Boundary Value Problem Using Series of Bessel Functions

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This Demonstration solves a Bessel equation problem of the first kind. The equation is for a thin elastic circular membrane and is governed by the partial differential equation in polar coordinates: ... . Here ... , a function of the coordinates and time, is the vertical displacement and ... , a constant independent of coordinates and time, which is determined by the density and tension in the membrane. The initial conditions are ... and ... , ... . In this example we assume circular symmetry. Thus the ... term can be removed from the equation, yielding the traditional form of Bessel's equation: ... . Using separation of variables with ... and the separation constant ... reduces the problem to two ordinary differential equations: ... , ... . The solution of these ODE equations is done using the techniques outlined in [1] for series solutions of ordinary differential equations. The general solution has the form: ... , ... . The boundary conditions that determine the constants ... , ... , ... , and ... are that ... , meaning that the function vanishes on the perimeter ... . The Bessel function of the first kind, ... , can be expressed by the series ... . Then with ... , ... , ... equal to the zeros of ... , the solution satisfying the boundary conditions is given by ... with ... .