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Using transfer-matrix methods, we investigate the response of a multilayeredmetamaterial system containing defects to an incident acoustic plane wave atnormal or oblique incidence. The transmission response is composed ofpass-bands with oscillatory behaviour, separated by band gaps and covers a widefrequency range. The presence of gain and loss in the layers leads to theemergence of symmetry breaking and re-entrant phases. In the general case, asystem containing defects will display a more general property,pseudo-Hermiticity (PH), of which $\mathcal{PT}$ systems are a subset. In thePH-symmetric phase, unidirectional responses of the reflection, accomplished byreversing the parity $\mathcal{P}$, can be found but the response sometimesdeviates from the predictions of simple scattering theory which call for apseudo-unitarity relation relating the transmission and the two directions ofreflections to hold. The converse of reversing the parity, reversing the timeoperator $\mathcal{T}$ in a spatially-asymmetric system within the PH-symmetricregime can lead to different transmissions: a pass-band versus a stop-band. Asregions of stable PH-symmetric pass-band transmission oscillations occur over awide spectral range, there is a large flexibility in system parameters such aslayer thicknesses, for leading to the desired unidirectional traits. Inaddition, we find that while defects in general lead to a near or complete lossof PH symmetry at all frequencies, they can be exploited to producehighly-sensitive responses, making such systems good candidates for sensorapplications.

Acoustic response in a a one-dimensional layered pseudo-Hermitian metamaterial containing defects