  Using transfer-matrix methods, we investigate the response of a multilayered
metamaterial system containing defects to an incident acoustic plane wave at
normal or oblique incidence. The transmission response is composed of
pass-bands with oscillatory behaviour, separated by band gaps and covers a wide
frequency range. The presence of gain and loss in the layers leads to the
emergence of symmetry breaking and re-entrant phases. In the general case, a
system containing defects will display a more general property,
pseudo-Hermiticity (PH), of which $\mathcal{PT}$ systems are a subset. In the
PH-symmetric phase, unidirectional responses of the reflection, accomplished by
reversing the parity $\mathcal{P}$, can be found but the response sometimes
deviates from the predictions of simple scattering theory which call for a
pseudo-unitarity relation relating the transmission and the two directions of
reflections to hold. The converse of reversing the parity, reversing the time
operator $\mathcal{T}$ in a spatially-asymmetric system within the PH-symmetric
regime can lead to different transmissions: a pass-band versus a stop-band. As
regions of stable PH-symmetric pass-band transmission oscillations occur over a
wide spectral range, there is a large flexibility in system parameters such as
layer thicknesses, for leading to the desired unidirectional traits. In
addition, we find that while defects in general lead to a near or complete loss
of PH symmetry at all frequencies, they can be exploited to produce
highly-sensitive responses, making such systems good candidates for sensor
applications.
