Random Wave Field Research Articles

Nonlinear random wave fields within a Boussinesq system

The dynamics of nonlinear random fields is important for understanding wave turbulence. In this work, we use a Boussinesq system to examine the distinctions between unidirectional and bidirectional waves. Our study demonstrates that in both scenarios, the wave spectra reach a stationary state. Moreover, we show that the occurrence of rogue waves is more probable in the unidirectional case. In the unidirectional case, the probability distribution of wave crests exceeds the one predicted by the Rayleigh distribution once the spectra reach the stationary state. Conversely, in the bidirectional case, the opposite trend is observed. The discovery of various types of rogue waves, including massive wave trains commonly known in the literature as “two sisters” and “three sisters” are found.

Statistics of Extreme Events in Integrable Turbulence.

We use the spectral kinetic theory of soliton gas to investigate the likelihood of extreme events in integrable turbulence described by the one-dimensional focusing nonlinear Schrödinger equation (fNLSE). This is done by invoking a stochastic interpretation of the inverse scattering transform for fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear wave field in terms of the spectral density of states of the corresponding soliton gas. We then apply the general result to two fundamental scenarios of the generation of integrable turbulence: (i)the asymptotic development of the spontaneous modulational instability of a plane wave, and (ii)the long-time evolution of strongly nonlinear, partially coherent waves. In both cases, involving the bound state soliton gas dynamics, the analytically obtained values of the kurtosis are in perfect agreement with those inferred from direct numerical simulations of the fNLSE, providing the long-awaited theoretical explanation of the respective rogue wave statistics. Additionally, the evolution of a particular nonbound state gas is considered, providing important insights related to the validity of the so-called virial theorem.

Evolution and Statistical Analysis of Internal Random Wave Fields within the Benjamin–Ono Equation

This study investigates the numerical evolution of an initially internal random wave field characterized by a Gaussian spectrum shape using the Benjamin–Ono (BO) equation. The research focuses on analyzing various properties associated with the random wave field, including the transition to a steady state of the spectra, statistical moments, and the distribution functions of wave amplitudes. Numerical simulations are conducted across different Ursell parameters, revealing intriguing findings. Notably, it is observed that the spectra of the wave field converge to a stationary state in a statistical sense, while exhibiting statistical characteristics that deviate from a Gaussian distribution. Moreover, as the Ursell parameter increases, the positive skewness of the wave field intensifies, and the kurtosis increases. The investigation also involves the computation of the probability of rogue wave formation, revealing deviations from the Rayleigh distribution. Notably, the study uncovers distinct types of internal rogue waves, specifically referred to as the “two sisters” and “three sisters” phenomena.

THE OCCURRENCE OF EXTREME WAVE HEIGHT IN A TWO-DIMENSIONAL RANDOM WAVEFIELD IN COASTAL AREA

As the water waves come to the coastal area, the variation in dispersion relation due to the spatial inhomogeneity brings about the deformation of the wave shape. With the local bathymetry effect, the propagation of the wave experiences a complicated evolution process, including the reflection, wave shoaling and breaking, etc. Considering the directional spreading effect and wave refraction becomes indispensable when it comes to a two-dimensional (2D) problem in the random wavefield. Mori et al. (2011) indicated that the distribution of maximum wave height in deep-water is decided by the directional dispersion. In this research, we aim to discuss the occurrence of maximum wave height in the 2D wave evolution in the coastal area. It helps to estimate better extreme events in the prevention work of coastal hazards. To simulate the 2D propagation, we develop a nonlinear model as an extension work of Lyu et al. (2021), which considers the high order harmonic interactions of irregular waves before the breaking in surf zone.

Comparison of schlieren-based techniques for measurements of a turbulent and wavy free surface

The free surface at an air–water interface can provide information regarding bathymetric complexities, as well as the subsurface flow. We present a comparison of the performance of two recent advances in light-based optical techniques for free surface measurements, total internal reflection-deflectometry and moon-glade background-oriented schlieren, with the more established method of free surface synthetic schlieren. We make use of an optical flow algorithm over the more traditional digital image correlation, in order to obtain higher spatial resolution data across the imaged free surface domain. The optical flow algorithm presents additional benefits, such as computational efficiency and robustness in capturing large displacements and straining of tracked features. The three optical techniques are assembled in synchronization to image two free surface conditions: (1) a free surface being impinged upon by an underlying turbulent, free-shear flow and (2) a random and irregular wave field induced by a free jet. Using the high-resolution measurements, we provide insight on the emergence of multiple free surface dynamics for a turbulent free surface. We present a comprehensive discussion on the benefits and drawbacks of each technique, including suggestions on the suitability of each technique for several experimental constraints.Graphical abstract

