On convexity and detachment of innermost wavefront-slowness sheet

Abstract

We have proved that the innermost wavefront-slowness sheet of a Hookean solid is convex, whether or not it is detached from the other sheets. This theorem is valid for the generally anisotropic case, and it is an extension of theorems whose proofs require the detachment of the innermost sheet. Although the Hookean solids that represent most materials encountered in seismology exhibit a detached innermost sheet, the positive definiteness of the elasticity tensor, which is its sole fundamental constraint, allows for the existence of both detached and nondetached sheets. Besides the foundational considerations, the omnipresence of computer methods requires that we investigate cases that, even if not commonly encountered, are within the realm of physical possibility, and can appear as the output of modeling. The theorem proved for a general Hookean solid, has been exemplified using a particular case of transverse isotropy. For that case, it has been shown that the innermost sheet exhibits a polarization of a quasicompressional wave. However, this need not be a general property of that sheet because the presented theorem refers to convexity of the innermost sheet, not to its polarization.