ESM: (A) auxiliary proofs; (B) application to polyatomic chains; (C) comparison with previously published results, and (D) effective description of nearby dispersion branches from A rational framework for dynamic homogenization at finite wavelengths and frequencies. 13 August 2018 12 February 2019

Appendix A - proofs of the featured lemmas and theorems; Appendix B - specialization of the mathematical framework to polyatomic chains; Appendix C - comparison of the numerical results with previously published data, and Appendix D - Detailed numerical analysis of nearby dispersion branches.