Dispersive effective medium for wave propagation in periodic composites

The problem of wave propagation through a periodic medium is considered. It is assumed that the ratio between the cell size and the shortest wavelength of the initial disturbance is small. Within this regime, an effective medium model of the propagation phenomena is obtained using the Bloch expansion. The effective medium obtained is shown to be dispersive. It is known that a dominant feature of the transient wave train in a periodic medium is that of dispersion. It is shown that our model recovers homogenization formally when the cell size approaches zero. Explicit formula for the constants appearing in the effective medium equation is derived for the case of one dimension. The accuracy of the resulting approximation is assessed in numerical simulations.