Dispersion Relation For Waves Research Articles

Electrically excited magnetoelastic waves and magnetoacoustic resonance in ferromagnetic films with voltage-controlled magnetic anisotropy

Ferromagnet-dielectric heterostructures may possess interfacial magnetic anisotropy, which varies with the electric field created in the dielectric interlayer. Such a voltage-controlled magnetic anisotropy (VCMA) represents an efficient tool for the excitation of spin waves without the use of microwave magnetic fields. In ferromagnets with a significant magnetoelastic coupling between spins and strains, however, the magnetization precession in the spin wave induces elastic vibrations. Here we report micromagnetoelastic simulations of the coupled spin and acoustic dynamics generated in the thick Fe film brought into contact with an MgO nanolayer subjected to a microwave voltage. It is found that the electrically induced periodic modulation of VCMA gives rise to the excitation of both spin and elastic waves in the magnetostrictive Fe film. Remarkably, the magnetoelastic coupling makes it possible to generate propagating spin waves having frequencies well below the ferromagnetic resonance frequency. Furthermore, when the voltage frequency approaches a value ${\ensuremath{\nu}}^{*}$ at which the dispersion relations of pure spin and transverse elastic waves cross, the excitation of traveling magnetoelastic waves with a specific hybrid dispersion occurs in the Fe film. To explain the variety of observed phenomena, we propose a generalized dispersion relation for magnetoelastic waves, which accounts for a possible difference between the wave numbers of their magnetic and elastic components. The analysis of the simulation data also reveals that frequency dependencies of the amplitudes of magnetic oscillations and elastic vibrations display significant anomalies around the crossing-point frequency ${\ensuremath{\nu}}^{*}$, which originate from the electrically induced magnetoacoustic resonance.

A simple approximation for the drift rates of rotating polygons on a free fluid surface

We report on laboratory experiments investigating the dynamics of a free fluid surface in a cylindrical tank with a rotating bottom plate. The shear instability in the system creates polygonal structures, which propagate around the domain, along the fixed vertical sidewalls. We analyze the wavelengths and drift rates of these patterns, which are known from previous literature to be created by a complex interplay between centrifugal effects and gravity wave propagation in this unique geometry. We find that with an empirical correction factor, the drift rates of the polygonal vortices can be approximated fairly well by a surprisingly simple formula derived from the dispersion relation of linear gravity waves using easily observable parameters.

Spin wave dispersion relation engineering by magnonic crystals with arbitrary symmetry

The use of metasurfaces to engineer the response of magnetic materials is of utmost importance in the field of magnon-spintronics. Here, we demonstrate a method to fabricate one- and two-dimensional magnonic crystals with arbitrary symmetry and use it to engineer the amplitude-frequency characteristic of magnetostatic surface spin waves excited in a magnetic material. The technique is based on the gentle microablation of the sample surface by focused femtosecond laser pulses. Tightly focused illumination allows using modest pulse energy while achieving micrometer precision. By raster scanning the incident laser spot on the sample surface, we control the shape and size of the building blocks constituting the unit cell of the crystal along with its symmetry and lattice parameter. Remarkable and controlled changes in the measured transmission characteristics reveal the strong and complex symmetry-dependent interaction of the spin waves with Bravais and non-Bravais lattices. The described single-step microfabrication method facilitates and speeds up the realization of integrated spintronics components and provides an efficient tool to explore complex magnetic dynamics in scattering lattices.

Observations of Waves and Structures by Frequency–Wavenumber Spectrum in Solar Wind Turbulence

A well-known shortcoming of single-spacecraft spectral analysis is that only the 1D wavenumber spectrum can be observed, assuming the characteristic wave propagation speed is much smaller than the solar wind flow speed. This limitation has motivated an extended debate about whether fluctuations observed in the solar wind are waves or structures. Multispacecraft analysis techniques can be used to calculate the wavevector independent of the observed frequency, thus allowing one to study the frequency–wavenumber spectrum of turbulence directly. The dispersion relation for waves can be identified, which distinguishes them from nonpropagating structures. We use magnetic field data from the four Magnetospheric Multiscale (MMS) spacecraft to measure the frequency–wavenumber spectrum of solar wind turbulence based on the k-filtering and phase differencing techniques. Both techniques have been used successfully in the past for the Earth’s magnetosphere, although applications to solar wind turbulence have been limited. We conclude that the solar wind turbulence intervals observed by MMS show features of nonpropagating structures that are associated with frequencies close to zero in the plasma rest frame. However, there is no clear evidence of propagating Alfvén waves that have a nonzero rest-frame frequency. The lack of waves may be due to instrument noise and spacecraft separation. Our results support the idea of turbulence dominated by quasi-2D structures.

