3d low-frequensy penetration of elastic waves through a doubly periodic array of cracks

The paper is devoted to the calculation of the reflection and transmission coefficients, when a plane wave is incident on a two-dimensional grating with a periodic array of rectangular cracks in the elastic material in 3-D. In the one-mode frequency range the problem is reduced to a single hipersingular integral equation, which leads to an explicit representation of the wave characteristics for various sizes of the cracks. New singularities along each coordinate of two-dimensional grating have been revealed for the obtained hipersingular integral equation of double singularity.