On 3d theory of acoustic metamaterials with a triple-periodic system of interior obstacles

In a three-dimensional (3-D) context an analytical approach is proposed to study the propagation of elastic plane wave through a system of any finite number of parallel doubly-periodic identical gratings of coplanar cracks. In the low frequency range the problem is reduced to a system of integral equations holding over the crack of a chosen elementary rectangular cell of the grating. The semi-analytical method previously introduced for scalar and elastic 2-D problems gives an explicit representations for the wave field and the scattering parameters - the reflection and transmission coefficients.