Willis couplings in one-dimensional and quasi-one-dimensional acoustic systems

Abstract

Since the seminal work of Willis in the 80’s, the eponymous materials have received an increasing attention, because of their analogy with bi-isotropic/anisotropic electromagnetic metamaterials. The Willis coupling parameters couple the potential and kinetic energy in the acoustic conservation relations, therefore enhancing the ability to control waves in metamaterials compared to other materials that do not exhibit such coupling. In this talk, I will present a general method to derive the closed form expressions of the effective properties, including the Willis coupling, of asymmetric and nonreciprocal one dimensional acoustic systems. This method relies on the Pade’s approximation of the matrix exponential, the latter being nothing but the transfer matrix that may relate the state vectors at both sides of a unit cell. The effective properties of various one-dimensional asymmetric resonant systems are first derived, numerically and experimentally validated, and analyzed. The nonlocal feature of the Willis coupling is then investigated in a simple fluid laminate system. The asymmetric and nonreciprocal Willis couplings are finally analytically derived and discussed in a system constituted of a periodic arrangement of thermoacoustic amplifiers.