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Good Vibrations: Vibrating Circular Membrane and Bessel Functions


In nature, vibrations occur in many forms. If it were not for them, we would not be able to hear anything, not even the beat of a drum. The vibrations of a bound circular membrane, like those of a drumhead, can be modelled with the two-dimensional wave equation, which is a partial differential equation. This partial differential equation is the subject of this report. The solution method for solving this equation required a coordinate transformation from Cartesian to polar coordinates and the use of the method of separation of variables to obtain three ordinary differential equations. The three resultant solutions were then combined to form the product solution of the original partial differential equation. The initial conditions governing the problem were then applied and various plots were produced. A drum was used as a model and the results of the model were compared with the corresponding plots of the vibrational modes.

Authors: Heather Cameron and Dana Mullins

Advisor: Raymond J. Spiteri

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