Nonlinear Wave Interactions Research Articles

Oscillations of a Floating Elastic Plate under Nonlinear Interaction of Bending-Gravitational Waves

Oscillations of a Floating Elastic Plate under Nonlinear Interaction of Bending-Gravitational Waves

A Multidimensional Superposition Principle: Classical Solitons IV

This article continues the series of the works of 1998–2007 years devoted to the Multidimensional Superposition Principle, the concept easily explaining both classical soliton and more complex wave interactions in nonlinear PDEs and allowing one, in particular, to construct the general Superposition Formulae for nonlinear wave interactions. In the present research the technique of multiexpansions with constraints is considered for finding the above SFs and investigation of the related solitons. (The simplest case of such expansions technically is analogous to the so-called invariant truncated singular expansions.) As the applications, the soliton SFs of the MKdV±, Kaup-Kupershmidt and new A± equation are obtained for the bell-shape exponential solitons of the various families, algebraic solitons, and the configuration of the two noninteracting kinks. The linearized, parameterized versions of these SFs are investigated then, and the related analysis of the interactions is presented. The obtained results allow one to consider the one soliton solutions mentioned as the strong bound states of the simpler solitons. Concerning the results for the above concrete nonlinear PDEs, the approach being developed made it possible both to obtain the new results and to reveal new moments for the already known ones.

Development of SRIAM Computation Module for Enhanced Calculation of Nonlinear Energy Transfer in 3rd Generation Wave Models

Because of the rapid development of computer technology in recent years, wave models can utilize parallel calculations for the high-resolution prediction of open sea and coastal areas with high accuracy. Parallel calculations also allow national agencies in the relevant sectors to produce marine forecasting data through massive parallel calculations. Meanwhile, the eastern coast of the Korean Peninsula has been increasingly damaged by swell-like high waves, and many researchers and scientists are continuing their efforts to anticipate and reduce the damage. In general, the short-term transformation of swell-like high waves can be reproduced relatively well in the third generation wave models, but the transformation of relatively long period waves needs to be simulated with higher accuracy in terms of the nonlinear wave interactions to gain a better understanding of the low-frequency wave generation and development mechanisms. In this study, we developed a calculation module to improve the calculation of the nonlinear energy transfer in the 3rd generation wave model and integrated it into the wave model to effectively consider the nonlinear wave interaction. First, the nonlinear energy transfer calculation module and third generation model were combined. Then, the combined model was used to reproduce the wave transformation due to the nonlinear interaction, and the performance of the developed operation module was verified.

The evolution of free and bound waves during dispersive focusing in a numerical and physical flume

Since the introduction of the NewWave theory (Lindgren, 1970), focused wave groups are used in physical and numerical studies to investigate the interaction of marine structures and ships with extreme waves. The propagation of such wave groups is associated with high order nonlinearities that can cause considerable deviations from linear and 2nd order predictions. Consequently, nonlinear numerical models or laboratory tests are needed to accurately describe the evolution of focused wave groups. In the present study, we validate a widely used two-phase Reynolds Averaged Navier-Stokes (RANS) solver realised in OpenFOAM with experimental results for the propagation of steep focused wave groups, using a newly developed methodology based on the separation of harmonics. This approach allows for accurate focusing of wave groups and in-detail examination of the individual evolution of the high order terms, as well as identifying the source of discrepancies between experiments and numerical models. The wave groups comprise long-crested broadbanded Gaussian spectra of increasing steepness propagating in intermediate water depth. The contribution of the nonlinear harmonics to the crest height and overall shape of the wave are also discussed, together with the effect of nonlinear wave interactions on the free-wave spectrum. The rapid growth of 3rd and 4th harmonics near focusing as well as the evolution of the free-wave spectrum, cause departures of up to 29% and 22% from analytic linear and 2nd order predictions. The present results demonstrate that RANS-VoF solvers constitute accurate models to propagate nearly breaking waves.

Rossby waves in the magnetic fluid dynamics of a rotating plasma in the shallow-water approximation

We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.

