Analysis Of Dispersion Relation Research Articles

Improved approximate dispersion relation analysis using deep neural network

In this work, we revisit and propose some improvements in the combined spatial-temporal approximate dispersion analysis (ADR). This analysis serves the purpose of quantifying spectral errors in the numerical solutions. The improvements we propose significantly enhance the predictive capabilities of ADR analysis, particularly in the context of multi-step and multi-level methods such as the Runge–Kutta (RK) and the Adam-Bashforth techniques. We deduce the equations governing numerical dissipation and group velocity, and we present corresponding plots for various multi-step methods based on our proposed approach. Additionally, we introduce a novel strategy for minimizing dissipation, optimizing it through the utilization of an evolutionary algorithm. While the linear analysis exhibits certain limitations when extended to nonlinear scenarios, we employ a deep neural network to determine stability equations for nonlinear problems. We conduct a comprehensive analysis to demonstrate that our proposed method exhibits minimal errors when compared to alternative advection equation schemes, both in one and two spatial dimensions.

Nonlinear dispersion relation of dust acoustic waves using the Korteweg–de Vries model

In this Brief Communication, we present an exact analytic nonlinear dispersion relation (NLDR) for the dust acoustic waves using the Korteweg–de Vries model. The NLDR agrees with the spectrum of spatiotemporal evolution obtained from an exact solution as in Mir et al. [Phys. Plasmas 27, 113701 (2020)]. The NLDR also shows a reasonable match with the experimental data of Thompson et al. [Phys. Plasmas 4, 2331 (1997)] in the long-wavelength limit (kλD≪1). We suggest that such nonlinear corrections should be incorporated in the dispersion relation along with damping, streaming, and correlation effects in order to provide a more realistic interpretation of experimental data.

Propagation characteristics of elastic longitudinal wave in a piezoelectric semiconductor metamaterial rod and its tuning

Elastic metamaterial structures have many distinct wave properties such as band gap structure and topological phase inversion. The coexistence and interaction of piezoelectricity and semiconductor behavior in piezoelectric semiconductor (PS) materials endow metamaterials with some new physical effects. We investigate the propagation characteristics of elastic longitudinal waves in a metamaterial rod made of n-type PS material and piezoelectric (PE) dielectric material. Based on one-dimensional rod model, we present the transfer matrixes for the PS and PE phases as well as the interfacial transfer matrix of unit cell. The analytical dispersion relations of the considered PS-PE metamaterial rod are obtained by utilizing Bloch's theorem. Numerical results show that band gap structures of the metamaterial rod are dependent on the initial electron concentration, length ratio of PE phase, interfacial condition, and capacitor connected with the PE phase. The selection of certain initial electron concentration, length ratio of PE phase, and capacitor can lead to the appearance of topological phase inversion, highlighting the potential for controlling and manipulating the wave propagation properties of PS metamaterial structures. This paper provides a theoretical guidance on the development of smart devices based on PS metamaterial structures. The novel combination of piezoelectricity and semiconductor behavior in these materials opens up new avenues for designing and engineering advanced devices with tailored wave properties and functionalities.

High-frequency dissipative MHD waves in straight magnetic cylindrical plasma: Coronal loops heating application

The theoretical model for analyzing the waves and oscillatory behavior in the structured solar corona using straight magnetic cylindrical geometry filled with uniform low-β plasma has been recognized as the most preferable classical model for the last few decades. A number of observations, since the first observation of the transition region and coronal explorer to the latest ones, have been adequately explained by adopting this model. In order to analytically formulate the oscillatory characteristics of magnetohydrodynamic (MHD) waves, most of the studies have considered the nature of plasma as an ideal fluid, particularly in the context of solar physics. However, a departure from ideal plasma consideration to non-ideal may lead to a number of modifications in the characteristics of the MHD waves, including its damping too. In what follows, we derive a more general analytical dispersion relation by extending the classical dispersion relation of [Edwin and Roberts, “Wave propagation in a magnetic cylinder,” Sol. Phys. 88, 179–191 (1983)] taking into account the effect of plasma viscosity as a non-ideal term in the existing formulations of the classical model. Consequently, the effects of viscosity on the damping of sausage and kink modes are examined in detail. Multiple trapped body waves of different frequencies exist for both kink and sausage modes in which trapped sausage body wave of comparatively high frequency is damped potentially to generate enough energy to balance the radiative losses of the coronal loop regions. For the coronal loop's plasma parameters, it is found that trapped first radial overtone body wave of sausage type is able to balance the radiative losses of coronal loop structure provided magnetic field strength does not exceed its value of more than 20G.

