In viscoelastic media, the slowness vector p of plane waves is complex-valued, p=P+iA. The real-valued vectors P and A are usually called the propagation and the attenuation vector, respectively. For P and A nonparallel, the plane wave is called inhomogeneous.
Three basic approaches to the determination of the slowness vector of an inhomogeneous plane wave propagating in a homogeneous viscoelastic anisotropic medium are discussed. They differ in the specification of the mathematical form of the slowness vector p. We speak of directional specification, componental specification and mixed specification of the slowness vector. Individual specifications lead to the eigenvalue problems for 3x3 or 6x6 complex-valued matrices.
In the directional specification of the slowness vector, the real-valued unit vectors N and M in the direction of P and A are assumed to be known. This has been the most common specification of the slowness vector used in the seismological literature. In the componental specification, the real-valued unit vectors N and M are not known in advance. Instead, the complex-valued vectorial component pSigma of slowness vector p into an arbitrary plane Sigma with unit normal n is assumed to be known. Finally, the mixed specification is a special case of the componental specification, with pSigma purely imaginary. In the mixed specification, plane Sigma represents the plane of constant phase, so that N=±n. Consequently, unit vector N is known, similarly as in the directional specification. Instead of unit vector M, however, the vectorial component d of the attenuation vector in the plane of constant phase is known.
The simplest, most straightforward and transparent algorithms to determine the phase velocities and slowness vectors of inhomogeneous plane waves propagating in viscoelastic anisotropic media are obtained, if the mixed specification of the slowness vector is used. These algorithms are based on the solution of an algebraic equation of the sixth degree. Alternatively, they are based on the solution of a conventional eigenvalue problem for 6x6 complex-valued matrices. The derived equations are quite general and universal. They can be used both for homogeneous and inhomogeneous plane waves, propagating in elastic or viscoelastic, isotropic or anisotropic media. Contrary to the mixed specification, the directional specification can hardly be used to determine the slowness vector of inhomogeneous plane waves propagating in viscoelastic anisotropic media. Although the procedure is based on 3x3 complex-valued matrices, it yields a cumbersome system of two coupled equations.
Viscoelastic anisotropic media, inhomogeneous plane waves, phase velocity, slowness vector.
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