Nonlinear coda wave interferometry: Characterizing damage in complex solids

In this talk, we report results on the nonlinear interactions of ultrasonic coda waves, in reverberating or multiple scattering mesoscopic solid media. Using the method of coda wave interferometry (CWI), we analyze the effect of mixing a coda wave with an additional lower frequency pump wave. The extracted CWI parameters, known to be highly sensitive to small geometric or elastic modifications of the tested medium, are shown to be pump-amplitude dependent and to capture finely the results of the nonlinear interactions. Although nonlinear self-action effects with coda waves have been reported in unconsolidated granular media, they are difficult to implement in cracked solids or concrete. Instead, the reported nonlinear CWI class of methods (NCWI) shows robustness, a high sensitivity, and has been applied successfully to various complex media and structures. We show through several examples on « model » media (cracked glass plates) and on concrete structures, that NCWI can be useful for the nondestructive evaluation of complex solids that are strongly scattering at usual probing frequencies. Preliminary results and prospects in nonlinear elastic properties imaging and quantitative evaluation with NCWI are discussed.

A Comparison of the Ocean Microbarom Recorded on the Ground and in the Stratosphere

AbstractThe, ocean microbarom is an acoustic signal generated via nonlinear interaction of ocean surface waves. It can propagate for thousands of kilometers and represent a significant infrasonic noise source for ground infrasound stations across the globe. However, wind noise often compromises detections at ground stations. Furthermore, the microbarom may travel in elevated acoustic ducts that do not transmit enough energy for detections on ground stations. Here the presence of the ocean microbarom on two high‐altitude balloon flights is investigated. A spectral peak consistent with the microbarom was observed on sensors in the stratosphere but not on those deployed on the ground near the flight path of the balloon. This is probably due to an elevated acoustic duct and/or a superior signal‐to‐noise ratio in the stratosphere. Thus, microbarom activity quantified solely with ground‐based sensors may underestimate the occurrence of the phenomenon. However, high levels of interference from flight system electronics and/other other payloads may have obscured other microbarom episodes during the balloon deployments.

Analysis of nonlinear acoustic wave propagation in HIFU treatment using Westervelt equation

Currently, the HIFU (High Intensity Focused Ultrasound) therapy method is known as one of the most advanced surgical techniques in tumor ablation therapy. Simulation of the non-linear acoustic wave and tissue interaction is essential in HIFU planning to improve the usefulness and efficiency of treatment. In this paper, linear, thermoviscous and nonlinear equations are applied using two different media, namely liver and water. Transducer power of 8.3-134 Watts with the frequency of 1.1 MHz is considered as the range of study to analyze the wave and tissue interaction. Results indicate that the maximum focal pressure of about 0.5-4.3 MPa can be achieved for transducer power of 8.3 to 134 W. Simultaneous solving of the acoustic pressure equation and Pennes’s bio-heat equation are used to determine the temperature rise at focal point and the ablated area. Finally, the linear and nonlinear simulations are compared, and the turning point of transition from linearity to nonlinearity is determined. The simulated results provide information about the behavior of the focalized ultrasound in interaction with liver tissue. The performance of phased array HIFU transducer can be improved for treatment considering the lesion size, temperature rise in tissue and choosing best range of operational power.

Simulating interaction of nonlinear spatial waves on a free surface of the shallow viscous liquid layer

This paper deals with the combined approach to describing the evolution of weakly nonlinear three-dimensional moderately long perturbations of free surface of viscous liquid. The initial system of hydrodynamic equations is reduced to the novel model system of equations. The first of them is integro-differential equation for nonlinear perturbation of the free surface, taking into account non-stationary shear stress on a weakly sloping bottom. Another equation is an auxiliary linear equation for determining the liquid horizontal velocity vector, averaged over the layer depth. This vector is present in the main equation only in the term of the second order of smallness. The proposed model is suitable for finite-amplitude waves, traveling in different directions in the horizontal plane. Some problems of interactions and collisions of such perturbations over the horizontal and weakly sloping bottom are solved numerically.

