Silver-mean canonical quasicrystalline-generated phononic waveguides

We investigate propagation of harmonic axial waves in a class of periodic two-phase phononic rods whose elementary cells are designed adopting the quasicrystalline silver mean Fibonacci substitution rule. The stop-/pass-band spectra of this family are studied with the aid of a trace-map formalism which provides a geometrical interpretation of the recursive rule governing traces of the relevant transmission matrices: the traces of two consecutive elementary cells can be represented as a point on a surface defined by an invariant function of the circular frequency, and the recursivity implies the description of an orbit on the surface. We show that, for a sub-class of silver mean-generated waveguides, the orbits predicted by the trace map at specific frequencies are periodic. The configurations for which this occurs, called canonical, are also associated with periodic stop-/pass-band diagrams along the frequency domain. Several types of periodic orbits exist and each corresponds to a self-similar portion of the dynamic spectra whose scaling law can be studied by linearising the trace map in the neighbourhood of the orbit. The obtained results provide both a new piece of theory to better understand the behaviour of classical two-phase composite periodic waveguides and an important advancement towards design and realisation of phononic quasicrystalline-based metamaterials.