TIME-AVERAGED AND TIME-DEPENDENT ENERGY-RELATED QUANTITIES OF HARMONIC WAVES IN INHOMOGENEOUS VISCOELASTIC ANISOTROPIC MEDIA

Abstract

The energy–flux vector and other energy-related quantities play an important role in vari-ous wave propagation problems. In acoustics and seismology, the main attention has been devoted to the time-averaged energy flux of time-harmonic wavefields propagating in non-dissipative, isotropic and anisotropic media. In this paper, we investigate the energy–flux vector and other energy-related quantities of wavefields propagating in inhomogeneous anisotropic viscoelastic media. These quantities satisfy energy-balance equations, which have, as we show, formally different forms for real-valued wavefields with arbitrary time dependence and for time-harmonic wavefields. In case of time-harmonic wavefields, we study both time-averaged and time-dependent constituents of the energy-related quantities. We show that the energy-balance equations for time-harmonic wavefields can be obtained in two different ways. First, using real-valued wavefields satisfying the real-valued equation of motion and stress–strain relation. Second, using complex-valued wavefields satisfying the complex-valued equation of motion and stress–strain relation. The former approach yields simple results only for particularly sim-ple viscoelastic models, such as the Kelvin–Voigt model. The latter approach is considerably more general and can be applied to viscoelastic models of unrestricted anisotropy and vis-coelasticity. Both approaches, when applied to the Kelvin–Voigt viscoelastic model, yield the same expressions for the time-averaged and time-dependent constituents of all energy-related quantities and the same energy-balance equations. This indicates that the approach based on complex-valued representation of the wavefield may be used for time harmonic waves quite universally. This study also shows importance of joint consideration of time-averaged and time-dependent constituents of the energy-related quantities in some applications.