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,θ;t)=AJn(kr)sinnθcoskνt, where (r,θ) 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,⋯ and m=1,2,3,⋯ where k is the mth zero of Jn.
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)
pylab.show()
