BESSELJ

Application:

Vibrating Circular Drumhead

Imagine a perfectly circular drumhead, fixed at its edge. When struck, it vibrates in specific modes. These modes are the standing waves on the drumhead. The amplitude of the displacement of the drumhead at a given point (r,θ) and time t can be described by a solution of the form:

$u(r,\theta ,t)=A{J}_n(k_{nm}r)\cos (n\theta -\alpha )\cos ({\omega }_{nm}t-\beta )$

Where:

• u is the displacement of the drumhead from its equilibrium position.
• A is the maximum amplitude.
• r is the radial distance from the center of the drumhead.
• n is a non-negative integer representing the number of nodal lines (lines of zero displacement) that pass through the center.
• knm​ is a constant related to the frequency of vibration and the size of the drumhead.
• Jn​(x) is the Bessel function of the first kind of order n.
• m is a positive integer representing the number of circular nodal lines.
• ωnm​ is the angular frequency of the vibration.

The values of knm​ are such that Jn​(knm​R)=0, where R is the radius of the drumhead. This condition ensures that the edge of the drumhead is fixed and does not move. The values of x that make Jn​(x)=0 are called the zeros of the Bessel function.

Example: The n=0 (Axially Symmetric) Mode

Let's consider the simplest case where the vibration is perfectly symmetrical around the center (n=0). This means the displacement only depends on the radial distance r, not the angle θ. The solution simplifies to:

$u(r,t)=A{J}_0(k_{0m}r)\cos ({\omega }_{0m}t-\beta )$

The function J0​(x) describes the shape of the drumhead at any given instant. The zeros of J0​(x) determine the circular nodal lines. The first zero of J0​(x) is approximately 2.4048. This means the first circular nodal line occurs at a radius r such that k01​r=2.4048. If the drumhead's radius is R, then k01​R=2.4048. The table below shows the values of J0​(x) for various values of x.

Table of J0​(x) Values for the First Mode (m=1)

This table shows the relative amplitude of the drumhead's displacement along a radius for the fundamental axially symmetric mode. The values are normalized such that the center of the drumhead (x=0) has an amplitude of 1.