Anisotropic Wave Equation Research Articles

A level set-based Eulerian approach for anisotropic wave propagation

The geometric optics approximation to high frequency anisotropic wave propagation reduces the anisotropic wave equation to a static Hamilton–Jacobi equation. This equation is known as the anisotropic eikonal equation and has three different coupled wave modes as solutions. We introduce here a level set-based Eulerian approach that captures all three of these wave propagations. In particular, our method is able to accurately reproduce the quasi-transverse, or quasi-S, waves with cusps, which form a class of multi-valued solutions. The level set formulation we use is borrowed from one for moving curves in three spatial dimensions, with the velocity fields for evolution following from the method of characteristics on the anisotropic eikonal equation. We present here our derivation of the algorithm and numerical results to illustrate its accuracy in different cases of anisotropic wave propagations related to seismic imaging.

Analytical solution of non-homogeneous anisotropic wave equations based on differential transfer matrices

We present a new analytical method for solution of non-homogeneous anisotropic optical systems, based on the extension of transfer matrices into differential form. This approach can be used for exact calculation of various functions including reflection and transmission coefficients, band structures and bound states. We discuss improvements of the method for infinite periodic structures and bounded modes. Then simplifications of the approach are considered for the TE and TM polarized isotropic structures, pseudo-isotropic and uniaxial non-magnetic media. Moreover a general variational representation of bound states is introduced. As application examples, we consider the reflection from inhomogeneous structures, the band structure of infinite periodic media and bounded modes in inhomogeneous anisotropic waveguides. Excellent agreement between the results from our differential transfer matrix method and other methods is observed where comparison is possible.

A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations

In this paper, a new kind of discrete non-reflecting boundary conditions is developed. It can be used for a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic wave equations and the equations for wave propagation in multi-phase and so on. In this kind of boundary conditions, the composition of all artificial reflected waves, but not the individual reflected ones, is considered and eliminated. Thus, it has a uniform formula for different wave equations. The velocity C A of the composed reflected wave is determined in the way to make the reflection coefficients minimal, the value of which depends on equations. In this paper, the costruction of the boundary conditions is illustrated and C A is found, numerical results are presented to illustrate the effectiveness of the boundary conditions.

ASYMPTOTIC SOLUTIONS OF THE ELASTIC EQUATION IN THE CASE OF TOTAL REFLECTION

In this paper we construct outgoing asymptotic solutions of the anisotropic elastic wave equation coinciding with given data on the boundary when total reflection happens. In this case the solutions consist of the usual hyperbolic parts and the elliptic parts exponentially decreasing in the normal direction to the boundary. The main assertion is that sum of those parts can cover any boundary data because of the elasticity. The proof is reduced to the complex analysis of the symbol of the elastic operator. By means of the asymptotic solutions we can make parametrices and study propagation of the singularities in the case of total reflection. *Partly supported by Grant-in-Aid for Sci. Research (c) 10640151, the Ministry of Educ., Japan.

Collective excitations of degenerate Fermi gases in anisotropic parabolic traps

The hydrodynamic low-frequency oscillations of highly degenerate Fermi gases trapped in anisotropic harmonic potentials are investigated. Despite the lack of an obvious spatial symmetry the wave-equation turns out to be separable in elliptical coordinates, similar to a corresponding result established earlier for Bose-condensates. This result is used to give the analytical solution of the anisotropic wave equation for the hydrodynamic modes.

Effects of the Biot and the squirt-flow coupling interaction on anisotropic elastic waves

Considering the velocity anisotropy of the solid/fluid relative motion and employment of the BISQ theory[1] based on the one-dimensional porous isotropic case, we establish a two-phase anisotropic elastic wave equation to simultaneously include the Biot and the squirt mechanisms in terms of both the basic principles of the fluid’s mass conservation and the elastic-wave dynamical equations in the two-phase anisotropic rock. Numerical results, while the Biot-flow and the squirt-flow effects are simultaneously considered in the transversely isotropic (TI) poroelastic medium, show that the attenuation of the quasi P-wave and the quasi SV-wave strongly depend on the permeability anisotropy, and the attenuation behavior at low and high frequencies is contrary. Meanwhile, the attenuation and dispersion of the quasi P-wave are also affected seriously by the anisotropic solid/fluid coupling additional density.

