Directions of degenerate plane‐wave propagation in orthorhombic media

Abstract

At each direction of propagation in an anisotropic elastic medium, there are three possible plane waves, each with its characteristic phase slowness. Points at which one or more of the sheets of the slowness surface are in contact are “degenerate” and specify directions for which the characteristic equation has repeated roots. For an orthorhomic medium, one with three mutually orthogonal mirror symmetry planes (and no further symmetry, for the purposes of this discussion), the degenerate directions are isolated. For degenerate directions that lie in any of the symmetry planes, the characteristic cubic equation in three variables, i.e., the squares of the three components of the slowness vector, is factorable, and the problem reduces to solving for the intersection of a straight line and a conic. The problem is much more complicated out of the symmetry planes as the characteristic equation is not factorable. However, from the fact that in a degenerate direction, the Christoffel equations' 3 × 3 matrix of coefficients, whose eigenvalues and eigenvectors give the slownesses and polarizations of the three waves, is rank one or less, one finds three linear equations on the squares of the three components of the repeated slowness vector. If the squares of the components are all positive, there is a single degenerate direction, not lying in any symmetry plane, in each symmetric octant, and explicit expressions for all relevant quantities are readily derived; if not, there is no degenerate direction that does not lie in a symmetry plane.