Weakly inhomogeneous plane waves in anisotropic, weakly dissipative media

Vlastislav Cerveny and Ivan Psencik


Weakly inhomogeneous time-harmonic plane waves propagating in homogeneous anisotropic, weakly dissipative media are studied using the perturbation method. Only dissipation mechanisms, which can be described in the framework of a linear viscoelasticity, are considered. As a reference (non-perturbed) case, plane waves with a real-valued slowness vector propagating in perfectly elastic anisotropic media are used. Simple approximate expressions for the complex-valued slowness and polarization vectors of weakly inhomogeneous plane waves propagating in anisotropic, weakly dissipative media are derived. A special attention is devoted to the imaginary part of the slowness vector, known as the attenuation vector, which is responsible for the amplitude attenuation of a plane wave. The derived approximate expression for the attenuation vector depends on the material (intrinsic) dissipation parameters as well as on the inhomogeneity of the plane wave. Its projection into the direction of the energy-velocity vector yields, however, the intrinsic attenuation factor, which does not depend on the inhomogeneity of the wave, and which thus represents a very suitable measure of the material dissipation. The derived expression for the intrinsic attenuation factor is valid for media of unrestricted anisotropy and weak dissipation and for homogeneous as well as weakly inhomogeneous plane waves. The intrinsic attenuation factor is inversely proportional to the scalar quantity, which, in isotropic viscoelastic media, corresponds to the well-known quality factor Q. Its generalization to anisotropic weakly viscoelastic media is directionally dependent. Numerical examples are presented, in which the accuracy of the approximate formulae based on the perturbation method is studied. The results indicate that the presented perturbation results are sufficiently accurate for the use in practical applications. Strong directivity of the intrinsic attenuation factor shows its great potential for the solution of inverse problems.


Anisotropy, attenuation, inhomogeneous plane waves, perturbation methods, viscoelasticity.

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In: Seismic Waves in Complex 3-D Structures, Report 17, pp. 149-172, Dep. Geophys., Charles Univ., Prague, 2007.
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