We consider the reflection/transmission (R/T) problem of inhomogeneous plane waves at a plane interface between two homogeneous isotropic viscoelastic halfspaces. A general orientation of the attenuation vector of the incident wave is allowed, even orientation away from the plane of incidence. A procedure of determining the complex-valued normal components of the slowness vectors of generated reflected/transmitted waves from the tangential components of these vectors, known along the interface, is proposed. Particular attention is devoted to the determination of the sign of complex-valued square roots, involved in the problem. Continuity criteria are proposed, which are based on the requirement that the vertical components of the slowness vectors of generated waves must be continuous along a parametric line in the complex-valued plane of tangential components of the slowness vector, along which we wish to calculate the R/T coefficients. For example, if we wish to compute the R/T coefficients versus the incidence propagation angle, we choose the parameter along the parametric line to represent the incidence propagation angle. The criteria remove the non-physical jumps of the vertical components of the slowness vectors of generated waves (and, consequently, even the nonphysical jumps in the R/T coefficients), caused by erroneous transition from one sheet of the Riemannian surface to the other.

For the general orientation of the attenuation vector of the incident
wave, the system of six boundary conditions at the interface cannot be
decomposed into two subsystems, corresponding to P-SV and SH waves.
Consequently, all these waves are mutually coupled at structural
interfaces due to viscoelasticity. In general, the R/T coefficients
are not symmetrical with respect to positive and negative incidence
propagation angles. Depending on the behaviour of the tangential
components of the attenuation vector, the amplitudes of generated R/T
waves may decay exponentially away from the interface for certain
incidence propagation angles, and grow exponentially
away from the interface for other incidence angles.
Similarly, the propagation vectors of generated waves may be oriented
towards the interface for certain incidence propagation angles.
Consequently, the R/T propagation angles may exceed 90^{o}.

Viscoelastic media, inhomogeneous waves, reflection, transmission, propagation vector, attenuation vector.

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In: Seismic Waves in Complex 3-D Structures, Report 12, pp. 213-239, Dep. Geophys., Charles Univ., Prague, 2002.