Band structure and effective properties of one-dimensional thermoacoustic Bloch waves

We investigate the dispersion characteristics and the effective properties of acoustic waves propagating in a one-dimensional duct equipped with periodic thermoacoustic coupling elements. Each coupling element consists of a classical thermoacoustic regenerator subject to a static spatial temperature gradient. When acoustic waves pass through the regenerator, thermal-to-acoustic energy conversion takes place and can either amplify or attenuate the wave, depending on the direction of propagation of the wave. The presence of the spatial gradient naturally induces a loss of reciprocity. This paper provides a comprehensive theoretical model as well as an in-depth numerical analysis of the band structure and of the propagation properties of this thermoacoustically coupled, tunable, one-dimensional metamaterial. Among the most significant findings, it is shown that the acoustic metamaterial is capable of supporting nonreciprocal thermoacoustic Bloch waves that are associated with a particular form of unidirectional energy transport. The nonreciprocal nature of the waveguide in the long wavelength limit is well understood by seeing the waveguide as an acoustic Willis material. The homogenized material properties following the Willis approach help both the analysis and the interpretation of the waveguide dynamic behavior in selected frequency ranges. Remarkably, the thermoacoustic coupling also allows achieving a zero refractive index that ultimately leads to phase-invariant propagating sound waves. This zero-index property is shown to have very interesting implications to attain acoustic cloaking.