Inverse problem for multi-body interaction of nonlinear waves

The inverse problem is studied in multi-body systems with nonlinear dynamics representing, e.g., phase-locked wave systems, standard multimode and random lasers. Using a general model for four-body interacting complex-valued variables we test two methods based on pseudolikelihood, respectively with regularization and with decimation, to determine the coupling constants from sets of measured configurations. We test statistical inference predictions for increasing number of sampled configurations and for an externally tunable temperature-like parameter mimicing real data noise and helping minimization procedures. Analyzed models with phasors and rotors are generalizations of problems of real-valued spherical problems (e.g., density fluctuations), discrete spins (Ising and vectorial Potts) or finite number of states (standard Potts): inference methods presented here can, then, be straightforward applied to a large class of inverse problems. The high versatility of the exposed techniques also concerns the number of expected interactions: results are presented for different graph topologies, ranging from sparse to dense graphs.

Role of continental shelf on non-linear interaction of storm surges, tides and wind waves: An idealized study representing the west coast of India

The study reports role and dependence of continental shelf geometry on the non-linear interaction of storm surges, tides, and wind waves. As a case study, the shelf geometry representing the west coast of India is considered and numerical simulations are executed using the standard representation of a tropical cyclone. The idealized experiments assume the west coast of India as a straight coastline with varying continental shelf width ranging from 35 km in the south to 330 km in the north. For the experiments, ADCIRC model in standalone mode and coupled ADCIRC+SWAN are used by considering 13 idealized cyclone tracks covering the study domain. It is noticed that an amplification of peak storm surge of ∼12 cm for every 10 km increase in the shelf width. During different phases of the tide, the surge-wave interaction modifies the water level elevation and its occurrence as the tidal range increases towards north. The interaction of tides on surges and wind waves is observed maximum at low-tide and minimum at flood tide over the wide shelf regions. In general, the non-linear interaction is found to be 15%–20% for any cyclone track or tidal phase. The wave setup caused by wind waves is marginally varies with change of amplitude and phase of the tide, however, its profile is significantly modified, particularly over the wider shelf widths.

Weak versus strong wave turbulence in the Majda-McLaughlin-Tabak model

Within the spirit of fluid turbulence, we consider the one-dimensional Majda-McLaughlin-Tabak (MMT) model that describes the interactions of nonlinear dispersive waves. We perform a detailed numerical study of the direct energy cascade in the defocusing regime. In particular, we consider a configuration with large-scale forcing and small scale dissipation, and we introduce three non- dimensional parameters: the ratio between nonlinearity and dispersion, {\epsilon}, and the analogues of the Reynolds number, Re, i.e. the ratio between the nonlinear and dissipative time-scales, both at large and small scales. Our numerical experiments show that (i) in the limit of small {\epsilon} the spectral slope observed in the statistical steady regime corresponds to the one predicted by the Weak Wave Turbulence (WWT) theory. (ii) As the nonlinearity is increased, the WWT theory breaks down and deviations from its predictions are observed. (iii) It is shown that such departures from the WWT theoretical predictions are accompanied by the phenomenon of intermittency, typical of three dimensional fluid turbulence. We calculate the structure-function as well as the probability density function of the wave field at each scale and show that the degree of intermittency depends on {\epsilon}.

Nonlinear wave interaction with curved front seawalls

The hydrodynamic performance of three prevalent curved front seawalls along with a vertical seawall has been investigated in the Ursell number ranging from 8 to 16. Experiments have been conducted to measure dynamic pressure on these seawalls under the action of regular monochromatic waves. Two different numerical models have been used to analyse the measured data: An in house Smoothed Particle Hydrodynamics (SPH) particle based model and the commercial Computational Fluid Dynamics (CFD) package ANSYS- FLUENT®. The numerical simulation of peak dynamic pressures on the seawall front surface differs with the laboratory measurements by less than 10% while, the maximum velocity magnitude reveals differences of the order of 20–30% between SPH and FLUENT during overtopping of the wave. The comparison between these models shows that the performance of improved SPH models is similar to that of FLUENT, as long as the flow is laminar. The SPH solution contains several important nonlinear aspects also revealed in the experiments. The comparison of measured and simulated pressures along the curved front proves the capability of the SPH models in predicting the nature of run-up and overtopping, if occurs.

