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

Abstract

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.