Expressions For Phase Velocity Research Articles

Frequency dependent qP-Lomax rays in three-dimensional weak TTI media

A simple and numerically effective approach for calculating frequency dependent qP-rays in threedimensional TTI media is presented in this paper. Our method involves propagating a locally plane fragment of wavefront which is sensitive to the distribution of the model parameters in some subvolume of the medium near a ray. The width of the sensitivity area depends on the wavelength in each point on the ray. For numerical realization we apply approximate expressions for phase and group velocities which are valid for weak TTI media. Numerical experiment proves the effectiveness of the proposed approach

An improved pure quasi-P-wave equation for complex anisotropic media

Abstract An accurate pure qP-wave equation in transverse isotropic (TI) media and its efficient and stable implementation are valuable for seismic imaging and inversion. Owing to the complexity of the qP-wave phase velocity expression in anisotropic media, it is difficult to construct such a pure qP-wave equation. In this paper, we combine the Taylor expansion and scalar operator methods to formulate an efficient and stable pure qP-wave equation in TI media. First, the Taylor expansion method is used to convert the square-root term into a fractional term in the qP-wave phase velocity expression. We further improve the approximation accuracy of the resulting equation by a correction technique. Then, the scalar operator is applied to scalarize the equivalent form of the fractional term in the approximated dispersion equation, deriving a simple and easy-to-implement pure qP-wave equation. We use the optical flow method to compute the direction of wave propagation, which improves the calculation accuracy of the scalar operators. Numerical experiments with representative models demonstrate that the new method has higher accuracy and better adaptability to models with strong anisotropy, complex structure, and rapid variation of the tilt angle than previous methods.

Two-dimensional wave motion in a viscous homogeneous liquid

The propagation of two-dimensional periodic capillary-gravitational waves in a viscous homogeneous liquid with a free surface in the frequency range from \(5 \cdot {{10}^{{ - 2}}}\) to \(5 \cdot {{10}^{4}}\) Hz is investigated. Dispersion relations are obtained, as well as expressions for phase and group velocities for surface waves in physically observable variables. It is shown that there is another class of solutions that is not observed in the ideal fluid model.

Periodic Waves and Ligaments on the Surface of a Viscous Exponentially Stratified Fluid in a Uniform Gravity Field

The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the 10−4<ω<103 s−1 frequency range, in a viscous continuously stratified fluid. Dispersion relations for flow constituents are given, as well as expressions for phase and group velocities for surface waves and ligaments in physically observable variables. When the wave-length reaches values of the order of the stratification scale, the liquid behaves as homogeneous. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In limiting cases, the expressions obtained are transformed into known wave dispersion expressions for an ideal stratified or actually homogeneous fluid.

Influence of initial stress on shear wave scattering in a functionally graded magneto-visco-elastic orthotropic multi-layered structure

A theoretical model is proposed to analyze the propagation behavior of SH-wave in a functionally graded magneto-visco-elastic orthotropic multi-layered composite structure. Haskell’s matrix technique has been used to derive the closed-form dispersion relation for the assumed model. It is found that under certain conditions, the acquired dispersion relation agrees with the classical Love wave equations. Also, a finite difference technique is adopted to deduce the expression for phase velocity and group velocity. Numerical computation has been done and presented graphically to show the effect of different parameters. Stability analysis for the finite difference scheme is carried out to reduce the growth of errors to make it stable and convergent.

