Prediction of sound transmission through plates using spectral Gaussian basis functions and application to plates with periodic acoustic black holes

In this study, we present a wave-based model for investigating sound transmission through infinite plates featuring periodically embedded acoustic black hole (ABH) cells. The model represents the transverse motion of the structure and its radiated acoustic pressures as a sum of harmonic components. The transverse displacement of the structure is periodic and modelled using spectral Gaussian basis functions. These basis functions then relate each displacement component to the periodic cells’ degrees of freedom. Expressions for the time-averaged kinetic and potential energy, as well as the external work done by the acoustic pressures, are derived. Subsequently, these energy formulations are used to establish the equation of motion of the system to determine its response to acoustic excitation. To validate the model and demonstrate its usefulness, two numerical examples of periodic plates with embedded rectangular cells, including those with flexible segments (FS) of constant thickness and ABH thickness profiles, are presented. The dispersion curves and sound transmission loss of the periodic plates with FS are calculated using the proposed model and verified by comparing the results with a well-established wave and finite element (WFE) method. The proposed model is further applied to calculate sound transmission through the ABH plate with a complex cross-section profile. The wave modes and vibration patterns induced by acoustic waves acting on the periodic plates are also analysed. It is found that the periodic plate with a thin FS demonstrates sound insulation capabilities comparable to, or even superior to, those of the ABH plate. The periodic plates’ low- and mid-frequency transmission loss is primarily influenced by global modes, whereas local modes predominantly govern the high-frequency performance. The findings from this study provide valuable insights for designing lightweight plates that effectively reduce sound transmission, particularly at and above coincidence frequencies.