Wave transmission characteristics in periodic media of finite length: multilayers and fiber arrays

Wave transmission characteristics in elastic media that have periodic microstructure over a finite spatial length are examined theoretically as well as numerically. Two classes of such media are demonstrated, namely, one-dimensional multilayered media with finite-length periodicity and two-dimensional composite media with square arrays of aligned fibers within a finite length. From these one-dimensional and two-dimensional analyses, the influence of the finite-length periodicity on the wave transmission characteristics is discussed. In these media, there are frequency bands (stop bands) where the energy transmission coefficient appears to vanish or takes very low values, while in pass bands it oscillates with the frequency due to the finite-length periodicity. It is theoretically demonstrated in the one-dimensional case of multilayers how the frequency intervals of the oscillation in the transmission spectrum depend on the repeating number of the periodic cells as well as other acoustic and geometrical parameters. The results of the two-dimensional fiber arrays, which are obtained numerically by solving the equations of the SH wave multiple scattering, are shown to fit well in the one-dimensional framework of multilayered structures up to a certain frequency encompassing the first stop band. This similarity between two classes of problems is demonstrated by appropriately identifying the one-dimensional reduced transfer matrix for a single cell that is representative of the periodic fiber array.