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Rev. Cubana Fis. 37, 55 (2020) PARA F´
ISICOS Y NO-F´
ISICOS
EXPLORING THE REASONS BEHIND THE CIRCULAR SHAPE OF
DRUMS
EXPLORANDO LAS RAZONES DETR ´
AS DE LA FORMA CIRCULAR DE LOS TAMBORES
P. Nishantha, K.M. Udayanandanb†
a) School of Pure and Applied Physics, Kannur University, Payyanur Campus, Payyanur, Kerala - 670 327, India.
b) Former HoD, Department of Physics, Nehru Arts and Science College, Kanhangad, Kerala - 671 314, India; udayanandan@gmail.com.
†corresponding author
Recibido 12/4/2020; Aceptado 13/5/2020
Many of the drums played all over the world are circular in shape.
There are many practical reasons for the instrument makers to
choose a circular shape for the drums. Apart from practical side,
there are some scientific bases for the circular shape of drums.
This paper investigates the physics behind the circular shape of the
drums.
La mayor´
ıa de los tambores utilizados alrededor del mundo tienen
forma circular. Hay muchas razones pr´
acticas para seleccionar esta
forma. Adem´
as the razones pr´
acticas, hay razones cient´
ıficas de la
forma circular de los tambores. Este art´
ıculo investiga la f´
ısica detr´
as
de esta forma circular.
PACS: Vibrations of membranes and plates (vibraciones de membranas y platos), 43.40.Dx; Music and musical instruments (m´
usica e
instrumentos musicales), 43.75.-z; Drums (tambores), 43.75.Hi.
I. INTRODUCTION
Musical instruments are an integral part of any visual or audio
performance. Among these instruments, drums are used for
producing either music or rhythm [1,2]. Most of the drums
are made with wood and animal skin. It is seen that almost
all drum heads made with animal skin are circular in shape.
The Fig. 1shows an ensemble of chenda, a temple musical
instrument played in Kerala (popularly known as God’s own
country), a small state in the southern part of India [3].
Figure 1. An ensemble of chenda played in Kerala, India.
The reason for the circular shape of drums is an interesting
research problem which has to be addressed by the tools of
Physics. With common reasoning ability one simple reason is
that the shaping of the wood and skin could be easily done
with minimum labor for making a circular form. Another
possible argument regarding the practical side of instrument
making is that, to stretch membrane uniformly on drum
head and to adjust uniform tension on the membrane, the
circular shape is the best one. The question of identification
of shape of drums from their eigenvalue spectrum was put
forward by Kac [4]. Kac and others studied the problem
mathematically by considering two drums with same set of
eigenfrequency spectrum and tried to prove that the drums
have the same shape [5,6]. Later investigations [7] found
that same set of eigenvalues can happen for drums with
different complex shapes also. But, for simple shaped drums
like circular, many information of the geometry of the drum
head can be identified from its eigenfrequency spectrum [8,9].
Both western and Indian drums have circular shapes and our
paper evaluates some of the physics behind the circular shape.
II. EIGENFREQUENCY SPECTRUM OF SOME
DIFFERENT SHAPED DRUMS
To differentiate and identify the characteristics of membranes,
we consider the modes of vibrations of rectangular, equilateral
triangular and circular membranes. Let us represent the
frequency of vibration of different modes as fnm where n,m
are the number of half waves in normal modes of vibration of
membranes in xand ydirection respectively. The frequencies
of different modes of vibration of drums may be different but
their frequency ratio remains the same. Hence the frequency
ratios of all three membranes are found for first 10 modes by
dividing frequency of each mode by the frequency of the first
or fundamental mode.
Rectangular membrane: For a rectangular membrane, the
frequencies of vibration of different modes fnm are given
REVISTA CUBANA DE F´
ISICA, Vol 37, No. 1 (2020) 55 PARA F´
ISICOS Y NO-F´
ISICOS (Ed. E. Altshuler)
by [10]:
fnm =v
2rn2
a2+m2
b2,(1)
where a,bare sides of the membrane and vis the velocity of
sound through the membrane . We calculated the frequencies
and the frequency ratio of modes of rectangular membrane
and the results obtained are given in Table 1. For the
calculation, the sides are chosen to be a=0.08m and b=0.1m.
Since vis taken as constant for all membranes and we are
more interested in the frequency ratio, the value of fnm/vis
tabulated.
Table 1. The frequency ratio of the rectangular membrane.
Mode of vibration Frequency/vfnm/vFrequency ratio
1.1 8.0039 1
1.2 11.7924 1.4733
2.1 13.4629 1.6820
2.2 16.0078 2
1.3 16.2500 2.0302
3.1 19.4052 2.4244
2.3 19.5256 2.4395
1.4 20.9538 2.6179
3.2 21.2500 2.6549
2.4 23.5849 2.9466
Equilateral membrane: The frequency of vibration of an
equilateral triangular membrane is given by [11]:
fnm =v
a√n2+m2+nm.(2)
For an equilateral triangular membrane all sides are equal. For
calculations the sides are assigned the value a=0.08m. The
frequency ratio is given in Table 2.
Table 2. The frequency ratio of the equilateral membrane.
