Nonreciprocal and even Willis couplings in periodic thermoacoustic amplifiers

Abstract

Thermoacoustic amplifiers are analyzed in the framework of nonreciprocal Willis coupling. The closed form expressions of the effective properties are derived, showing that an applied temperature gradient causes the appearance of a nonreciprocal Willis coupling. Even and nonreciprocal Willis couplings are exhibited already in the first-order Taylor expansion of the solution and are of equal modulus but opposite sign, thus suggesting that the even Willis coupling is a reaction to the nonreciprocity introduced by the temperature gradients. These Willis couplings cause a coalescence point in the k space, which deviates from Re(k) = 0 (with k the wave number) and is thus a zero-group-velocity point, as well as the opening of an amplification gap at low frequency. Effective parameters and scattering properties are found in excellent agreement with experimental results. This article paves the way to further control the acoustic waves at very low frequencies with nonreciprocal systems.

Non-locality of the Willis coupling in fluid laminates

“…Since the seminal work of Willis in the 80's [1], the eponymous materials have received an increasing attention, because of their analogy with bi-isotropic electromagnetic metamaterials [2]. The Willis coupling parameters couple the potential and kinetic energies in the acoustic conservation relations, therefore enhancing the ability to control waves in asymmetric [3,4,5,6,7,8,9,10] or non-reciprocal [11,12,13,14,15] systems. Willis couplings effectively appear along the diagonal of the propagation matrix, i.e., the Hamiltonian, either with opposite signs to account for the structural asymmetry, i.e., the even coupling, or with identical signs to account for the non-reciprocity and/or nonlocality, i.e., the odd coupling [6].…”