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Extreme waves, also known as ‘rogue waves’, have posed considerable challenges to maritime traffic over some time. Efforts have been directed at investigating the mechanisms governing these extreme energy localizations in oceanic environments. Modulational instability, also known as sideband instability, is one such mechanism that has been proposed to explain the occurrence of such phenomena in the framework of non-linear theory. The current work is aimed at better understanding the effects of sideband modulations on the propagation of unidirectional waves. To achieve this, a numerical wave tank (NWT) has been constructed using Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) to investigate the different parameters associated with the generation and propagation of plane, modulated waves. General Process Graphics Computing Unit (GPGPU) computing has been utilized to accelerate the computational process and improve the computational efficiency. The chosen numerical scheme has been validated by carrying out irregular waves focusing simulations to compare with available experimental data. Additionally, a Peregrine-type breather experiment has also been performed as part of the validation studies to look at energy localization within the NWT. The effects of the different parameters associated with the modulations to a plane propagating wave have been investigated using a blend of surface elevation data, eigenvalue, and frequency spectra. The effect of water depth on the perturbations to plane waves has been also investigated. The observations from these experiments can help shed light into the effects of modulations in the propagation of plane waves and help in the study of oceanic energy localization studies in future. 

Rogue waves, which are defined as waves with a wave height, or alternatively a crest height, exceeding the significant wave height by a certain factor, continue to endanger ships and offshore infrastructure. Hence, reliable rogue wave forecasting is of utmost importance to increase the safety for maritime operations. While the occurrence of rogue waves is widely acknowledged, their emergence remains unpredictable due to the lack of a well-accepted basis for explaining their occurrence. In fact, two popular mechanisms explaining the formation of rogue waves lead to considerably different conclusions about their predictability. On the one hand, a rogue wave could be formed by a superposition of wave trains with unknown phases. With this generation mechanism, rogue wave prediction is not viable. On the other hand, nonlinear focusing leading to the Benjamin-Feir instability gives rise to slowly developing rogue waves. Hence, this rogue wave formation could be detected with significant advance time. Given this background, there is an imperative need to address the basic question: Are rogue waves predictable? In this article, the authors explore the predictability of rogue waves by constructing and parameterizing neural networks. The networks are trained on available buoy data, which allows not only for an assessment under the most realistic conditions but also for indicating the sufficiency of current ocean measurements for rogue wave prediction. 

Freak waves, waves significantly higher than neighboring waves, are a serious threat to ships and marine infrastructure. Despite significant refinement of operational wave models and recent progress in studying the theoretical foundations of such extreme events, the emergence of these events remains unpredictable. In this work, the authors propose a data-driven wave forecasting approach by combining the essence of common wave models, rapid oscillations, and slowly changing spectrum with data-driven techniques such as recurrent neural networks. A judicious minimization procedure is developed, wherein the sea surface elevation is first decomposed into harmonic functions with varying amplitudes. Then, the amplitude variations are forecasted by fitting universal, black-box models. This approach, which can be used to forecast wave crests and troughs in real time, is tested on available buoy data. Overall, the developed models and fitting strategies outperform simple benchmarks indicating the approach’s potential for operational, real-time wave forecasting. 

In this work, the authors explore the influence of noise on the dynamics of coupled nonlinear oscillators. Numerical studies based on the Euler–Maruyama scheme and experimental studies with finite duration noise are undertaken to examine how the response can be moved from one response state to another by using noise addition to a harmonically forced system. In particular, jumps from a high amplitude state of each oscillator to a low amplitude state of each oscillator and the converse are demonstrated along with noise-influenced localizations. These events are found to occur in a region of multi-stability for the system, and the corresponding noise levels are reported. A method for recognizing how much noise is required to induce a change the system dynamics is developed by using the response basins of attraction. The findings of this work have implications for weakly coupled, nonlinear oscillator arrays and the manner in which noise can be used to influence energy localization and system dynamics in these systems. 

Energy localization, which are spatially confined response patterns, have been observed in turbomachinery applications, micro-electromechanical systems, and atomic crystals. While confined energy can reduce a device’s life-span, in sensing and energy harvesting applications, it can be beneficial to steer a system’s response into a localized mode. Building on earlier studies, in this article, the authors extend the research on localization by considering an array of coupled Duffing oscillators arranged in a circle. The system is composed of multiple nonlinear oscillators each connected to two neighboring oscillators via springs. Due to the periodic boundary conditions waves can propagate through the boundaries. These oscillators are hardening in most of the considered cases, and softening in the others. In the studied parameter range, the system is characterized by multi-stable behavior and a localized mode as well as a unison-low-amplitude motion coexist. The possibility that white noise can drive the system response from the localized mode to the low amplitude mode and thus suppresses energy localization is investigated. For different noise levels, the duration needed to stop energy localization as well as the probability to suppress localization within a certain time is numerically studied. In addition, the effects of linear coupling and nonlinear coupling between the oscillators on the strength of localization and the minimum noise addition needed to suppress energy localization are examined in depth. Moreover, modeling of large array dynamics with smaller subsystems is explored and dynamics with non-Gaussian noise is also considered. 

Generation and propagation of waves in a numerical wave tank constructed using Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) are considered here. Numerical wave tank simulations have been carried out with implementations of different Wendland kernels in conjunction with different numerical dissipation schemes. The simulations were accelerated by using General Process Graphics Processing Unit (GPGPU) computing to utilize the massively parallel nature of the simulations and thus improve process efficiency. Numerical experiments with short domains have been carried out to validate the dissipation schemes used. The wave tank experiments consist of piston-type wavemakers and appropriate passive absorption arrangements to facilitate comparisons with theoretical predictions. The comparative performance of the different numerical wave tank experiments was carried out on the basis of the hydrostatic pressure and wave surface elevations. The effect of numerical dissipation with the different kernel functions was also studied on the basis of energy analysis. Finally, the observations and results were used to arrive at the best possible numerical set up for simulation of waves at medium and long distances of propagation, which can play a significant role in the study of extreme waves and energy localizations observed in oceans through such numerical wave tank simulations.