Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as Willis materials

Abstract

As building blocks of acoustic metamaterials, resonant scatterers have demonstrated their ability to modulate the effective fluid parameters, which subsequently possess extreme properties such as negative bulk modulus or negative mass density. Promising applications have been shown such as extraordinary absorption, focusing, and abnormal refraction for instance. However, acoustic waves can be further controlled in Willis materials by harnessing the coupling parameters. In this work, we derive the closed forms of the effective parameters from the transfer matrix in three asymmetric and reciprocal one-dimensional resonant configurations and exhibit the differences in terms of coupling coefficients. The way in which Willis coupling occurs in spatially asymmetric unit cells is highlighted. In addition, the analysis shows the absence of odd Willis coupling for reciprocal configurations. These effective parameters are validated against experimental and numerical results in the three configurations. This article paves the way of a novel physical understanding and engineering use of Willis acoustic materials.

Nonreciprocal and even Willis couplings in periodic thermoacoustic amplifiers

“…The expressions of the effective parameters then reduce to those provided in Ref. [30]. In addition, for symmetric systems t 11 = t 22 and as a consequence χ a e = 0.…”

“…Following Ref. [30] the propagation matrix is correctly approximated by the inversion of the first-order Padé's approximation of the matrix exponential,…”

“…Second, the thermoacoustic amplifier appears as a Willis material and exhibits both even and nonreciprocal coupling parameters at the first-order Taylor expansion. This constitutes a major difference with respect to laminate or resonant Willis materials, which require second-order Taylor expansion to exhibit Willis coupling [26,30].…”

“…Fourth, when Ḡ2 and Ḡ4 are both equal to zero, the effective parameters collapse to those of a usual laminate fluid in the first-order expansion, with χ a e = χ nr e = 0. Note that the Willis even coupling term does not vanish in that case when second-order expansions are used because the unit cell involves different materials and is asymmetric [30].…”

“…In this article we derive the closed form expressions of the effective properties of a one-dimensional periodic arrangement of a thermoacoustic amplifier following Ref. [30]. The procedure relies on the Padé's approximation of the total transfer matrix that links the state vectors at both sides of the unit cell and directly provides both even and nonreciprocal Willis coupling terms.…”