On one class of theoretically constructed isotropic single negative continuous acoustic metamaterials

We consider a class of complex isotropic linear elastic media. A point-body of the medium is characterized by various generalized co-ordinates. We suppose the principle of the material objectivity to be satisfied. One of the generalized co-ordinates is "special": the strain energy depends on it but does not depend on its gradient. Apart from this, we make some requirements for inertial and elastic characteristics. Such a medium may have resonances or anti-resonances at the partial frequency of the special co-ordinate. Then we consider an isotropic medium whose point-body is characterized by two vectorial generalised co-ordinates, one of them special, coupled to another one ("bearing" co-ordinate). Under some restrictions for inertial and elastic constants there exist a forbidden frequency zone at least for one dispersion branch, limited by the partial frequency of the special co-ordinate. In this zone the medium is an acoustic single metamaterial respectively to this kind of waves. Several examples are considered.