Dispersion Relation For Waves Research Articles

Rayleigh wave propagation in an initially stressed orthotropic elastic solid half-space lying under inviscid liquid and viscous liquid layers

PurposeThe purpose of this article is the propagation of Rayleigh waves in a homogeneous, isotropic, initially stressed orthotropic elastic solid half-space lying under a homogeneous, viscous, inviscid liquid layer of finite thickness.Design/methodology/approachIn the presence of both viscous and inviscid liquids, it derives the phase velocity, initial stress, and wave number-dependent frequency equation for an orthotropic elastic solid. With the help of the MATLAB program, the thickness effects of liquid layers, initial stress, and viscosity on the phase velocity and attenuation coefficient of the Rayleigh wave are explained for a particular model.FindingsThe phase velocity-dependent dispersion relation of Rayleigh waves at the interface of viscous liquid and solid half-space is a function of initial stress and wave number. Rayleigh waves along the free surface of an orthotropic elastic half-space are also derived as a particular case. The classical results of an inviscid liquid are achieved when the thickness of a viscous liquid approaches zero. Well-known classical results for initially stressed orthotropic elastic solids were also derived.Originality/valueSo far, many researchers have looked into the propagation of surface waves at the interfaces of solid–inviscid liquid, solid–solid, and multilayer interfaces. But in this article, the dispersion behavior of Rayleigh wave propagation in an initially stressed homogeneous orthotropic elastic solid half-space under a double layer of viscous liquid and inviscid liquid is studied.

Ultra-low frequency and broadband flexural wave attenuation using an inertant nonlinear metamaterial beam

Attenuation of elastic waves in the extremely low frequency range is a considerable challenge. While linear acoustic metamaterials can manipulate elastic waves, their effectiveness is often limited to a narrow frequency range. The present paper proposes a novel inertant nonlinear metamaterial beam to address the challenging problem of the ultra-low frequency broadband flexural wave attenuation. The amplitude-frequency response and dispersion relations of the flexural waves are obtained using the first-order harmonic balance method, where four different resonant units are investigated and the resulting band gaps are compared. Finite element (FE) simulations are also conducted to evaluate the transmission spectra of the flexural vibration through a finite metamaterial beam. The good consistency between the band gaps predicted by the analytical model and the FE simulation verifies the correctness and effectiveness of the developed analytical approach. The results of this study demonstrate that introducing the inertance and softening nonlinearity into the resonant units can significantly widen the flexural wave band gaps within the low-frequency range. This finding provides a viable solution for engineering applications that require low-frequency vibration suppression and isolation.

Asymptotic integrability of nonlinear wave equations.

We introduce the notion of asymptotic integrability into the theory of nonlinear wave equations. It means that the Hamiltonian structure of equations describing propagation of high-frequency wave packets is preserved by hydrodynamic evolution of the large-scale background wave so that these equations have an additional integral of motion. This condition is expressed mathematically as a system of equations for the carrier wave number as a function of the background variables. We show that a solution of this system for a given dispersion relation of linear waves is related to the quasiclassical limit of the Lax pair for the completely integrable equation having the corresponding dispersionless and linear dispersive behavior. We illustrate the theory with several examples.

Anisotropic Magnetized Asteroseismic Waves

We solve for waves in a polytropic, stratified plasma with a spatially varying background magnetic field that points along the horizontal x-direction, and with gravity that is directed along the vertical z-direction. Force balance determines the magnitude of the background magnetic field, B02∼zn+1 , where n is the polytropic index. Using numerical and asymptotic methods, we deduce an explicit dispersion relation for fast pressure-driven waves: Ω2∼K2m+n[1+1/MA2(4−2γ + cos2θ−3cos4θ)/4] . Here, Ω is the frequency, K is the wavenumber, θ is the angle the wavevector makes with the background magnetic field, M A is the Alfvénic Mach number, and m is an integer representing the eigenstate. We discuss the roles of such an explicit formula in asteroseismology.

