Determining the Right Scattering Coefficients for an Interface Between a Composite and Homogeneous Medium

Metamaterials are artificially designed composites with desired properties. These properties are generally achieved by dynamic homogenization. Mostly the homogenization is performed for an infinite periodic composite and the results are applied to finite domains. This approximation that the homogenization of infinite domains could be applied to finite samples is not always true. One such problem is that in a finite setting the composite supports evanescent waves, and the dynamic homogenization for an infinite medium does not account for such evanescent waves.

At an interface between a homogeneous medium and a metamaterial (periodic composite), the displacement and traction continuity need to be satisfied. To satisfy the displacement and traction continuity at the interface, evanescent waves are necessary. For this work, we consider the problem of transmission and reflection of waves from an interface between a periodic bi-layer composite and a homogenous medium with the interface perpendicular to the layers of the laminate. An incoming wave from the homogenous side is incident at the interface and a finite number of reflected and transmitted propagating modes are generated, with an infinite number of evanescent modes. Hence, if the scattering coefficients for an interface between a homogeneous medium and a composite are determined by requiring the satisfaction of displacement and traction continuity using propagating modes only, then the scattering coefficients so obtained will violate the conservation of energy flux.

The question we address in this work is, how to determine the scattering coefficients correctly that satisfy the conservation of energy flux and what appropriate jump conditions should be imposed to obtain these coefficients? First, we consider the second order accurate dispersion relation obtained from the homogenization of a periodic bilayer composite, and then invert it to the space-time domain to obtain a dynamic higher-order pde. The homogenization is done for the High Wavenumber-High Frequency regime so that we can look at wavenumbers close to the edge of the band gaps or the edge of the first Brillouin zone, while also looking at higher frequencies. Proceeding with this pde, we consider an interface between two periodic composites, and determine the jump conditions that need to be satisfied at the interface. Now, the scattering coefficients are obtained by imposing the higher order jump conditions at the interface and then check the conservation of energy flux.

Next, we also perform numerical calculations to test performance of the higher order jump conditions, by comparing the results obtained from solving the wave equation with the ones obtained by solving the higher order pde.