A simple example of the tunnelling effect in periodic elastic structures

The phenomenon of tunnelling manifests itself in many branches of physics. It is generated by the presence of turning points, where the wave undergoes a transition from cut-on to cut-off and backwards. Despite the enormous interest in wave propagation in periodic media in general, its existence in continuous periodic elastic structures remains obscured. This paper is concerned with demonstrating this effect in, probably, the simplest possible model. The closed-form analytical solutions obtained in the Floquet theory framework and by means of modal analysis are shown to be in perfect agreement with conventional calculations of insertion losses.