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A spatially periodic structure of heterogeneous elastic rods that periodically oscillate along their axes is proposed as a time-modulated phononic crystal. Each rod is a bi-material cylinder, consisting of periodically distributed slices with significantly different elastic properties. The rods are imbedded in an elastic matrix. Using a plane wave expansion, it is shown that the dispersion equation for sound waves is obtained from the solutions of a quadratic eigenvalue problem over the eigenfrequency ω. The coefficients of the corresponding quadratic polynomial are represented by infinite matrices defined in the space spanned by the reciprocal lattice vectors, where elements depend on the velocity of translation motion of the rods and Bloch vector k. The calculated band structure exhibits both ω and k bandgaps. If a frequency gap overlaps with a momentum gap, a mixed gap is formed. Within a mixed gap, ω and k acquire imaginary parts. A method of analysis of the dispersion equation in complex ω−k space is proposed. As a result of the high elastic contrast between the materials in the bi-material rods, a substantial depth of modulation is achieved, leading to a large gap to midgap ratio for the frequency, momentum, and mixed bandgaps. 

Propagation and attenuation of sound through a layered phononic crystal with viscous constituents is theoretically studied. The Navier–Stokes equation with appropriate boundary conditions is solved and the dispersion relation for sound is obtained for a periodic layered heterogeneous structure where at least one of the constituents is a viscous fluid. Simplified dispersion equations are obtained when the other component of the unit is either elastic solid, viscous fluid, or ideal fluid. The limit of low frequencies when periodic structure homogenizes and the frequencies close to the band edge when propagating Bloch wave becomes a standing wave are considered and enhanced viscous dissipation is calculated. Angular dependence of the attenuation coefficient is analyzed. It is shown that transition from dissipation in the bulk to dissipation in a narrow boundary layer occurs in the region of angles close to normal incidence. Enormously high dissipation is predicted for solid–fluid structure in the region of angles where transmission practically vanishes due to appearance of so-called “transmission zeros,” according to El Hassouani, El Boudouti, Djafari-Rouhani, and Aynaou [Phys. Rev. B 78, 174306 (2008)]. For the case when the unit cell contains a narrow layer of high viscosity fluid, the anomaly related to acoustic manifestation of Borrmann effect is explained. 

It is demonstrated that acoustic transmission through a phononic crystal with anisotropic solid scatterers becomes non-reciprocal if the background fluid is viscous. In an ideal (inviscid) fluid, the transmission along the direction of broken P symmetry is asymmetric. This asymmetry is compatible with reciprocity since time-reversal symmetry ( T symmetry) holds. Viscous losses break T symmetry, adding a non-reciprocal contribution to the transmission coefficient. The non-reciprocal transmission spectra for a phononic crystal of metallic circular cylinders in water are experimentally obtained and analysed. The surfaces of the cylinders were specially processed in order to weakly break P symmetry and increase viscous losses through manipulation of surface features. Subsequently, the non-reciprocal part of transmission is separated from its asymmetric reciprocal part in numerically simulated transmission spectra. The level of non-reciprocity is in agreement with the measure of broken P symmetry. The reported study contradicts commonly accepted opinion that linear dissipation cannot be a reason leading to non-reciprocity. It also opens a way for engineering passive acoustic diodes exploring the natural viscosity of any fluid as a factor leading to non-reciprocity.