Propagation number in periodic structures considering losses

The development of absorbing or reflecting panels is of interest for underwater acoustic stealth. Most of the time, losses are not considered in the structures studied, specifically in the propagation number determination, despite the fact that most materials used, such as polymer matrices, have non negligible viscous losses. So, for a better understanding of the acoustic properties of these structures and to allow their optimization, simulations should consider the material properties, including losses. In order to obtain more realistic results from simulation, two numerical tools based on the finite element method (FEM) are proposed, with the help of the ATILA software. One is based on a differential method, the other on the transfer matrix. The two methods are first validated in the lossless case, then when losses are taken into account. Both methods give results in good agreement and give the propagation number where losses are taken into account.