The interactions of the elliptical instability and convection

Elliptical instability is an instability of elliptical streamlines, which can be excited by large-scale tidal flows in rotating fluid bodies and excites inertial waves if the dimensionless tidal amplitude (ε) is sufficiently large. It operates in convection zones, but its interactions with turbulent convection have not been studied in this context. We perform an extensive suite of Cartesian hydrodynamical simulations in wide boxes to explore the interactions of elliptical instability and Rayleigh–Bénard convection. We find that geostrophic vortices generated by the elliptical instability dominate the flow, with energies far exceeding those of the inertial waves. Furthermore, we find that the elliptical instability can operate with convection, but it is suppressed for sufficiently strong convection, primarily by convectively driven large-scale vortices. We examine the flow in Fourier space, allowing us to determine the energetically dominant frequencies and wavenumbers. We find that power primarily concentrates in geostrophic vortices, in convectively unstable wavenumbers, and along the inertial wave dispersion relation, even in non-elliptically deformed convective flows. Examining linear growth rates on a convective background, we find that convective large-scale vortices suppress the elliptical instability in the same way as the geostrophic vortices created by the elliptical instability itself. Finally, convective motions act as an effective viscosity on large-scale tidal flows, providing a sustained energy transfer (scaling as ε2). Furthermore, we find that the energy transfer resulting from bursts of elliptical instability, when it operates, is consistent with the ε3 scaling found in prior work.

Kelvin wave in miscible two-component Bose-Einstein condensates

We study the dispersion relation of Kelvin waves propagating along single- and half-quantum vortices in miscible two-component Bose-Einstein condensates based on the analysis of the Bogoliubov--de Gennes equation. With the help of the interpolating formula connecting the dispersion relations in low- and high-wave-number regimes, we reveal the nontrivial dependence of the dispersion relation on the intercomponent interaction through the change in the vortex-core size of the vortical component. We also find the splitting of the Kelvin wave dispersion into gapless and gapped branches when both components have overlapping single-quantized vortices.

Multimode theory of electron hole transverse instability

We present Vlasov–Poisson three-dimensional linear stability analysis of an initially planar electron hole structure, solving for the distribution function by integration along unperturbed orbits. The non-sinusoidal potential perturbation shape (parallel to$B$) is expanded in eigenfunctions of the adiabatic Poisson operator, generalizing the prior assumption of a rigid shift of the equilibrium. We show that the shiftmode is then modified by a second discrete mode plus an integral over a continuum of wave-like modes. A rigorous treatment shows that the continuum can be approximated effectively by a single mode that satisfies the external wave dispersion relation, thus making the perturbation a weighted sum of three modes. We find numerically the solution for the complex instability frequency, and the corresponding three mode amplitudes determining the perturbation eigenmode. This multimode analysis refines the accuracy of the prior single-mode results, giving slightly higher growth rates at most parameters, as expected from the extra mode shape freedom. Oscillating modes near stability boundaries have larger mode distortions which help explain particle-in-cell simulations that observe instability up to${\sim }20$ % beyond the prior shiftmode thresholds, and narrowing of the perturbation. At high magnetic field, the multimode analysis predicts a reduction of the already small growth rate.

Microstructural design for elastic wave attenuation in 3D printed nacre-like bioinspired metamaterials lightened with hollow platelets

In this work, the wave propagation properties of lightened bioinspired composite materials are investigated. A new lightened nacre-like microstructure is proposed demonstrating that the incorporation of hollow platelets inside the soft matrix and the application of specific prestress states represent a viable way for tuning the complete band gaps range without varying the geometry length scale and also improving the mechanical properties in terms of homogenized tangent moduli. The wave dispersion relations are obtained by adopting the Bloch-Floquet theory to investigate the appearance of complete band gaps for several geometrical and material parameters (platelets aspect ratio, matrix joints thickness, void volume fraction, and shear modulus contrast). The obtained results reveal new combinations of geometrical and material parameters for which excellent wave absorption capabilities in nacre-like periodic composite microstructure are obtained thanks to the application of prestress states which may provide new opportunities in the microstructural design of 3D printed composite metamaterials.