Decoupling approximation of P- and S-wave phase velocities in orthorhombic media

Characterizing the kinematics of seismic waves in elastic orthorhombic media involves nine independent parameters. All wave modes, P-, S1-, and S2-waves, are intrinsically coupled. Because the P-wave propagation in orthorhombic media is weakly dependent on the three S-wave velocity parameters, they are set to zero under the acoustic assumption. The number of parameters required for the corresponding acoustic wave equation is thus reduced from nine to six, which is very practical for the inversion algorithm. However, the acoustic wavefields generated by the finite-difference scheme suffer from two types of S-wave artifacts, which may result in noticeable numerical dispersion and even instability issues. Avoiding such artifacts requires a class of spectral methods based on the low-rank decomposition. To implement a six-parameter pure P-wave approximation in orthorhombic media, we have developed a novel phase velocity approximation approach from the perspective of decoupling P- and S-waves. In the exact P-wave phase velocity expression, we find that the two algebraic expressions related to the S1- and S2-wave phase velocities play a negligible role. After replacing these two algebraic expressions with the designed constant and variable, respectively, the exact P-wave phase velocity expression is greatly simplified and naturally decoupled from the characteristic equation. Similarly, the number of required parameters is reduced from nine to six. We also derive an approximate S-wave phase velocity equation, which supports the coupled S1- and S2-waves and involves nine independent parameters. Error analyses based on several orthorhombic models confirm the reasonable and stable accuracy performance of our phase velocity approximation. We further derive the approximate dispersion relations for the P-wave and the S-wave system in orthorhombic media. Numerical experiments demonstrate that the corresponding P- and S-wavefields are free of artifacts and exhibit good accuracy and stability.

Decoupled approximation and separate extrapolation of P- and SV-waves in transversely isotropic media

Characterizing the kinematics of seismic waves in elastic vertical transversely isotropic (VTI) media involves four independent parameters. To reduce the complexity, the acoustic approximation for P-waves reduces the number of required parameters to three by setting the vertical S-wave velocity to zero. However, because only the SV-wave phase velocities parallel or perpendicular to the symmetry axis are indirectly set to zero, the acoustic approximation leads to coupled P-wave components and SV-wave artifacts. The new acoustic approximation suggests setting the vertical S-wave velocity as a phase angle-dependent variable so that the SV-wave phase velocity is zero at all phase angles. We find that manipulating this parameter is a valid approach for P-wave approximation but doing so inevitably leads to zero- or nonzero-valued spurious SV-wave components. Thus, we have developed a novel approach to efficiently approximate and thoroughly separate the two wave modes in VTI media. First, the exact P- and SV-wave phase velocity expressions are rewritten by introducing an auxiliary function. After confirming the insensitivity of this function, we construct a new expression for it and obtain simplified P- and SV-wave phase velocity expressions, which are three and four parameters, respectively. This approximation process leads to the same reasonable error for both wave modes. Accuracy analysis indicates that, for the P-wave, the overall accuracy performance of our approach is comparable to that of some existing three-parameter approximations. We then derive the corresponding P- and SV-wave equations in tilted transversely isotropic (TTI) media and provide two available solutions, the hybrid finite-difference/pseudospectral scheme and the low-rank approach. Numerical examples illustrate the separability and high accuracy of the proposed P- and SV-wave simulation methods in TTI media.

Acoustic Propagation in a Medium with Spatially Distributed Relaxation Processes and a Possible Explanation of a Frequency Power Law Attenuation

A possible medium is considered with a large number of diverse but possibly similar relaxation processes. The equations for single spatially located relaxation processes of general type are developed and extended to a large spatially extended system. Each process is characterized by a relaxation time and a relaxation strength and is excited by the local pressure in an incident acoustic wave. A constant frequency model is initially assumed with complex amplitudes associated with acoustic pressure and relaxation responses. A smearing process results in the collective relaxation responses being regarded as a continuous function of relaxation times. An expression for entropy perturbation is developed in terms of the relaxation responses. The equations of fluid mechanics then lead to an expression for the pressure perturbation in terms of the dilatation and the relaxation processes. The extended result yields a time-domain wave equation for propagation through an inhomogeneous medium with distributed relaxation processes. Expressions for frequency-dependent attenuation and phase velocity are derived in which relaxation is characterized by a single function giving strength as a continuous function of relaxation time. Possible choices for this function are discussed, and it is shown that some choices lead to an attenuation varying nearly as a power of the frequency over any fixed and possibly large range of frequencies. A model set of parameters leads to good agreement with attenuation and phase velocity measurements for a suspension of human red blood cells in saline solution, as communicated to the authors by Treeby and analogous to those reported by Treeby, Zhang, Thomas, and Cox.

