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Multiscale modelling aims to systematically construct macroscale models ofmaterials with fine microscale structure. However, macroscale boundaryconditions are typically not systematically derived, but rely on heuristicarguments, potentially resulting in a macroscale model which fails toadequately capture the behaviour of the microscale system. We derive themacroscale boundary conditions of the macroscale model for longitudinal wavepropagation on a lattice with periodically varying density and elasticity. Wemodel the macroscale dynamics of the microscale Dirichlet, Robin-like,Cauchy-like and mixed boundary value problem. Numerical experiments test thenew methodology. Our method of deriving boundary conditions significantlyimproves the accuracy of the macroscale models. The methodology developed herecan be adapted to a wide range of multiscale wave propagation problems.

Boundary conditions for macroscale waves in an elastic system with microscale heterogeneity