Drumheads

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

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

The constant k_m,n is the n^th zero of the m^th 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 = 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
This Mathematica notebook shows the normal modes.

Superpositions of Modes
The following are various superpositions of the modes

z = a f_0,1 + b f_1,1 + c f_2,1 + d f_0,2 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 f_0,1 + J₁(k_1,1r) (b cos(θ) cos(k_1,1t )+ c sin(θ) sin(k_1,1t ) ) +d f_2,1 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 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 - r²) (a + b r² + c r⁴ ). I then used Mathematica to find the best fit for the coefficients a, b, and c. For J₀(k_0,2 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 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).

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