Osculations of spectral lines in a layered medium

When the dense web of spectral curves for the normal modes of wave propagation in a layered medium are examined carefully, it is often observed that pairs of curves approach each other to the point that they appear to intersect, yet a closer examination reveals that they do not. In the technical literature these are referred to as avoided crossings, curve veering, mode-splitting, double cusps, diabolical points, or osculation points . We consider herein their presence–or absence–in a homogeneous elastic layer underlain by an elastic half-space, and include the two extreme cases of a homogeneous stratum where the half-space is infinitely rigid, and a free plate when it is infinitely flexible. First we show rigorously that the spectral curves for generalized Love waves (SH waves) in a layered medium will not ever intersect, even if in some cases they may come close and osculate. We then demonstrate that the spectral lines for generalized Rayleigh waves (SV-P waves) in a homogeneous stratum will only intersect when Poisson’s ratio is either exactly View the MathML source, or is in some other fractional (but not continuous) ratio below View the MathML source, in which case we provide the intersection points in closed form. We also provide the general conditions that must be satisfied by the SVP spectral lines for double roots to exist in a layered medium, whether they are plates, strata, half-spaces of full spaces. We then posit that intersections of SVP spectral lines (i.e. double roots) can only occur when the layered medium exhibits material symmetry with respect to some horizontal plane. Although at first sight the stratum case might seem to contradict this principle we show that ultimately it does not. Thus, neither a non-symmetric free plate nor a half-space will ever have intersecting lines. By contrast, a symmetric full space does have such intersections, as do also symmetrically layered plates, among which the homogeneous free plate and the homogeneous doubly-clamped plate exhibit an abundance of double roots.