Explicit Q expressions for inhomogeneous P- and SV-waves in isotropic viscoelastic media

We derive explicit expressions for the dissipation factors of inhomogeneous P and SV-waves in isotropic viscoelastic media. The Q^-1 values are given as concise and simple functions of material parameters and the wave inhomogeneity parameter using two different definitions. Unlike homogenous waves, inhomogeneous waves may have significant differences in the values of dissipation factors because of different definitions. For example, under one of the three dissipation factor definitions that Q^-1 is equal to the time-averaged dissipated-energy density divided by twice the time-averaged strain-energy density, it is found and proved that the dissipation factor of SV-waves is totally independent of the inhomogeneity parameter. For materials in which P-waves are normally more dissipative than S-waves (e.g. a porous reservoir), the dissipation factors of P-waves tend to decrease with increasing degree of inhomogeneity. Based on Buchan's classic real value energy balance equation, a parallel investigation is conducted for each step similar to that based on the Carcione equations, including derivation of explicit formulas (with inhomogeneity angle representing the degree of inhomogeneity of a plane wave), and dissipation curves calculations. We also obtain an inhomogeneity independent formula of $Q-{SV} { - 1}$, and exactly the same phase velocity and attenuation dispersion results for the example material.