In this thesis, elastic wave propagation in metamaterials and strongly scattered random media is studied in two dimensions.

Based on dispersed-inclusion geometry, an effective medium theory (EMT) is developed in the long-wavelength limit for certain elastic metamaterials, which are composed of isotropic scatterers embedded in a solid host. The effective bulk modulus, shear modulus and mass density determined by the EMT can turn negative near resonances with properly chosen resonant scatterers. The EMT is capable of predicting both the dispersion relations in the case of triangular lattices and the angle-averaged dispersion relations in the case of square lattices.

By using the multiple-scattering theory (MST) method and the radiative transfer equation (RTE), the elastic wave propagation through a strongly scattered random media is numerically studied. The equilibration of the shear and compressional energy densities due to the mode conversions is clearly observed in both calculations. However, the ratio of the shear energy density to the compressional energy density obtained from MST is higher than that obtained from the RTE, which has the value predicted by the principle of the equipartition of wave modes. The discrepancy is due to a negative interference energy inside the sample, which contradicts the common belief that the interference energy should always average to zero inside a random medium. The negative interference energy occurs close to the boundary of the cylinder and therefore cannot be averaged to zero. Various distribution functions of the transmitted fields before the establishment of the energy equilibration are studied. They are described by a random-phasor-sum model and exhibit crossover behavior from ballistic to diffusive transport.