The present study investigates the wavespace of Highly Contrasted Structures (HCS) and Highly Dissipative Structures (HDS) by wave-based models. The Asymptotic Homogenization Method (AHM), exploits the asymptotic Zig-Zag model and homogenization technique to compute the bending wavenumbers via a 6th-order equation. The General Laminate Model (GLM) employs Mindlin’s displacement field to establish displacement-constraint relationships and resolves a quadratic Eigenvalue Problem (EVP) of the dispersion relation. The Wave Finite Element (WFE) scheme formulates the Nonlinear Eigenvalue Problem (NEP) for waves in varying directions and tracks complex wavenumbers using Weighted Wave Assurance Criteria (WWAC). Two approaches are introduced to estimate the Damping Loss Factor (DLF) of HDS, with the average DLF calculated by the modal density at various angles where non-homogeneity is present. Evaluation of robustness and accuracy is made by comparing the wavenumbers and DLF obtained from AHM and GLM with WFE. WFE is finally extended to a sandwich metastructure with a non-homogeneous core, and the Power Input Method (PIM) with Finite Element Method (FEM) data is employed to assess the average DLF, demonstrating an enhanced DLF compared to layered configurations with the same material portion, indicating increased energy dissipation due to the bending-shear coupling effects.