  The in-plane acoustic behavior of non-centrosymmetric lattices having nodes
endowed with mass and gyroscopic inertia and connected by massless ligaments
with asymmetric elastic properties has been analysed through a discrete model
and a continuum micropolar model. In the first case the propagation of harmonic
waves and the dispersion functions have been obtained by the discrete
Floquet-Bloch approach. It is shown that the optical branch departs from a
critical point with vanishing group velocity and for the considered cases this
branch is decreasing for increasing the norm of the wave vector from the long
wave limit. A micropolar continuum model, useful to approximate the discrete
model, has been derived through a continualization method based on a
down-scaling law based on a second-order Taylor expansion of the generalized
macro-displacement field. It is worth noting that the second order elasticity
tensor coupling curvatures and micro-couples turns out to be negative defined
also in the general case of non-centrosymmetric lattice. The eigenvalue problem
governing the harmonic propagation in the micropolar non-centrosymmetric
continuum results in general characterized by a hermitian full matrix that is
exact up to the second order in the wave vector. Examples concerning square and
equilateral triangular lattices and their acoustic properties have been
analysed from both the exact Lagrangian model (within the assumed hypotheses)
and the micropolar approximate model. Finally, in consideration of the negative
definiteness of the second order elastic tensor of the micropolar model, the
loss of strong hyperbolicity of the equation of motion has been investigated.
