Energy flux in viscoelastic anisotropic media

Abstract

SUMMARY We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P, SV and SH waves. 1 I NTR O DUCTION In this paper, we study the properties of the energy flux of homogeneous and inhomogeneous time-harmonic plane waves propagating in an unbounded viscoelastic anisotropic medium. The complexvalued energy-flux vector F (also called the Poynting or Umov‐ Poynting vector) is used as the quantity which fully characterizes the energy flux and the dissipation of energy of the wave. The real part of the vector F represents the time-averaged energy flux S. Both real, Re F = S, and imaginary, Im F, parts are closely related to the density of the time-averaged dissipated energy W d : W d equals twice the scalar product of −Im F and propagation vector P (real part of the slowness vector, perpendicular to the wave front) or twice the scalar product of Re F and attenuation vector A (imaginary part of the slowness vector, perpendicular to the plane of constant amplitude). The aim of this paper is to present a concise derivation of the expression for the complex-valued energy-flux vector F of homogeneous and inhomogeneous plane waves propagating in unbounded media of unrestricted anisotropy and viscoelasticity, to propose an algorithm for its calculation, and to perform analytical and numerical studies necessary for correct understanding of energy propagation and dissipation of the relevant plane waves. Considerable attention in seismology has been devoted to the energy flux of plane waves propagating in elastic anisotropic and in viscoelastic isotropic media. For elastic anisotropic media, see, for