Waves and Vibrations

Abstract

This course material presents the fundamental principles of waves and vibrations for undergraduate students in engineering and applied sciences. The objective of the course is to develop a solid theoretical and analytical understanding of oscillatory systems and wave propagation phenomena commonly encountered in physical and engineering applications. The material begins with an introduction to the Lagrangian formalism, providing the theoretical framework required to derive equations of motion for mechanical systems with single and multiple degrees of freedom. It then explores free and forced oscillations, including undamped and damped systems, resonance phenomena, and steady-state responses. The course further investigates multi-degree-of-freedom systems, modal analysis, and coupled oscillators. These concepts are extended to the study of wave propagation, where the wave equation is derived and analyzed in one- dimensional and three-dimensional media. Applications to vibrating strings, elastic waves in solids, and acoustic waves in fluids are presented, emphasizing practical engineering contexts such as structural dynamics, mechanical vibrations, and sound propagation. Overall, this course provides students with the mathematical tools and physical insight necessary to analyze oscillatory systems and wave phenomena in engineering and scientific disciplines.