Roots of Bessel functions¶
In the theory of vibrations of a circular drum, the displacement of the drumhead can be expressed in terms of pure harmonic modes,
Jm(ωk,mr)cos(mθ)cos(cωk,mt),
where (r,θ) are polar coordinates, 0≤r≤1, t is time, m is a positive integer, c is a material parameter, and Jm is a Bessel function of the first kind. The quantity ωk,m is a resonant frequency and is a positive root of the equation
Jm(ωk,m)=0,
which states that the drumhead is clamped around the rim. Tabulating approximations to the zeros of Bessel functions has occupied countless mathematician-hours throughout the centuries.
From the graph we see roots near 6, 10, 13, 16, and 19. We use nlsolve
from the NLsolve
package to find these roots accurately. (It uses vector variables, so we have to adapt it for use with scalars.)
root estimate | function value |
---|---|
6.380161895923975 | 2.4702462297909733e-15 |
9.761023129981334 | -8.375244942016025e-14 |
13.015200721696122 | 5.048461648726743e-13 |
16.223466160318768 | 6.938893903907228e-18 |
19.40941522643161 | 6.126904539272005e-13 |