Despite the growing capacity of computer codes, analytical solutions are still of great interest. As a rule, they are based on certain asymptotic approximations. In our work, we use a two-scale asymptotic procedure. Anti-plane shear waves in a layered medium are studied. To clarify the basics of the methodology, we restrict ourselves with a layered membrane. For long-wave case we obtained solutions for periodic and anti-periodic modes. We analyse them in the low- and high-contrast cases. The results obtained can be generalized for complex multiscale heterogeneous media and structures. They are useful for bridging the gap between mathematically rigorous and phenomenological approaches in dynamics of heterogeneous materials. They also can be implicated as benchmarks for numerical modelling.