Influence of electron temperature anisotropy on wave mode propagation and power deposition characteristics in helicon plasma

As the core issue in helicon discharge, the physical mechanism behind the high ionization rate phenomenon is still not fully understood. Based on the warm plasma dielectric tensor model which contains both the particle drift velocity and temperature anisotropy effect, by employing the general dispersion relation of electromagnetic waves propagating in magnetized and uniform plasma with typical helicon discharge parameter conditions, wave mode propagation characteristic and collisional, cyclotron and Landua damping induced wave power deposition properties of azimuthally symmetric mode are theoretically investigated. Systematic analysis shows the following findings. 1) Under typical helicon plasma parameter conditions, i.e. wave frequency <i>ω</i>/(2π)=13.56 MHz, ion temperature is one tenth of the electron temperature, and for a given magnetic field <i>B</i><sub>0</sub> (or wave frequency <i>ω</i>), there exists a critical wave frequency <i>ω</i><sub>cr</sub> (or magnetic field <i>B</i><sub>0,cr</sub>), above which (or below <i>B</i><sub>0,cr</sub>) the damping of the <i>n = </i>1, 2, 3 cyclotron harmonics begins to increase sharply. 2) For the electron temperature isotropic case, the attenuation constants of different harmonics start to increase significantly and monotonically at different thresholds of magnetic field, while the phase constant abruptly increases monotonically from the beginning of the parameter interval. On the other hand, for the electron temperature anisotropic case, both the phase constant and attenuation constant have peaking phenomenon, i.e. the attenuation constant begins to increase sharply at a certain value of <i>B</i><sub>0</sub> and meanwhile the phase constant presents a maximum value near the same value of magnetic field, thus the phase constant starts to keep constant at a certain value of <i>B</i><sub><i>0</i></sub> and meanwhile the attenuation constant has a maximum value near this same value of magnetic field. 3) For the wave power deposition properties, under electron temperature anisotropy conditions, power deposition due to collisional damping of Trivelpiece-Gould (TG) wave plays a dominant role in a low field (<i>B</i><sub>0 </sub>= 48 Gs) (1 Gs = 10<sup>–4</sup> T); by considering the electron finite Larmor radius (FLR) effect, the power deposition of TG wave presents a maximum value at a certain point of parallel electron temperature<i> T</i><sub>e,</sub><sub>//</sub>; with the decrease of <i>T</i><sub>e,⊥</sub>/<i>T</i><sub>e,</sub><sub>//</sub>, the maximum value of power deposition increases gradually. All these findings are very important in further revealing the physical mechanism behind the high ionization rate in helicon plasma.

Synergistic current drive of helicon wave and lower hybrid wave in HL-2M

Non-inductive current drive plays a crucial role in tokamak, especially for its steady state operations. Recently, the helicon wave (HW) has been regarded as a promising tool for driving off-axis plasma current in reactor-grade machine. The lower-hybrid wave (LHW) is the most effective radio-frequency current drive method, however, it has the drawback, which is limited by the conditions of wave accessibility in the high parameter tokamak, making the wave power usually damped at the plasma edge. HW can spiral towards the plasma centre directly under a high electron density. To obtain a long pulse steady state operation of reactor tokamak, the complementarity of HW and LHW in the aspect of driven current distribution in the high parameter tokamak is considered. The synergy current drive of the HW and the LHW is studied numerically in the steady-state scenario of HL-2M. According to the fast wave dispersion relation of plasma, the HW parameters, including its wave frequency and launched parallel refractive index, are obtained firstly. Results of GENRAY code simulation show that a single pass wave power absorption of the HW can be obtained generally through the electron Landau damping and transit time magnetic pumping effects. On the other hand, the LHW parameters are adopted from the equipped system on the machine. Results of single pass wave absorption are also obtained in the case of LHW. And then, the synergy effects of HW and LHW are studied numerically based on the GENRAY/CQL3D models. The cooperation of these two waves results in a broad plasma current distribution along the radial direction (<inline-formula><tex-math id="M2">\begin{document}$\rho = $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231077_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20231077_M2.png"/></alternatives></inline-formula>0.2－0.9) in the machine. Taking the electron distribution functions of these waves into account, it is clear that the electrons are accelerated by the HW in the parallel magnetic field direction, resulting in more electrons entering the region of LHW resonance area. As the consequence, a net plasma current appears. Furthermore, a fine-grained parametric scan is performed by changing the launched parallel refractive index of HW, and the results indicate that positive synergy effects can be generally observed once the related wave current drive profiles are overlapped. Finally, the synergy factor is shown to be proportional to this overlap and reaches its maximum value of 1.18 in HL-2M.

