Spatially periodic structures exhibit intriguing dynamic characteristics, contributing to their growing applications as phononic crystals, acoustic metamaterials and lightweight lattice materials. A striking feature, employed in many engineering applications, is their filtering effect, whereby waves can propagate only in specific frequency intervals known as pass bands. Other frequency components (stop bands) are spatially attenuated as they propagate through the structure.

This thesis studies nonlinear wave transmission in periodic structures of finite extent in the presence of dissipative forces and externally induced nonlinear forces. Perfectly periodic structures with identical units are considered, as well as nearly periodic structures with small deviations from periodicity extended throughout the structure.

At high amplitudes of motion, nonlinear forces gain significance, generating qualitatively new dynamic phenomena such as supratransmission. Supratransmission is an instability-driven transmission mechanism that occurs when a periodic structure is driven harmonically at one end with a frequency within its stop band. The ensuing enhanced transmission contrasts the vibration isolation characteristic of the same structure operating in the linear regime.

In the context of engineering applications, three factors play a significant role:​ dissipative forces, symmetry-breaking imperfections induced by manufacturing constraints (disorder) and the finite size of the structure. This thesis systematically investigates the influence of these parameters on supratransmission in a onedimensional periodic structure, studying the competition between the effects of dispersion, dissipation, nonlinearity and disorder-borne wave localization (Anderson localization).

We identify the mechanism underlying supratransmission using direct numerical simulations and numerical continuation. Based on this insight, we obtain analytical expressions for the onset of supratransmission for weakly coupled structures using asymptotic analysis. Particularly, we highlight the non-trivial effects of damping on supratransmission in finite structures. We demonstrate that, regardless of the type of nonlinearity, dissipative forces can delay the onset of supratransmission, and high levels of damping can eliminate it.

Given that the spectral contents of transmitted energies fall within the pass band, we expect a competition between supratransmission and Anderson localization. Using direct numerical simulations and continuation techniques, we demonstrate that disorder reduces the transmitted wave energy in the ensemble-average sense. However, the average force threshold required to trigger supratransmission remains unchanged.