Upper and lower bounds of low-frequency band gaps

Anton Krynkin

It is known that periodic structures (PS) exhibit frequency intervals where sound waves do not propagate (i.e. band gaps, BG). The existence of the band gaps is attributed to the periodicity (i.e. the Bragg BG) and the properties of a periodic element. The later substantially enhances the performance of the PS if it supports resonances that generate additional BG (resonant BG). By changing geometrical and physical properties of the periodic element the resonances can be observed below the first Bragg BG (associated with the half of sound wavelength) that makes PS effective treatment in the low frequency regime. The aim of this paper is to approximate the limiting frequencies (lower and upper bounds) of the low-frequency resonant BGs. The PS is represented by an array of thin elastic shells exhibiting multiple low-frequency resonances. In the vicinity of lower bound the approximation is found by means of the Rayleigh Identity which leads to the Foldy- type equation. The upper bound of the resonant BG is approximated with the help of matched asymptotic expansions. This gives an accurate approximation for the upper bound approaching the first Bragg BG where the contribution of higher modes has to be taken into account.

Mots clés: Bloch-Floquet theorem; Periodic; Band gaps; Matched Asymptotic Expansions

Domaines: Acoustique [physics.class-ph]

Anton Krynkin. Upper and lower bounds of low-frequency band gaps. Acoustics 2012, Apr 2012, Nantes, France. ⟨hal-00810790⟩