ULTRASONIC INTERROGATION TECHNIQUE FOR DETECTING HATCH ANGLE VARIATIONS IN ADDITIVE MANUFACTURING BY PHONONIC LATTICE STRUCTURES

Abstract Additive Manufacturing/3D Printing (AM/3DP) has revolutionized part production by enabling the creation of intricate internal structures and complex geometries from diverse materials directly from digital design files. Among powder-based metal AM/3DP methods, Selective Laser Melting (SLM) is widely used in advanced applications such as biomedical devices and aerospace parts. Despite considerable progress in AM/3DP and SLM, at present, challenges in print quality persist, and vast resources for post-production quality assessment are allocated. The quality of SLM prints is influenced by various process and design parameters, such as the accuracy of hatch angle deposition, laser intensity/power, scanning speed of the laser beam, print line spacing, layer depth, printing chamber conditions, and the material's physical and chemical properties. Direct ultrasonic Non-Destructive Evaluation (NDE) offers comprehensive internal inspection and real-time data acquisition ability; however, in AM/3DP it faces severe limitations due to a build's intricate internal and external geometric features. In the current study, we present a Phononic Crystal Artifact (PCA)-based real-time ultrasonic NDE quality monitoring framework and show offline its utility in detecting and evaluating hatch angle variations, a critical process parameter. A PCA is substantially simpler and smaller than the actual build but represents its critical geometric and structural intricacies and mechanical properties. The current offline study demonstrates that hatch angle variations can be monitored from ultrasonic responses' spectral modal frequency peaks and wave dispersion relations.

Topological interface and corner states of flexural waves in metamaterial plates with glide-symmetric holes

This study investigates the topological localization and propagation of flexural waves in metamaterial plates with perforated holes exhibiting glide symmetry. The dispersion relations of flexural waves in this mechanical system exhibit four-fold Dirac degeneracies at the M point of the Brillouin zone. By properly adjusting the rotation angles of the perforated holes, these degeneracies can be lifted up to create omnidirectional bandgaps, enabling nontrivial topological phases. Both numerical and experimental results demonstrate multiple topological states of flexural waves in the developed elastic metaplates, including dual-band helical interface states and anisotropic interface/corner states. The simple perforated metaplates with glide-symmetric holes provide novel, manufacturable platforms for the flexible manipulation of topological flexural waves, greatly facilitating potential applications such as energy harvesting and signal enhancement.

Electromagnetic Surface Waves in a Relativistic Magnetized Plasma Propagating in a Cylindrical Column

Dispersion relations of electromagnetic surface waves in a magnetized relativistic plasma propagating in a cylindrical column surrounded by vacuum are derived by means of the relativistic fluid equations. The zero order relativistic beam is assumed to stream along the direction of the axial magnetic field. The coupled equations of the E- and B-wave modes are solved analytically in terms of the cylindrical coordinates in weak or intermediate values of the magnetic field in which the cyclotron frequency is much smaller than the relativistic Doppler-shifted frequency. The entangled field components of the E- and B-waves are decoupled by assuming that the flow is incompressible. The surface wave dispersion relations of the E- and B-wave modes show a strong similarity and symmetry. The static magnetic field splits the E- and B-waves, thus doubling the modes in unmagnetized plasma. By taking the radius of the cylinder infinite, the dispersion relations in a semi-infinite plasma are derived.

Nonlinear Ion Acoustic Waves in Multicomponent Plasmas with Nonthermal Electrons-positron and Bipolar Ions

Abstract This paper studied the propagating characteristics of (2+1) dimensional nonlinear ion acoustic waves in a multicomponent plasma with nonthermal electrons, positrons and bipolar ions. The dispersion relations are initially explored by using the small amplitude wave's dispersion relation. Then, the Sagdeev potential method is employed to study large amplitude ion acoustic waves. The analysis involves examining the system's phase diagram, Sagdeev potential function, and solitary wave solutions through numerical solution of an analytical process in order to investigate the propagation properties of nonlinear ion acoustic waves under various parameters. It is found that the propagation of nonlinear ion acoustic waves is subject to the influence of various physical parameters, including the ratio of number densities between the unperturbed positrons, electrons to positive ions, nonthermal parameters, the masses ratio of positive ions to negative ions, and the charge numbers ratio of negative ions to positive ions, the ratio of the electrons’ temperature to positrons’ temperature. In addition, the multicomponent plasma system has a compressive solitary waves with amplitude greater than zero or a rarefactive solitary waves with amplitude less than zero, in the meantime, compressive and rarefactive ion acoustic wave characteristics depend on the charge numbers ratio of negative ions to positive ions.

