Vibrations of Indian musical drums regarded as composite membranes

Raman first observed that the overtones of the Indian musical drums, Mridanga and Thabala, form a sequence of five natural harmonics, and that they result from nine modes of vibration, some of which are (approximately) degenerate. After a brief review of the previous work on the subject, a theory of these drums is developed in the present paper on the basis that the drumheads of these instruments can be regarded as circularly symmetric membranes with a radial step discontinuity the density. The eigenvalues and eigenfunctions of such a composite membrane are obtained, and it is then shown from a numerical calculation that for suitable ratios between the densities and radii of the two parts, the frequencies and the modes of vibration are in accordance with Raman's observations. Measured values of the frequencies of the first nine modes confirm the approximate degeneracies and the harmonicity indicated by the theory and establish the adequacy of the composite membrane as a mathematical idealization of these drumheads