# Engineering Acoustics/Bessel Functions and the Kettledrum

# Abstract

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 of the kettledrum works. More specifically, the kettledrum will
be introduced as a circular membrane and its solution will be discussed
in visual (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.

# What is a kettledrum

A kettledrum is a percussion instrument with a circular drumhead mounted
on a \"kettle-like\" enclosure. When one strikes the drumhead with a
mallet, it vibrates which produces its sound. The pitch of this sound is
determined by the tension of the drumhead, which is precisely tuned
before playing. The sound of the kettldrum (called the Timpani in
classical music) is present in many forms of music from many difference
places of the world.

![](myKettledrum.jpeg "myKettledrum.jpeg")

# The math behind the kettledrum:the brief version

When one looks at how a kettledrum produces sound, one should look no
farther than the drumhead. 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 situation for the 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
drumhead using a helmholtz equation. The next step (solved in detail
below) is to assume that the displacement of the drumhead (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 make 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
$\partial^2 \Psi/ \partial r^2 + 1/r \partial^2\Psi/ \partial r +1/r^2 \partial^2 \Psi /\partial \theta^2$

Now lets assume that$$\Psi (r,\theta) = R(r) \Theta(\theta)$$

This assumption follows the method of separation of variables. (see
Reference 3 for more info) Substituting this result back into our trusty
Helmholtz equation gives the following:

$r^2 / R (d^2 R/dr^2 + 1/r dR/dr) + k^2 r^2 = -1/\Theta d^2 \Theta /d\theta^2$

Since we separated the variables of the solution into two
one-dimensional functions, the partial derivatives become ordinary
derivatives. Both sides of this result must equal the same constant. For
simplicity, I will use $\lambda$ as this constant. This results in the
following two equations:

$d^2 \Theta / d\theta^2 = -\lambda^2 \Theta$

$d^2 R / dr^2 + 1/r dR/dr + (k^2 - \lambda^2 / r^2) R = 0$

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 that
are unbounded at when kr (for the function of the second kind)
approaches zero and diminish as kr get larger. (For more information on
what these functions look like see References 1,2, and 3)

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 drumhead 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 differencial 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

Wikibooks sister cite, references under Timpani: Wikipedia
reference

# References

1.Eric W. Weisstein. \"Bessel Function of the First Kind.\" From
MathWorld---A Wolfram Web Resource.
<http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html>

2.Eric W. Weisstein. \"Bessel Function of the Second Kind.\" From
MathWorld---A Wolfram Web Resource.
<http://mathworld.wolfram.com/BesselFunctionoftheSecondKind.html>

3.Eric W. Weisstein. \"Bessel Function.\" From MathWorld---A Wolfram Web
Resource. <http://mathworld.wolfram.com/BesselFunction.html>

4.Eric W. Weisstein et al. \"Separation of Variables.\" From
MathWorld---A Wolfram Web Resource.
<http://mathworld.wolfram.com/SeparationofVariables.html>

5.Eric W. Weisstein. \"Bessel Differential Equation.\" From
MathWorld---A Wolfram Web Resource.
<http://mathworld.wolfram.com/BesselDifferentialEquation.html>

6\. Kinsler and Frey, \"Fundamentals of Acoustics\", fourth edition,
Wiley & Sons

7\. Haberman, \"Applied Partial Differential Equations\", fourth
edition, Prentice Hall Press

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