Plasma elongation effects on energetic particle-induced geodesic acoustic modes in tokamaks

Plasma elongation effects on energetic particle-induced geodesic acoustic modes (EGAMs) are theoretically investigated by using gyro-kinetic equations and the Miller local equilibrium model. Including an arbitrary elongation κ and a finite radial derivative sκ=r∂rκ/κ , a general EGAM dispersion relation is obtained for an arbitrary energetic particle (EP) distribution. In particular, we obtain analytical EGAM dispersion relations for both the double-shifted Maxwellian distribution and the standard slowing-down distribution of EPs. In both cases, the frequency of the unstable EGAM branch decreases slowly with increasing elongation, while its growth rate decreases rapidly with κ when the ratio of the EP to the bulk ion density, nh/ni⩾0.1 . These trends agree well with previous GENE and ORB5 simulations (Di Siena et al 2018 Nucl. Fusion 58 106014), but differ significantly from the elongation effects on geodesic acoustic modes (GAMs) (Gao et al 2009 Nucl. Fusion 49 045014). The portion of the EGAM dispersion relation accounting for the first-order finite-orbit-width shows greater sensitivity to frequency compared to that of GAM, which explains the smaller variations in the frequency of EGAM as κ changes. When the EP number is small (typically, nh/ni≈5% ) in the double-shifted Maxwellian case, the growth rate of EGAMs first increases with the increasing elongation and then decreases, while it monotonically increases with κ in the slowing-down case. Furthermore, the effects of s κ on EGAMs are similar to the elongation κ effects but weaker.

Low-frequency band gaps in quasi-zero stiffness locally resonant metamaterial shaft

The suppression of low-frequency torsional vibration has always been a difficult problem in the field of vibration control. Recently, metamaterials have attracted considerable attention due to their superior vibration insulation properties, at least in some targeted and tunable band gaps ranges. In particular, the position of the local resonance band gaps is related to the resonance frequency of the local resonator (LR). In this paper, a novel quasi-zero-stiffness torsional LR is proposed and a metamaterial shaft is devised by periodically attaching this LR onto a uniform shaft. The statics analysis for the torsional LR is conducted, and the propagation characteristics of the torsional vibration in the metamaterial shaft is studied with the transfer matrix theory and the Galerkin method. The analytical dispersion relation and the band gap structure indicate the existence of low frequency band gaps. And the location of the band gaps and the attenuation ability of torsional vibration can be effectively controlled by tuning the structural parameters of LR and the material of shaft, which provide a theoretical basis and guidance for addressing the challenge of low-frequency torsional vibration suppression.

Mathematical Analysis on Dispersion Relation of Rayleigh Wave for an Initially Stressed Magneto-Poroelastic Material with Corrugated and Impedance Boundary

This study showcases the Mathematical modelling of elastic surface waves and the analysis of dispersion relations of Rayleigh-type of wave in an initially stressed homogeneous porous magneto-elastic material having corrugated and impedance boundary exhibitions. Adoption of the classical harmonic method of wave analysis, non-dimensionalization of the resulting equations of motion, and corrugated-impedance boundary conditions produced by the modeled problem are also captured. The dispersion equation was analytically and graphically presented. Effects of the contributing physical quantities, such as impedance and corrugated boundary parameters, on Rayleigh waves for the chosen material are analyzed. The magnetic field’s influences and the wavenumber associated with the corrugated boundary surfaces on the material increase the dispersion relations of the Rayleigh wave profile on the material. In addition, we noted that the dispersion relations of the wave increases for increasing phase velocity and some parameters of voids. Also, a void parameter triggered a decrease on the dispersion profile of the Rayleigh wave when increased while noting some uniform exhibitions from other voids coefficients. This work holds the potential to provide more insights into studies involving displacement distributions in earthquake sciences, stress-strain analysis in Structural Engineering materials, among others.

