Anisotropy and dispersion in periodically layered media

Abstract

Elastic waves propagating in a periodically layered medium exhibit transverse isotropy, provided the wavelength is long compared to the spatial period of the layer sequence. The elastic constants of the long‐wave equivalent transversely isotropic medium can be calculated in several ways, all of which are based on the low‐frequency limit: one either determines from the outset the macroscopic (in average homogeneous) strain response of a representative block of the periodically layered medium to (in average homogeneous) stresses, or one determines the dispersion equation for such media and evaluates this equation for low frequencies. Both approaches yield the same replacement medium. The replacement of a layered medium by a homogeneous transversely isotropic medium is justified if all wavelengths are sufficiently long. High‐resolution techniques, the increasing use of shear waves, and attention to stratigraphic detail require a quantitative evaluation of what is sufficiently long, as well as a study of what happens for wavelengths between that limit and the limit of resolution. Such information can be obtained through a numerical evaluation of the general dispersion equation: one obtains the frequency as a function of the spatial wave vector k. The phase velocity is the v=ωk/(k⋅k) and the group velocity [Formula: see text]. In general, both dispersion and anisotropy are to be expected. This method has been applied to SH-waves in periodically layered media in general, and the dispersion equation has been evaluated numerically for two representative media. The most significant result is that the long‐wavelengths approach, i.e., a nondispersive transversely isotropic replacement medium, is strictly valid for wavelengths larger than three times the spatial period of layering. For small angles against the vertical, dispersion for shorter wavelength is significant. However, for directions making an angle of more than about 30 degrees with the vertical, dispersion sets in at much shorter wavelengths and is in general much more gentle.