Understanding Vibration of Membranes

This course covers the comprehensive study of the vibration of membranes, both rectangular and circular, and their applications in various fields. The course begins with an introduction to membranes and their significance in engineering and human anatomy. For instance, the course explains how membranes function in musical instruments, the human ear, and microphones. The course then delves into the mathematical aspects, including the derivation of the equation of motion for the membrane, eigen solutions, and the concept of degenerate modes. The course also explores the free vibration response of the membrane and the method to solve the initial value problem. Furthermore, the course discusses the vibration of circular membranes, the eigenvalue problem, the Laplacian operator in polar coordinates, and the use of Bessel functions to solve the differential equation of motion. The course concludes with a detailed discussion on mode shapes and nodal lines or circles.