Wave shadowing effects on phase resolved wave spectra estimated from RADAR backscatter: Part 1: An investigation of the errors due to wave shadowing in the spectral averaging approach

This is the first part of a short two part series concerned which is acknowledged as the primary source of error involved in the estimation of phase resolved sea wave spectra from RADAR backscatter measurement. A mathematical model is produced of what is termed here as spectral averaging which is the most commonly used technique for such estimation purposes. It is found that the spectral averaging technique can be described by a wave-number domain convolution integral. The kernel of the convolution must be a good approximation to a delta function for good estimates to be obtained. Using simulations of the wave shadowing process it is shown that good approximations can be obtained to the delta function requirement using approximately one tenth of the data employed in typical implementations of spectral averaging. This large data requirement is due to the need to impose the wave dispersion relationship as a constraint during the estimation process. Two new estimation methods are briefly introduced to resolve this situation and can operate on small data sets, one of which, termed conflation, is developed in detail in Part 2.

Wave shadowing effects on phase resolved wave spectra estimated from RADAR backscatter: Part 2: The conflation technique and the deconvolution techniques for the mitigation of wave shadowing effects

This is the second part of a short two series concerned with wave shadowing which is acknowledged as the primary source of error involved in the estimation of phase resolved sea wave spectra from RADAR backscatter measurement. A new technique termed conflation is introduced that uses multiple RADAR scan data sets to build a multi-segment data set together with a corresponding theoretic model. Using simulation the conflation methodology is compared against the so called spectral averaging technique that is the standard approach for estimating phase resolved wave spectra. Conflation is found to provide superior estimates, in particular making it possible to avoid the need to impose the wave dispersion relationship though an additional computational step as is necessary in spectral averaging. A second technique based upon deconvolution is also introduced for mitigating wave shadowing errors. However because deconvolution is a very mature process well covered in the signal process and linear system theory literature the discussion of this second method is very brief.

Dispersive staggered grid finite difference modelling of Rayleigh waves

Non-destructive surface wave-based ground exploration techniques such as multichannel analysis of surface wave (MASW) require a rigorous numerical model to predict the soil properties accurately. The staggered grid finite difference (SGFD) method is widely used to simulate elastic wave propagation due to its stability and ease of implementation. However, in the presence of low-velocity layers, the SGFD method requires computationally expensive high spatial grid resolution to accurately model the surface waves. On the other hand, velocity dependent grid approaches become unstable for the highly irregular soil profile. In order to overcome these challenges, a vertically non-uniform dispersive staggered grid scheme is proposed that significantly reduces the computing time without sacrificing accuracy. The proposed non-uniform discretization strategy is based on the Rayleigh wave dispersion relation, where the penetration depth of each frequency is assumed to be roughly one and half of the wavelength. Based on the above assumption, an exponential relationship between grid size and grid number is developed. The proposed method calculates continuously increasing grid length while ensuring grid points at the exact layer interface. Five different soil profiles: a homogenous elastic half-space, an irregularly dispersive crustal level profile, a low velocity layer at a deeper depth model, a real-world model, and a lateral heterogeneous model are used to demonstrate the efficacy of the proposed approach. For all the cases, the results of the proposed dispersive grid match those of the uniform grid with a correlation coefficient value (R) > 0.99. Compared to the uniform and velocity dependent grid approaches, the proposed dispersive grid scheme requires ½ to 1/4th of elements and CPU time. Furthermore, unlike the velocity dependent grid scheme, the proposed method does not become unstable for a highly irregular profile. Results indicate that the proposed method is superior to the traditional uniform and velocity dependent grid approaches for modelling elastic waves in layered media.