Cosmological study in Myrzakulov [formula omitted] quasi-dilaton massive gravity

This study explores the cosmological implications of the Myrzakulov F(R,T) quasi-dilaton massive gravity theory, a modification of the de Rham-Gabadadze-Tolley (dRGT) massive gravity theory. Our analysis focuses on the self-accelerating solution of the background equations of motion, which are shown to exist in the theory. Notably, we find that the theory features an effective cosmological constant corresponding to the massive graviton, which has important implications for our understanding of the universe’s accelerated expansion. To assess the validity of the Myrzakulov F(R,T) quasi-dilaton massive gravity theory, we employ two datasets: the Union2 Type Ia Supernovae (SNIa) dataset, consisting of 557 observations, and the Pantheon SNIa data, which includes 1048 SNe I-a events gathered from diverse SN I-a samples. Our results demonstrate that the theory is capable of explaining the accelerated expansion of the universe without requiring the presence of dark energy. This finding supports the potential of the Myrzakulov F(R,T) quasi-dilaton massive gravity theory as an alternative explanation for the observed cosmic acceleration. Moreover, we investigate the properties of tensor perturbations within the framework of this theory and derive a novel expression for the dispersion relation of gravitational waves. Our analysis reveals interesting features of the modified dispersion relation in the Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology, providing new insights into the nature of gravitational waves in the context of the Myrzakulov F(R,T) quasi-dilaton massive gravity theory.

Constitutive modelling and wave propagation through a class of anisotropic elastic metamaterials with local rotation

Elastic metamaterials are typically periodic materials possessing unit cells endowed with engineered architecture much smaller than the typical phenomenological length scale. The development of continuum models capable of accurately representing the effects of this aforementioned architecture is extremely challenging, and hence a sparsely developed area. This paper develops a novel 2-D continuum model capable of capturing the dynamic behaviour of a class of anisotropic elastic metamaterials with local rotational elements in the long wavelength limit. A constitutive relation incorporating these local rotational effects is proposed, and ratified using a representative discrete model using linear Hookean springs and identical rigid disks. The new continuum model is used to generate a dispersion relation for harmonic plane waves propagating in an arbitrary direction, which is subsequently compared to the dispersion behaviour of the original discrete model. The general behaviour of this continuum when subjected to 2-D planar harmonic wave propagation in the anisotropic medium is then analyzed, with specific attention given to the effect of material anisotropy and wave propagation direction. This work is the first of its kind to create a new continuum model of a class of anisotropic elastic metamaterials with local rotational effects.

Estimating the In Situ Stratification via Remotely Sensed Internal Wave Speeds

Abstract This work tests a methodology for estimating the ocean stratification gradient using remotely sensed, high temporal and spatial resolution field measurements of internal wave propagation speeds. The internal wave (IW) speeds were calculated from IW tracks observed using a shore-based, X-band marine radar deployed at a field site on the south-central coast of California. An inverse model, based on the work of Kar and Guha, utilizes the linear internal wave dispersion relation, assuming a constant vertical density gradient is the basis for the inverse model. This allows the vertical gradient of density to be expressed as a function of the internal wave phase speed, local water depth, and a background average density. The inputs to the algorithm are the known cross-shore bathymetry, the background ocean density, and the remotely sensed cross-shore profiles of IW speed. The estimated density gradients are then compared to the synchronously measured vertical density profiles collected from an in situ instrument array. The results show a very good agreement offshore in deeper water (∼50–30 m) but more significant discrepancies in shallow water (20–10 m) closer to shore. In addition, a sensitivity analysis is conducted that relates errors in measured speeds to errors in the estimated density gradients. Significance Statement The propagation speed of ocean internal waves inherently depends on the vertical structure of the water density, which is termed stratification. In this work, we evaluate and test with real field observations a technique to infer the ocean density stratification from internal wave propagation speeds collected from remote sensing images. Such methods offer a way to monitor ocean stratification without the need for extensive in situ measurements.

Aurorally Driven Supersonic Gravity Waves in Saturn's Atmosphere

AbstractSimulations with the Saturn Thermosphere‐Ionosphere General Circulation Model have revealed global‐scale gravity waves in Saturn's upper atmosphere that have not been observed or predicted before. They are forced by diurnally varying zonally non‐uniform distributions of Joule heating and ion drag at the high‐latitude auroral regions in both hemispheres and propagate outward from the source region. The supersonic zonal phase speed imposed by Saturn's rotation and the subsonic meridional phase velocity of about 800 m form a distinct spiral wave structure. They exhibit features of gravity waves: vertical phase progression opposite to the propagation of wave energy and long vertical wavelengths consistent with the dispersion relation for gravity waves. The amplitudes of the perturbations grow with height, reaching 6%–10% for relative temperature variations and up to 350 m for the meridional velocity perturbations. The main effect of these waves is to accelerate the retrograde westerly jets.