Nonlinear interaction of electromagnetic waves with 3-component relativistic quantum plasma

The interaction of intense circularly polarized electro-magnetic (CPEM) wave with 3-component relativistic-quantum plasma consisting of relativistic-degenerate electrons and positrons, and dynamic degenerate ions is theoretically studied. A mathematical model is structured by coupling Klein-Gordon equations for the electrons and positrons, and Schrödinger equation for the ions with Maxwell equations through Poisson equations. The solutions of the dispersion relation are plotted for relativistic quantum plasma in the density-range of ∼1030→1036m−3 for several positron concentrations. Three wave modes are observed: electrons, ions, and positrons. The pair branch mode having a possible association with the positron states stays unaltered by variation in the positron concentration but varies significantly with a change in the quantum parameter defined in terms of the particles number density. The addition of positron to the plasma and increasing the positron concentration suggest enhancement of the opacity of the relativistic quantum plasma. The nonlinear interaction of large amplitude CPEM waves with the plasma leads to self-induced transparency. The transparency decreases with increasing positron concentration. The model so developed is then applied to study stimulated Raman scattering, modulational instability, and stimulated Brillouin scattering of intense CPEM waves in such plasmas. The results show that the growth rates are affected by the positron concentration, the quantum parameter of the plasma, as well as by the amplitude of the incident electromagnetic wave.

Simulating global atmospheric microbaroms from 2010 onward

Microbaroms are atmospheric pressure oscillations radiated from non-linear ocean surface wave interactions. Large regions of interacting high-energetic ocean waves, e.g., ocean swell and marine storms, radiate almost continuously acoustic energy. Microbaroms dominate the infrasound ambient noise field, which makes them a preferred source for passive atmospheric probing. Microbarom are simulated using a two-fluid model, representing an atmosphere over a finite-depth ocean and a coupled ocean-wave model providing the sea state. Air-sea coupling is crucial due to the two-way interaction between surface winds and ocean waves. In this study, a detailed overview is given on how global microbarom simulations are obtained, including a sensitivity analysis of the various model input data and parameterizations. Simulations are validated by infrasound array observations of the International Monitoring Systems (IMS) of the Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO). An brief demonstration is given on ...

A wind-driven noise model in deep water

Surface processes, including non-linear surface wave interactions and bubble oscillations, dominate the deep water ambient noise in the absence of shipping and wildlife. A single-parameter noise model, determined by the wind speed above the ocean surface, is proposed to interpret the deep ocean acoustics from 1 Hz to 4 kHz. Below hundreds of hertz, the sound is primarily generated by surface wave interactions, which is the function of surface wave spectrum. A theoretical expression of wave spectrum is proposed to contain the gravity and capillary waves, which is in accord with the features of measured wave spectra. The noise spectrum for frequency above hundreds of hertz due to the effective bubble oscillation within the bubble cloud is proposed. The noise spectrum has been estimated as a function of frequency and wind speed based on available information of the bubble distribution. The model/data comparison shows that the proposed single-parameter noise model are in reasonable agreement with the data. [Work supported by National Natural Science Foundation of China, Grant No. 11125420.]

