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ON TYMBAL VIBRATIONS IN CICADA
K. Kishan Rao, A. Waheedullah, A. Ahmad
To cite this version:
K. Kishan Rao, A. Waheedullah, A. Ahmad. ON TYMBAL VIBRATIONS IN CICADA. Journal de Physique Colloques, 1990, 51 (C2), pp.C2-885-C2-888. �10.1051/jphyscol:19902206�. �jpa-00230523�
COLLOQUE DE PHYSIQUE
Colloque C2, suppl6ment au n02, Tome 51, F6vrier 1990 ler CongrQs F r a n ~ a i s dlAcoustique 1990
ON TYMBAL VIBRATIONS IN CICADA
K. KISHAN RAO, A. WAHEEDULLAH and A. AHMAD
Biophysics Laboratory, Department of Physics, Nizam College, ~yderabad 500001, India
Abstract: The paper reports the frequencies of the different modes of vibrations of the tymbal in cicada theoretically. The two-dimensional wave equation in elliptical coordinates is set up. Since the tymbal is elliptical the solution of the modified Mathieu's equation is considered. In order to obtain the different frequencies the pulsatance equation is used. The study suggests that elliptic membrane produces vibration of frequencies a m f o r m = 1,2,3,..
...
The theoretical frequencies are in agreement with those determined experimentally by taking sonograms of the sound produced by cicada.1 - INTRODUCTION
The songs of cicadas are produced by a specialized acoustic apparatus and these songs are used for specific communication /I/. The acoustic apparatus in cicada is called tymbal. This is a stretched elastic membrane which when vibrates produces sound. This tymbal is supported by a thick circular rim and a series of ribs on each dorsolateral region of the first abdominal segment. In the interior of the tymbal there is a fibriller muscle and due to contraction of this muscle- the tymbal vibrates. In addition to this there is the tensor muscle which increases the curvature of the tymbal. Idhen the tensor muscle contracts. it causes an inward buckling of the tymbal producing a click. Upon relaxation of this muscle the tymbal returns to its original position and thus produces a second click. The frequency of the double click sound is determined by the natural frequency of the tymbal whereas the natural frequency of the tymbal depends upon the tension in the membrane /2,3/.
Each cicada has two tymbals. These tymbals work synchronously due to simultaneous nervous stimulation of the muscle. In addition to tymbals. there are air sacs. The presence of air sacs allow the tymbal to vibrate with minimum damping.
2 - THEORETICAL CONSIDERATION
Since the boundary of the tymbal is elliptical, we transform the two dimensional wave equation into elliptical coordinates. Consider the two dimensional wave equation.
L
where =
T / e
Here Z is the uniform tension per unit length, is the mass per unit area.
r
is thedisplacement normal to the plane of equilibrium of the membrane. L e t 5 vary sinusoidally with time t, so that
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902206
C2-886 COLLOQUE DE PHYSIQUE
Substituting in equation (1) we get
t
where K: = f' W /
Here K t is a constant depending upon the properties of the medium and the pulsatance of the disturbance in it. Transforminq (3) into elliptical coordinates we get
solving equation (4) by the method of separation of variables, we get
where 'a' is a separation constant. Equations (6) and (5) are called the canonical forms of Mathieu's equatio~and the modified Mathieu's equation. The completion of equation (1) is
TCF,7,t) = ,F0
cmC ~ ~ ( ~ ~ ? I ) C ~ ~ C ~ , V )
CCJS(C%~+
E l u )In order to study the vibrations of the tymbals of cicada.. we are interested in the solution of equation ( 5 ) . Each individual solution of (5) form = 0, 1, 2,
...
corresponds to a different mode of vibration. When q has its appropriate value. the dynamic deformation surface of the membrane and the pulsatance differe for each m. The maximum displacement of the surface for a particular mode is governed by the configuration of and the normal velocity distribution over the membrane at t = 0. Here it is considered to be pulled from the centre into a conoidal shape and released at t = 0.The boundary condition for all
rf
is that 3 = 0 at the clamping rings where'5
=f,.
Thuswe have the pulsatance equations
By fixing
5 , ,
we find the positive value of q from (8) and (9). These are the parametric zeros of the functions Ce (E,,, q) and S% (q,, q). The frequencies of the modes of vibration are obtained as 8 1 1 0 ~ s :Let -al
b' the semi-major and semi-minor axes of the ellipse. The eccentricity e =
.
The first step is to calculate qm, the lowest parametric zero for which (8) and (9) hold. Since ~ o s h a 5 , = l/e, 5, is obtained from the tables of hyperbolic functions.Now consider the pulsatance equation (8) ie.,
Ce, ( 5 , ,?I) =
0We truncate the series C& upto q3
.
This gives a cubic equation in q. The solution of this equation gives the values of q. The frequencies of different modes of vibrations is obtained from the e uation.h
P
6 3 = 2
9
(10)3 - EXPERIMENTAL VERIFICATION
The dimensions 'a' and 'b' of the elliptical tymbal were measured by using a comparator in order to determine the frequencies of different modes or vibration produced by it. Due to low intensity of vibrations. the direct verification of theoretical model is rather difficult. Hence sonogram of cicada songs recorded in the field are used to examine the validity of the theoretical model to some extent /4/.
4 - RESULTS AND DISCUSSION
By taking the semi-major and semi-minor axes of a cicada as a = 0.177 cm and b = 0.063 cm.
we have obtained frequencies of different modes of vibration given in Table 1. The experi- mental frequencies obtained frori the sonograms of cicada songs are also presented in this table. Since the mode due to Se2 has the lowest frequency. we have considered this as the fundamental frequency. Relative frequencies of the other modes are computed with reference to this fundamental frequency. It is seen that the frequencies of the modes corresponding to the functions Se2, Cel, Se4 and Se5
,
Ce4 and Se6 are in agreement with those obtained experimentally. These frequencies do not form harmonic series in contrast to circular membrane. The presence of higher frequency band in the sonogram may be attributed due to the vibrations of air sacs apart from the tymbal vibrations.5 - CONCLUSIONS
1. The vibrations of tymbal are complex in nature.
2. The frequencies of different modes do not form harmonic series 3. Different modes can have the same frequency.
REFERENCES
/1/ Pierce, G.W., The songs of insects, Harward University Press, Massachussets (1948).
/2/ Pringle, J.W.S., J.Exp.Biol.,
3
(1954) 525/3/ Pringle, J.W.S.. Proc. Linn. Soc., London,
167
(1957) 144./4/ Siddiqui, M.A., Studies on Bioacoustics of insects, Ph.D. Thesis, Osmania University, Hyderabad, India (1988).
TABLE-1: THEORETICAL AND EXPERIMENTAL FREQUENCIES OF DIFFERENT MODES OF VIBRATION
Cosine e l l i p t i c (Ce,)
T h e o r e t i c a l
Mode o f F u n c t i o n q m Frequency R e l a t i v e
O s c i l l a t i o n Frequency
Sine E l l i p t i c (Se,)
Experimental
Mode Frequency R e l a t i v e
(Hz) Frequency
Fundarnenta 1 500 1
1 s t o v e r t o n e 1000 2 2nd overtone 4500-6000 9 t o 12
( a band o f f r e q u e n c y )
