## Representation theorem for viscoelastic waves
with a non-symmetric stiffness matrix

**Ludek Klimes**
### Summary

In an elastic medium,
it was proved that the stiffness tensor is symmetric with respect
to the exchange of the first pair of indices
and the second pair of indices,
but the proof does not apply to a viscoelastic medium.
In this paper, we thus derive the representation theorem
for viscoelastic waves in a medium with a non-symmetric stiffness matrix.
The representation theorem expresses the wave field at a receiver,
situated inside a subset of the definition volume
of the viscoelastodynamic equation,
in terms of the volume integral over the subset
and the surface integral over the boundary of the subset.
For the given medium, we define the complementary medium
corresponding to the transposed stiffness matrix.
We define the frequency-domain complementary Green function
as the frequency-domain Green function in the complementary medium.
We then derive the provisional representation theorem as the relation
between the frequency-domain wave field in the given medium
and the frequency-domain complementary Green function.
This provisional representation theorem yields the reciprocity relation
between the frequency-domain Green function and
the frequency-domain complementary Green function.
The final version of the representation theorem
is then obtained by inserting the reciprocity relation
into the provisional representation theorem.

### Keywords

Anisotropic viscoelastic media, stiffness tensor, wave propagation,
Green function, representation theorem, reciprocity relation.

### Whole paper

The reprint is available in
PDF (97 kB).

*Stud. geophys. geod.*, **65** (2021), 53-58.