A uniqueness theorem for the time-domain elastic-wave scattering in inhomogeneous, anisotropic solids with relaxation

• Published in: The Journal of the Acoustical Society of America 2004-06, Vol.115 (6), p.2711-2715
• Main Author: DE HOOP, Adrianus T
• Format: Article
• Language: English
• Subjects: Acoustics; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Other topics; Physics
• ISSN: 0001-4966; 1520-8524
• DOI: 10.1121/1.1710876

A uniqueness theorem for the (analytic or computational) time-domain modeling of the elastic wave motion in a scattering configuration that consists of inhomogeneous, anisotropic solids with arbitrary relaxation properties, occupying a bounded subdomain in an unbounded homogeneous, isotropic, perfectly elastic embedding, is presented. No direct time-domain uniqueness proof seems to exist for this kind of configuration. As an intermediate step, the one-to-one correspondence between the causal time-domain wavefield components and the constitutive material response functions on the one hand, and their time Laplace-transform counterparts for (a sequence of) real, positive values of the transform parameter on the other hand, seems a necessary tool. It is shown that such an approach leads to simple, explicit, sufficiency conditions on the inertial loss and compliance relaxation tensors describing the solid’s constitutive behavior for uniqueness to hold. In it, the property of causality plays an essential role. In Christensen [Theory of Viscoelasticity—An Introduction (Academic, New York, 1971)] a similar approach is applied to the problem of uniqueness of the elastodynamic initial-/boundary-value problem associated with a viscoelastic object of bounded extent, the surface of which is subject to an admissible set of explicit boundary values. In the scattering configuration of unbounded extent, no explicit boundary values occur and the far-field compressional and shear wave radiation characteristics at “infinity” in the embedding play a key role in the proof.