Mode of vibration Frequency/vfnm/vFrequency ratio
1.1 21.6506 1
1.2 33.0718 1.5275
2.1 33.0718 1.5275
2.2 43.3012 2
1.3 45.0693 2.0816
3.1 45.0693 2.0816
2.3 54.4862 2.5166
3.2 54.4862 2.5166
1.4 57.2821 2.6457
4.1 57.2821 2.6457
Circular membrane: The frequency of vibration of a circular
membrane is given by [12]:
fnm =xnmv
2πa.(3)
Here xnm are the roots of Bessel function of order n. Here nand
mrepresents the number of half waves of modes of vibration
in θand rdirection since polar coordinate is used for circular
membrane problem. The radius of the membrane is chosen as
a=0.08m for calculation. The obtained values are given in
Table 3.
Table 3. The frequency ratio of the circular membrane.
Mode of vibration Frequency/vfnm/vFrequency ratio
0.1 4.7866 1
1.1 7.6267 1.5933
2.1 10.2221 2.1355
0.2 10.9874 2.2954
3.1 12.6994 2.6531
1.2 13.9641 2.9173
4.1 15.1041 3.1554
2.2 16.7539 3.5001
0.3 17.2247 3.5985
3.2 19.4287 4.0589
From the Table 1,2and 3the ratio of frequencies of different
membrane gives some important insights. They are:
•It is found that the tenth mode of rectangular membrane
produces 2.9466 times fundamental frequency and for
equilateral triangular membrane same mode produces
a frequency 2.6457 times the fundamental frequency. But
for a circular membrane, the tenth mode produces 4.0589
times the fundamental frequency. This shows that with
same number of modes circular membrane can produce
wide range of frequency than other membranes. So
a player must have to excite less number of modes
to obtain higher frequencies compared other shaped
membranes. This reduces the strain of the player if one
uses the circular shaped drum.
•It is also seen that the frequency set of all membranes
is different. The pitch, tone color and amplitude are
interrelated and all depends on fundamental frequency,
intensity of sound and overtone structure [13]. Hence
the tone color of the sound produced by the drums of
these membranes will be heard differently.
III. ISOPERIMETRIC THEOREM AND ITS EFFECTS
In two dimension, out of many shapes with same perimeter,
circle has the largest area and this is called isoperimetric
theorem [14]. For a drum, if the shape of the membrane
is circular then the area of vibration will be more than
other shapes. This increase the sound intensity or amplitude.
Isoperimetric theorem has deeper effects on the sound
produced by drum which was found by Lord Rayleigh [15].
He studied about membranes of different shapes such as
rectangle, equilateral triangle, circle and many more of same
area and found that the pitch or fundamental frequency of
the deepest tone is smallest for circle. In their paper Z. Lu
and J. M. Rowlett [16] found mathematically that a listener
could identify the corners of a drum . This indicates that the
sound produced by a circular membrane and other shaped
membranes with different number of corners such as rectangle
or triangle will be heard differently. The isoperimetric theorem
and other works [4,17] suggest following ideas. Circular
shaped drums:
•Produce more sound compared to other drums
•Produce low pitched sounds or bass sound
•Can maintain the particular tone quality or timbre.
REVISTA CUBANA DE F´
ISICA, Vol 37, No. 1 (2020) 56 PARA F´
ISICOS Y NO-F´
ISICOS (Ed. E. Altshuler)
IV. SYMMETRY AND SHAPE OF THE DRUM
In our daily life we find many natural objects with beautiful
symmetry such as flowers, leafs and the physics behind the
symmetry of objects is a vast and promising field of research
[18]. If some transformation such as rotation or reflection
is performed on an object with any shape and if the shape
remains unaffected, then the object with that particular shape
is said to be symmetric. For a 2D shape, a line of symmetry is a
line passing through the centre which divides it into identical
halves. As the symmetry of the object increases the number of
lines will also increase. For the rectangle there are two lines
of symmetry, for the equilateral triangle it is three and for the
square the number is four and so on. For circular shape there is
infinite number of lines of symmetry and hence it is the most
symmetrical shape in 2D [19]. From Group theory, the group
formed by circular shape have infinite number of rotation
and reflection symmetry [20]. The symmetry and degeneracy
are interrelated. The circular drum has large number of
degenerate modes represented by sine and cosine solutions
of the circular membrane problem. In real circular drums, the
tampering of the circular symmetry of the rim, application
of varying tension on the membrane and the change in the
thickness of the membrane creates a shift in frequency of the
degenerate modes and the beats produced can be removed by
the player [21]. So the circular symmetry creates the following
effect:
•The same sound is produced by the drum played from
any side of the circular head.
V. CONCLUSIONS
The paper discussed some features of physics behind the
circular shape for the drums. The practical easiness in
construction is one aspect to choose the circular shape for
the drums. The circular shape gives the drum most low pitch
or bass sound compared with other shapes. The particular
symmetry helps in tuning of the drum and even distribution
of the tension on the membrane.
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ISICA, Vol 37, No. 1 (2020) 57 PARA F´
ISICOS Y NO-F´
ISICOS (Ed. E. Altshuler)
