Group Velocity Surfaces Research Articles

On the singularity point in acoustic orthorhombic media

Abstract The acoustic orthorhombic model is widely used in seismic modeling and processing of P-wave data. However, the anisotropic acoustic models have so called S-wave artifacts (1 artifact in transversely isotropic acoustic medium and two artifacts in orthorhombic acoustic medium). I show that S-wave artifacts can have one singularity point that results in complications in polarization field and the group velocity surface. The conditions of the existence of this point are defined in terms of anellipticity parameters. This singularity point and its group velocity image are the objects of my analysis.

Anisotropic media with singular slowness surfaces

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

Model anisotropic materials with ellipsoidal wave surfaces and their application to determination of elastic constants by acoustical methods

A new class of elastic materials with orthorhombic symmetry is introduced. For such materials, group velocity surfaces of quasi-longitudinal waves have ellipsoidal shapes. Relations between elastic constants that guaranty ellipsoidal shapes of the group velocity surfaces are indicated. The model materials are used for approximate determination of effective elastic constants of heterogeneous materials by acoustical methods. Examples of reconstruction of elastic constants from acoustical data are presented, and possible errors of the reconstructions are discussed.

Behavior of S waves in vicinity of singularity point in elliptic orthorhombic media

In low-symmetry anisotropic models, the S waves are generally coupled and have very complex behavior due to triplications and singularity points. The presence of singularity point affects S-wave phase and group velocity surface in a simple elliptical orthorhombic model. This model has no triplications but has a singularity point in one of the symmetry planes. We find that the slowness ellipse (defined by the zero Gaussian curvature) is the image of a singularity point on the group velocity surface, and the group velocity ellipse (also known as the internal refraction cone) is the image of a singularity point on the slowness surface. The equations for the slowness and group velocity ellipses are explicitly defined. The square of S1- and S2-wave lacunae also is computed.

Yet another modification of relativistic magnetohydrodynamic waves: Electron thermal inertia

This study investigates the properties of waves in relativistic extended magnetohydrodynamics (RXMHD), which includes Hall and electron thermal inertia effects. We focus on the case when the electron temperature is ultrarelativistic, and thus, the electron thermal inertia becomes finite at near the proton inertial scale. We derive the linear dispersion relation of RXMHD and find that the Hall and electron thermal inertia effects couple with the displacement current, giving rise to three superluminous waves in addition to the slow, fast, and Alfvén waves. We also show that the phase- and group-velocity surfaces of fast and Alfvén waves are distorted by the Hall and electron thermal inertia effects. There is a range of scales where the group velocity of fast wave is smaller than that of the Alfvén and slow waves. These findings are applicable to a region near the funnel base of low-luminosity accretion flows where electrons can be ultrarelativistic.

Group velocity of light in biaxial crystals.

Based on the wave vector surface, we have derived the analytical expressions for the group velocity of light traveling in an optically biaxial crystal in an arbitrary direction. Our formulas are in terms of either the wave vector direction or the ray direction. An axis dispersion vector is introduced to specify the rotation of the dielectric frame in monoclinic and triclinic crystals, which is found to be responsible for the asymmetry of the group velocity surfaces of these two crystal systems when compared to their phase and ray velocity surfaces. Our formulas confirm that the group velocity of light in a nondissipative medium equals the ratio between the Poynting vector and the total energy density of the electromagnetic field. The group index of light is found to be decomposable into three parts: the ray index, the group index due to principal refractive index dispersion, and that due to axis dispersion. Numerical calculations are performed for the group velocity of light in ${\rm Sn}_2{\rm P}_2{\rm S}_6$ crystal at 550 nm wavelength. Our results show that the dispersion of the principal refractive indices has a considerable contribution to the group index.

Asymmetry of the group velocity of light in monoclinic crystals.

We have proved that compared to phase velocity and ray velocity surfaces, the group velocity surface of a monoclinic crystal has a reduced symmetry, due to the loss of the two mirror planes that contain the crystallographic b axis. We have derived a formula for calculation of the group velocity of the extraordinary light traveling in a principal plane of a biaxial crystal, which takes into account the rotation of the dielectric frame due to frequency dispersion. The maximum asymmetry of the group velocity of light traveling in the a-c plane is found to be 2.4% at 365 nm in BiB3O6 and 1.4% at 550 nm in Sn2P2S6.

