Vibrational modes of cymbals and Chladni’s law

Abstract

In 1830, Chladni proposed an empirical law relating the modal frequencies of a flat circular plate to the number of nodal diameters m and the number of nodal circles n. This law leads to a mathematical relationship for the modal frequencies: fm,n=C(m+2n)2, where C is an empirical constant. An earlier analysis showed that a modified Chladni’s law fm,n=Cn(m+2n)Pn could be fitted to modal frequencies in a wide variety of flat and nonflat plates, including cymbals. Using electronic holography, as well as scanning the near-field sound, some 300 modes have been observed in a 16-in.-diam cymbal. These data can be fitted to a modified Chladni’s law fm,n=Cn(m+2n)Pn [T. D. Rossing, Am. J. Phys.50 , 271–274 (1982)] or to a similar equation with three parameters fm,n=C(m+bn)P [Perrinet al., J. Sound Vib. 102, 11–19 (1985)], where b is also an empirical constant. Both modifications of Chladni’s law appear to have some advantages in understanding the acoustics of cymbals.