Harmonic waves in the simplest reduced Kelvin’s and gyrostatic media under an external body follower torque

We consider two types of homogeneous reduced elastic media with finite dynamic spin. One is Kelvin's medium, whose particles have axial symmetry and may freely rotate about their axes. Its equations are analogous to those of ferromagnetic insulators near the state of magnetic saturation. Another one is a gyrostatic medium, consisting of spherical body-points containing rotors. Gyroscopic term prevails in the dynamics of both continua. Axes in the reference configuration are collinear. We consider reduced media, whose elastic energy does not depend on the gradient of turn, under the follower body torque, proportional to the rotational displacement. We obtain that there exist polarised shear harmonic waves. The only exception is for the wave vector orthogonal to the axes: in this case we have two shear plane waves, one has a band gap and another one is classical. For the direction of wave propagation parallel to the unit vector, polarisation is circular and does not depend on frequency. In all other cases, polarisation is elliptic and depends on the frequency and on the direction of the wave propagation relatively to the axes. If the external follower torque is positive (reduces the rotational displacement of the body-point), one of the polarised waves has a band gap and another one not. Changing the external loading, we change the band gap. If we apply a negative torque, depending on the direction of the wave propagation and parameters we may obtain band gaps for both branches and a decreasing part of dispersion curve for one of them just above the cut-off frequency. Thus this media behave as smart acoustic metamaterials, single or double negative, depending on the wave polarisation, external torque, parameters and the frequency domain.