Shear wave propagation in a periodically layered medium - an asymptotic theory

The propagation of low frequency or long wavelength disturbances in periodically layered media is considered. An asymptotic series is derived for the frequency of the first branch of the Bloch wave spectrum. The expansion is in dimensionless wavenumber and is developed explicitly for SH waves traveling obliquely through layerings with arbitrary periodic stratification. The first dispersive term is discussed in detail for a two phase medium and numerical results are presented showing that the asymptotic approximation to the dispersion equation can accurately approximate pulse dispersion. The asymptotic theory presented here may be used to define a dynamic effective medium, as opposed to an equivalent static effective medium.