  The complex band structures calculated using the Extended Plane Wave
Expansion (EPWE) reveal the presence of evanescent modes in periodic systems,
never predicted by the classical \omega(\vec{k}) methods, providing novel
interpretations of several phenomena as well as a complete picture of the
system. In this work we theoretically and experimentally observe that in the
ranges of frequencies where a deaf band is traditionally predicted, an
evanescent mode with the excitable symmetry appears changing drastically the
interpretation of the transmission properties. On the other hand, the
simplicity of the sonic crystals in which only the longitudinal polarization
can be excited, is used to interpret, without loss of generality, the level
repulsion between symmetric and antisymmetric bands in sonic crystals as the
presence of an evanescent mode connecting both repelled bands. These evanescent
modes, obtained using EPWE, explain both the attenuation produced in this range
of frequencies and the transfer of symmetry from one band to the other in good
agreement with both experimental results and multiple scattering predictions.
Thus, the evanescent properties of the periodic system have been revealed
necessary for the design of new acoustic and electromagnetic applications based
on periodicity.