Group velocity of light in uniaxial crystals.

Using the wave vector surface, we have derived the analytical expressions for the group velocities of the extraordinary light and the ordinary light traveling in an optically uniaxial crystal. Our formulas are in terms of either the wavevector direction or the ray direction, and they use only the principal refractive indices and their frequency dispersions. The algebraic equation for the group velocity surface of the crystal is also derived. While the group velocity is in the same direction as the ray velocity, numerical calculation shows that in the visible region the group velocity is slower than the ray velocity, and the difference between them becomes significant at short wavelengths.

On-axis triplications in elastic orthorhombic media

SUMMARY Triplicated traveltime curve has three arrivals at a given distance with the bowtie shape in the traveltime-offset curve. The existence of the triplication can cause a lot of problems such as several arrivals for the same wave type, anomalous amplitudes near caustics, anomalous behaviour of rays near caustics, which leads to the structure imaging deviation and redundant signal in the inversion of the model parameters. Hence, triplication prediction becomes necessary when the medium is known. The research of the triplication in transversely isotropic medium with a vertical symmetry axis (VTI) has been well investigated and it has become clear that, apart from the point singularity case, the triplicated traveltime only occurs for S wave. On contrary to the VTI case, the triplication behaviour in the orthorhombic (ORT) medium has not been well focused due to the model complexity. In this paper, we derive the second-order coefficients of the slowness surface for two S waves in the vicinity of three symmetry axes and define the elliptic form function to examine the existence of the on-axis triplication in ORT model. The existence of the on-axis triplication is found by the sign of the defined curvature coefficients. Three ORT models are defined in the numerical examples to analyse the behaviour of the on-axis triplication. The plots of the group velocity surface in the vicinity of three symmetry axes are shown for different ORT models where different shapes: convex or the saddle-shaped (concave along one direction and convex along with another) indicates the existence of the on-axis triplication. We also show the traveltime plots (associated with the group velocity surface) to illustrate the effect of the on-axis triplication.

Geometrical characteristics of phase and group velocity surfaces in anisotropic media

ABSTRACTThe phase and group velocity surfaces are essential for wave propagation in anisotropic media. These surfaces have certain features that, especially, for shear waves result in complications for modelling and inversion of recorded wavefields. To analyse wave propagation in an anisotropic model, it is important to identify these features in both the phase and group domains. We propose few characteristics for this analysis: the energy flux angle, decomposed in the polar and azimuth angle correction angles and enhancement factor, which is able to characterize both singularity points and triplication zones. The very simple equation that controls the triplications is derived in the phase domain. The proposed characteristics are illustrated for elastic and acoustic anisotropic models of different symmetry classes.

Thermal transport and energy dissipation in two-dimensional Bi2O2Se

Thermal transport and energy dissipation are important for a material in both thermoelectric and electronic devices. Here, we investigate the lateral and interfacial thermal transport of two-dimensional (2D) Bi2O2Se by Raman spectroscopy. It is found that thin Bi2O2Se flakes have a low in-plane thermal conductivity while maintaining an appropriate interfacial thermal conductance. The in-plane thermal conductivity of Bi2O2Se decreases with decreasing thickness, to as low as 0.92 ± 0.18 W⋅m−1⋅K−1 at a thickness of ∼8 nm. Such a low thermal conductivity is derived from the low phonon group velocity, strong anharmonicity, and large surface scattering of acoustic phonons of the Bi2O2Se thin layer. Simultaneously, thinner Bi2O2Se presents a higher thermal dissipation to the substrate than the thicker counterparts in the device. The interfacial thermal conductance increases with decreasing thickness, and reaches ∼21 MW⋅m−2⋅K−1 at ∼8 nm. These results provide critical information for the design of thermoelectric devices with high figures of merit and electronics with low-power consumption based on 2D materials.

Dispersion of a surface plasmon polaritons in a thin dielectric films surrounded by a two graphene layers

In this work, we investigated the features of the propagation of surface plasmon-polariton modes with a frequencies ω=(1014..1015)rad/s with a symmetric and antisymmetric distribution of the electric field in the planar structure of a «graphene-thin dielectric film-graphene». They are received the dispersion characteristics for these types of modes. Also we found a spectral range, where the dispersion waves, phase and group velocity surface waves are most sensitive to change of a chemical potential of a graphene layers.

