Vibration characteristics of a bi-density drumhead

The dayan tabla is a percussion instrument used as an accompaniment to classical Indian singing or music. In this work, a mathematical model was developed to describe the response of this instrument and to analytically determine the areal density distribution required to obtain overtones that were rational multiples of each other. The model was verified by comparing the exact Bessel function solution of the constant density drum with the results obtained using the approximate method outlined in this paper. The maximum error obtained for the first 10 natural frequencies was less than 1% while the associated mode shapes differed by less than 5%. The bi-density model of the tabla, with a mass ratio of 4.1 and a radius ratio of 0.49, predicted natural frequency ratios of approximately 1.1:2:3 (twice):4 (twice) for the fundamental and the first few overtones. This compares well with previously reported experimental data.