Acoustics/Bessel Functions and the Kettledrum

Introduction
In class, we have begun to discuss the solutions of multidimensional wave equations. A particularly interesting aspect of these multidimensional solutions are those of bessel functions for circular boundary conditions. The practical application of these solutions is the kettledrum. This page will explore in qualitative and quantitative terms how the kettledrum works. More specifically, the kettledrum will be introduced as a circular membrane and its solution will be discussed with visuals (e.g. visualization of bessel functions, video of kettledrums and audio forms (wav files of kettledrums playing. In addition, links to more information about this material, including references will be included.

The math behind the kettledrum: the brief version
When one looks at how a kettledrum produces sound, one should look no farther than the drum head. The vibration of this circular membrane (and the air in the drum enclosure) is what produces the sound in this instrument. The mathematics behind this vibrating drum are relatively simple. If one looks at a small element of the drum head, it looks exactly like the mathematical model for a vibrating string (see:). The only difference is that there are two dimensions where there are forces on the element, the two dimensions that are planar to the drum. As this is the same situation, we have the same equation, except with another spatial term in the other planar dimension. This allows us to model the drum head using a helmholtz equation. The next step (solved in detail below) is to assume that the displacement of the drum head (in polar coordinates) is a product of two separate functions for theta and r. This allows us to turn the PDE into two ODES which are readily solved and applied to the situation of the kettledrum head. For more info, see below.

The math behind the kettledrum: the derivation
So starting with the trusty general Helmholtz equation:


 * $$\nabla^2\Psi+k^2\Psi=0.$$

Where $$k$$ is the wave number, the frequency of the forced oscillations divided by the speed of sound in the membrane.

Since we are dealing with a circular object, it makes sense to work in polar coordinates (in terms of radius and angle) instead of rectangular coordinates. For polar coordinates the Laplacian term of the Helmholtz relation ($$\nabla^2$$) becomes $$\frac{\partial^2 \Psi}{\partial r^2} + \frac{1}{r} \frac{\partial\Psi}{\partial r} +\frac{1}{r^2} \frac{\partial^2 \Psi}{\partial \theta^2}$$

Using the method of separation of variables (see Reference 3 for more info), we will assume a solution of the form
 * $$\Psi (r,\theta) = R(r) \Theta(\theta).$$

Substituting this result back into our trusty Helmholtz equation, then multiplying through by $$r^2/(R\Theta)$$ gives


 * $$\frac{1}{R} \left(r^2\frac{d^2 R}{dr^2} + r \frac{dR}{dr}\right) + k^2 r^2 = -\frac{1}{\Theta} \frac{d^2 \Theta}{d\theta^2},$$

where we moved the $$\theta$$-dependent terms to the right hand side. Since we separated the variables of the solution into two one-dimensional functions, the partial derivatives become ordinary derivatives. In order for the above equality to hold regardless of changes in $$r$$ and $$\theta$$, both sides must be equal to some constant. For simplicity, I will use $$\lambda^2$$ as this constant. This results in the following two equations:


 * $$ \frac{d^2 \Theta}{d\theta^2} = -\lambda^2 \Theta, $$


 * $$ r^2\frac{d^2 R}{dr^2} + r \frac{dR}{dr} + (k^2 r^2 - \lambda^2) R = 0. $$

The first of these equations readily seen as the standard second order ordinary differential equation which has a harmonic solution of sines and cosines with the frequency based on $$ \lambda $$. The second equation is what is known as Bessel's Equation. The solution to this equation is cryptically called Bessel functions of order $$ \lambda $$ of the first and second kind. These functions, while sounding very intimidating, are simply oscillatory functions of the radius times the wave number. Both sets of functions diminish as $$kr$$ becomes large, but are unbounded as $$kr$$ goes to zero for the Bessel functions of the second kind.

Now that we have the general solution to this equation, we can now model a infinite radius kettledrum head. However, since i have yet to see an infinite kettle drum, we need to constrain this solution of a vibrating membrane to a finite radius. We can do this by applying what we know about our circular membrane: along the edges of the kettledrum, the drum head is attached to the drum. This means that there can be no displacement of the membrane at the termination at the radius of the kettle drum. This boundary condition can be mathematically described as the following:


 * $$ R(a) = 0 $$

Where a is the arbitrary radius of the kettledrum. In addition to this boundary condition, the displacement of the drum head at the center must be finite. This second boundary condition removes the bessel function of the second kind from the solution. This reduces the $$R$$ part of our solution to:


 * $$ R(r) = AJ_{\lambda}(kr) $$

Where $$ J_{\lambda} $$ is a bessel function of the first kind of order $$ \lambda $$. Apply our other boundary condition at the radius of the drum requires that the wave number $$k$$ must have discrete values, ($$j_{mn}/a$$) which can be looked up. Combining all of these gives us our solution to how a drum head behaves (which is the real part of the following):


 * $$ y_{\lambda n}(r,\theta,t) = A_{\lambda n} J_{\lambda n}(k_{\lambda n} r)e^{j \lambda \theta+j w_{\lambda n} t} $$

The math behind the kettledrum: the entire drum
The above derivation is just for the drum head. An actual kettledrum has one side of this circular membrane surrounded by an enclosed cavity. This means that air is compressed in the cavity when the membrane is vibrating, adding more complications to the solution. In mathematical terms, this makes the partial differential equation non-homogeneous or in simpler terms, the right side of the Helmholtz equation does not equal zero. This result requires significantly more derivation, and will not be done here. If the reader cares to know more, these results are discussed in the two books under references 6 and 7.

Sites of interest
As one can see from the derivation above, the kettledrum is very interesting mathematically. However, it also has a rich historical music tradition in various places of the world. As this page's emphasis is on math, there are few links provided below that reference this rich history.


 * A discussion of Persian kettledrums: Kettle drums of Iran and other countries
 * A discussion of kettledrums in classical music: Kettle drum Lit.
 * A massive resource for kettledrum history, construction and technique" Vienna Symphonic Library