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

.