Freak wave in a two-dimensional directional wavefield with bottom topography change. Part 1. Normal incidence wave

In the propagation and evolution of sea waves, previous studies pointed out that the occurrence of the freak wave height is significantly related to the quasi-resonant four-wave interaction in the modulated waves. From numerical--experimental study over an uneven bottom, the nonlinear effect caused by the bathymetry change also contributes to the occurrence of extreme events in unidirectional waves. To comprehensively analyse the two-dimensional wavefield, this study develops an evolution model for a directional random wavefield based on the depth-modified nonlinear Schrödinger equation, which considers the nonlinear resonant interactions and the wave shoaling the shallow water. Through Monte Carlo simulation, we discuss the directional effect on the four-wave interaction in the wave train and the maximum wave height distribution from deep to shallow water with a slow varying slope. The numerical result indicates that the directional spreading has a dispersion effect on the freak wave height. In a shallow-water environment, this effect becomes weak, and the bottom topography change is the main influencing factor in the wave evolution.

Non-Gaussian Statistics of Mechanically Generated Unidirectional Irregular Waves

An experimental study is presented on the non-Gaussian statistics of random unidirectional laboratory wave fields described by JONSWAP spectra. Relationships between statistical parameters indicative of the occurrence of large-amplitude waves are discussed in the context of the initial steepness of the waves combined with the effect of spectral peakedness. The spatial evolution of the relevant statistical and spectral parameters and features is also considered. It is demonstrated that over the distance the spectra exhibit features typical for developing nonlinear instabilities, such as spectral broadening and downshift of the peak, along with lowering of the high-frequency tail and decrease of the peak magnitude. The wave fields clearly show an increase of third-order nonlinearity with the distance, which can be significant, depending on the input wave environment. The steeper initial conditions, however, while favouring the occurrence of extremely large waves, also increase the chances of wave breaking and loss of energy due to dissipation, which results in lower extreme crests and wave heights. The applied Miche-Stokes-type criteria do confirm that some of the wave extremes exceed the limiting individual steepness. Eventually, this result agrees with the observation that the largest number of abnormal waves is recorded in sea states with moderate steepness.

Efficient computation of temporal exceeding probability of ship responses in a random wave field

In this work, we develop a computational framework to efficiently quantify the temporal exceeding probability of ship responses in a random wave field generated by a narrow-band spectrum, i.e., the fraction of time that ship responses exceed a given threshold. In particular, we consider large thresholds so that the response exceedance needs to be treated as rare events. Our computational framework builds on the parameterization of wave field into wave groups and efficient sampling in the group parameter space through Bayesian experimental design (BED). Previous works following this framework exclusively studied extreme statistics of group-maximum response, i.e., the maximum response within a wave group, which is however not straightforward to interpret in practice (e.g., its value depends on the subjective definition of the wave group). In order to adapt the framework to the more robust measure by the temporal exceeding probability, novel developments need to be introduced to the two components of BED (on surrogate model and acquisition function), which are detailed in the paper. We validate our framework for a case of ship roll motion (without capsizing) calculated by a nonlinear roll equation, in terms of the agreement of our results to the true solution and the independence of our results to the criterion of defining groups. Finally, we demonstrate the coupling of the framework with CFD models to handle more realistic and general ship response problems.

Effects of finite non-Gaussianity on evolution of a random wind wave field.

We examine the long-term evolution of a random wind wave field generated by constant forcing, by comparing numerical simulations of the kinetic equationand direct numerical simulations (DNS) of the dynamical equations. While the integral characteristics of the spectra are in reasonably good agreement, the spectral shapes differ considerably at large times, the DNS spectral shape being in much better agreement with field observations. Varying the number of resonant and approximately resonant wave interactions in the DNS numerical scheme, we show that when the ratio of nonlinear and linear parts of the Hamiltonian tends to zero, the DNS spectral shape approaches the shape predicted by the kinetic equation. We attribute the discrepancies between the kinetic equationmodeling, on one side, and the DNS and observations, on the other, to the neglect of non-Gaussianity in the derivation of the kinetic equation.

