Nonlinear Wave Interactions Research Articles

The spectral representation method: A framework for simulation of stochastic processes, fields, and waves

The Spectral Representation Method (SRM) was developed in the 1970s to simulate Gaussian stochastic processes and fields from a Fourier series expansion according to the Spectral Representation Theorem. Since those early developments, the SRM has continuously evolved into a comprehensive framework for the simulation of stochastic processes, fields, and waves with a rigorous theoretical foundation. Its major advantages are conceptual simplicity and computational efficiency. In the 1990s, much of the theory for simulation of Gaussian stochastic processes, fields, and waves was firmly established and early methods for simulation of non-Gaussian processes, fields, and waves were introduced. In the 2000s and 2010s, methods that coupled the SRM with Translation Process Theory were improved to enable efficient and accurate simulations of stochastic processes, fields, and waves with strongly non-Gaussian marginal probability distributions. More recently, the SRM was extended for higher-order non-Gaussian processes, fields, and waves by extending the Fourier stochastic expansion to include non-linear wave interactions derived from higher-order spectra. This paper reviews the key theoretical developments related with the SRM and provides the relevant algorithms necessary for its practical implementation for the simulation of stochastic processes, fields, and waves that can be either stationary or non-stationary, homogeneous or non-homogeneous, one-dimensional or multi-dimensional, scalar or multi-variate, Gaussian or non-Gaussian, or any combination thereof. The paper concludes with some brief remarks addressing the open research challenges in SRM-based theory and simulations.

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

Mathematical and Numerical Explanation of the Nonlinear Acoustic Wave Interaction in Acousto-Optical Cells

Mathematical and Numerical Explanation of the Nonlinear Acoustic Wave Interaction in Acousto-Optical Cells

Nonlinear Crack-Wave Modulations in Shear Horizontal Wave Propagation for Fatigue Crack Detection

The nonlinear interaction of shear horizontal waves with fatigue cracks is investigated for structural damage detection. The focus is on the non-classical modulation effect in the presence of damage. The major principle of the method is explained theoretically and supported by a semi-analytical solution. The method is tested experimentally in aluminium beams with fatigue cracks. A combination of high-frequency ultrasonic and low-frequency vibration shear excitations are introduced to monitored beam components using low-profile piezoceramic shear actuators. Nonlinear vibro-acoustic responses are gathered using laser vibrometry. Vibration excitation frequencies are selected using experimental and numerical modal analysis. A series of vibro-acoustic experiments is performed for fatigue crack detection. The results show that the non-classical nonlinear vibro-acoustic effect – based on shear horizontal wave propagation – can be used for crack detection. In addition, the method also allows for damage localisation.

Optical pattern formation of laser fields in the Rydberg atomic gases

We study the phenomena of long-wave and short-wave modulation instabilities (LMI and SMI) in Rydberg atomic gases within the framework of the nonlocal nonlinear Schrödinger equations in the local and nonlocal regions (characterized by the ratio of beam radius to radius of Rydberg blockade sphere), respectively. We further reveal the rich soliton dynamics arising from interactions of nonlinear waves triggered by LMI. We also show the variety of spatial self-organization pattern formations and their active manipulation due to SMI. Through a synthesis of theoretical analysis and numerical simulations, the self-organized optical structures reported here provide a way for realizing what we believe to be novel optical patterns and solitons based on Rydberg atomic gases.

Transition delay in a Mach 6 boundary layer using steady blowing and suction strips

Direct numerical simulations (DNS) were carried out to investigate flow control for transition delay using steady blowing/suction strips at the wall of a flared cone at Mach 6 and zero angle of attack. For the numerical investigations of the transition control strategy, the flared cone geometry and the flow conditions of the experiments in the Boeing/Air Force Office of Scientific Research (AFOSR) Mach 6 Quiet Tunnel (BAM6QT) at Purdue University were chosen. For the DNS, transition was initiated by introducing random disturbances at the inflow of the computational domain, emulating ‘natural’ transition in wind-tunnel experiments caused by free-stream noise. In both wind-tunnel experiments and numerical simulations, streamwise ‘hot’ streaks were found on the surface of the flared cone, which are caused by a nonlinear interaction of an axisymmetric second-mode wave and a pair of oblique waves of the same frequency (‘fundamental resonance’). The objective of the flow control strategy proposed here is to delay the transition onset, and thus mitigate the negative consequences associated with the nonlinear transition stages, i.e. the development of hot streaks and large wall-pressure amplitudes that were observed in experiments and DNS. Our previous so-called ‘controlled’ transition simulations have shown that flow control using steady blowing and suction strips can lead to a significant delay of the hot streak development on the surface of the flared cone. The simulation results presented in this paper show that this flow control strategy remains effective, even in a natural transition scenario characterized by broadband disturbances.

