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 in a classical thermoacoustic regenerator subject to a 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 study 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 non-reciprocal thermoacoustic Bloch waves that are associated with a particular form of unidirectional energy transport. Remarkably, the thermoacoustic coupling also allows achieving effective zero compressibility and zero refractive index that ultimately lead to the phase invariance of the propagating sound waves. This single zero effective property is also shown to have very interesting implications in the attainment of acoustic cloaking.
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