Swirling instability of coaxial liquid jet in gas surroundings

Linear instability analysis of an inviscid coaxial swirling jet is carried out by deriving an analytical dispersion relation of perturbation growth. The azimuthal Rankine vortex and the axial discontinuous velocity distribution are utilized as the jet basic flow. Due to the existence of double interfaces, the instability mechanisms of the coaxial swirling jet are much more complex than those of the single-layered swirling jet. The effects of control parameters (including the swirling ratio, the Weber number, the jet radius ratio, the velocity ratios between different fluids, and the azimuthal velocity jump at the inner interface) on the temporal instability of coaxial swirling jet with different azimuthal modes are studied. By comparing the growth rate of different azimuthal modes, the predominant mode that determines the jet breakup is identified. The results indicate that an increase in the swirling ratio, the Weber number, and the radius ratio can lead to predominant mode transition to larger azimuthal wavenumbers. The velocity ratio between the inner jet and the annular jet and that between the surrounding fluid and the annular jet mainly affect the axial Kelvin–Helmholtz (KH) instability. An enhancement of the KH instability leads to the jet breakup with smaller azimuthal wavenumbers. The azimuthal velocity jump affects the azimuthal KH instability, the centrifugal instability, and the Coriolis instability simultaneously, thus leading to a multiple influence on modes transition. The phase-diagram of the predominant modes is further given, showing that the relative importance between the centrifugal force and the interfacial tension plays a significant role on the transition of predominant modes.

On the Propagation of Whistler‐Mode Waves in the Magnetic Ducts

AbstractThis paper studies extremely low frequency (ELF) whistler‐mode waves' behavior within small‐scale magnetic field irregularities in the Earth's magnetosphere, known as magnetic ducts. Based on the magnetic fields' magnitude inside and outside these ducts, they are categorized as high‐magnetic ducts and low‐magnetic ducts (LBD). Using the whistler‐mode dispersion relation analysis, our primary focus is to show that LBDs are prone to leak electromagnetic energy outside the duct. We further investigate the hypothesis that whistlers can propagate within LBDs without any signal loss when the width of the duct corresponds to an integer multiple of the perpendicular wavelengths of the waves inside it. This condition offers a straightforward and effective method for identifying non‐leaking eigenmodes of LBDs. Our analysis of this non‐leaking condition reveals that every LBD possesses a finite number of non‐leaking eigenmodes directly proportional to the duct's width and the magnitude of the ambient magnetic field within it. The analytical results are then validated using two‐dimensional, time‐dependent simulations of the electron‐Magnetohydrodynamics model. Also, we model the non‐leaking propagation of an ELF whistler‐mode wave observed inside the LBD by the NASA Magnetospheric Multiscale mission satellites.

The influence of an outer bath on the dewetting of an ultrathin liquid film

We report a theoretical and numerical investigation of the linear and nonlinear dynamics of a thin liquid film of viscosity μ sandwiched between a solid substrate and an unbounded liquid bath of viscosity λμ. In the limit of negligible inertia, the flow depends on two non-dimensional parameters, namely, λ and a dimensionless measure of the relative strengths of the stabilizing surface tension force and the destabilizing van der Waals force between the substrate and the film. We first analyze the linear stability of the film, providing an analytical dispersion relation. When the viscosity of the outer bath is much larger than that of the film, λ≫1, the most amplified wavenumber decreases as km∼λ−1/3, indicating that very slender dewetting structures are expected when λ becomes large. We then perform fully nonlinear simulations of the complete Stokes equations to investigate the spatial structure of the flow close to rupture revealing that the flow becomes self-similar with the minimum film thickness scaling as hmin=K(λ)τ1/3 when τ→0, where τ is the time remaining before the singularity. It is demonstrated that the presence of an outer liquid bath affects the self-similar structure obtained by Moreno-Boza et al. [“Stokes theory of thin-film rupture,” Phys. Rev. Fluids 5, 014002 (2020)] through the prefactor of the film thinning law, K(λ), and the opening angle of the self-similar film shape, which is shown to decrease with λ.