Transient and steady state heat transport in layered materials from molecular dynamics simulation

In this paper, both transient and steady state heat transfer in graphite are investigated with molecular dynamics (MD) simulations. The simulation results demonstrate that elastic anisotropy controls heat transfer in the transient state, which makes the outermost isothermal surface of temperature distribution similar to the phonon group velocity surface that has a shape of very flat ellipse in layered materials. In steady state, with the help of phonons engaging in sufficient scattering, the basal plane phonon modes determine the thermal conductivity along all directions except those very close to the cross plane. Our simulation results confirm that the classical theoretical model can accurately predict the thermal conductivity along arbitrary directions in layered materials.

Modification of magnetohydrodynamic waves by the relativistic Hall effect.

This study shows that a relativistic Hall effect significantly changes the properties of wave propagation by deriving a linear dispersion relation for relativistic Hall magnetohydrodynamics (HMHD). Whereas, in nonrelativistic HMHD, the phase and group velocities of fast magnetosonic wave become anisotropic with an increasing Hall effect, the relativistic Hall effect brings upper bounds to the anisotropies. The Alfvén wave group velocity with strong Hall effect also becomes less anisotropic than the nonrelativistic case. Moreover, the group velocity surfaces of Alfvén and fast waves coalesce into a single surface in the direction other than near perpendicular to the ambient magnetic field. It is also remarkable that a characteristic scale length of the relativistic HMHD depends on ion temperature, magnetic field strength, and density while the nonrelativistic HMHD scale length, i.e., ion skin depth, depends only on density. The modified characteristic scale length increases as the ion temperature increases and decreases as the magnetic field strength increases.

Elastic Properties, Phonon Focusing and Electronic Structures of Typical Long-Period Superstructures Al<sub>5</sub>Ti<sub>2</sub> and Al<sub>11</sub>Ti<sub>5</sub>

The first-principles calculations were performed to systematically study the elastic properties, phonon focusing and electronic structures of typical long-period superstructures Al5Ti2 and Al11Ti5. The obtained lattice parameters are in good agreement with the experimental data. The elastic constants were calculated, and bulk modulus B, shear modulus G, Young’s modulus E and Poison’s ration ν were further obtained. Both of the numerical indicators and the three dimensional images show that Al5Ti2 and Al11Ti5 are anisotropic, and the anisotropy is slightly high for Al5Ti2. Three slowness surfaces were constructed for Al5Ti2 and Al11Ti5 to investigate the phonon focusing patterns, and the group velocity surfaces were also obtained to gain more insight into the elastic anisotropy. Due to elastic anisotropy, both of the slowness surfaces and the group velocity surfaces are nonspherical, and the anisotropy is highest for slow transverse mode, and then followed by fast transverse mode and longitudinal mode. The electronic density of states and charge density distribution of Al5Ti2 and Al11Ti5 indicate that due to strong hybridization between Al-2p and Ti-3d, there is a strong directional bonding between the Al and Ti atoms. Electronic structures reveal the underling mechanism of elastic properties and phonon focusing for Al5Ti2 and Al11Ti5.

Algebraic degree of a general group-velocity surface

Three algebraic surfaces — the slowness surface, the phase-velocity surface, and the group-velocity surface — play fundamental roles in the theory of seismic wave propagation in anisotropic elastic media. While the slowness (sometimes called phase-slowness) and phase-velocity surfaces are fairly simple and their main algebraic properties are well understood, the group-velocity surfaces are extremely complex; they are complex to the extent that even the algebraic degree, [Formula: see text], of a system of polynomials describing the general group-velocity surface is currently unknown, and only the upper bound of the degree [Formula: see text] is available. This paper establishes the exact degree [Formula: see text] of the general group-velocity surface along with two closely related to [Formula: see text] quantities: the maximum number, [Formula: see text], of body waves that may propagate along a ray direction in a homogeneous anisotropic elastic solid [Formula: see text] and the maximum number, [Formula: see text], of isolated, singularity-unrelated cusps of a group-velocity surface [Formula: see text].

