One class of single negative acoustic metamaterials

We consider linear elastic complex media the point-body motion of which is described by two vectorial generalized coordinates and the elastic energy contains a term coupling them. All constraints are holonomic and ideal. The dynamics of this continuum is determined by modified Lagrange equations. We specify a class of media whose strain energy depends only on one of the vectorial generalized coordinates, but does not depend on its gradient. We call this generalized coordinate "special." On the partial frequency of the special generalized coordinates under some conditions for the inertial and elastic characteristics, the medium behaves as a system of independent harmonic oscillators, and under other conditions, only the trivial solution exists. It is shown that independently of the nature of generalized coordinates, if the inertial and elastic tensors satisfy certain requirements, there exists a band gap of frequencies for most of the dispersion curves; i.e., the medium is a single negative acoustic metamaterial in this band of frequencies. The partial frequency of the "special" coordinate limits the band gap. For a specific class of parameters, apart from this, there exists a decreasing part of the dispersion curve; i.e., the medium is also a double negative acoustic metamaterial in a certain domain of frequencies.