An accurate method for dispersion characteristics of surface waves in layered anisotropic semi-infinite spaces

In this paper, we develop a novel method for the dispersion characteristics of surface waves in layered anisotropic semi-infinite spaces. Compared to solving the algebraic transcendental eigenequations directly, this method can find all surface waves accurately. For solving accurately, the first-order state equation is formed by introducing the dual variable. Some properties of the complex Hamiltonian matrix are presented and proved, then based on these properties, the radiation condition for the homogeneous anisotropic half-space is given. And, based on the scaling and squaring algorithm and mixed energy matrix, the precise integration method (PIM) is presented for solving accurately the eigenvalue problem of ordinary differential equations (ODEs) corresponding to surface waves. All eigenfrequencies can be found with certainty by using the concept of the eigenvalue count in the Wittrick-Williams (W-W) algorithm. Numerical examples show that the proposed method is highly accurate and efficient for solving dispersion relations of surface waves in layered anisotropic semi-infinite spaces, which involves the comparison of other previously published methods.

Love-type waves in thermoelastic solid with double porosity structure

In the present work, we consider a model consisting of an elastic layer of finite thickness resting over an elastic half-space in perfect contact. The layer and the half-space both are assumed to be composed of different thermoelastic materials possessing different double porosity structures. The dispersion relation for Love-type surface waves has two factors, one of which matches exactly with the dispersion relation of Love waves of classical elasticity, while the second factor arises due to the presence of double porosity and thermal field in the model. Each of these factors (dispersion relation) provides two types of Love waves called ‘first front’ and ‘second front’ of Love waves. For a specific model, these two dispersion relations are computed numerically to obtain the phase speed of corresponding front of propagating Love waves. It is found that the speed of second front is much faster than the speed of first front. The phase speed of fundamental mode of the second front increases with the inclusion of thermal property, while it decreases with the inclusion of either type of voids/porosities in the model. Some earlier known results are also recovered from the present formulation as special cases.

Polarizations of gravitational waves in the bumblebee gravity model

Lorentz violation modifies the dispersion relation of gravitational waves (GWs), and induces birefringence and anisotropy in propagation. Our study shows that Lorentz violation can also activate multiple polarizations of GWs. We use the gauge invariants to investigate the polarizations of GWs in the bumblebee gravity model, and obtain the following results. (i) For a vector background ${b}^{\ensuremath{\mu}}$ with only a nonzero temporal component ${b}^{t}$, there are five independent propagating degrees of freedom (d.o.f.), which is similar to the Einstein-\ae{}ther theory. (ii) The presence of a spatial component in the background defines a preferred spatial direction which breaks rotational symmetry. We denote $\stackrel{^}{\mathbit{b}}$ as the direction of the spatial part of the background and ${b}_{s}$ as its length. If GWs propagate along $\stackrel{^}{\mathbit{b}}$, the polarization content is similar to the purely timelike case. (iii) If the propagation direction of GWs is separated by an angle $\ensuremath{\beta}$ to $\stackrel{^}{\mathbit{b}}$, and $\ensuremath{\beta}=\mathrm{arccos}({b}^{t}/{b}_{s})$, there are only two tensor polarizations. (iv) If $\ensuremath{\beta}\ensuremath{\ne}\mathrm{arccos}({b}^{t}/{b}_{s})$, there are only two independent d.o.f., and the vector and scalar modes degenerate with the tensor modes. The tensor perturbations can activate a mixture of all six polarizations simultaneously. Finally, we point out the difference in GWs between the bumblebee gravity model and the minimal Standard-Model Extension framework in the linearized regime. Current observations have placed stringent constraints on the anisotropy induced by the background, while our theoretical study still reveals some novel phenomena and provides more understanding about the interaction between the Lorentz-violating vector field and gravity.

