Abstract: This paper takes circular membranes as the research object, and uses the separation of variables method and impulse theorem method to separately study their free vibration and forced vibration problems. The free vibration and forced vibration solutions in the form of Bessel functions are obtained, and the results are visualized and analyzed. Research has found that when a circular membrane vibrates freely, for the same order, the higher the resonant circular frequency, the more wrinkles it will have; The higher the order, the more the number of nodal diameters, and the more diverse the fold morphology. When a circular membrane is forced to vibrate, due to the set external force being independent of the angle, under the conditions of fixed boundaries, no initial displacement, and no initial velocity, the forced vibration solution of the circular membrane that satisfies the initial conditions is also independent of the angle, resulting in a centrally axisymmetric distribution of the vibration shape. At the same circular frequency and different times, the number of folds is basically the same. As time changes, the central wave packet oscillates up and down around the central axis. At the same time, as the circular frequency changes, the vibration pattern of the circular membrane also changes. At the same circle frequency and at the same time, as the root value of the Bessel function in the external force increases, the number of folds in the circular membrane vibration increases.

Key words: circular membrane, forced vibration, separation of variables method, theorem of impulse, Bessel function