Whitham modulation theory and Riemann problem for the Kundu–Eckhaus equation

In this paper, the Riemann problem for the defocusing Kundu–Eckhaus equation is investigated by Whitham modulation theory. First, we study the dispersion relation for linear waves. Then, the zero-phase and one-phase periodic solutions of the Kundu–Eckhaus equation along with the corresponding Whitham modulation equations are derived by the finite-gap integration method. Further, employing the Whitham equations parametrized by the Riemann invariants, the main fundamental wave structures induced by the discontinuous initial data are found. Analytical and graphic methods are utilized to provide the wave structures of rarefaction waves and dispersive shock waves, and thus for a complete classification of solutions under general step-like conditions of initial discontinuity.

Rayleigh Waves in a Thermoelastic Half-Space Coated by a Maxwell–Cattaneo Thermoelastic Layer

This paper investigates the propagation of in-plane surface waves in a coated thermoelastic half-space. First, it investigates a special case where the surface layer is described by the Maxwell–Cattaneo thermoelastic approach, while the half-space is filled by a thermoelastic material described by the classical Fourier law for the heat flux. The contact between the layer and the half-space is assumed to be welded, i.e., the displacements and the temperature, as well as the stresses and the heat flux are continuous through the interface of the layer and the half-space. The boundary and continuity conditions of the problem are formulated and then the exact dispersion relation of the surface waves is established. An illustrative numerical simulation is presented for the case of an aluminum thermoelastic layer coating a thermoelastic copper half-space, highlighting important aspects regarding the propagation of Rayleigh waves in such structures. The exact effective boundary conditions at the interface are also established replacing the entire effect of the layer on the half-space. The general case of the problem is also investigated when both the surface layer and the half-space are described by the Maxwell–Cattaneo thermoelasticity theory. This study helps to further understand the propagation characteristics of elastic waves in layered structures with thermal effects described by the Maxwell–Cattaneo approach.

Trigger mechanism for a singing cavitating tip vortex

The discrete tone radiated from tip vortex cavitation (TVC), known as ‘vortex singing’, was recognized in 1989, but its triggering remains unclear for over thirty years. In this study, the desinent cavitation number and viscous correction are applied to describe the dynamics of cavitation bubbles and the dispersion relation of cavity interfacial waves. The wavenumber-frequency spectrum of the cavity radius from the experiment in CSSRC indicates that singing waves predominantly consist of the stationary double helical modes (kθ = 2- and -2+) and the breathing mode (kθ = 0-), rather than standing waves as assumed in previous literatures. Moreover, two trigger mechanisms, expressed by two triggering lines, are proposed: the twisted TVC, initially at rest, is driven into motion through the corrected natural frequency (fn) due to the step change of the far-field pressure. Subsequently, the frequency associated with the zero-group-velocity point (ῶzgv) at kθ = 0- is excited through ῶi, the frequency at the intersection of dispersion curves at kθ = 0- and -2+, or ῶj, the frequency at the intersection of dispersion curves at kθ = 0- and 2-, corresponding to two types of the vortex singing triggering. These solutions, without empirical parameters, are validated using singing conditions provided by CSSRC and G.T.H., respectively. Furthermore, the coherence and the cross-power spectral density spectrum indicates a large-scale breathing wave propagating along the singing cavity surface and travelling from downstream to hydrofoil tip, providing us a comprehensive understanding for the triggering of vortex singing.

