enables fast and accurate predictions of one-dimensional waves over a time horizon that spans multiple peak periods.

Observation of Bohm trajectories and quantum potentials of classical waves

In 1952 David Bohm proposed an interpretation of quantum mechanics, in which the evolution of states results from trajectories governed by classical equations of motion but with an additional potential determined by the wave function. There exist only a few experiments that test this concept and they employed weak measurement of non-classical light. In contrast, we reconstruct the Bohm trajectories in a classical hydrodynamic system of surface gravity water waves, by a direct measurement of the wave packet. Our system is governed by a wave equation that is analogous to the Schrödinger equation which enables us to transfer the Bohm formalism to classical waves. In contrast to a quantum system, we can measure simultaneously their amplitude and phase. In our experiments, we employ three characteristic types of surface gravity water wave packets: two and three Gaussian temporal slits and temporal Airy wave packets. The Bohm trajectories and their energy flows follow the valleys and bounce off the hills in the corresponding quantum potential landscapes.

Soliton Solution for the Korteweg-de Vries Equation

Korteweg de Vries (KdV) model is considered quintessential in modeling the surface gravity water waves in shallow water. In this project, we are interested in starting from the Elliptic Jacobian Functions, and performing a complete analysis of these functions to discover that one can recover the soliton in the particular case, when m approaches 1, wherem is a parameter between 0 and 1 in the definition of the Elliptic Jacobian Functions. This analysis will provide us with an understanding of cnoidal periodic waves and how, through them, we can derive the soliton solution. Finally, this project grants readers a deeper understanding of the origin of solitons and their applications in water wave theory.

Periodic Wave Trains in Nonlinear Media: Talbot Revivals, Akhmediev Breathers, and Asymmetry Breaking

We study theoretically and observe experimentally the evolution of periodic wave trains by utilizing surface gravity water wave packets. Our experimental system enables us to observe both the amplitude and the phase of these wave packets. For low steepness waves, the propagation dynamics is in the linear regime, and these waves unfold a Talbot carpet. By increasing the steepness of the waves and the corresponding nonlinear response, the waves follow the Akhmediev breather solution, where the higher frequency periodic patterns at the fractional Talbot distance disappear. Further increase in the wave steepness leads to deviations from the Akhmediev breather solution and to asymmetric breaking of the wave function. Unlike the periodic revival that occurs in the linear regime, here the wave crests exhibit self acceleration, followed by self deceleration at half the Talbot distance, thus completing a smooth transition of the periodic pulse train by half a period. Such phenomena can be theoretically modeled by using the Dysthe equation.

Step approximation on oblique water wave scattering and breaking by variable porous breakwaters over uneven bottoms

In this study, the scattering of oblique water surface gravity waves by variable porous breakwaters over uneven bottoms is investigated using the eigenfunction matching method (EMM). In the EMM procedure, the variable breakwaters and bottom profiles were sliced into shelves separated by steps. The solutions on the shelves are composed of eigenfunctions. Then, by applying the conservations of mass and momentum, a system of linear equations is obtained and can be solved with a sparse-matrix solver. The effects of wave breaking and dissipation were also included in the EMM formulation using energy-dissipation factors. In order to validate the proposed EMM method, the numerical solutions were compared with the results in the literatures for non-breaking wave scattering by rectangular and parabolic porous breakwaters. The convergence of the proposed solutions was studied. Then some discussions were provided when the EMM with energy-dissipation factors was applied to evaluate the surface elevations of breaking and/or dissipated waves. The numerical results illustrated the potential application of the EMM for problems with wave breaking and dissipation if the empirical formulas for the energy-dissipation factors can be well calibrated in future researches. Some further applications were provided, including wave scattering by multiple porous breakwaters.

Spatial deterministic wave forecasting for nonlinear sea–states

We derive a simple algebraic form of the nonlinear wavenumber correction of unidirectional surface gravity waves in deep water, based on temporal measurements of the water surface and the spatial Zakharov equation. This allows us to formulate an improvement over linear deterministic wave forecasting with no additional computational cost. Our new formulation is used to forecast both synthetically generated as well as experimentally measured seas and shows marked improvements over the linear theory.

Diffractive Guiding of Waves by a Periodic Array of Slits.

We show that in order to guide waves, it is sufficient to periodically truncate their edges. The modes supported by this type of wave guide propagate freely between the slits, and the propagation pattern repeats itself. We experimentally demonstrate this general wave phenomenon for two types of waves: (i)plasmonic waves propagating on a metal-air interface that are periodically blocked by nanometric metallic walls, and (ii)surface gravity water waves whose evolution is recorded, the packet is truncated, and generated again to show repeated patterns. This guiding concept is applicable for a wide variety of waves.

