A method to compute the complex-valued slowness vector of a plane wave, propagating
in a viscoelastic anisotropic medium in an arbitrarily specified direction **N**, is proposed.
The method is quite universal and can be applied to homogeneous and inhomogeneous plane waves,
propagating in isotropic and anisotropic, perfectly elastic and viscoelastic media. This is, in
general, impossible with the well-known conventional approaches, used in isotropic media. The
method is based on the so-called mixed specification of the slowness vector, and leads to the
solution of a complex-valued algebraic equation of the sixth degree. Standard methods can be used
to solve this equation. Once the roots of the algebraic equation have been found, the phase velocities,
exponential decays of amplitudes, attenuation angles, polarisation vectors, etc., of qP, qS1 and qS2
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 qP, qSV and SH plane waves, propagating in a plane of symmetry of a monoclinic (orthorhombic, hexagonal) viscoelastic medium is discussed in greater detail. In this case, the algebraic equation of the sixth degree is factorized into one quartic equation (qP and qSV waves) and one quadratic equation (SH waves). The analytical discussion of these waves offers a simple physical insight into the problem of inhomogeneous plane waves, and valuable comparisons of advantages and drawbacks of different approaches.

In this paper (Part 1),
the qP, qSV and SH plane waves, propagating
in the plane of symmetry of
anisotropic
viscoelastic media, are studied as functions of propagation direction **N**
and the
inhomogeneity parameter *D* from (-infinity, +infinity).
For inhomogeneous plane waves, *D* is not equal 0, and for
homogeneous plane waves, *D* equals 0.
The inhomogeneity parameter *D* offers many advantages in comparison with the conventionally used
attenuation angle γ.
In the **N**, *D* domain, all combinations of **N** and *D* are physically acceptable.
This is, however, not the case in the **N**, γ domain,
where certain combinations of **N** and γ yield non-physical slowness vectors. Another
advantage of the used 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*.
Attenuation angle γ cannot exceed its boundary value, γ^{*}.
The boundary attenuation
angle γ^{*} is, in general, different from *90*^{0}, and depends both on the direction of
propagation **N** and on the
sign of
inhomogeneity parameter *D*.
The polarization of qP and qSV 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 Part 2, see
Cerveny and Psencik (2004).

Viscoelastic media, inhomogeneous and homogeneous plane waves, phase velocity, attenuation angle, amplitude decay, slowness vector, inhomogeneity parameter.

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In: Seismic Waves in Complex 3-D Structures, Report 14, pp. 123-147, Dep. Geophys., Charles Univ., Prague, 2004.

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