Surface Gravity Wave Interferometry and Ocean Current Monitoring With Ocean‐Bottom DAS

AbstractThe cross‐correlation of a diffuse or random wavefield at two points has been demonstrated to recover an empirical estimate of the Green's function under a wide variety of source conditions. Over the past two decades, the practical development of this principle, termed ambient noise interferometry, has revolutionized the fields of seismology and acoustics. Yet, because of the spatial sparsity of conventional water column and seafloor instrumentation, such array‐based processing approaches have not been widely utilized in oceanography. Ocean‐bottom distributed acoustic sensing (OBDAS) repurposes pre‐existing optical fibers laid in seafloor cables as dense arrays of broadband strain sensors, which observe both seismic waves and ocean waves. The thousands of sensors in an OBDAS array make ambient noise interferometry of ocean waves straightforward for the first time. Here, we demonstrate the application of ambient noise interferometry to surface gravity waves observed on an OBDAS array near the Strait of Gibraltar. We focus particularly on a 3‐km segment of the array on the continental shelf, containing 300 channels at 10‐m spacing. By cross‐correlating the raw strain records, we compute empirical ocean surface gravity wave Green's functions for each pair of stations. We first apply beamforming to measure the time‐averaged dispersion relation along the cable. Then, we exploit the non‐reciprocity of waves propagating in a flow to recover the depth‐averaged current velocity as a function of time using a waveform stretching method. The result is a spatially continuous matrix of current velocity measurements with resolution <100 m and <1 hr.

Automated acoustic arrival matching using a convolutional neural network approach informed by statistics of acoustic scattering from internal waves

A machine learning model was developed to automatically align underwater acoustic measurements taken at various depths and ranges from a transmitting source in the Philippine Sea to a reference model of long range acoustic arrival structure, simultaneously determining source-receiver range and travel-time offsets associated with multipath arrivals. Ocean sound-speed variability complicates the task as the measured arrivals may exhibit scattering not present in range-independent predictions. Monte Carlo style broadband parabolic equation simulations through random internal wave fields consistent with the Garrett-Munk internal wave energy spectrum were used to generate a large data set of simulated acoustic receptions including scattered multipath arrivals with known source-receiver ranges and imposed travel time offsets. These simulated receptions were used to train and evaluate the machine learning model for arrival pattern matching to the reference model. The inclusion of various data dimensions, such as peak amplitude and width, and contextual information, such as range and depth, were also explored as input to the model. Ranging results for the machine learning model were compared to a programmatic solution engineered for the same task.

Estimating space–time wave statistics using a sequential sampling method and Gaussian process regression

The traditional Monte Carlo sampling of waves requires generating tens of thousands of random waves to achieve stable statistics in the tail of the distribution, which is computationally costly for numerical simulations and time-consuming and often impractical for experiments. To improve the sampling efficiency, we present a sequential sampling method, which predicts the wave statistics based on the nonlinear response of deterministic wave groups. This data-driven method starts with parameterising the linear random wave field with a series of Gaussian wave groups. The nonlinear system response from the water wave system is then approximated by observations through the nonlinear evolution of wave groups with carefully designed initial conditions. A sequential sampling method is introduced during this sampling procedure, which determines the next best sampling point based on the previous observations and the distance metric. We examine the performance of the proposed method with Monte Carlo simulation of random wave fields as well as a grid search sampling strategy. The results show the proposed method can achieve computational cost savings over several orders of magnitude for numerical simulations. The results also suggest such a sequential sampling method can be used to determine the test matrix of the wave experiments with better coverage over the parameter space.

Wave action on a vertical wall with a submerged horizontal plate: Analysis of phase variation of forces and probability of exceedance

In this study, experiments on the wave forces and moments acting on a Vertical Wall (VW) and a VW attached to a horizontal plate (VWHP) subjected to regular and random wave fields were conducted. In the experiments, the phase lag between the horizontal and vertical wave forces was analyzed for the regular waves and the probability density was analyzed for the random waves. The experimental results showed that attaching a horizontal plate with a length equal to the water depth to a VW at a position corresponding to 28.5% of the water depth reduced the 95% non-exceedance value of the normalized horizontal wave force from 1.05 to 0.8 for a relative water depth of d/Lp = 0.1, where d is the water depth and Lp is the wavelength of the peak period. For d/Lp = 0.5, the corresponding reduction was from 0.6 to 0.17. The results of this study can be used to optimize the design of VW- and VWHP-type marine structures.

