Second-order homogenization of boundary and transmission conditions for one-dimensional waves in periodic media

We consider the homogenized boundary and transmission conditions governing the mean-field approximations of 1D waves in finite periodic media within the framework of two-scale analysis. We establish the homogenization ansatz (up to the second order of approximation), for both types of problems, by obtaining the relevant boundary correctors and exposing the enriched boundary and transmission conditions as those of Robin type. Rigorous asymptotic analysis is performed for boundary conditions, while the applicability to transmission conditions is demonstrated via numerical simulations. Within this framework, we also propose an optimized second-order model of the homogenized wave equation for 1D periodic media, that follows more accurately the exact dispersion relationship and generally enhances the performance of second-order approximation. The proposed analysis is applied toward the long-wavelength approximation of waves in finite periodic bilaminates, subject to both boundary and transmission conditions. A set of numerical simulations is included to support the mathematical analysis and illustrate the effectiveness of the homogenization scheme.