On Transversely Isotropic Elastic Media with Ellipsoidal Slowness Surfaces

Abstract

We consider inhomogeneous elastic media with ellipsoidal slowness surfaces. We describe all classes of transversely isotropic media for which the sheets associated to each wave mode are ellipsoids. These media have the property that elastic waves in each mode propagate along geodesic segments of certain Riemannian metrics. In particular, we study the intersection of the sheets of the slowness surface. In view of applications to the analysis of propagation of singularities along rays, we give pointwise conditions that guarantee that the sheet of the slowness surface corresponding to a given wave mode is disjoint from the others. We also investigate the smoothness of the associated polarization vectors as functions of position and direction. We employ coordinate and frame-independent methods, suitable to the study of the dynamic inverse boundary problem in elasticity.

On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode

“…(We say that the system for elastodynamics has a disjoint wave mode if the corresponding sheet of the slowness surface is always distinct from the other sheets, or equivalently, if the corresponding eigenvalue of the determinant of the principal symbol is always distinct from the other eigenvalues.) For transverse isotropy the disjoint-mode condition does not appear restrictive; in the case of the 204 transversely isotropic materials listed in [9, pages 39-49] for which elasticity parameters have been computed completely, we find in [10] that each has a disjoint mode.…”

“…We present examples (CM1) and (CM2) of anisotropic elastic media with a disjoint mode in [10]. The classes (CM1) and (CM2) consist exactly of the transversely isotropic elastic media with GWP and a disjoint mode, and so satisfy the conditions of our main result and its corollary if Ω is strictly convex with respect to each of g, g.…”

“…We emphasize that all transversely isotropic elastic materials listed in [9, pages 39-49] have a disjoint mode. (See [10]. )…”

“…This so-called elliptic transverse isotropy has been studied, for example, by Rudzki [35], Helbig [36,37], Payton [38], Chadwick and Norris [39], Burridge, Chadwick, and Norris [40], and Bakker [41]. (See [10] for references on this topic.) In the case of elliptic transverse isotropy the system of differential equations that models the behavior of the corresponding elastic waves is more tractable due to the possibility of decoupling the system into linear, scalar wave equations for each of the wave modes.…”

“…The conditions for (CM2) may be derived from Payton [38, p. 693-695], who shows that the quasi-shear and quasi-compressional sheets of the slowness surface are distinct when ( We observe in [10] that the metrics describing wave propagation for transversely isotropic elastic media with GWP all have the form g −1 = pI +q(k ⊗k), where k is the fiber direction, and conclude in Section 3.4 that for (CM1) and (CM2) two of the five (density and elasticity) parameters are determined by the Dirichlet-to-Neumann map (up to certain factors associated with the boundary rigidity).…”