Dissipative Instability of Magnetohydrodynamic Sausage Waves in a Compressional Cylindrical Plasma: Effect of Flow Shear and Viscosity Shear

The shear flow influences the stability of magnetohydrodynamic (MHD) waves. In the presence of a dissipation mechanism, flow shear may induce an MHD wave instability below the threshold of the Kelvin–Helmholtz instability, which is called dissipative instability. This phenomenon is also called negative energy wave instability because it is closely related to the backward wave, which has negative wave energy. Considering viscosity as a dissipation mechanism, we derive an analytical dispersion relation for the slow sausage modes in a straight cylinder with a discontinuous boundary. It is assumed that the steady flow is inside and dynamic and bulk viscosities are outside the circular flux tube under photospheric condition. When the two viscosities are weak, it is found that for the slow surface mode, the growth rate is proportional to the axial wavenumber and flow shear, consistent within the incompressible limit. For a slow body mode, the growth rate has a peak at a certain axial wavenumber, and its order of magnitude is similar to surface mode. The linear relationship between the growth rate and the dynamic viscosity established in the incompressible limit develops nonlinearly when the flow shear and/or the two viscosities are sufficiently strong.

Porosities-dependent wave propagation in bi-directional functionally graded cantilever beam with higher-order shear model

The contribution provided in this paper is to investigate the dynamic response of bidirectional graded material beams with porosities. Higher order shear deformation beam theory for wave propagation in porous functionally graded cantilevered beam are developed, taking into account the bidirectional distribution, which is mainly represented in the density and the Young’s modulus. The material properties of the porous FG beam are graded through the thickness and width using a power law distribution. The governing equations of the wave propagation in the porous FG beam are derived by employing the Hamilton’s principle. The analytic dispersion relation of the bidirectional porous FG beam is obtained by solving an eigenvalue problem. Two porosities models approximating the even and uneven distributions of porosity are considered. The results obtained for bidirectional FG beam were compared with those reported in the literature, and the results were very agreement. The effects of the volume fraction distributions, the number of wave propagation, various parameters and the porosity type on dynamic of functionally graded beam are discussed in details. The numerical results show that the power law index, number of wave, wave velocity, and porosity distribution model affect the dynamic behavior of the beam significantly. The results show that the uneven porosity predicts higher natural frequencies than the even porosity.

Improved analytic modeling of the linear Rayleigh–Taylor instability with plasma transport

The Rayleigh–Taylor (R–T) instability with transport effects has long been studied analytically in the neutral fluid regime. How this picture may differ in plasmas has been given less consideration, however. Leveraging previous numerical simulations with plasma viscosity and diffusion of R–T at a binary plasma interface, we show how plasma transport and kinetic effects alter the linear R–T dispersion relation. Our insight informs a much improved analytical dispersion relation, which better matches the simulation data than previously established models. Additionally, we demonstrate the importance of employing asymptotically correct viscosity coefficients in fluid codes for accurately capturing plasma R–T growth.

A two-dimensional wave and Rayleigh–Ritz method for complex dispersion in periodic arrays of circular damped acoustic black holes

Two-dimensional arrays of circular acoustic black holes (ABHs) on plates are known to exhibit many surprising properties such as bandgap formation, scattering, topological edge states and negative refraction and bi-refraction, to name a few. These effects can be deduced from the analysis of dispersion relations. However, to date, most of the work on periodic ABH arrays has only dealt with real dispersion curves that do not adequately reproduce their behavior. The role of the damping layer and intrinsic losses, which is essential to ABHs, is ignored. This can be accounted for by complex dispersion. In this paper, we extend the wave and Rayleigh–Ritz method (WRRM) previously developed for periodic ABHs in beams to address complex dispersion in two-dimensional periodic ABH arrays. This poses several challenges. On the one hand, in the WRRM the solution of the equations of motion is expanded in terms of the basis of the nullspace determined by the periodic boundary conditions. Although this only involves point connections for beams, in the two-dimensional case continuous line constraints are present and care must be taken to find the appropriate nullspace. To obtain the complex dispersion surface and complex equi-frequency contours of two-dimensional periodic ABH arrays we then need to set the eigenvalue problem in the k(ω) approach. For a fixed angular frequency ω∈R and a direction of propagation, θ, whose tangent is the quotient of the unknown wavenumber components in the y and x directions, we derive polynomial and transcendental eigenvalue problems for the complex wavenumber k∈ℂ×ℂ as a function of θ. The complex dispersion surface and equi-frequency contours of the periodic ABH arrays can be obtained by solving these eigenvalue problems with an iteration method. The two-dimensional WRRM is validated by comparison with finite element simulations. Finally, we apply it to calculate the complex dispersion of an infinite periodic strip and of an infinite periodic array of circular ABHs. In the latter case, collimation effects are observed. The transmissibility of finite ABH strips and arrays is also presented for comparison.

