Propagation of inhomogeneous plane waves in anisotropic viscoelastic media

A new technique is explained to study the propagation of inhomogeneous waves in a general anisotropic medium. The harmonic plane waves are considered in a viscoelastic anisotropic medium. The complex slowness vector is decomposed into propagation vector and attenuation vector for the given directions of propagation and attenuation of waves in an unbounded medium. The attenuation is further separated into the contributions from homogeneous and inhomogeneous waves. A non-dimensional inhomogeneity parameter is defined to represent the deviation of an inhomogeneous wave from its homogeneous version. Such a partition of slowness vector of a plane wave is obtained with the help of an algebraic method for solving a cubic equation and a numerical method for solving a real transcendental equation. Derived specifications enable to study the 3D propagation of inhomogeneous plane waves in a viscoelastic medium of arbitrary anisotropy. The whole procedure is wave-specific and obtains the propagation characteristics for each of the three inhomogeneous waves in the anisotropic medium. Numerical examples analyze the variations in propagation characteristics of each of the three waves with propagation direction and inhomogeneity strength.     

Propagation of inhomogeneous waves in anisotropic piezo-thermoelastic media

A mathematical model for the propagation of harmonic plane waves in an anisotropic piezo-thermoelastic medium is explained through three relations. Two of them relate the stress-induced harmonic variations in temperature and electric potential to mechanical displacement of material particles. The third is a system that defines modified Christoffel equations for wave propagation in the medium. The solution of this system is ensured by a quartic equation whose complex roots explain the existence and propagation of four attenuating waves in the medium. The effects of piezoelectricity and thermoelasticity on the wave propagation are analyzed in the discussion of special cases. An angle between propagation direction and direction of maximum attenuation defines the attenuated wave as inhomogeneous wave. The complex slowness vector for each of the four attenuated waves in the medium is resolved to calculate the phase velocity and the attenuation factor for its propagation as an inhomogeneous wave along a general direction in three-dimensional space. The variations in phase velocities and attenuation factors with propagation direction are computed, for a realistic numerical model.     

Acoustic reflection from the boundary of anisotropic thermoviscoelastic solid with fluid

Vertical slownesses of waves at a boundary of an anisotropic thermoviscoelastic medium are calculated as roots of a polynomial equation of degree eight. Out of the corresponding eight waves, the four, which travel towards the boundary are identified as upgoing waves. Remaining four waves travel away from the boundary and are termed as downgoing waves. Reflection and refraction of plane harmonic acoustic waves are studied at a plane boundary between anisotropic thermoviscoelastic solid and a non-viscous fluid. At this fluid-solid interface, an incident acoustic wave through the fluid reflects back as an attenuated acoustic wave and refracts as four attenuating waves into the anisotropic base. Slowness vectors of all the waves in two media differ only in vertical components. Complex values of vertical slowness define inhomogeneous refracted waves with a fixed direction of attenuation, i.e. perpendicular to the interface. Energy partition is calculated at the interface to find energy shares of reflected and refracted waves. A part of incident energy dissipates due to interaction among the attenuated refracted waves. Numerical examples are considered to study the variations in energy shares with the direction of incident wave. For each incidence, the conservation of incident energy is verified in the presence of interaction energy. Energy partition at the interface seems to be changing very slightly with the azimuthal variations of the incident direction. Effects of anisotropy, elastic relaxation and thermal parameters on the variations in energy partition are discussed. The acoustic wave reflected from isothermal interface is much significant for incidence around some critical directions, which are analogous to the critical angles in a non-dissipative medium. The changes in thermal relaxation times and uniform temperature of the thermoviscoelastic medium do not show any significant effect on the reflected energy.     

Attenuation and quality factor surfaces in anisotropic-viscoelastic media

We obtain expressions of the attenuation vector and quality factor of the three possible wave modes propagating in a linear anisotropic medium. The theory assumes, in principle, a general stiffness matrix. Probing the medium with a time-harmonic homogeneous plane wave gives the attenuations and quality factors as simple forms of the propagation direction, complex stiffnesses and mass density. As an application, we introduce a new constitutive relation, based on four complex moduli, for which the values of the quality factor along three preferred directions can be matched with experimentally pre-determined values. The rheology is causal and allows an arbitrary frequency-dependence of the stiffnesses based on the generalized standard linear solid model. Two examples are explicitly worked out. The first is clay shale, a material of hexagonal symmetry. Since, by Neumann's principle, the attenuation symmetries are determined by the crystal class, the medium presents isotropic attenuation in a plane normal to the symmetry axis. For instance, in materials with c₁₁ > c₃₃, it is found that the quasi-compressional wave attenuates more along the symmetry axis direction than in the plane of isotropy. The second medium is tellurium dioxide, a strongly anisotropic material of tetragonal symmetry. In this case, the diagrams show that strong attenuation is associated with high slowness values, as at around 45° in the horizontal plane. Both case studies show that the features of the attenuation surfaces strongly depend on the values of the elasticities.     

