HARMONIC PROPERTIES OF THE ANNULAR MEMBRANE

Properties of a vibrating annular membrane are analyzed mathematically and numerically to investigate the relative harmonicity of the higher-order modes compared with the fundamental mode. The harmonicity depends on the zeros of a cross-product of Bessel functions. For a ratio of external-to-internal diameter up to about 1.2, the higher modes have frequencies which are very nearly exact integer multiples of the fundamental frequency. For circularly symmetric modes, the ratio of all higher radial mode frequencies to exact harmonic values is less than 1.0006 (a logarithmic cent), a ratio imperceptible to a listener. Angle-dependent modes also give good harmonicity, especially if some lower radial modes are suppressed. Thus the modes of an annular membrane whose external-to-internal diameter ratio is less than 1.2 are much more harmonic than those of the full circular membrane of a conventional drum. The theoretical harmonicity is at least as good as reported for stringed melodic instruments (piano, violin). The analysis suggests characteristics for construction of an annular drum-head which is supported on concentric rims of two cylinders; to sound the drum, a braced hoop might be used to excite radial modes, or strikers with grooved faces might be used to excite angular modes without exciting lower radial modes. © 1979, American Association of Physics Teachers. All rights reserved.