The role of the thermal properties of electrons on the dispersion properties of Alfvén waves in space plasmas

Context. The transition from left-hand to right-hand polarised Alfvén waves depends on the wavenumber, the ratio of kinetic to magnetic pressure, β, the temperature anisotropy, and the ion composition of the plasma. Along with the temperature anisotropy, the electron-to-proton temperature ratio, Te/Tp, is of great relevance for the characterisation of the thermal properties of a plasma. This ratio varies significantly between different space plasma environments. Thus, studying how variations in this ratio affect the polarisation properties of electromagnetic waves becomes highly relevant for our understanding of the dynamics of space plasmas. Aims. We present an extensive study on the effect of the thermal properties of electrons on the behaviour and characteristics of Alfvénic waves in fully kinetic linear theory, as well as on the transition from the electromagnetic ion-cyclotron wave to the kinetic Alfvén wave. Methods. We solved the fully kinetic dispersion relation for oblique electromagnetic waves of the Alfvén branch in a homogenous Maxwellian electron-proton plasma. We quantified the effect of the thermal properties of electrons by varying the electron-to-proton temperature ratio for different configurations of the propagation angle, βp = 8πnkTp/B2, and wavenumber. Results. We show that the temperature ratio, Te/Tp, has strong and non-trivial effects on the polarisation of the Alfvénic modes, especially at kinetic scales (k⊥ρL > 1, where k⊥ = k sin θ, θ is the propagation angle, and ρL = cs/Ωp, with cs the plasma sound speed and Ωp the proton’s gyrofrequency) and βe + βp > 0.5. We conclude that electron inertia plays an important role in the kinetic scale physics of the kinetic Alfvén wave in the warm plasma regime, and thus cannot be excluded in hybrid models for computer simulations.

The best whistler: A cavitating tip vortex

The discrete tone radiated from a cavitating tip vortex, known as “vortex singing,” was first recognized in 1989, but its sound generation mechanism has remained a mystery for over 30 years. In this Letter, by means of the correction for the cavitation bubble dynamics and the dispersion relation of cavity interfacial waves, we found that after the far-end disturbances propagate upstream, the whistling vortex should be triggered by near-end sound sources, the breathing mode waves. Further utilizing the theoretical solutions for singing lines and the potential singing cavitation number with frequency, we accurately identified all available tests for seeking the vortex singing over the past three decades, which not only demonstrates that the vortex singing frequency is only determined by the measurable mean cavity radius (rc), cavitation number (σ), and desinent cavitation number (σd) in experiments, but also responses to a long-standing perplexity: why such a best whistler is able to appear only within a narrow range of the cavitation number.

The steady-state solution of wave–current interaction based on the third-order Stokes wave theory

This manuscript reports on the interaction of a current-free monochromatic surface wave field with a wave-free uniform current field. The existing reasonable theories of wave–current interactions are primarily based on weak current assumptions and derived from linear theory, resulting in calculation bias in the analysis of nonlinear wave–current interactions. Moreover, experimental data on high-order wave–current interactions still need to be collected. Thus, steady-state solutions named the third-order wave–current theory based on the third-order wave dispersion relationship and the principle of wave–current energy conservation were derived. The wave–current interaction experiment was set up to cover 164 sets of experimental conditions, including 33 types of periodic waves from the second to the fifth order and six different current velocities. The effects of water depth, current velocity, wave period, and height on the wave height and wavelength in the wave–current interaction field were investigated. A comparison of the mean relative error (MRE) and the determination coefficient (R2) of the wavelength with the experimental data revealed that the third-order wave–current theory outperformed the traditional linear theory, with an optimal reduction of 75% and an enhancement of 25%, respectively. Additionally, the third-order wave–current theory reduces the MRE by 25%–40% in the wave height calculation, with R2 consistently outperforming the linear theory. The third-order wave–current theory can significantly improve the calculation accuracy of the theoretical method in solving nonlinear wave–current interactions.

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

Methodological notes on gauge invariance in the treatment of waves and oscillations in plasmas via the Einstein-Vlasov-Maxwell system: Fundamental equations.

The theory of gauge transformations in linearized gravitation is investigated. After a brief discussion of the fundamentals of the kinetic theory in curved spacetime, the Einstein-Vlasov-Maxwell (EVM) system of equationsin terms of gauge-invariant quantities is established without neglecting the equationsof motion associated with the dynamics of the nonradiative components of the metric tensor. The established theory is applied to a noncollisional electron-positron plasma, leading to a dispersion relation for gravitational waves in this model system. The problem of Landau damping is addressed and some attention is given to the issue of the energy exchanges between the plasma and the gravitational wave. In a future paper, a more complete set of approximate dispersion relations for waves and oscillations in plasmas will be presented, including the dynamics of nonradiative components of the metric tensor, with special attention to the problems of the Landau damping and of the energy exchanges between matter, the electromagnetic field and the gravitational field.