Caustic spin wave beams in soft thin films: Properties and classification

In the context of wave propagation, caustics are usually defined as the envelope of a finite-extent wavefront; folds and cusps in a caustic result in enhanced wave amplitudes. Here, we tackle a related phenomenon, namely, the existence of well-defined beams originating solely from the geometric properties of the corresponding dispersion relation. This directional emission, termed caustic beam, is enabled by a stationary group velocity direction, and has been observed first in the case of phonons. We propose an overview of this ``focusing'' effect in the context of spin waves excited in soft, thin ferromagnetic films. Based on an analytical dispersion relation, we provide tools for a systematic survey of caustic spin wave beams. Our theoretical approach is validated by time-resolved microscopy experiments using the magneto-optical Kerr effect. Then, we identify two cases of particular interest both from fundamental and applicative perspectives. Indeed, both of them enable broadband excitations (in terms of wave vectors) to result in narrow-band beams of low divergence.

Pipe vibration attenuation through internal damping and optimal design of vibro-impact systems

Pipelines periodically supported by rack structures (PPRs) are common in chemical and petrochemical plants, among others, and conventional tools such as dampers and hysteretic absorbers are commonly used to mitigate large vibrations in these systems. In this study, we explore two alternative strategies: (i) enhancing the attenuation rate of PPR vibrations through structural internal damping, and (ii) using nonlinear vibro-impact systems (VIS) to reduce seismic vibrations in a PPR. To shed light on the first strategy, we develop analytical dispersion relations for a PPR and show how damping can improve the mitigation capabilities of the periodic system. As for the second strategy, we consider a 9-node beam, i.e., a single span (SS) of a PPR equipped with a VIS, and combine the central composite design (CCD) and Kriging metamodelling to maximize dissipation energy and minimize the number of impacts. This multi-objective optimization problem aims to find the most effective design solution for the VIS in terms of gap and coefficient of restitution (COR). Additionally, we consider the stochastic nature of seismic input and the possible chaotic behavior of the VIS. To account for the sensitive variability of the number of impacts in seismic records, we perform incremental dynamic analyses and calculate fragility functions for various engineering demand parameters, including the number of impacts. We define a 3D surface for selecting the optimal gap-COR pair. When impacts occur, transient results can be chaotic, and we compute the largest Lyapunov exponents of a few representative trajectories.

Influence of a random phase plate on the growth of the backward stimulated Brillouin scatter.

We derive the analytical dispersion relation of a high-energy laser beam's backward stimulated Brillouin scattering (BSBS) in a hot plasma, that accounts both for the random phase plate (RPP) induced spatial shaping and its associated phase randomness. Indeed, phase plates are mandatory in large laser facilities where a precise control of the focal spot size is required. While the focal spot size is well controlled, such techniques produce small scale intensity variations that can trigger laser-plasma instabilities such as BSBS. Quantifying the resulting instability variability is shown to be crucial for understanding accurately the backscattering temporal and spatial growth as well as the asymptotic reflectivity. Our model, validated by means of a large number of three-dimensional paraxial simulations and experimental data, offers three quantitative predictions. The first one addresses the temporal exponential growth of the reflectivity by deriving and solving the BSBS RPP dispersion relation. A large statistical variability of the temporal growth rate is shown to be directly related to the phase plate randomness. Then, we predict the portion of the beam's sectionthat is absolutely unstable, thus helping to precisely assess the validity of the vastly used convective analysis. Finally, a simple analytical correction to the plane wave spatial gain is extracted from our theory giving a practical and effective asymptotic reflectivity prediction that includes the impact of phase plates smoothing techniques. Hence, our study sheds light on the long-time studied BSBS, deleterious to many high-energy experimental studies related to the physics of inertial confinement fusion.

