1.Introduction
Everyone knows the sound of a drum and you can't tell what is definitely the pitch that matches to a piano key. It is because the sound of a drum is inharmonic which means that the higher frequency is not an integer multiple of a fundamental frequency. The sound of drum is a composite of all its frequency together and transfers through air media to your ears. The drumhead is made of a stretched circular membrane and set into vibration by the impact of a drumstick. You can see the surface of drumhead in Fig 1. The sound of drum is depended on the tension of membrane, the air pressure, the damping, the air volume inside the drum and the hitting point of drumhead. In this project, I develop a model of the drum based on the wave equation to investigate the sound made when hitting different points on a drumhead.
2.Governing Equation
The vibration of a damped circular membrane is governed by the following second order linear differential equation:
where u is the amplitude of vibration of the membrane, r and 
are polar coordinates of membrane, a is the damping factor, c is the speed of a
wave on the membrane.
The initial conditions are given by
With boundary conditions given by
where A and 
are constant, R is the radius of the drumhead,
and 
is the polar
coordinate
of hitting point of the drumhead, 0 <= r <= 1 and 0 <=
<= 2
. Without
loss of generality,
we take = 0
and A=10000.
In order to make the amplitude of vibration more local at hitting point, we
take = 0.05.
3.General Solution of Governing Equation
After using separate variable method to solve the Governing Equation which satisfies the initial and boundary conditions, the general solution is
Now using the orthogonality relationship of cosine, sine and bessel function to derive the coefficient of general solution, then the complete description of the drumhead vibration is provided by the sum
,
With
Where
is the angular velocity of vibrating drumhead and 
is n th roots of order m of bessel function. We let 
,
and solve for 
and 
by
Trapezoidal rule.
4.Procedure to make a drum sound
First, I set the amplitude of sound of drumhead
and the frequency of sound of drumhead
Lastly, we need to calculate a decay time for vibrating drumhead. This is all we need to input to Csound to make a drum sound. Csound is computer language can design a musical instrument.
5.Results
5.1The motion of vibrating drumhead
You can see movie that wave disturbance is traveling on a drumhead shortly after hitting at different points of drumhead by clicking the picture below. This phenomenon you can watch similar waves by releasing drops from an eyedropper into a full teacup. The drumhead is capable of carrying a wave disturbance makes the surface of drumhead vibrate, the adjoining air will be disturbed, too. This creates a sound wave that can travel to your ear. The fact is that striking a drumhead at any given point excites each natural mode in proportion to how much that mode involves motion of that particular point. For instance, the first mode of a drumhead is most efficient excited by striking it in center and as you strike closer and closer to the edge, you will get vibrations involving smaller and smaller proportions of that mode. Therefore different sound are made when hitting at different points of drumhead.
| center(r = 0) | offcenter (r = 0.2) | offcenter (r = 0.4) | offcenter (r = 0.6) | offcenter(r = 0.8) |
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5.2 Sound of drumhead
You can click different points in drum to see what sound is made and all the sound samples are au format.
For audio Plug in only,click here to have more wonderful drum sound.
Email: ma_hkx@stu.ust.hk
@copyright 1998 by Hon Kin.
8 June 1998
