This document contains the slides of the second part of my course at CISM, 2023. It addresses the two-scale, long-wavelength homogenization method for scalar waves in 2D periodically architectured media (typically acoustic or antiplane shear waves in lattices of obstacles / phononic crystals), and the topological optimisation of these media to obtain certain properties regarding wave propagation (typically dispersion and attenuation in chosen directions). It is based on my works on these topics, mostly with Marc Bonnet, Bojan Guzina, Cédric Bellis and Marie Touboul (see references in the last slides). Hope you enjoy, all remarks are welcome. RC, 06/2023