This work theoretically and experimentally reports the evanescent connections between propagating bands in periodic acoustic materials. The complex band structures obtained by solving for the k(ω) problem reveal a complete interpretation of the propagation properties of these systems. The prediction of evanescent modes, non predicted by classical ω(~k) methods, is of interest for the understanding of these propagation properties. Complex band structures provide an interpretation of the evanescent coupling and the level repulsion states showing the possibility to control of evanescent waves in periodic materials.