  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.
