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, tau0, of an
infinitely long slab with the same configuration. The determination of
tau0 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 which could prove useful in narrow band filters and resonators.