The scattering of a time harmonic acoustic wave by an array of N identical baffled membranes is considered in the limit as N→∞. The method of matched asymptotic expansions is used to construct an inner and outer expansion; the inner expansion contains the physics of the local periodic structure and the outer expansion has the proper behavior in the far field. The matching process is used to determine a set of unknown coefficients in the analysis and the result is a far-field pattern for the structure that blends the physics of both expansions. In particular, the far-field pattern is highly localized about the Bragg or mode angles of the inner representation. The maximum value of this function is directly proportional to the reflection coefficient determined numerically from the inner problem. Several numerical examples are presented illustrating these features. Moreover, it is observed that the asymptotic theory agrees quite well with the exact answer (obtained by using a finite difference scheme) when N=3.