Harmonics from a Drum

Project 2: Bongo Drum Harmonics

Drew Lehe

1/30/2005

1 Introduction

In this project, we analyze the waves emitted from a traditional Western bongo drum.

We use the wave equation to determine the shape of the soundwaves emitted from

the instrument, then we analyze the eigenvalues to determine if the soundwaves are

harmonic.

2 Reduced Wave Equation

The initial conditions for our wave equation are

u(x, y, t0 ) = u0 (x, y)

(1)

ut (x, y, 0) = v0 (x, y)

(2)

for the initial shape, and

for the initial velocity.

The wave equation for a 2-dimensional surface is as follows:

(δ 2 u)/(δt2 ) = a2 (δ 2 u/δx2 + δ 2 u/δy 2 ) = a2 ∆u

(3)

3 Is the Bongo Harmonic?

The short answer to this question is ”no.” The reason the bongo is an aharmonic instrument lies in what a harmonic tone is. All soundwaves contain a shape and when a

harmonic instrument is played, it emits a lot of soundwaves that all follow one shape. If

the waves emitted from it are all even multiples of the original waveform, then it will be

a harmonic tone. A bongo drum’s soundwaves are irregular, inconsistent, and therefore

non-harmonic.

1

4 Bongo Eigenmodes

In this section we take a look at a set of solutions to the wave equation, a hyperbolic

partial differential equation. Below we print out the values of l, the eigenvalues of the

function

∆U = λdU

(4)

>> l = l(1:10)

l =

2.8844

7.3149

7.3344

13.1623

13.1626

15.2082

20.3235

20.3239

24.5461

24.6096

Here MATLAB prints the first ten eigenvalues for us. An eigenvalue in a 3D array is

the axis of rotation upon which the graph is centered. Most of our early eigenvalues are

doubled. The next script below simply converts the eigenvalues into frequencies from a

circular plane (the drum’s membrane).

>> f = sqrt(l)/(2*pi)

f =

0.2703

0.4305

0.4310

0.5774

0.5774

0.6207

0.7175

0.7175

0.7885

0.7895

And next we print a list of the eigenvalues divided by the frequencies. I’ve done this

to check whether the...