Calculation displacement Vibration of Circular Membranes and Insulated Circular Disk Temperature by Using Bessel Functions

Abstract

This study investigates the mathematical formulation and physical applications of the Bessel differential equation in analyzing the displacement of vibrating circular membranes and the temperature distribution in insulated circular disks. The Bessel equation, a second-order linear differential equation with variable coefficients, naturally emerges in problems exhibiting cylindrical or radial symmetry in applied mathematics and engineering. The research begins by deriving the Bessel equation using the Frobenius series method and then explores the Bessel functions of the first and second kinds, emphasizing their orthogonality, oscillatory nature, and significance in modeling wave and heat phenomena.

To solve practical problems involving circular geometry, the study employs the Fourier–Bessel series expansion, which extends classical Fourier analysis to circular domains. This method enables the determination of the membrane’s displacement as a function of radial position and time, as well as the calculation of temperature variations within a circular disk subjected to thermal insulation. Through separation of variables, both the wave equation and the heat equation are reduced to Bessel-type forms, whose solutions satisfy the required boundary and initial conditions.

The findings demonstrate that the Bessel function framework provides accurate analytical solutions to complex two-dimensional physical problems. These solutions are highly relevant to fields such as mechanical vibration analysis, thermal engineering, acoustics, and electromagnetic modeling, where cylindrical symmetry plays a critical role in predicting physical behavior.