Dispersive behavior is studied in detail. It is shown that there are critical contrasts at which different stop-bands disappear, as well as a critical angle at which all stop-bands disappear. This characterization of the higher order stop-bands at various contrasts is vital for a complete understanding of the effective behavior of layered media across the entire frequency spectrum.

The sensitivity of long-wave effective anisotropy with respect to the individual material properties is examined in detail, reinforcing previous research in this area. Long-wave anisotropy is also examined with respect to anisotropic layers, and fractured layered media. The dependence of the dispersive limit on material contrast is demonstrated and a new approach is used to quantify what is meant by long-wave approximation.

Effective wave behavior in periodic media can be both anisotropic and dispersive. These behaviors depend on two factors, the wavelength of the insonifying field and the relative material properties of the individual layers. This research takes a quantitative look at those two factors and asks how they control the anisotropic and dispersive behavior. The fields of research that this work draws from are diverse in scale, from the quantum with respect to solid state physics to solid earth scales in the case of earthquake seismology; however, the problem is very similar.