Negative refraction in a laminate

This work is concerned with the reflection and transmission of waves at a plane interface between a homogeneous elastic half-space and a half-space of elastic material that is periodically laminated. The lamination is always in the direction of the x1-coordinate axis and the displacement is always longitudinal shear, so that the only non-zero displacement component is u3(x1,x2,t). After an initial discussion of Floquet-Bloch waves in the laminated material, brief consideration is given to the reflection-transmission problem, when the interface between the two media is the plane x1=0. Nothing unusual emerges: there are just a single reflected wave and a single transmitted wave, undergoing positive group-velocity refraction. Then, the problem is considered when the interface between the two media is the plane x2=0. The periodic structure of the interface induces an infinite set of reflected waves and an infinite set of transmitted waves. All need to be taken into account, but most decay exponentially away from the interface. It had previously been recognized that, if the incident wave had appropriate frequency and angle of incidence, a propagating transmitted wave would be generated that would undergo negative group-velocity refraction - behaviour usually associated with a metamaterial. It is established by an example in this work that there is, in addition, a propagating transmitted wave with smaller wavelength but larger group velocity that undergoes positive group-velocity refraction. The work concludes with a brief discussion of this finding, including its implications for the utility (or not) of “effective medium” theory.