Low-frequensy penetration of elastic waves through a periodic array of cracks

Introduction. The investigation of elastic wave penetration through periodic gratings is an important problem in the fields of the ultrasonic quantitative evaluation of materials, sound propagation and electromagnetic waveguides with diaphragms. In practice, analytical results can be obtained under the assumption of low frequency, with a weak interaction regime where some approximated results can be established in an analytical form. Materials and Methods. In the previous papers, the 3-D normal penetration of a wave through a plane screen with an infinite doubly periodic system of cracks with a low frequency assumption and the 2-D normal penetration of a wave through a couple of the systems with an infinite periodic array of cracks are studied. The present study objective is to generalize the results obtained before extending the explicit analytical expressions of he considered coefficients for the system of parallel plane screens based on the context of the in-plane problem for the wave propagation through elastic solids with a periodic array of cracks. Research Results. The present work continues to study the 2-D problem for three such parallel arrays which form a doubly-periodic system. The investigation is devoted to the derivation of the analytic expressions for coefficients of reflection and transmission when a longitudinal plane wave falls on the system of three identical two-dimensional gratings. In the one-mode range, the approximation of the problem is reduced to a system of the hypersingular integral equations whose solution gives these coefficients and an explicit representation of the wave field inside the structure. Discussion and Conclusions. The desired control of the acoustic filtering in the considered grating can be arranged by the appropriate choice of the crack’s length, respective frequency interval, and finally - by the distance between two vertical arrays containing periodic systems of cracks.