The modes of vibration of a drumhead are described in polar coordinates (r, θ) by Bessel functions times sines or cosines:

The constant k

m = 0, n = 1 k _{0,1} = 2.404 |
m = 1, n = 1 k _{1,1} = 3.832 |
m = 2, n = 1 k _{2,1} = 5.135 |
m = 0, n = 2 k _{0,2} = 5.520 |
m = 3, n = 1 k _{3,1} = 6.379 |

**Superpositions of Modes**

The following are various superpositions of the modes

a = 0, b = 1.65, c = 1, and d = 0 or a = 0.5, b = c = 0, d = 0.5.

Here is a superposition of some different modes:

**Reference**

My source for this is "Mathematical Physics," by Eugene Butkov: section
9.7, entitled "Bessel Functions."

**Polynomial Approximations of Bessel Functions**

Here's how I got the polynomial approximations for J_{m}(k_{m,n} r).
I did *not* use the Taylor expansions, except for the fact that the expansion has
the form J_{m}(r) = c_{m} r^{m} + c_{m+2} r^{m+2} +
c_{m+4} r^{m+4} + ... .
The approximations I used are polynomials
p(r) = r^{m} (1 - r^{2}) (a + b r^{2} + c r^{4} ).
I then used *Mathematica* to find the best fit for the coefficients a, b, and c.
For J_{0}(k_{0,2} r) I also put in the factor (1-5.27 r^{2})
to put the zero at the correct position.

Most graphs were produced by
DPGraph,
which is a **fast** program for
viewing 3D objects. The plot of the Bessel funtions J_{m}(r) was made
with *Mathematica*, which is a **slow** program for doing just about everything (except implicit 3-D plots, real-time animations, and real-time rotations of 3-D objects: all the things DPGraph does so well).

e-mail: Jim.Swift@nau.edu