Modes of vibration and sound radiation from tuned handbells

TLDR: The tonal quality of a handbell depends upon the clapper strike point, the hardness of the clappers, and the force of the blow in addition to the design and construction of the bell itself as discussed by the authors.

Abstract: Although handbells can vibrate in many different modes, only the two modes of lowest frequency are normally tuned by the bellcrafters. The frequency of the second mode is generally tuned to be three times that of the first. Each of these modes also radiates sound at twice the modal frequency, however, so that the radiated sound spectrum includes the second and sixth harmonics as well as the first and third. The amplitude of the second harmonic partial is proportional to the amplitude of the fundamental, and a similar relationship relates the sixth harmonic to the third. The first and third harmonics are radiated most strongly at right angles to the bell axis; the second and sixth harmonics are maximum along the axis. Small asymmetries in the bell often create warble due to beats between the two components of a split modal doublet. The tonal quality of a handbell depends upon the clapper strike point, the hardness of the clapper, and the force of the blow in addition to the design and construction of the bell itself.