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).

Attenuation, seismic anisotropy, seismic waves, viscoelasticity.

The reprint is available in PostScript (1005 kB !), GZIPped PostScript (357 kB), and PDF (156 kB).

SW3D - main page of consortium