Algebraic degree of a general group-velocity surface

Abstract

Three algebraic surfaces — the slowness surface, the phase-velocity surface, and the group-velocity surface — play fundamental roles in the theory of seismic wave propagation in anisotropic elastic media. While the slowness (sometimes called phase-slowness) and phase-velocity surfaces are fairly simple and their main algebraic properties are well understood, the group-velocity surfaces are extremely complex; they are complex to the extent that even the algebraic degree, [Formula: see text], of a system of polynomials describing the general group-velocity surface is currently unknown, and only the upper bound of the degree [Formula: see text] is available. This paper establishes the exact degree [Formula: see text] of the general group-velocity surface along with two closely related to [Formula: see text] quantities: the maximum number, [Formula: see text], of body waves that may propagate along a ray direction in a homogeneous anisotropic elastic solid [Formula: see text] and the maximum number, [Formula: see text], of isolated, singularity-unrelated cusps of a group-velocity surface [Formula: see text].