Nonlinear spatial evolution of degenerate quartets of water waves

In this manuscript we investigate the Benjamin–Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. We show that the maximal energy transfer consistently occurs for smaller side-band separation than the fastest linear growth rate. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.

Nonlinear Wave Interaction of Nanorods Embedded in a Viscoelastic Medium

Abstract Purpose Nonlinear interactions between two acoustic waves in nanorods traveling at various wave numbers, group velocities, and frequencies are examined in this study. Methods The nonlinear equation of the nanorod in a viscoelastic medium is obtained using the theory of nonlocal elasticity. Furthermore, the multiple-scale expansion method is applied to study strongly dispersive, weakly nonlinear waves in a nonlocal viscoelastic medium. Using this expansion technique, we can derive the coupled nonlinear Schrödinger equations as the governing equations, which we solve as differential equations of some parameters by expanding the field quantities into an asymptotic series of the smallness parameter. Results We give the nonlinear plane wave solutions to these equations in several special cases. The plane wave solutions show how the wave amplitude affects the frequencies of nonlinear plane waves. Additionally, we show numerically how the real and imaginary parts of the group velocities and natural frequency of the system for a carbon nanotube in a viscoelastic medium are affected by the nonlocal, damping, and stiffness parameters.

Mechanisms of Fundamental Electromagnetic Wave Radiation in the Solar Wind

Large-scale and long-term two-dimensional particle-in-cell simulations performed for parameters relevant to type III solar radio bursts have provided new results on the generation mechanisms of fundamental electromagnetic waves radiated at the plasma frequency ω p . The paper first considers the nonlinear wave interaction process of electromagnetic decay (EMD) in a homogeneous solar wind plasma with an electron-to-ion temperature ratio T e /T i > 1. The dynamics of ion-acoustic waves (dispersion, spectra, growth/damping) is studied, and signatures confirming the three-wave interactions (cross-bicoherence, correlations between waves’ phases and between waves' growths, resonance conditions) are provided. The decisive role played in EMD by the backscattered Langmuir waves coming from the electrostatic decay (ESD) is demonstrated. EMD can be triggered by ion acoustic waves coming from the two cascades of the faster and more intense ESD. The same study is then performed in a solar wind plasma with random density fluctuations. In this case, EMD is not suppressed but develops only within plasma regions of reduced or quasi-uniform density. It coexists with linear mode conversion (LMC) of Langmuir waves into electromagnetic radiation, which is the fastest and most prominent process, as well as with ESD. LMC can lead to enhanced occurrence of EMD in the early stage. Moreover, the impact of T e /T i on electromagnetic energy growth and saturation is shown to be rather weak. Ion-acoustic waves are heavily damped at T e ∼ T i , so that EMD is overcome by nonlinear induced scattering on thermal ions. In actual solar wind plasmas, EMD should be more easily observed in plasma regions weakly perturbed by the background density turbulence and where ion temperature is decreased.

Fully nonlinear wave interaction with 2D floating body including nonlinear sloshing tank

Two-dimensional fully-nonlinear numerical sloshing tank (FN-NST) simulation program is developed based on boundary element method (BEM), potential theory with artificial free-surface damping, and space-fixed (global) coordinate system. The free surface in sloshing tank is updated by the mixed Eulerian-Lagrangian (MEL) method with material node approach (Full Lagrangian approach). The developed potential-based nonlinear model was validated through comparisons with available experimental and CFD (computational fluid dynamics) results and they show good agreements. The nonlinear sloshing model is then further implemented in the fully nonlinear wave and floating-body simulation program to observe the effects of liquid sloshing motion. In the fully nonlinear program, the entire nonlinear fluid pressure is integrated over the instantaneous wetted body surface and position, which is subsequently compared with the corresponding linear theory and results. The second-order effects are also identified from the fully-nonlinear simulation. The floater Response Amplitude Operators (RAOs) for frozen- and liquid-cargo cases are also compared for various wave frequencies and amplitudes to better identify the role of liquid cargo in the stability of rotational motions. The effects of the vertical location of liquid tank and artificial free-surface damping are also investigated.

