In this study, the performance of an acoustic metamaterial based on a membrane-coupled dual-Helmholtz resonator is investigated. This metamaterial has the potential to be applied in domains such as ventilation systems or household appliances, such as vacuum cleaners. A key feature of the proposed structure is that it attenuates sound without posing an obstruction to the airflow path. The main innovation of this research lies in investigating the direct correlation between the membrane mode shape, the volume variation of the Helmholtz resonator cavity, and the generation of sound transmission loss (STL) peaks within a dual-Helmholtz coupled membrane structure. Simulations were carried out using COMSOL software in two stages, consisting of a simplified model followed by a unit layer of the proposed structure. The results demonstrated that the peaks in the transmission loss curve predominantly occur at mode frequencies where the deformation of the membrane induces a volume change in the Helmholtz cavity; three effective modes of this type were identified in the simplified model, and five in the proposed structure. The frequency discrepancies between the STL peaks and their corresponding eigenfrequencies across the two stages were approximately 2% and 1.5%, respectively. Furthermore, calculating 55 eigenfrequencies in the range of 0 to 2000 Hz took 9 seconds, whereas the frequency-domain analysis with a 1 Hz step required approximately 630 seconds. Consequently, the eigenfrequency analysis provided acceptable accuracy for the targeted identification of sound transmission loss peaks while reducing computation time by roughly 70-fold.