Pole distribution in finite phononic crystals: Understanding Bragg-effects through closed-form system dynamics

Bragg band gaps associated with infinite phononic crystals are predicted using wave dispersion models. This paper departs from the Bloch-wave solution and presents a comprehensive dynamic systems analysis of finite phononic systems. Closed form transfer functions are derived for two systems where phononic effects are achieved by periodic variation of material property and boundary conditions. Using band structures, differences in dispersion characteristics are highlighted and followed by an analytical derivation of the eigenvalues. The latter is used to derive the end-to-end transfer function of a finite phononic crystal as a function of any given parameters. The analysis reveals intriguing features that explain the evolution of Bragg band gaps in the frequency response. It quantifies how the split of eigenvalues into sub- and super-band-gap natural frequencies contribute to band gap formation. The unique distribution of poles allows the closely packed sub-band gap natural frequencies to achieve maximum attenuation in the Bode response. At that point, the impact of the super-band-gap frequencies on the opposing side becomes significant causing the attenuation to fade and the band gap to come to an end. Finally, the effect of splitting the poles further apart is presented in both phononic systems, with material and boundary condition periodicities.