Investigations on the Complex Band Diagram of Flexural Wave through the Fluid-Loaded Phononic Plate

This paper investigates the complex band diagram of flexural waves in the phononic plate with semi-infinite heavy fluid loading. The system under examination is a square plate lattice with two-dimensional periodicity immersed in a fluid domain with infinite height. The numerical models based on the wave field transformation and the Galerkin method combined with the finite element discretization technique are developed to investigate the real and imaginary parts of the dispersion relation of flexural waves propagating through the phononic plate incorporating the fluid-loading effects. A perfect agreement is found between the location and width of stop bands from the real band diagram and the attenuation diagram, which supports the validity of the numerical models. Moreover, the complex band diagram is verified by the transverse vibration transmittance of the finite phononic plate. The results demonstrate that the external fluid loading is able to adjust the location, bandwidth, and decaying level as well as affect the degree of attenuation anisotropy of the complete and directional band gaps.

UBathy (v2.0): A Software to Obtain the Bathymetry from Video Imagery

UBathy is an open source software developed for bathymetry estimation from video images. The proposed scheme is based on extracting the wave modes from videos of the nearshore surface wave propagation. These videos can be formed either from raw camera images, which must have been previously calibrated, or from georeferenced planviews. For each wave mode extracted from the videos, the wave frequency and the spatially dependent wavenumbers are obtained. The frequencies and wavenumbers from different videos are used to estimate the bathymetry by adjusting the dispersion relationship for linear surface water waves. The bathymetry at different times can further be weighted and aggregated using the Kalman filter. The new software is suitable for Argus-type video monitoring stations and for moving cameras mounted on drones or satellites, and it is meant for users familiar with coastal image processing and suitable for non-experienced users. The software and an application example are available on the GitHub platform.

A Multi‐Layer Linear Rossby Wave Dispersion Relation for Vertical Tilt of Mesoscale Eddies

AbstractVertical tilt of mesoscale eddies has been frequently observed in world oceans. The observations of composite eddies, individual eddies from an oceanic reanalysis product, and idealized numerical experiments of mesoscale eddies all show that the eddy vertical tilt is gradually increased with time in eddy mature stage, implying that the eddy translation speed changes with depth. However, eddy vertical tilt cannot be explained by the classic linear Rossby wave dispersion relation, since the classic zonal translation speed () of linear Rossby waves is depth‐independent from the quasi‐geostrophic potential vorticity (QGPV) equation. To deduce a depth‐dependent linear Rossby wave dispersion relation, we solved the QGPV equation by assuming that the stratified water column in oceans can be separated into a number of fine layers with constant stratification in each layer, whereas the stratification changes among layers. A multi‐layer linear Rossby wave (MLRW) dispersion relation in stratified oceans is then deduced. The MLRW dispersion relation shows that changes with depth. In each layer, is proportional to stratification and layer thickness but inversely proportional to Earth's rotation. The estimated structure from the MLRW dispersion relation is consistent with the structure of eddies in numerical experiments and the observed composite eddies. Noticeably, there are some discrepancies between the theoretical estimates and those from observations and numerical simulations, primarily due to the sensitivity of the theoretical estimates to layer thickness. The study attempts to provide an approach to describe and predict mesoscale eddy vertical tilt.

Multi-mode propagation and diffusion analysis using the three-dimensional second strain gradient elasticity

The multi-mode propagation and diffusion properties are crucial informations when studying complex waveguides. In this paper, firstly, the three-dimensional modeling of micro-sized structures is introduced by using the second strain gradient theory. The constitutive relation is deduced while the six quintic Hermite polynomial shape functions are employed for the displacement field. The weak formulations including element stiffness and mass matrices and the force vector are calculated through the Hamilton’s principle and the global dynamic stiffness matrix of a unit cell is assembled. Then, free wave propagation characteristics are analyzed by solving eigenvalue problems within the direct wave finite element method framework. The dispersion relations of positive going waves considering the size effects are illustrated. Furthermore, the effects of higher order parameters on the dispersion curves are discussed and the forced responses with two boundary conditions are expounded. Eventually, the wave diffusion including reflection and transmission coefficients are illustrated through simple and complex coupling conditions, respectively. The dynamic analysis of coupled waveguides through the wave finite element method equipped with the second strain gradient is a novel work. The results show that the proposed approach is of significant potential for investigating the wave propagation and diffusion characteristics of micro-sized structures.