Statistical distributions of nonlinear waves from random laboratory wave fields

The research focuses on the probabilistic structure of individual wave parameters from deep water laboratory experiments. The studied sea states are characterized by the JONSWAP frequency spectra with different energy content and steepness which offer favourable conditions for intense quasi-resonant modulations resulting in enhanced probability for unusually large wave events. Analysis of the results at different stages of the instability process, as the waves propagate along the tank, confirms the increasing influence of third-order nonlinearity on the extreme wave statistics. In particular, the tails of the empirical distributions over the last gauges consist of a growing number of waves fulfilling the common criteria for abnormality. Moreover, non-skewed surface profiles corresponding to initially large coefficients of kurtosis continue to show substantial deviations from the Rayleigh distribution as a result of the nonlinear exchange of energy between free wave modes. The uncertainty when predicting the empirical tails supports the use of statistical models for the wave amplitudes and heights based on the Gram-Charlier series expansions which demonstrate a generally good agreement, particularly with the crest-to-trough wave heights from the moderately steep sea states.

Development of a Probability Model for the Transmission Coefficient of Low Crested Breakwaters (LCB)

Park, S.-H.; Choi, J.-G., and Cho, Y.-J., 2021. Development of a probability model for the transmission coefficient of Low Crested Breakwaters (LCB). In: Lee, J.L.; Suh, K.-S.; Lee, B.; Shin, S., and Lee, J. (eds.), Crisis and Integrated Management for Coastal and Marine Safety. Journal of Coastal Research, Special Issue No. 114, pp. 484–488. Coconut Creek (Florida), ISSN 0749-0208. Low Crested Breakwaters (LCB) have been the most preferred structural type among many countermeasures against beach erosion. The extent of shore protection by LCB depends on transmittance coefficients, which are random due to the variability inherent in the marine environment. However, a probability model for transmittance coefficients is hard to find in the literature, indispensable for the reliability-based optimal design of LCB. In this rationale, first, a probability model of transmission coefficients is empirically developed. In doing so, transmittance characteristics of LCB were examined using the situ-wave data at the up-wave and down-wave sides of LCB at Bong-Po and Sok-Cho from 2019.8.6 to 2019.8.21 in order to identify the pertinent random variables affecting LCB transmittance. It turns out that sea wave conditions significantly affect the transmittance coefficients, and the roughness of sea wave conditions can be quantified in terms of wave height and its associated wave slope. Wave height and its associated wave slope are highly correlated in the mild sea, but they behave independently as wave heights get more enhanced. These complicated interrelations between wave height and wave slope can be described using the joint distribution of wave amplitude and its associated period by Longuet-Higgins (1983). In this rationale, we analytically derived a probability model for transmittance coefficients of LCB from the joint distribution of wave amplitude and its associated period by Longuet-Higgins (1983) and d'Angremond, van der Meer, and de Jong (1996) model using the standard technique of transformation of random variables (Papoulis, 1984). Numerical simulation shows that as sea wave conditions are getting harsh, a non-negligible probability mass shifts toward the low transmittance coefficients due to long waves appearing in the random wave field by sub-harmonic resonance wave-wave interaction, which is well-known after the studies of Hasselmann(1967) and Phillips(1980) on the development of wind waves. It was also shown that the probability mass shift toward the low transmission coefficients is accompanied by the probability mass shift toward higher transmission coefficients, which also comply with the studies of Hasselmann (1967) and Phillips (1980).

Freak waves caused by reflection

We study the temporal distribution of wave energy in a wave field that is generated by the reflection of a wave spectrum from a vertical wall. Weakly nonlinear wave fields over finite, constant depth are considered, and the reflection induces large correlations between different wave components of the wave field. The nonlinear time evolution of such an inhomogeneous random wave field is studied by means of an equation developed by Crawford, Saffman and Yuen in 1980. We show that, depending on the spectrum and the water depth, there is a significant increase in the probability of freak waves, whose height is more than twice the significant wave height, created by the reflection off the wall.

