We study behavior of attenuation (inhomogeneity) angles γ, i.e., angles between real and imaginary parts of the slowness vectors of inhomogeneous plane waves propagating in isotropic or anisotropic, perfectly elastic or viscoelastic, unbounded media. The angle γ never exceeds the boundary attenuation angle γ*. In isotropic viscoelastic media γ* = 90°; in anisotropic viscoelastic media γ* may be greater than, equal to, or less than 90°. Plane waves with γ > γ* do not exist. Because γ* in anisotropic viscoelastic media is usually not known a priori, the commonly used specification of an inhomogeneous plane wave by the attenuation angle γ may lead to serious problems. If γ is chosen close to γ* or even larger, indeterminate, unstable or even nonphysical results are obtained. We study properties of γ* and show that the approach based on the mixed specification of the slowness vector fully avoids the problems mentioned above. The approach allows exact determination of γ* and removes instabilities known from the use of the specification of the slowness vector by γ. For γ = γ*, the approach yields zero phase velocity, i.e., the corresponding wave is a nonpropagating wave mode. The use of the mixed specification leads to the explanation of the deviation of γ* from 90° as a consequence of different orientations of energy-flux and propagation vectors in anisotropic media. The approach is universal; it may be used for isotropic or anisotropic, perfectly elastic or viscoelastic media, and for homogeneous and inhomogeneous waves, including strongly inhomogeneous waves, like evanescent waves.