Analytical results for wave propagation through elastic solids with defects of arbitrary shape periodically distributed

In the context of wave propagation through damaged (elastic) solids, an analytical approach to study the normal penetration of a scalar plane wave into a periodic array of defects with arbitrary shape is developed. Starting from an integral representation of the wave field and the scattering parameters already used in the literature for purely numerical treatments, we apply uniform approximations valid in the assumed one-mode regime so as to derive some auxiliary integral equations independent on frequency. The problem is thus reduced to a 13 x 13 (or 22 x 22) linear system, whose solution leads to explicit analytical formulas for the above field and parameters. Numerical resolution of those integral equations for assigned shapes provide values for some constants, so that several graphs are set up showing comparison with previous and exact (numerical) results.