Synthetic spectral structure of the seismic ambient vibrations generated by a distribution of superficial random sources with a finite extension

A model of seismic ambient vibrations is presented in the form of a random wavefield generated by a uniform continuous distribution of correlated aleatory forces, located at the surface of a flat layered weakly-dissipative Earth. The frequency-dependent spatial correlation among these sources is assumed to be representative of their spatial extension and has been parametrized by considering the role played by sea-wave dynamics in generating ambient vibrations. This enables a realistic description of the average power spectral density function of ambient vibrations observed worldwide at reference soil conditions. Effectiveness of the model is tested by reproducing observed average Horizontal-to-Vertical Spectral Ratios obtained from ambient vibration measurements at a well documented site in Northern Italy.

Assessment of global wave forces and moments on porous vertical barriers in random wave fields

Experimental investigations were carried out to assess the global wave forces and wave induced moments on slotted vertical barriers (SVB). Fourty two different wave barrier configurations (5%, 10%, 20%, 30%, 40%, 50% and 60% porosities and 1 to 6 number of slotted walls) were tested in random wave fields of JONSWAP spectra for wide range of significant wave heights and peak periods. It is found that the wave force is very sensitive to the change in porosity of the SVB. It is also found that relatively long waves and low porosity on SVB results in the highest wave force and short waves and high porosity on the SVB results in the lowest wave force. For most of the conditions, the wave force on SVB is less than the wave force on a single impervious vertical wall and force reduction to an extent of 20% to 80% is possible for the range of porosity and number of porous walls studied. A predictive equation to estimate the wave induced significant moment is provided with high regression coefficient. The average lever arm for assessing the wave induced moment is 0.6145 times the local water depth.

Retrieval of wave period from altimetry: Deep learning accounting for random wave field dynamics

As an extension of wave period inversion models based on altimetry data by an artificial neural network approach, a Deep Neural Network approach (DNN) mean wave period model is developed by introducing new parameters as DNN inputs. In addition to conventional altimeter-derived parameters such as significant wave height (SWH) and the normalized radar cross-section (NRCS) sigma0, the spatial (along-track) SWH gradient and SWH standard deviation (STD) for standard 1-s altimetry data are assumed to be responsible for random wave field dynamics and, thus, for the observed characteristic mean wave period. A comparison with in situ measurements of wave buoys shows higher accuracy of the novel DNN models by using these new parameters. The wave period estimation from DNN model is also consistent with the latest wave reanalysis and indicates less bias when compared to the buoys. The global mean wave steepness distribution from the DNN model shows good agreement with those provided by the wave reanalysis. The sensitivity of input variables sigma0, SWH, SWH gradient, and SWH STD on the results of the DNN model are also investigated. Perspectives on the DNN method for developing universal mission-independent wave period models are discussed. Objectives of the workThe mean wave period is an important parameter for characterizing wave properties and can be retrieved from altimeter observations. Considering random wave field dynamics, we present a deep neural network (DNN) model for mean wave period retrieval by introducing the SWH gradient and standard deviation as the inputs. The mean wave period estimation of the DNN model shows good agreement with the latest wave reanalysis and less bias compared to buoys. In an attempt to make the DNN mean wave period retrieval model universal, a method of applying DNN model cross-altimetry missions is presented.

Sound Field Reconstruction in Rooms with Deep Generative Models

The characterization of Room Impulse Responses (RIR) over an extended region in a room by means of measurements requires dense spatial with many microphones. This can often become intractable and time consuming in practice. Well established reconstruction methods such as plane wave regression show that the sound field in a room can be reconstructed from sparsely distributed measurements. However, these reconstructions usually rely on assuming physical sparsity (i.e. few waves compose the sound field) or trait in the measured sound field, making the models less generalizable and problem specific. In this paper we introduce a method to reconstruct a sound field in an enclosure with the use of a Generative Adversarial Network (GAN), which s new variants of the data distributions that it is trained upon. The goal of the proposed GAN model is to estimate the underlying distribution of plane waves in any source free region, and map these distributions from a stochastic, latent representation. A GAN is trained on a large number of synthesized sound fields represented by a random wave field and then tested on both simulated and real data sets, of lightly damped and reverberant rooms.