  Multiscale 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 problems.