爆破地震波穿越不同介质结构面的衰减效应

为探索爆破地震波穿越不同充填介质结构面的振动速度、频率和能量分布的衰减规律,采用混凝土边坡相似模型进行爆破振动试验。通过分析爆破地震波的峰值振动速度、边际能量谱和各频带能量百分比,获得了爆破地震波穿越无结构面与结构面充填泥、细沙、水和空气的衰减规律。结果表明:当充填介质波阻抗越小,地震波的峰值能量和峰值振动速度衰减越快,频率上表现出高频滤波作用越强,在频带间能量分布变窄,主频带有往低频带0～20 Hz聚集的趋势;值得注意的是,当水作为充填介质时,不仅水的波阻抗会影响地震波的衰减,而且水的微压缩连续性也会使峰值能量和峰值振动速度衰减更慢,能量在频带间分布更广,水的流动性则会造成爆破地震波的主频带在各传播方向上分布不一。在相同充填介质条件下,随着结构面倾角增大,峰值振动速度在各传播方向上均减小。     

爆破地震波穿越不同介质结构面的衰减效应

为探索爆破地震波穿越不同充填介质结构面的振动速度、频率和能量分布的衰减规律,采用混凝土边坡相似模型进行爆破振动试验。通过分析爆破地震波的峰值振动速度、边际能量谱和各频带能量百分比,获得了爆破地震波穿越无结构面与结构面充填泥、细沙、水和空气的衰减规律。结果表明:当充填介质波阻抗越小,地震波的峰值能量和峰值振动速度衰减越快,频率上表现出高频滤波作用越强,在频带间能量分布变窄,主频带有往低频带0~20 Hz聚集的趋势;值得注意的是,当水作为充填介质时,不仅水的波阻抗会影响地震波的衰减,而且水的微压缩连续性也会使峰值能量和峰值振动速度衰减更慢,能量在频带间分布更广,水的流动性则会造成爆破地震波的主频带在各传播方向上分布不一。在相同充填介质条件下,随着结构面倾角增大,峰值振动速度在各传播方向上均减小。     

Inhomogeneous waves in transversely isotropic dielectrics

Summary The propagation of inhomogeneous transverse plane waves in a transversely isotropic dielectric solid is investigated within the theory of generalized piezo-thermo-elasticity. Geometrical restrictions on the real and imaginary parts of the wave vector are obtained as a consequence of the particular anisotropy of the solid. Dispersion relations for the most significant propagation modes are derived and discussed. Allowing for the inhomogeneity yields a generalization of previous results on piezo-thermo-elastic waves.     

Some comments on the dynamic properties of anisotropic and strongly anisotropic pre-stressed elastic solids

The effect of pre-stress on the propagation of small amplitude waves in an incompressible, transversely isotropic elastic solid is discussed in respect of the most general appropriate strain energy function. A simple set of sufficient conditions is noted which ensures that two real wave speeds exist for all directions of propagation. In the case of bi-axial primary deformations, and for propagation within each principal plane of the homogeneous primary deformation, the propagation condition is factorised and conditions which are both necessary and sufficient to ensure the existence of two real wave speeds are established. The paper also includes some graphical illustrations of the associated slowness and wave surfaces and discussion of the strongly anisotropic case, for which the extensional modulus along the preferred fibre direction is much larger than other material parameters.     

Dispersion of elastic waves in periodically inhomogeneous media

Propagation of time-harmonic elastic waves through periodically inhomogeneous media is considered. The material inhomogeneity exists in a single direction along which the elastic waves propagate. Within the period of the linear elastic and isotropic medium, the density and elastic modulus vary either in a continuous or a discontinuous manner. The continuous variations are approximated by staircase functions so that the generic problem at hand is the propagation of elastic waves in a medium whose finite period consists of an arbitrary number of different homogeneous layers. A dynamic elasticity formulation is followed and the exact phase velocity is derived explicitly as a solution in closed form in terms of frequency and layer properties. Numerical examples are then presented for several inhomogeneous structures.     

