Modes of Vibration of the Indian Drum Dugga or Left-Hand Thabala

Abstract

The dugga or the left-hand thabala is a companion drum to the right-hand thabala and the two together are widely used in India. The vibration characteristics of the latter which has a symmetrically loaded membrane were discussed previously where it was shown how this drum produces harmonic overtones. The left-hand thabala also has a loaded drumhead, but the loading is here eccentric. The boundary value problem of this drumhead is not separable, and hence there is no readily available method for computing its characteristic frequencies. An approximate theory for computing the frequencies of the lower modes is here developed under the assumption that for these modes the effect of the (unloaded) annular region on the loaded region may be replaced by a suitably determined elastic support around its boundary. This leads to the boundary value problem of a membrane with an elastic boundary support whose stiffness constant per unit length is a function of the polar angle. The problem is still unseparable, but the eigenvalues can be now computed by a boundary perturbation technique from those of a membrane with a boundary support of constant stiffness. Computed and measured frequency ratios agree within seven percent or better.

Effects of air loading on the acoustics of an Indian musical drum

“…Our interest again is to investigate the effect of air loading on the acoustics of the drum, in particular with respect to making the frequencies more harmonic and sustaining musically important modes longer than others. To be able to compare our results directly with Ramakrishna, 4 we take the geometric parameters as a ¼ 0.028575 m, b ¼ 0.1016 m, and d ¼ 0.01905 m. The frequency ratios are calculated with respect to the frequency of the g o11 mode. The length of the air column is taken to be 0.25 m, as measured from an actual tabla.…”

“…We note that the idealized model for an eccentric composite membrane, as proposed by Ramakrishna 4 and adapted in Sec. II B, is independent of mass density of either the loading or the base membrane.…”

“…On the other hand, a composite membrane with an eccentrically located region of extra mass does not yield harmonic overtones. 4 The inharmonicity of the ideal membrane can alternatively be (partially) removed by incorporating air loading and a suitably sized cavity backing the membrane; this is well understood in the context of kettledrum. 5,6 The present work is motivated by the lack of detailed investigations into the effects of coupling air loading and cavity with the composite membrane, both with central and eccentric mass distribution.…”

The eigenspectra of Indian musical drums

“…An exact solution for the eigenmodes in terms of known functions is not available. Two approximate solutions [4,5] have been presented, but the agreement with experimental values is generally poor. Little, therefore, is known about the eigenspectrum of the eccentrically loaded drum head.…”

“…It is worthwhile comparing our numerical results with two prior approximate analytical calculations. In Table II we compare the eigenvalues as obtained from experiment, an approximate calculation based on the composite membrane model [4], and the present model. The agreement with experimental values is very good, except for eigenmodes which have one nodal circle.…”