## Weakly inhomogeneous plane waves in anisotropic,
weakly dissipative media

**Vlastislav Cerveny** and
**Ivan Psencik**
### Summary

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.

### Keywords

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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