Fundamental elastodynamic solutions for anisotropic media with ellipsoidal slowness surfaces

Abstract

When the slowness surface S of an anisotropic elastic medium consists of three concentric ellipsoids, solutions of the displacement equations of motion can be generated from functions satisfying scalar wave equations and the problem of constructing the fundamental, or Green’s, tensor G for an infinite region becomes tractable. This paper has two aims: first, to find all the conditions on the linear elastic moduli under which S is ellipsoidal (that is the union of concentric ellipsoids), and, second, to determine G for each case in which S simplifies in this way. The two stages of the investigation have a key idea in common. The ellipsoidal form of S requires the eigenvalues of the acoustical tensor Q ( n ) to be quadratic forms in the unit vector argument n : at least two of the associated eigenvectors are either constant or linear in n and the squared moduli of the linear eigenvectors are divisors of eigenvalue differences. These algebraic properties provide a classification of media with ellipsoidal slowness surfaces and aid in characterizing the membership of each class. The first stage culminates in four sets of conditions, labelled A, B, C(i) and C(ii): case C(i) is a restriction of transverse isotropy and the others are specializations of orthorhombic symmetry. At the second stage n is replaced by the gradient ∂ with respect to spatial position and polynomials in n become differential operators. The construction of G involves two canonical problems of classical type, an initial-value problem for a scalar wave equation and a potential problem for a pair of ‘ charged ’ ellipsoids. The divisibility property indicated above implies that the ellipsoids are confocals carrying equal and opposite charges and these characteristics render the fundamental solution causal in the sense that the entire disturbance excited by the point impulse begins with the first and ends with the last of the wavefront arrivals. The structures of the fundamental solutions in cases A, B, C(i) and C(ii) are described and the latter solution is shown to reduce to a standard result of Stokes in the degenerate case of isotropy. Mention is also made of a specialization of case B, appropriate to a transversely isotropic medium which is inextensible in the direction of the symmetry axis.