Multiscale analysis of elastic waves in soft materials: From molecular chain networks to fiber composites

We provide a multiscale analysis of elastic waves in soft microstructured materials including the underlying molecular chain network mechanisms and micromechanics of hyperelastic fiber composites. First, we examine the interplay between the crosslinked and entangled polymer chains and elastic wave characteristics in finitely deformed materials. Next, we study the shear wave propagation in a class of heterogeneous hyperelastic composites. In particular, we consider the transversely isotropic fiber composites with phases characterized by the stiffening behavior stemming from the non-Gaussian statistics of polymer chains. By employing a micromechanics-based approach, we derive explicit expressions for phase velocities in terms of material properties and volume fraction of the phases. Our results indicate the significant influence of the variety of length-scales mechanisms on the elastic wave characteristics. We provide examples to illustrate the influence of the mechanisms under different loading conditions, wave propagation directionality, and material microstructure parameters.

Explicit expression for complex velocity of Rayleigh wave in dissipative poroelastic solid

ABSTRACT This study considers to solve the complex dispersion equation for propagation of Rayleigh waves in a dissipative porous composite. This porous medium is a viscoelastic porous solid frame saturated with a viscous fluid. Seepage of fluid at the boundary is controlled through pore-fluid pressure and its normal gradient. A complex irrational equation governs the propagation, attenuation and dispersion of the Rayleigh waves. A complex analysis technique is used to solve this equation into an explicit expression for complex phase velocity. Effects of porosity and seepage from surface pores are analyzed numerically on the velocity of Rayleigh wave in a saturated poroelastic material.

Fractional derivative group shear wave speeds for model-based viscoelastic characterization

We present a method for model-based viscoelastic characterization using shear wave elastography. The shear wave displacement signal is differentiated to a series of fractional derivative orders, and the group speed of the signal at each derivative order is calculated. These speeds are compared to a look up table built from simulations for a given viscoelastic material model. The best-fit material parameters are chosen to minimize the mean square difference between the group speeds from the look up table and experimental measurements across all fractional derivative orders. Various two- and three-parameter viscoelastic material models are tested. The technique is validated in simulations with ultrasonic tracking and demonstrated in three viscoelastic phantoms with material properties that match those associated with different degrees of liver fibrosis. Comparatively, phase velocity curves are computed, and parameters determined by fitting analytic expressions for phase velocity for a given material model. The fractional derivative group speed-based technique gives model parameters with lower mean square error than fitting material model parameters directly to phase velocities. Additionally, we conclude these viscoelastic phantoms are sufficiently characterized by a two-parameter material model and are better characterized by the Linear Attenuation model than the more commonly used Voigt model.

Torsional waves of infinite fully saturated poroelastic cylinders within the framework of Biot viscosity-extended theory

The repercussion of understanding how waves propagate in poroelastic media is enormous. Poroelastic cylinders have been used in several applications of different fields. For example, geophysics, structural mechanics, medicine, aerodynamic, nanoscience, among others. This work, based on Biot viscosity-extended theory, investigates the torsional vibrations of infinite fully saturated, axial-symmetric, poroelastic cylinders. Two complex frequency equations are obtained when the stresses-free boundary conditions are taking into account: one associated with the fast S wave and another to the slow S wave. One of the challenges is finding the analytical expressions for phase velocity and attenuation from the frequency equation. In this work, these expressions were obtained and compared with those corresponding to the Biot theory. When comparing the results with both theoretical frameworks, as expected, the torsional waves were attenuated more in the Biot viscosity-extended theory framework. The influence of the slow S wave causes the torsional waves to travel slower than Biot’s theoretical framework.