Staggered mesh for the anisotropic and viscoelastic wave equation

Computation of the spatial derivatives with nonlocal differential operators, such as the Fourier pseudospectral method, may cause strong numerical artifacts in the form of noncausal ringing. This situation happens when regular grids are used. The problem is attacked by using a staggered pseudospectral technique, with a different scheme for each rheological relation. The nature and causes of acausal ringing in regular grid methods and the reasons why staggered‐grid methods eliminate this problem are explained in papers by Fornberg (1990) and Özdenvar and McMechan (1996). Thus, the objective here is not to propose a new method but to develop the algorithm for the viscoelastic and transversely isotropic (VTI) wave equation, for which the technique can be implemented without interpolation. The algorithm is illustrated for one physical situation that requires very high accuracy, such as a fluid‐solid interface, where very large contrasts in material properties occur. The staggered‐grid solution is noise free in the dynamic range where regular grids generate artifacts that may have amplitudes similar to those of physical arrivals.

An inverse approach to an anisotropic wave equation in a stratified half-space

An inverse approach to an anisotropic wave equation in a stratified half-space

First‐arrival traveltime calculation for anisotropic media

Huygens’s principle is used to derive a method for calculating first arrival traveltimes in anisotropic media. Numerical results for several transverse isotropic (TI) models show that calculated Huygens traveltimes are in good agreement with traveltimes computed by a finite‐difference solution to the anisotropic wave equation. This Huygens method is stable and accurate for the test models and so it may be useful in anisotropic data inversion and wave propagation visualization.

Spherical wave functions and dyadic Green's functions for homogeneous elastic anisotropic media.

A spherical-wave-function theory of bounded homogeneous elastic anisotropic media is developed. The anisotropic elastodynamic wave equations are solved exactly by using the method of plane-wave angular spectrum expansions. The series, integral representations, and addition theorems of the spherical wave functions of the first, second, third, and fourth kind for homogeneous elastic anisotropic media (HEAM) are presented. Weyl's method for the scalar Green's functions in isotropic media is generalized to the study of the dyadic Green's functions in HEAM. The series representations of Green's functions are of the form of separated variables. These representations are well suited to imposing a boundary condition when dealing with waves in spherically layered HEAM.

Homogenisation of parametrised families of hyperbolic problems

SynopsisThis paper is concerned with homogenisation processes for parametrised families of transport equations in ℝn symmetric hyperbolic systems in several space dimensions and anisotropic wave equations. The main tools for carrying out the homogenisation are the Young measures and Radon transform combined with the integral representation of holomorphic functions of Nevanlinna-Pick's type.

Acoustic reflection from transversely isotropic consolidated sediments

Measurements of an excess of horizontal to vertical velocity (averaging 10% for shales) have been made in consolidated marine sediments and in sedimentary rocks. In this paper, the proposed causes of such anisotropies are reviewed, and the reflection coefficient for a homogeneous, solid, transversely isotropic seafloor is derived from plane‐wave solutions of the anisotropic wave equations. Bottom loss calculations are used to show that anisotropy can affect the reflectivity of exposed consolidated sediments at the ocean floor. Studies of bottom loss are also used to develop a method for estimating the elastic parameter C13 (frequently not measured for sedimentary material) for consolidated materials.

The effect of anisotropy on the integral representation of a cylindrical pulse

The infinite medium Green's function for a two dimensional anisotropic scalar wave equation is obtained in closed form using a technique developed by De Hoop1). The effect of anisotropy on the complex contour integral representation of this Green's function is explicitly exhibited.