Rossby waves on stellar equatorial β planes: Uniformly rotating radiative stars

Context. Rossby waves arise due to the conservation of total vorticity in rotating fluids and may govern the large-scale dynamics of stellar interiors. Recent space missions have collected a lot of information about the light curves and activity of many stars, which triggered observations of Rossby waves in the stellar surface and interiors. Aims. We aim to study the theoretical properties of Rossby waves in stratified interiors of uniformly rotating radiative stars with a sub-adiabatic vertical temperature gradient. Methods. We used the equatorial β plane approximation and linear vertical gradient of temperature to study the linear dynamics of equatorially trapped Rossby and inertia-gravity waves in interiors of radiative stars. The governing equation was solved by the method of separation of variables in the vertical and latitudinal directions. Results. Vertical and latitudinal solutions of the waves are found to be governed by Bessel functions and Hermite polynomials, respectively. Appropriate boundary conditions at the stellar surface and poles define analytical dispersion relations for Rossby, Rossby-gravity, and inertia-gravity waves. The waves are confined in surface layers of 30–50 H0, where H0 is the surface density scale height, and they are trapped between the latitudes of ±600. Observable frequencies (normalised by the angular frequency of the stellar rotation) of Rossby waves with m = 1 (m = 2), where m is the toroidal wavenumber, are in the interval of 0.65–1 (1.4–2), depending on the stellar rotation, radius, and surface temperature. Conclusions. Rossby-type waves can be systematically observed using light curves of Kepler and TESS (Transiting Exoplanet Survey Satellite) stars. Observations and theory then can be used for the sounding of stellar interiors.

A scale-aware dispersion-relation-preserving finite difference scheme for computational aeroacoustics

The numerical schemes for computational aeroacoustics (CAA) should have minimal dispersion and proper dissipation in order to accurately capture the amplitude and phase of waves. In this paper, we propose a scale-aware dispersion-relation-preserving (SA-DRP) finite difference scheme based on an improved scale sensor and a new dispersion control strategy. The scale sensor quantifies the local length scale of the solution in the form of the effective scaled wavenumber. The new feature of this scale sensor is the accurate prediction of the wavenumber for a pure sine wave. The new dispersion control strategy determines the dispersion parameter of the scheme in terms of the scale sensor. In contrast to the traditional dispersion-relation-preserving (DRP) scheme that minimizes the integral dispersion error, the new strategy directly solves the dispersion parameter by requiring the numerical dispersion relation to be equal to the exact one. As a result, precise dispersion relation can be realized within a very broad wavenumber range. The approximate dispersion relation analysis shows that the SA-DRP scheme maintains an accurate dispersion relation up to the scaled wavenumber k = 2.5. Moreover, the overshoot in the dispersion relation of the DRP scheme is not presented in that of the SA-DRP scheme. To suppress nonphysical oscillations, we also add proper dissipation that is adjusted automatically according to the effective scaled wavenumber. Several CAA benchmark test cases are presented to demonstrate the higher resolution and higher efficiency achieved by the proposed scheme compared with the conventional spectrally optimized schemes.

Dyakonov surface polaritons in antiferromagnet film

Dyakonov surface polaritons (DSPs) in a single antiferromagnetic (AF) film are investigated in the parallel geometry where the AF anisotropic axis is parallel to the film and no external magnetic field is applied. The analytical dispersion relation is obtained and four surface modes are found, among which one symmetric mode and an anti-symmetric mode are exist in the AF reststrahlen frequency band, and another symmetric and anti-symmetric modes are situated out the reststrahlen band. The existence of symmetric and anti-symmetric modes is caused by the mirror symmetry and finite thickness of the AF film. A pair of modes in the band have magnetostatic limits but the other pair out the band do not. The dispersion properties, field and energy-flux density distributions of each DSP are numerically calculated based on the FeF2 film for various propagating directions and film thicknesses. It is very interesting that the energy flux of each DSP can seriously deviates from its propagation direction, and the largest walk-off angle can reach 200°. In addition, we briefly discuss dispersion properties of Dyakonov odd and even guided modes, they also are symmetric and antisymmetric and have multiple solutions.