Efficient mixed-domain pure P-wave equations for 2D and 3D tilted transversely isotropic media

In transversely isotropic (TI) media, accurate and efficient pure P-wave extrapolation is the basis of seismic imaging and inversion algorithms. For pure P-wave equations in TI media, combining the finite-difference (FD) method with the pseudo-spectral (PS) method is an effective solution. However, if the mixed-domain pure P-wave equation contains multiple wavenumbers, using the hybrid approach will involve multiple Fourier transforms, which will result in high computational costs. Referring to the weak anisotropy approximation, we propose a new approximate P-wave phase velocity expression. We then use an optimization strategy to reduce the number of wavenumber terms in the corresponding dispersion relations and derive the mixed-domain pure P-wave equations in 2D and 3D tilted transversely isotropic (TTI) media. Through numerical experiments in 2D and 3D TTI models, we verify the feasibility and efficiency of the proposed mixed-domain P-wave equations.

Scattering and Backscattering Study of Mechanical Plane Wave in Composite Materials Plates (Earth model 1066B and LiNbO3)

Reflection and refraction phenomenon pattern of elastic plane wave at the interface between anisotropic monoclinic elastic half-space and isotropic elastic half-spaces is studied. Closed-form expression for phase velocity is obtained. Reflection and transmission coefficients are obtained using the method of Cramer's rule in determinant form. Also, the energy ratios are calculated in terms of reflection and transmission coefficients. Numerical examples are considered to exhibit all the findings graphically. The energy conservation law is implemented at each angle of incidence to validate the numerical results, and it is found that energy ratios are in good agreement with the energy conservation law.

Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

This study extends the second-order attenuation (SOA) model for elastic waves in texture-free inhomogeneous cubic polycrystalline materials with equiaxed grains to textured polycrystals with ellipsoidal grains of arbitrary crystal symmetry. In term of this work, one can predict both the scattering-induced attenuation and phase velocity from Rayleigh region (wavelength >> scatter size) to geometric region (wavelength << scatter size) for an arbitrary incident wave mode (quasi-longitudinal, quasi-transverse fast or quasi-transverse slow mode) in a textured polycrystal and examine the impact of crystallographic texture on attenuation and phase velocity dispersion in the whole frequency range. The predicted attenuation results of this work also agree well with the literature on a textured stainless steel polycrystal. Furthermore, an analytical expression for quasi-static phase velocity at an arbitrary wave propagation direction in a textured polycrystal is derived from the SOA model, which can provide an alternative homogenization method for textured polycrystals based on scattering theory. Computational results using triclinic titanium polycrystals with Gaussian orientation distribution function (ODF) are also presented to demonstrate the texture effect on attenuation and phase velocity behaviors and evaluate the applicability and limitation of an existing analytical model based on the Born approximation for textured polycrystals. Finally, quasi-static phase velocities predicted by this work for a textured polycrystalline copper with generalized spherical harmonics form ODF are compared to available velocity bounds in the literature including Hashin–Shtrikman bounds, and a reasonable agreement is found between this work and the literature.

Study on the generation mechanism of bright and dark solitary waves and rogue wave for a fourth-order dispersive nonlinear Schrödinger equation

In this paper, we study the generation mechanism of bright and dark solitary waves and rogue wave for the fourth-order dispersive nonlinear Schrödinger (FODNLS) equation, which can not only model the nonlinear propagation and interaction of ultrashort pulses in the high-speed optical fiber transmission system, but also govern the nonlinear spin excitations in the onedimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole-dipole interaction. Firstly, via the phase plane analysis, we obtain both the homoclinic and heteroclinic orbits for the two-dimensional plane autonomous system reduced from the FODNLS equation. Further, we derive the bright and dark solitary wave solutions under the corresponding conditions, which reveals the relationship between the homoclinic (heteroclinic) orbit and solitary wave. Secondly, based on the exact first-order breather solution of the FODNLS equation over a nonvanishing background, we give the explicit expressions of group and phase velocities, and reveal that there exists a jump in both the velocities. Finally, in order to verify that the breather becomes a rogue wave at the jumping point, we obtain the first-order rogue wave solution by taking the limit of the breather solution at such point, which confirms the relationship of the generation of rogue wave with the velocity discontinuity.

