Mathematical modeling of acoustic wave propagation in elastic materials with multi-periodic hierarchical structure of heterogeneities

Abstract

An asymptotic homogenization procedure is applied for acoustic waves' propagation in heterogeneous elastic medium with multi-periodic hierarchical structure of heterogeneities. We assume the existence of two periodic cells with characteristic sizes l_2 and l_1 respectfully. The ratio ε=l_2/l_1 is considered to be small and of the same order of magnitude as the ratio l_1/L, where L is the macroscopic characteristic size of a system. The solution of the problem will be largely determined by the relation between the wavelength λ and the characteristic sizes of the system. For the case when λ/l_1 ~1 we derive homogenized macroscopic equation for the displacement and obtain approximations to the displacement and frequency. Several illustrative examples are considered to show the effect of peculiarities of the structure on two different spacial scales on energy distribution in propagating wave.