Equations for the Momentum and Energy of an X-ray Wave Field in a Crystal

A new approach for describing the interaction of X rays with a crystalline medium is proposed. General relations for a change in the electromagnetic-field momentum and energy have been derived for a nonmagnetic medium with variable permittivity. A special local coordinate system for an inhomogeneous medium is introduced, in which the Maxwell tension tensor has a canonical form determined by the energy and momentum densities. Main relations for changes in the energy and momentum densities have been obtained in the coordinate system related to the propagation direction of a plane electromagnetic wave pulse in a homogeneous medium.     

The effect of inhomogeneity and anisotropy on Stoneley waves

Summary In this paper we study the dispersion equation of Stoneley waves that are travelling in an inhomogeneous elastic half-space over an anisotropic homogeneous elastic half-space.The phase velocity is calculated as a function of the wave number. The results indicate that the effect of anisotropy on such waves is small and can be neglected, while the effect of inhomogeneity is very pronounced. The results show that Stoneley waves do not exist after some cut-off value of the wave number.     

Complex unitary vectors for the direction of propagation and for the polarization of electromagnetic waves in uniaxial and absorbing dielectric media

The propagation of inhomogeneous and elliptically polarized plane waves in absorbing uniaxial anisotropic media is described using complex unitary vectors to represent the direction of propagation and the direction of polarization. Detailed expressions for electric displacement, electric field, and magnetic field vectors are obtained for the ordinary and extraordinary waves, and their geometry is discussed. According to the complex direction of propagation, three particular cases are studied: the real case (homogeneous wave), the case perpendicular to the optical axis, and the case coplanar with the optic axis. The case of isotropic media is also analyzed.     

Existence of longitudinal waves in anisotropic poroelastic solids

In anisotropic fluid-saturated porous solids, four waves can propagate along a general phase direction. However, solid particles in different waves may not vibrate in mutually orthogonal directions. In the propagation of each of these waves, the displacement of pore–fluid particles may not be parallel to that of solid particles. The polarization for a wave is the direction of aggregate displacement of the particles of the two constituents of a porous aggregate. These polarizations, for different waves, are not mutually orthogonal. Out of the four waves in anisotropic poroelastic medium, two are termed as quasi-longitudinal waves. The prefix ‘quasi’ refers to their polarization being nearly, but not exactly, parallel to the direction of propagation. The existence of purely longitudinal waves in an anisotropic poroelastic medium is ensured by the stationary characters of two expressions. These expressions involve the elastic (stiffness and coupling) coefficients of a porous aggregate and the components of phase direction. Necessary and sufficient conditions for the existence of longitudinal waves are discussed for different anisotropic symmetries. Conditions are also discussed for the existence of the apparent longitudinal waves, i.e., the propagation of wave motion with the particle displacement parallel to the ray direction instead of the phase direction. A graphical solution of a numerical example is shown to check the existence of these apparent longitudinal waves for general directions of phase propagation.     

Velocity of shear waves in an initially stressed incompressible anisotropic medium

The paper aims at the study of propagation of shear waves in an initially stressed anisotropic medium. The velocity equation is obtained and it is seen that the velocity of propagation depends upon the direction of propagation, the anisotropic factor and also on the initial stresses. Numerical computation indicates that the initial compressive stress diminishes the velocity of a shear wave which propagates along the direction 0° to 45° but increases it along the direction 45° to 90°. The reverse effect is obtained when initial stress is tensile. The anisotropy increases the velocity and even in the absence of initial stresses the velocity of the wave depends on the direction of propagation.     

A spectral finite element for axial-flexural-shear coupled wave propagation analysis in lengthwise graded beam

A new spectral element (SE) is formulated to analyse wave propagation in anisotropic inhomogeneous beam. The inhomogeneity is considered in the longitudinal direction. Due to this particular pattern of inhomogeneity, the governing partial differential equations (PDEs) have variable coefficients and an exact solution for arbitrary variation of material properties, even in frequency domain, is not possible to obtain. However, it is shown in this work that for exponential variation of material properties, the equations can be solved exactly in frequency domain, when the same parameter governs the variation of elastic moduli and density. The SE is formed using this exact solution as interpolating polynomial. As a result a single element can replace hundreds of finite elements (FEs), which are essential for all wave propagation analysis and also for accurate representation of the inhomogeneity. The developed element is used for eliciting several advantages of the gradation, including mode selection, mode blockage and smoothening of stress waves.     