A 3-D minimum-enstrophy vortex in stratified quasi-geostrophic flows

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

Frequency shift of parametric sound by face-to-face pair of sources in relative motion.

This paper reports an acoustic phenomenon regarding a parametric sound source (also referred to as a parametric array): a secondary sound wave is generated from the nonlinear interaction of multiple primary sound waves with varied frequency components, particularly when two relatively moving sound sources face each other. It was found that the frequency of the secondary wave fluctuated according to the source movement and provided a theoretical explanation for this phenomenon. It is experimentally demonstrated that this frequency shift was approximately proportional to the velocity of the moving source toward the fixed source and to the driving frequency of the moving source. This phenomenon has much in common with the Doppler's effect, but its unique property is that the frequency shift depends on neither the observation position nor the source velocity toward the observer. These sound generation principles enable measurement of the velocity of slowly moving sound sources while maintaining a low-modulation frequency band and a short measurement time. This phenomenon can potentially be applied to an alternative approach for acoustic noncontact velocimetry of moving objects.

Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt [Formula: see text] model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical [Formula: see text] equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding [Formula: see text] model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.

CFD analysis of wave loading on a 10 MW TLP-type offshore floating wind turbine in regular waves

This paper aims to study the higher-harmonics wave-induced surge force on the CENTEC-TLP platform in regular head waves with non-breaking assumptions, using a CFD-based numerical wave tank model, which is a useful tool for the analysis of the nonlinear interaction of waves and structures. Moreover, the influence of change in the wave diffraction and the incident wave angle on the surge force subject to operational and extreme wave conditions is investigated. The numerical analysis is conducted using the CFD toolbox OpenFOAM. The link between Secondary Load Cycle and the wave scattering Type2 is apparent and it is concluded that they are all inertia-driven and that 2nd harmonics play an important role.

Analytical and numerical modeling of nonlinear lamb wave interaction with a breathing crack with low-frequency modulation

To characterize fatigue crack, an analytical calculation and finite element (FE) simulation of Lamb wave propagating through the region of a breathing crack in a two-dimensional(2D) isotropic plate were studied. Contact surface boundary conditions between the two surfaces of the vertical crack were considered to study contact acoustic nonlinearity (CAN) from the breathing contact crack in conjunction with the modal decomposition method, Fourier transform, and variational principle-based algorithm. Reflection and transmission coefficients in the fundamental frequency and second harmonic frequency were calculated and analyzed quantitatively. Different ratios of incident wave amplitude to crack width were studied to calculate CAN results related to micro-crack width. In addition, a low-frequency (LF) vibration(10 Hz) excitation was introduced to perturb the free surface vertical crack to close, and an interrogating Lamb wave(1 MHz) was used to study crack-related CAN in different conditions for interpreting the modulation mechanism. The contact boundary conditions between two surfaces of vertical crack were set which were dynamically changed due to the low frequency modulation. The clapping effects when the crack closed due to the modulation of the contact boundary conditions between the crack surfaces were studied and analyzed to get the quantitative correlation between CAN and LF modulation. The results obtained from the analytical model were compared with those from the FE simulation, showing good consistency. Knowledge of these effects is essential to correctly gauge the severity of surface cracks in the plate, which can be spotlighted in its application to quantitative evaluation of micro fatigue cracks in structural health monitoring(SHM).

The evolution of second harmonic waves in oblique transition

The evolution of second harmonic waves in oblique transition

Effect of non-local near-resonant interactions of Rossby waves on formation of large-scale zonal flows in unforced two-dimensional turbulence on rotating sphere

This study investigates the effect of nonlinear interactions of Rossby waves on large-scale zonal flow formation in two-dimensional turbulence on a rotating sphere. The coefficients of nonlinear interactions are first calculated. Then, the non-local, near-resonant, and non-local near-resonant interactions are investigated in detail. The results show that the formation of large-scale westward circumpolar zonal flows is directly caused by non-local energy transfer due to the three-wave near-resonant interactions of Rossby waves.

