Weak spatial dispersion and the unfolding of wave arrival singularities in the elastodynamic Green’s functions of solids

This paper is concerned with the influence of spatial dispersion on transient acoustic waves in solids, and in particular with how the first onset of spatial dispersion leads to the unfolding of wave arrival singularities into wave trains known as pseudo-wave arrivals. Spatial dispersion is the dependence of wave speed on spatial frequency, and arises when the wavelength approaches the natural length scale of the medium, which might for example be the lattice spacing of a crystal or the repeat distance of a layered solid or fiber composite. In centrosymmetric crystals, which are treated here, the initial onset of spatial dispersion is represented by a correction to the limiting wave speed which is of second order in the spatial frequency, and the augmentation of the wave equation with fourth order spatial derivatives. Using Fourier transform techniques, general formulas are derived here for the line and point force elastodynamic Green’s functions of centrosymmetric anisotropic solids subject to weak spatial dispersion. Numerical examples are provided of isotropic solids exhibiting wave arrival singularities, and the analytic forms are established of the pseudo-arrival unfoldings of the step function, delta function and ramp function arrivals of point force Green’s functions, and the 1∕√τ and −√τ arrivals of line force Green’s functions, where τ is time after arrival.