On the Theory of Acoustic Tomography of Stresses in Solids

We consider a mathematical model of propagation of weak elastic perturbations in acoustically inhomogeneous solids. The acoustic inhomogeneity and anisotropy of the body are induced by a field of initial strains. A linearized system of dynamic equations obtained for the Murnagan cubic potential of elasticity includes coefficients depending on the components of initial strains. An iterative approach proposed for the solution of this system of equations whose coefficients depend on space coordinates enables us to reduce the problem to the solution of a sequence of inhomogeneous wave equations with constant coefficients. Even in the zero-order approximation, this approach enables us to establish relations connecting the mean phase velocities of plane monochromatic waves of various types with linear integrals of the components of strains for arbitrary directions of propagation of waves. The corresponding relations are presented for the case of plane deformation.     

层理面对水压爆破应力波传播规律的影响

水压爆破应力波的传播过程决定了爆破的作用效果及影响范围,层状岩体中的层理面特征对水压爆破应力波的传播路径及衰减具有直接影响.为明确水压爆破应力波在层状岩体内的传播规律及影响因素,采用理论分析和数值模拟相结合的方法,研究了层理面位置及其介质属性对水压爆破应力波传播的影响规律.结果表明:水压爆破相对于常规爆破产生的初始应力...     

Theoretical study of surface acoustic waves in (n11) GaAs-cuts

The propagation characteristics of true and leaky or pseudo surface acoustic waves (TSAW and PSAW=LSAW) on (n11) GaAs-cuts, n=1, 2, 3 and 4, are theoretically calculated as a function of propagation direction. They include phase velocity (V), electromechanical coupling constant (K(2)), and attenuation factor (alpha) of wave propagation on a metallized surface. The results show that PSAW mode velocities are significantly higher than corresponding TSAW velocities, and for certain propagation directions the attenuation factor is extremely small (10(-5) dB/lambda). Highly coupled PSAW modes exist for propagation directions where the TSAW are very poorly coupled. For certain isolated directions, attenuation of the wave is null (alpha=0), PSAW becoming a non-leaky SAW with partial polarization. And in this case the corresponding TSAW are decoupled from the surface electric excitation. Analysis of relations between various modes (TSAW, PSAW and SSBW, surface skimming bulk wave) is made with the help of the effective surface permittivity function and the generalized slowness diagram. A coupling constant definition different from the usual 2DeltaV/V is used, its validity and application conditions are discussed.     

Anomalous propagation of elastic waves through a boundary between liquid and magnetoacoustic material

We consider a longitudinal acoustic wave incident onto a plane boundary between a liquid and a magnetoacoustic medium representing an antiferromagnetic material with anisotropy of the easy plane type, occurring in the vicinity of an orientational phase transition with respect to magnetic field. The directions of propagation and the amplitudes of reflected and transmitted longitudinal and transverse waves are determined. The possibility of an effective field control for the refraction angle and the wave type transformation is demonstrated. Beginning with a certain critical angle of incidence, the longitudinal and, eventually, the transverse waves in the magnetic medium become inhomogeneous and slide along the interface. If the magnetic material is sufficiently close to the phase transition point, the waves can be reirradiated into the liquid medium.     

An eigen theory of electromagnetic waves based on the standard spaces

The Maxwell’s equations are studied here based on the standard spaces of the physical presentation, and the modal electromagnetic wave equations of anisotropic media are deduced. Several novel theoretical results were obtained: the number of electromagnetic waves in anisotropic media is equal to that of eigen-spaces of anisotropic media; the velocity of propagation of electromagnetic waves is dependent on the dielectric eigen-permittivity and magnetic eigen-permeability; the direction of propagation of electromagnetic waves is related on the electromagnetic eigen-operator in the corresponding eigen-space; the direction of polarization of electromagnetic waves is relevant to the eigen-electromagnetic quantities in the corresponding eigen-space. Based on these laws, we discuss the propagation behaviour of electromagnetic wave in anisotropic media.