This document contains the slides of the first part of my course at CISM, 2023. It addresses the two-scale, long-wavelength homogenization method for waves in 1D periodic media (typically acoustic or antiplane shear waves at normal incidence in laminated materials). The dispersive nature of the medium is captured tanks to second-order homogenization, and particular attention is paid to boundaries and interfaces. In the end, a "hybrid" model combining bulk homogenization and boundary/transmission correctors is proposed to simulate waves in bounded periodic media. It is based on my works on these topics, mostly with Bojan Guzina and Bruno Lombard (see references in the last slides). Hope you enjoy, all remarks are welcome. RC, 06/2023