Effect of spatial dispersion on transient acoustic wave propagation in 3D

Spatial dispersion is the variation of wave speed with wavelength. It sets in when the acoustic wavelength approaches the natural scale of length of the medium, which could, for example, be the lattice constant of a crystal, the repeat distance in a superlattice, or the grain size in a granular material. In centrosymmetric media, the first onset of dispersion is accommodated by the introduction of fourth order spatial derivatives into the wave equation. These lead to a correction to the phase velocity which is quadratic in the spatial frequency. This paper treats the effect of spatial dispersion on the point force elastodynamic Green's functions of solids. The effects of dispersion are shown to be most pronounced in the vicinity of wave arrivals. These lose their singular form, and are transformed into wave trains known as quasi-arrivals. The step and ramp function wave arrivals are treated, and it is shown that their unfolded quasi-arrival forms can be expressed in terms of integrals involving the Airy function.