  The acoustic dispersion properties of monodimensional waveguide filters can
be assessed by means of the simple prototypical mechanical system made of an
infinite stack of periodic massive blocks, connected to each other by elastic
joints. The linear undamped dynamics of the periodic cell is governed by a two
degree-of-freedom Lagrangian model. The eigenproblem governing the free
propagation of shear and moment waves is solved analytically and the two
dispersion relations are obtained in a suited closed form fashion. Therefore,
the pass and stop bandwidths are conveniently determined in the minimal space
of the independent mechanical parameters. Stop bands in the ultra-low frequency
range are achieved by coupling the stacked material with an elastic half-space
modelled as a Winkler support. A convenient fine approximation of the
dispersion relations is pursued by formulating homogenised micropolar continuum
models. An enhanced continualization approach, employing a proper Maclaurin
approximation of pseudo-differential operators, is adopted to successfully
approximate the acoustic and optical branches of the dispersion spectrum of the
Lagrangian models, both in the absence and in the presence of the elastic
support.
