Wave arrival singularities at cuspidal points in the acoustic wave surfaces of anisotropic solids and their unfolding under weak spatial dispersion

Abstract

This paper is concerned with wave arrival singularities in the elastodynamic Green's functions of infinite anisotropic elastic solids, and their unfolding into smooth wave trains, known as quasi-arrivals, through spatial dispersion. The wave arrivals treated here are those occurring in (i) the displacement response to a suddenly applied point force or three-dimensional Green's function,, and (ii) the displacement response to an impulsive line force or two-dimensional Green's function,. These arrivals take on various analytical forms, including step function and logarithmic and power-law divergences. They travel outwards from the source at the group velocities in each direction, and their locus defines the three- and two-dimensional acoustic wave surfaces, respectively. The main focus of this paper is on the form of the wave arrivals in the neighbourhood of cuspidal points in the wave surfaces, and how these arrivals unfold into quasi-arrivals under the first onset of spatial dispersion. This regime of weak spatial dispersion, where the acoustic wavelength,λ, begins to approach the natural length scale,l, of the medium, is characterized by a correction to the phase velocity, which is quadratic in the wavevector,k, and the presence of fourth-order spatial derivatives of the displacement field in the wave equation. Integral expressions are established for the quasi-arrivals near to cuspidal points, involving the Airy function in the case ofand the Scorer function in the case of. Numerical results are presented, illustrating the oscillatory nature of the quasi-arrivals and the interference effects that occur near to cuspidal points in the wave surface.