Stabilization of Unsteady Nonlinear Waves by Phase-Space Manipulation

We introduce a dynamic stabilization scheme universally applicable to unidirectional nonlinear coherent waves. By abruptly changing the waveguiding properties, the breathing of wave packets subject to modulation instability can be stabilized as a result of the abrupt expansion a homoclinic orbit and its fall into an elliptic fixed point (center). We apply this concept to the nonlinear Schrödinger equation framework and show that an Akhmediev breather envelope, which is at the core of Fermi-Pasta-Ulam-Tsingou recurrence and extreme wave events, can be frozen into a steady periodic (dnoidal) wave by a suitable variation of a single external physical parameter. We experimentally demonstrate this general approach in the particular case of surface gravity water waves propagating in a wave flume with an abrupt bathymetry change. Our results highlight the influence of topography and waveguide properties on the lifetime of nonlinear waves and confirm the possibility to control them.

Projectile motion of surface gravity water wave packets: An analogy to quantum mechanics

We study phase contributions of wave functions that occur in the evolution of Gaussian surface gravity water wave packets with nonzero initial momenta propagating in the presence and absence of an effective external linear potential. Our approach takes advantage of the fact that in contrast to matter waves, water waves allow us to measure both their amplitudes and phases.

Phase modulated domain walls and dark solitons for surface gravity waves

We report theoretical prediction of localized solutions for dynamics of surface gravity waves, at the critical point kh≈1.363, modelled by higher-order nonlinear Schrödinger equation. The model possesses domain walls (kink solitons) and dark solitons modulated through different phase profiles. The parametric domains are delineated for the existence of soliton solutions. The effects of wave parameters have been discussed on the amplitude of surface gravity waves. Our work is motivated by Tsitoura et al. [1], on experimental and analytical observation of phase domain walls for deep water surface gravity waves modelled by nonlinear Schrödinger equation.

Analog Hawking effect: BEC and surface waves

We take into account two further physical models which play an utmost importance in the framework of Analogue Gravity. We first consider Bose--Einstein condensates (BEC) and then surface gravity waves in water. Our approach is based on the use of the master equation we introduced in a previous work. A more complete analysis of the singular perturbation problem involved, with particular reference to the behavior in the neighbourhood of the (real) turning point and its connection with the WKB approximation, allows us to verify the thermal character of the particle production process. Furthermore, we can provide a simple scheme apt to calculate explicitly the greybody factors in the case of BEC and surface waves. This corroborates the improved approach we proposed for studying the analogue Hawking effect in the usual limit of small dispersive effects.

On the distribution of maximum crest and wave height at intermediate water depths

We report new descriptions for the (probability) distributions of hourly maximum crest and wave height of water surface gravity waves for intermediate water depths. Estimated distributions are based on analysis of laboratory-scale measurements at the DHI wave basin. For a given sea state, the distribution of both hourly maximum crest and hourly maximum wave height, normalised by sea state significant wave height, is found to follow a generalised extreme value (GEV) distribution. Variation of the three parameters of the GEV distribution across sea states, is expressed in terms of a response surface model as a function of non-dimensional sea state Ursell number and wave steepness, and wave directional spreading angle. For inference, conventional Monte Carlo wave basin measurements are supplemented with measurements selected by means of a novel “pre-selection” sampling scheme using numerical simulations. This scheme effectively guarantees that extreme events from tails of distributions are produced, and reduces uncertainties associated with the estimated distributions. Estimation is performed using Bayesian inference, allowing uncertainties to be quantified, and providing estimates of posterior predictive tail distributions for sea states with arbitrary characteristics within the domain of sea state characteristics covered by the model.

A Unified Breaking Onset Criterion for Surface Gravity Water Waves in Arbitrary Depth

AbstractWe investigate the validity and robustness of the Barthelemy et al. (2018, https://doi.org/10.1017/jfm.2018.93) wave‐breaking onset prediction framework for surface gravity water waves in arbitrary water depth, including shallow water breaking over varying bathymetry. We show that the Barthelemy et al. (2018) breaking onset criterion, which they validated for deep and intermediate water depths, also segregates breaking crests from nonbreaking crests in shallow water, with subsequent breaking always following the exceedance of their proposed generic breaking threshold. We consider a number of representative wave types, including regular, irregular, solitary, and focused waves, shoaling over idealized bed topographies including an idealized bar geometry and a mildly to steeply sloping planar beach. Our results show that the new breaking onset criterion is capable of detecting single and multiple breaking events in time and space in arbitrary water depth. Further, we show that the new generic criterion provides improved skill for signaling imminent breaking onset, relative to the available kinematic or geometric breaking onset criteria in the literature. In particular, the new criterion is suitable for use in wave‐resolving models that cannot intrinsically detect the onset of wave breaking.

Stabilization of uni-directional water wave trains over an uneven bottom

We study the evolution of nonlinear surface gravity water wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approximation and validated by NLSE simulations. Our result will contribute to understand the dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media.

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces , , to a class of nonlinear, dispersive evolution equations of the formwhere the dispersion is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse and the nonlinearity is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves and Wahlén on a class of equations which includes Whitham’s model equation for surface gravity water waves featuring the exact linear dispersion relation. Tools involve constrained variational methods, Lions’ concentration-compactness principle, a strong fractional chain rule for composition operators of low relative regularity, and a cut-off argument for which enables us to go below the typical regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when is nonnegative, and provide a nonexistence result when is too strong.