Perpendicular transmission of acoustic waves between two substrates connected by sub-wavelength pillars

We discuss theoretically the acoustic resonant transmission and zeros of transmission between two substrates connected by sub-wavelength pillars. The features of the transmission coefficient are explained in terms of the coupling of the incident waves with the Fabry-Perot oscillations inside the pillars and with the surface waves of both substrates. We discuss the dependence of the selective and zero transmission frequencies, in particular Fano resonances resulting from the proximity of a resonance to a zero of transmission, on the geometrical and physical parameters of the materials constituting the pillars and the substrate. These phenomena are studied in both one-and two-dimensional periodicities where the substrates are connected by a series of parallel plates or by a square lattice of cylindrical pillars, respectively. Finally, the calculation is extended to a periodic stacking of slabs and pillars that constitute a type of three-dimensional phononic crystal.