Elastic anisotropy and phonon focusing for tetragonal crystals: Application to γ-TiAl

A theoretical formalism to investigate the elastic anisotropy and phonon focusing for tetragonal crystals is described in detail in this paper. This is applied for γ-TiAl and both of the numerical indicators and the three dimensional images of elastic anisotropy are derived. The percentage of bulk and shear anisotropies indicate γ-TiAl exhibits relatively high degree of anisotropy in shear but small anisotropy in compressibility. Anisotropy indexes A∗ and AU demonstrate γ-TiAl possesses high degree of anisotropy, and this feature is also reflected in the three dimensional Young’s modulus, shear modulus and Poisson’s ratio. Three slowness surfaces are constructed for γ-TiAl to investigate the phonon focusing patterns, and the group velocity surfaces are also obtained to gain more insight into the elastic anisotropy. Due to elastic anisotropy, both of the slowness surfaces and the group velocity surfaces are nonspherical. The anisotropy is highest for slow transverse mode, and then followed by fast transverse mode and longitudinal mode. Interestingly, only transverse phonons cause caustics for γ-TiAl. Furthermore, the phase velocity and group velocity are not collinear for transverse mode in some directions, resulting in cuspidal edges along these directions in the group velocity surfaces.

Analytical calculation of electron group velocity surfaces in uniform strained graphene

Electron group velocity for graphene under uniform strain is obtained analytically by using the tight-binding (TB) approximation. Such closed analytical expressions are useful in order to calculate the electronic, thermal and optical properties of strained graphene. These results allow to understand the behavior of electrons when graphene is subjected to strong strain and nonlinear corrections, for which the usual Dirac approach is no longer valid. Some particular cases of uniaxial and shear strain were analyzed. The evolution of the electron group velocity indicates a break-up of the trigonal warping symmetry, which is replaced by a warping consistent with the symmetry of the strained reciprocal lattice. To do this, analytical expressions for the shape of the first Brillouin zone (BZ) of the honeycomb strained reciprocal lattice are provided. Finally, the Fermi velocity becomes strongly anisotropic, i.e., for a strong pure shear strain (20% of the lattice parameter), the two inequivalent Dirac cones merge and the Fermi velocity is zero in one of the principal axis of deformation. We found that nonlinear terms are essential to describe the effects of deformation for electrons near or at the Fermi energy.

Novel composites with asymmetrical elastic wave properties

Abstract The traditional structural composites are known as materials with a spatial order. The temporal order is introduced into structural composites in this paper. We theoretically describe and numerically demonstrate the architecture for a structural composite which exhibits tunable and nearly “full-banded” asymmetrical elastic wave property. Given its spatiotemporal order and unconventional wave properties, such composite can be named as spatiotemporal metamaterial (STMM). Firstly, the effective material parameters, which include some new material parameters, are obtained based on improved multi-scale homogenization techniques. Secondly, the unconventional wave properties are theoretically analyzed. It is found that the elastic wave group velocity surfaces do not satisfy the centrosymmetry in STMM, i.e. the time-reversal symmetry of linear elastic wave is violated. In certain instances, the waves even propagate unidirectionally. Finally, we present an example of the STMMs, which includes homogenized calculations and numerical simulations where the shear waves are forbidden in some directions. The unconventional wave properties of the STMMs proposed in this paper own attractive potential applications, such as acoustic absorbing, acoustic information processing, energy-saving, and energy-harvesting.

Shear-wave group-velocity surfaces in low-symmetry anisotropic media

Shear waves excited by natural sources constitute a significant part of useful energy recorded in downhole microseismic surveys. In rocks, such as fractured shales, exhibiting symmetries lower than transverse isotropy (TI), the shear wavefronts are always multivalued in certain directions, potentially complicating the data processing and analysis. This paper discusses a basic tool — the computation of the phase and group velocities of all waves propagating along a given ray — that intends to facilitate the understanding of geometries of the shear wavefronts in homogeneous anisotropic media. With this tool, arbitrarily complex group-velocity surfaces can be conveniently analyzed, providing insights into possible challenges to be faced when processing shear waves in anisotropic velocity models that have symmetries lower than TI. Among those challenges are complicated multipathing and the presence of cones of directions, known as internal refraction cones, in which no fast shear waves propagate and the entire shear portion of the body-wave seismic data consists of several branches of the slow shear wavefronts.