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

z = fm,n(r, θ, t) = Jm(km,n r) sin(m θ) sin(km,n t)
where m = 0, 1, 2, ... , n = 1, 2, 3, ... and Jm is the Bessel function of order m. The first four Bessel functions are plotted below: J0(r), J1(r), J2(r), and J3(r).

The constant km,n is the nth zero of the mth Bessel function. This choice makes z = 0 at r = 1. The "sin" functions can be replaced by "cos" functions. The following figures show the five modes with the lowest frequency:

 m = 0, n = 1k0,1 = 2.404 m = 1, n = 1k1,1 = 3.832 m = 2, n = 1k2,1 = 5.135 m = 0, n = 2k0,2 = 5.520 m = 3, n = 1k3,1 = 6.379
This Mathematica notebook shows the normal modes.

Superpositions of Modes
The following are various superpositions of the modes

z = a f0,1 + b f1,1 + c f2,1 + d f0,2
each oscillating at the correct frequency. You can change a, b, c, and d with the scrollbar, but click on the link to get a dpgraph with these the initial consants:
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:

z = a f0,1 + J1(k1,1r) (b cos(θ) cos(k1,1t )+ c sin(θ) sin(k1,1t ) ) +d f2,1
with the initial constants a = 0, b = 1.65, c = 1.65, and d = 0.

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 Jm(km,n r). I did not use the Taylor expansions, except for the fact that the expansion has the form Jm(r) = cm rm + cm+2 rm+2 + cm+4 rm+4 + ... . The approximations I used are polynomials p(r) = rm (1 - r2) (a + b r2 + c r4 ). I then used Mathematica to find the best fit for the coefficients a, b, and c. For J0(k0,2 r) I also put in the factor (1-5.27 r2) 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 Jm(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).

Jim Swift's DPGraph page     Jim Swift's home page     Department of Mathematics     NAU
e-mail: Jim.Swift@nau.edu