Generalized Sagdeev potential theory for shock waves modeling

In this paper, we develop an innovative approach to study the shock wave propagation using the Sagdeev potential method. We also present an analytical solution for Korteweg de Vries Burgers (KdVB) and modified KdVB equation families with a generalized form of the nonlinearity term which agrees well with the numerical one. The novelty of the current approach is that it is based on a simple analogy of the particle in a classical potential with the variable particle energy providing one with a deeper physical insight into the problem and can easily be extended to more complex physical situations. We find that the current method well describes both monotonic and oscillatory natures of the dispersive-diffusive shock structures in different viscous fluid configurations. It is particularly important that all essential parameters of the shock structure can be deduced directly from the Sagdeev potential in small and large potential approximation regimes. Using the new method, we find that supercnoidal waves can decay into either compressive or rarefactive shock waves depending on the initial wave amplitude. Current investigation provides a general platform to study a wide range of phenomena related to nonlinear wave damping and interactions in diverse fluids including plasmas.

Experimental study of head-on and rear-end collisions of two unequal solitary waves

The features of two unequal solitary waves during an interaction were experimentally investigated by optical and particle-tracer methods. To estimate the temporal colliding-wave profiles and phase shift during the head-on collision, the water surface variations were measured using two wave gauges. In addition, the spatial surface profiles were analyzed by combining particle mask correlation (PMC) with an image-thresholding method that detects the air–water interface as a set of locally extreme luminance values. The experimental surface displacement of the colliding wave was compared with the corresponding shape of a third-order perturbation approximation. Applying a particle image velocimetry (PIV) method, the kinetic features of right-running, left-running, and colliding waves were measured in head-on collisions, and in cases of shorter, taller, and compound waves in rear-end collisions. The PIV technique accurately measured the water velocity spatially induced by the nonlinear solitary wave interactions. The paths of the water particles were also successfully tracked by this method. Finally, to understand the effects of the interactions, the dynamic pressure was measured by tiny pressure transducers placed at horizontal locations throughout the water depth.

Reduced-order prediction of rogue waves in two-dimensional deep-water waves

We consider the problem of large wave prediction in two-dimensional water waves. Such waves form due to the synergistic effect of dispersive mixing of smaller wave groups and the action of localized nonlinear wave interactions that leads to focusing. Instead of a direct simulation approach, we rely on the decomposition of the wave field into a discrete set of localized wave groups with optimal length scales and amplitudes. Due to the short-term character of the prediction, these wave groups do not interact and therefore their dynamics can be characterized individually. Using direct numerical simulations of the governing envelope equations we precompute the expected maximum elevation for each of those wave groups. The combination of the wave field decomposition algorithm, which provides information about the statistics of the system, and the precomputed map for the expected wave group elevation, which encodes dynamical information, allows (i) for understanding of how the probability of occurrence of rogue waves changes as the spectrum parameters vary, (ii) the computation of a critical length scale characterizing wave groups with high probability of evolving to rogue waves, and (iii) the formulation of a robust and parsimonious reduced-order prediction scheme for large waves. We assess the validity of this scheme in several cases of ocean wave spectra.

Analysis of self-consistent nonlinear wave-particle interactions of whistler waves in laboratory and space plasmas

Whistler mode chorus is one of the most important emissions affecting the energization of the radiation belts. Recent laboratory experiments that inject energetic electron beams into a cold plasma have revealed several spectral features in the nonlinear evolution of these instabilities that have also been observed in high-time resolution in situ wave-form data. These features include (1) a sub-element structure which consists of an amplitude modulation on time-scales slower than the bounce time, (2) closely spaced discrete frequency hopping that results in a faster apparent frequency chirp rate, (3) fast frequency changes near the sub-element boundaries, and (4) harmonic generation. In this paper, we develop a finite dimensional self-consistent Hamiltonian model for the evolution of the resonant beam of electrons. We analyze a single wave case and demonstrate that the instability occurs due to a Krein collision, which manifests as a coupling between a negative and positive energy mode. This analysis revealed that the nonlinear evolution of the spectrally stable fixed-points of the self-consistent Hamiltonian develop a sub-packet structure similar to that of space observations. We then analyze the case of two whistler waves to show that the model reproduces the nonlinear harmonic generation and leads to a hypothesis for the closely spaced frequency hopping observed in laboratory experiments and space data.