Dynamics of quasi-seismic wave in an isotropic layer between monoclinic materials overburden by self-reinforced crustal zone

Present work exhibits an analytical approach for determination of the reflection/transmission coefficients of quasi-longitudinal/shear vertical seismic wave in dual sandwiched isotropic-monoclinic layers between self-reinforced overburdened Earth crust and monoclinic subspace. Closed-form expressions for phase velocity, reflection/transmission coefficients of all the scattered waves are obtained and certain published works are derived as a particular case of the current study. Numerical simulation with lithium niobate type monoclinic material and fiber epoxy resin composites self-reinforced material based on free oscillation frequencies are used to observe the response spectra of relevant parameters on scattered velocities, angles, and coefficients. Findings of present work may be used in modeling the propagation lineaments of quasi-waves rendered through man-made detonation such as those by earth-penetrator weapon (EPW), shallow earthquake impact analysis on the reinforced building designs, exploration of hydrocarbon in the upper crustal region etc.

Consistent upwinding for sequential fully implicit multiscale compositional simulation

There is strong interest to design sequential fully implicit (SFI) methods for compositional flow simulations with convergence properties that are comparable to fully implicit (FI) methods. SFI methods decompose the fully coupled system into a pressure equation and a transport system of the components. During the pressure update, the compositions are frozen, and during the transport calculations, both the pressure and total velocity are kept constant. The two systems are solved sequentially, and the solution, which is a fully implicit one, is obtained by controlling the splitting errors due to the decoupling. Having an SFI scheme that enjoys a convergence rate similar to FI makes it possible to design specialized numerical methods optimized for the different parabolic and the hyperbolic operators, as well as the use of high-order spatial and temporal discretization schemes. Here, we use the multiscale restriction-smoothed basis (MsRSB) method for the parabolic operator. We also show that phase-potential upwinding is incompatible with the total velocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong gravity or capillary pressure, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. In this work, we employ phase upwinding (PU) as well as implicit hybrid upwinding (IHU) with a SFI method. IHU determines the upwinding direction differently for the viscous, buoyancy, and capillary pressure terms in the phase velocity expressions. The use of IHU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong buoyancy or capillarity. We demonstrate the robustness of the IHU-based SFI algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.

Analysis on scattering characteristics of Love-type wave due to surface irregularity in a piezoelectric structure.

Surface irregularity in piezoelectric sensors causes scattering of Love-type waves, which may lead to significant signal insertion loss. Also, as the reflected wave field in the piezoelectric sensor is increased due to surface irregularity, the input interdigital transducer (IDT) can be damaged. The present analysis confers the characteristics of Love-type waves scattered through surface irregularity in piezoelectric composite structures with different shaped surface irregularity, e.g., rectangular shaped, parabolic shaped, and triangular notch shaped. The expressions of phase and group velocities of scattered Love-type waves are derived for electrically open and short conditions. Also, expressions of mechanical displacement and electrical potential function due to incident and scattered Love-type waves are deduced for both electrical conditions. It is reported from the analysis that the increase in vertical irregularity parameter causes stronger reflected wave fields. Also, with the increase in piezoelectricity of the superficial layer, signal insertion loss and strength of reflected wave field diminish.

Modeling of SH-Wave Propagation in a Pre-stressed Highly Anisotropic Layered Structure

The present study presents the propagation of horizontally polarized shear waves (SH-waves) in highly anisotropic layered media under the effect of horizontal and vertical initial stresses. Following Biot’s theory of incremental stresses, the present problem has been formulated. The dispersion equation has been derived using the Thomson–Haskell matrix method. Closed form expressions for phase and group velocities have been obtained using finite difference schemes. Also, the stability condition of using this finite difference scheme has been achieved. Substantial influence of the Courant number (stability ratio), tensile and compressive stress on both phase and group velocities has been observed and depicted through graphs.