Plane waves in viscoelastic anisotropic media -- I. Theory

Vlastislav Cerveny & Ivan Psencik

Summary

Properties of homogeneous and inhomogeneous plane waves propagating in an unbounded viscoelastic anisotropic medium in an arbitrarily specified direction N are studied analytically. The method used for their calculation is based on the so-called mixed specification of the slowness vector. It is quite universal and can be applied to homogeneous and inhomogeneous plane waves propagating in perfectly elastic or viscoelastic, isotropic or anisotropic media. The method leads to the solution of a complex-valued algebraic equation of the sixth degree. Standard methods can be used to solve the algebraic equation. Once the solution has been found, the phase velocities, exponential decays of amplitudes, attenuation angles, polarization vectors, etc., of P, S1 and S2 plane waves, propagating along and against N, can be easily determined.

Although the method can be used for an unrestricted anisotropy, a special case of P, SV and SH plane waves, propagating in a plane of symmetry of a monoclinic (orthorhombic, hexagonal) viscoelastic medium is discussed in greater detail. In this plane the waves can be studied as functions of propagation direction N and of the real-valued inhomogeneity parameter D. For inhomogeneous plane waves, D.ne.0, and for homogeneous plane waves, D= 0. The use of the inhomogeneity parameter D offers many advantages in comparison with the conventionally used attenuation angle gamma. In the N, D domain, any combination of N and D is physically acceptable. This is, however, not the case in the N, gamma domain, where certain combinations of N and gamma yield non-physical solutions. Another advantage of the use of inhomogeneity parameter D is the simplicity and universality of the algorithms in the N, D domain.

Combined effects of attenuation and anisotropy, not known in viscoelastic isotropic media or purely elastic anisotropic media, are studied. It is shown that, in anisotropic viscoelastic media, the slowness vector and the related quantities are not symmetrical with respect to D=0 as in isotropic viscoelastic media. The phase velocity of an inhomogeneous plane wave may be higher than the phase velocity of the relevant homogeneous plane wave, propagating in the same direction N. Similarly, the modulus of the attenuation vector of an inhomogeneous plane wave may be lower than that for the relevant homogeneous plane wave. The amplitudes of inhomogeneous plane waves in anisotropic viscoelastic media may increase exponentially in the direction of propagation N for certain D. The attenuation angle gamma cannot exceed its boundary value, gamma*. The boundary attenuation angle gamma* is, in general, different from 90°, and depends both on the direction of propagation N and on the sign of the inhomogeneity parameter D. The polarization of P and SV plane waves is, in general, elliptical, both for homogeneous and inhomogeneous waves. Simple quantitative expressions or estimates for all these effects (and for many others) are presented. The results of the numerical treatment are presented in a companion paper (Paper II, this issue).

Keywords

Attenuation, seismic anisotropy, seismic waves, viscoelasticity.

Whole paper

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Geophys. J. int., 161 (2005), 197-212.
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