In this article, we have studied acoustic transmission theoretically using two analytical methods, the transfer matrix method (TMM) and Sylvester’s theorem. The system studied consists of a combination of two kinds of closed and open resonators grafted perpendicularly in a main waveguide. The aim is to study the propagation of acoustic waves in a perfect periodic system and in a structure with defects. We have studied two different kinds of geometric defects: the first is a variable-length segment inserted without branched resonators, and the second is an opened resonator with large heights. In the analytical procedure, the linear acoustic modulus is taken into account and higher-order modes are ignored. In comparison with experimental works, we have shown that the periodicity of the regular structure formed by the combination of closed and open resonators leads to the creation of large forbidden frequency intervals in which the wave transmission rate is not allowed. In these intervals, the acoustic wave is totally reflected, which means that there is no absorption of the wave in these frequency zones. This makes it easy to design large acoustic/elastic filters to increase acoustic attenuation and solve environmental noise problems. The presence of defects in the middle of the structure also leads to the creation of defect modes in the band gaps used to design acoustic selective filters. These results are entirely consistent with experimental research based on transmission spectra using acoustic resonators systems.