.

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

.