  We consider the weighted isotropic relaxed micromorphic model and provide an
in depth investigation of the characteristic dispersion curves when the
constitutive parameters of the model are varied. The weighted relaxed
micromorphic model generalizes the classical relaxed micromorphic model
previously introduced by the authors, since it features the Cartan-Lie
decomposition of the tensors $P_{,t}$ and Curl $P$ in their dev, dev sym, skew
and spheric part. It is shown that the split of the tensor $P_{,t}$ in the
micro-inertia provide an independent control of the cut-offs of the optic
benches. This is crucial for the future calibration of the relaxed micromorphic
model on real band-gap metamaterials.
  Even if the physical interest of the introduction of the split of the tensor
Curl $P$ is less evident than in the previous case, we discuss in detail which
is its effect on the dispersion curves. Finally, we also provide a complete
parametric study involving all the constitutive parameters of the introduced
model, so giving rise to an exhaustive panorama of dispersion curves for the
relaxed micromorphic model.
