Effective wave transmission at interfaces between metamaterials.

Generally, dynamic homogenization techniques are performed for infinite periodic composites and the results are applied to finite domains. This approximation is not always true. At an interface between 2 composites, the displacement and traction continuity need to be satisfied, and evanescent waves play an important role. Without the evanescent waves, the displacement and traction continuity conditions are not satisfied exactly. We consider the problem of transmission and reflection of waves from an interface between a bilayer composite and an homogenous medium. Our aim is to determine the scattering coefficients that satisfy the conservation of energy flux and find the appropriate jump conditions to be satisfied at the interface. We use a rational function approximation of the exact dispersion relation, and then invert it to the space-time domain to obtain a dynamic higher-order pde and interface jump conditions. The scattering coefficients are obtained by imposing the higher order jump conditions at the interface. Numerical calculations are performed to test performance of the higher order jump conditions, by comparing with the results obtained from solving the exact wave equation. We present results for both 1d and 2d cases.