Drum vibrations with Bessel functions

The vibrations of a thin circular membrane stretched across a rigid circular frame (such as a drum head) can be described as normal modes written in terms of Bessel functions: $$ z(r, \theta; t) = A J_n(kr) \sin n\theta \cos k\nu t, $$ where $(r, \theta)$ describes a position in polar co-ordinates with the origin at the centre of the membrane, $t$ is time and $v$ is a constant depending on the tension and surface density of the drum. The modes are labelled by integers $n = 0, 1, \cdots$ and $m = 1, 2, 3, \cdots$ where $k$ is the $m$th zero of $J_n$.

The following program produces a plot of the displacement of the membrane in the $n=3, m=2$ normal mode at time $t=0$.

import numpy as np
from scipy.special import jn, jn_zeros
import pylab

# Allow calculations up to m = mmax
mmax = 5

def displacement(n, m, r, theta):
    Calculate the displacement of the drum membrane at (r, theta; t=0)
    in the normal mode described by integers n >= 0, 0 < m <= mmax.


    # Pick off the mth zero of Bessel function Jn
    k = jn_zeros(n, mmax+1)[m]
    return np.sin(n*theta) * jn(n, r*k)

# Positions on the drum surface are specified in polar co-ordinates
r = np.linspace(0, 1, 100)
theta = np.linspace(0, 2 * np.pi, 100)

# Create arrays of cartesian co-ordinates (x, y) ...
x = np.array([rr*np.cos(theta) for rr in r])
y = np.array([rr*np.sin(theta) for rr in r])
# ... and vertical displacement (z) for the required normal mode at
# time, t = 0
n, m = 3, 2
z = np.array([displacement(n, m, rr, theta) for rr in r])

pylab.contour(x, y, z)

Vibrational normal mode for n=3, m=2 of a circular drum