Chen, C.Roberts, A.J.Bunder, J.E.2020-04-082020-04-082018IMA Journal of Applied Mathematics, 2018; 83(3):347-3790272-49601464-3634http://hdl.handle.net/2440/124089Advance Access Publication on 19 March 2018Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments, potentially resulting in a macroscale model which fails to adequately capture the behaviour of the microscale system. We derive the macroscale boundary conditions of the macroscale model for longitudinal wave propagation on a lattice with periodically varying density and elasticity. We model the macroscale dynamics of the microscale Dirichlet, Robin-like, Cauchy-like and mixed boundary value problem. Numerical experiments test the new methodology. Our method of deriving boundary conditions significantly improves the accuracy of the macroscale models. The methodology developed here can be adapted to a wide range of multiscale wave propagation problemsen© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.micro; macro; multiscale; boundary condition; wave; centre manifold; macroscale modelling; heterogeneity; heterogeneous modelsJournal article003009053910.1093/imamat/hxy0040004397080000012-s2.0-85054537175421970Roberts, A.J. [0000-0001-8930-1552]Bunder, J.E. [0000-0001-5355-2288]