Unfolding of wave-arrival singularities in the elastodynamic Green’s functions of anisotropic solids under weak spatial dispersion

Abstract

In this paper we examine the effects on transient wave propagation of weak spatial dispersion, i.e., the first onset of dispersion as the acoustic wavelength approaches the natural scale of length of a medium. In centrosymmetric solids, and also away from acoustic axes in noncentrosymmetric crystals, this takes the form of a correction to the phase velocity which is quadratic in the wave vector $\mathbf{k}$. Our analysis is developed for centrosymmetric solids for which weak spatial dispersion expresses itself through the presence of fourth order spatial derivatives of the displacement field in the wave equation. We examine the effect that spatial dispersion has on wave-arrival singularities in the elastodynamic Green's functions of anisotropic solids. These are singular features that propagate at the group velocities in each direction, and are interpreted on the basis of the stationary phase approximation. The focus of this paper is on wave arrivals associated with elliptic and hyperbolic points on the acoustic slowness surface. In the response to a suddenly applied point force, the wave arrival pertaining to an elliptical point takes the form of a step function, or where symmetry intervenes, a ramp function. The wave arrival for a hyperbolic point takes the form of a logarithmic singularity, or where symmetry intervenes, a softer logarithmic singularity. We report how, under the influence of weak spatial dispersion, these arrivals unfold into wave trains known as quasiarrivals, which conform to integral expressions involving the Airy function. For positive dispersion, i.e., the more common situation of downward curving dispersion relation, $\ensuremath{\omega}(\mathbf{k})$, the ripples of the wave train follow the arrival, whereas for negative dispersion they precede the arrival. The results presented here are of relevance in the broad context of transient acoustic waves in any situation where the characteristic wavelength approaches the natural scale of length of the medium, whether that be the interatomic spacing in a crystal, the repeat distance in a layered solid, or whatever. They are expected to be of particular interest in the topical field of picosecond laser ultrasonics.