Metamaterials are artificial, engineered materials designed to manipulate wave propagation in ways not possible with conventional materials. An Introduction to Waves in Mechanical Periodic Structures and Metamaterials presents a systematic and rigorous treatment of wave propagation in periodic structures, with a particular focus on mechanical and acoustic metamaterials. This book bridges fundamental concepts and advanced topics, offering a unified framework to analyze discrete and continuous periodic systems. Key topics include:Comprehensive mathematical formulation of wave propagation in 1D and 2D periodic structures In-depth discussion on spectral element formulation, transfer matrix method, and bandgap formationCoverage of advanced concepts such as energy transportation, meta-damping, non-reciprocity, active control, and nonlinear metamaterialsUnique focus on damping evaluation in metamaterials, critical for shock resistance, and transient dynamic applicationsThis book serves as an accessible yet rigorous resource for senior undergraduate and graduate students in structural engineering, mechanical engineering, applied physics, and materials science, as well as for researchers seeking to deepen their understanding of wave mechanics in metamaterials.