  A recent study [H. Hao and F. Semperlotti, Phys. Rev. B 104, 104303 (2021)]
investigated the dynamic behavior of an infinite one-dimensional (1D)
thermoacoustic waveguide (TAWG) and illustrated its ability to sustain
nonreciprocal and near zero-index sound propagation behavior; these properties
can be very beneficial in the design of acoustic devices, including acoustic
diodes, amplifiers, and cloaks. Nevertheless, it is critical to realize that
when this concept is implemented in a finite-length waveguide, dynamic
instabilities may occur and either drastically reduce or completely hinder the
ability of the TAWG to control and manipulate sound. In this work, we uncover
and investigate the occurrence of either evanescent or intrinsic thermoacoustic
(ITA) modes in a 1D TAWG with anechoic terminations. The stability analysis
clearly distinguishes these two types of evanescent modes and highlights their
different origin rooted in either acoustic or thermoviscous effects. Numerical
results reveal that ITA modes in anechoic-terminated TAWG are strictly
connected to the acoustic-driven evanescent modes, and evolve towards unstable
modes as the TA coupling strength is increased. This study may have important
implications for the practical design of novel acoustic manipulating devices
enabled by TA coupling elements. The conclusions drawn in this study may also
shed lights on the effective suppression of instabilities in TAWGs.
