A Comparative Analysis of Wave Properties of Finite and Infinite Cascading Arrays of Cracks

Reflection and transmission coefficients in the problems of the normal plane wave incidence on the system of finite and infinite periodic arrays of cracks in an elastic body are determined. We propose a method permitting to solve the scalar diffraction problem for both single crack and any finite number of cracks with arbitrary lattice geometry. Under the condition of one-mode frequency regime the problem is reduced to a discretization of the basic integral equation holding on the boundary of the scatterers located in one horizontal waveguide. A semi-analytical method developed earlier for diffraction problems on infinite periodic crack arrays permits a comparative analysis of the properties of the main external parameters for a finite periodic system of cracks, where the solution of the boundary integral equations is numerically constructed, and we obtain explicit analytical representations for the wave field at the boundary of the obstacles. The analysis of the properties of the scattering coefficients depending on the physical parameters is carried out for three diffraction problems: a finite periodic system in a scalar formulation, an infinite periodic system in a scalar formulation, an infinite periodic system in a plane problem of the elasticity theory.