## Time-averaged and time-dependent energy-related quantities of
harmonic waves in inhomogeneous viscoelastic anisotropic media

**Vlastislav Cerveny** ** & **
**Ivan Psencik**
### Summary

The energy-flux vector and other energy-related
quantities play an important role in various
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 simple
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 viscoelasticity.
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.

### Keywords

Inhomogeneous media, seismic anisotropy, seismic waves, viscoelasticity.

### Whole paper

The reprint is available in
PDF (122 kB).

*Geophys. J. int.*, **170** (2007), 1253-1261.

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