  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 $k(\omega)$ problem reveal a complete
interpretation of the propagation properties of these systems. The prediction
of evanescent modes, non predicted by classical $\omega(\vec{k})$ 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.
