Complete transmission through a periodically perforated rigid slab

Abstract

The propagation of a normally incident plane acoustic wave through a three-dimensional rigid slab with periodically placed holes is modeled and analyzed. The spacing of the holes $A$ and $B$ , the wavelength $\lambda$ , and the thickness of the slab $L$ are order one parameters compared to the characteristic size $D$ of the holes, which is a small quantity. Scattering matrix techniques are used to derive expressions for the transmission and reflection coefficients of the lowest mode. These expressions depend only on the transmission coefficient, ${\tau }_0$ , of an infinitely long slab with the same configuration. The determination of ${\tau }_0$ requires the solution of an infinite set of algebraic equations. These equations are approximately solved by exploiting the small parameter $D∕\sqrt{AB}$ . Remarkably, this structure is transparent at certain frequencies and opaque for all others. Such a structure may be useful in constructing narrow-band filters and resonators.