On the dynamics of soliton interactions in the stellar environments

The effects of trapping of relativistically degenerate electrons are studied on the formation and interaction of nonlinear ion-acoustic solitary waves (IASWs) in quantum plasmas. These plasmas are detected in high-density astrophysical entities and can be created in the laboratory by interacting powerful lasers with matter. The formula for the number density of electrons in a state of relativistic degeneracy is provided, along with an analysis of the non-relativistic and ultra-relativistic scenarios. While previous studies have delved into specific aspects of relativistic effects, there needs to be a more detailed and systematic examination of the fully relativistic limit, which is essential for gaining a holistic perspective on the behavior of solitons in these extreme conditions. The aim of this work is to comprehensively investigate the fully relativistic limit of the system to fill this gap. The reductive perturbation technique is utilized to deduce the Korteweg–de Vries (KdV) equation, which is used to analyze the properties of the IASWs. Hirota bilinear formalism is applied to obtain single- and multi-soliton solutions for the KdV equation. The numerical analysis is focused on the plasma properties of the white dwarf in the ongoing investigation. The amplitude of the IASWs is found to be maximum for the non-relativistic, intermediate for the ultra-relativistic, and minimum for the fully relativistic limit. Most importantly, it is found that the fastest interaction occurs in the non-relativistic limit and the slowest in the fully relativistic limit.

Nonlinear interaction of head-on solitary waves in integrable and nonintegrable systems

This study numerically investigates the nonlinear interaction of head-on solitary waves in a granular chain (a nonintegrable system) and compares the simulation results with the theoretical results in fluid (an integrable system). Three stages (the pre-in-phase traveling stage, the central-collision stage, and the post-in-phase traveling stage) are identified to describe the nonlinear interaction processes in the granular chain. The nonlinear scattering effect occurs in the central-collision stage, which decreases the amplitude of the incident solitary waves. Compared with the leading-time phase in the incident and separation collision processes, the lagging-time phase in the separation collision process is smaller. This asymmetrical nonlinear collision results in an occurrence of leading phase shifts of time and space in the post-in-phase traveling stage. We next find that the solitary wave amplitude does not influence the immediate space-phase shift in the granular chain. The space-phase shift of the post-in-phase traveling stage is only determined by the measurement position rather than the wave amplitude. The results are reversed in the fluid. An increase in solitary wave amplitude leads to decreased attachment, detachment, and residence times for granular chains and fluid. For the immediate time-phase shift, leading and lagging phenomena appear in the granular chain and the fluid, respectively. These results offer new knowledge for designing mechanical metamaterials and energy-mitigating systems.

Bragg scattering of nonlinear surface waves by sinusoidal sandbars

Based on the multiple-scale expansion technique, a new set of extended nonlinear Schrödinger (ENLS) equations up to the third order is derived to account for the additional high-order bottom and dispersion effects as well as the nonlinear wave interaction on wave transformation over periodic sandbars of sinusoidal geometry. By employing the small-amplitude wave assumption, a closed-form analytical solution for Bragg scattering is obtained from the linearised ENLS equations, which demonstrates that a downshift of wave frequency of the maximum reflection is mainly due to the inclusion of the high-order bottom effect. The factors that affect the downshift of the resonant frequency are identified and a theoretical expression in parabolic form is derived to quantify the downshift magnitude. The fully ENLS equations are further analysed to reveal the additional wave nonlinear effects on Bragg scattering characteristics. Under the condition of infinitesimal sandbar amplitude, the ENLS equations render a theoretical expression of the critical value of $kh$ when the nonlinear wave self-modulation effect and the nonlinear wave cross-modulation effect are equal, whereas the former effect is responsible for wavenumber upshifting and the latter downshifting. When $kh$ is larger than the critical value, the increase of wave nonlinearity will enhance the downshift magnitude of the Bragg resonance, and vice versa. For finite amplitude of the bottom sandbar, the ENLS equations are solved numerically to examine the influence of both wave nonlinearity and sandbar amplitude on the characteristics of Bragg resonance. The results reveal that as the increase of sandbar amplitude, the critical $kh$ increases monotonically.