## Asymptotic Green's function in homogeneous
anisotropic viscoelastic media

**Vaclav Vavrycuk**
### Summary

An asymptotic Green's function in homogeneous
anisotropic viscoelastic media is derived.
The Green's function in viscoelastic media is
formally similar to that in elastic media, but
its computation is more involved. The stationary
slowness vector is, in general, complex
valued and inhomogeneous. Its computation
involves finding two independent real-valued
unit vectors which specify the directions of
its real and imaginary parts and can be done
either by iterations or by solving a system
of coupled polynomial equations. When the
stationary slowness direction is found, all
quantities standing in the Green's function such
as the slowness vector, polarization vector,
phase and energy velocities and principal
curvatures of the slowness surface can readily
be calculated.

The formulae for the exact and asymptotic
Green's functions are numerically checked
against closed-form solutions for isotropic
and simple anisotropic, elastic and viscoelastic
models. The calculations confirm that the
formulae and developed numerical codes are
correct. The computation of the P-wave Green's
function in two realistic materials with a
rather strong anisotropy and absorption
indicates that the asymptotic Green's function is
accurate at distances greater than several
wavelengths from the source. The error in the
modulus reaches at most 4% at distances greater
than 15 wavelengths from the source.

### Keywords

Anisotropy, attenuation, Green's function, viscoelasticity.

### Whole paper

The reprint is available in
PDF (282 kB).

*Proc. R. Soc. A*, **463** (2007), 2689-2707.

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