In Search of the Missing Fundamental: by Richard K. Jones
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Vibrating Circular Membranes

Harmonic partials (harmonic overtones) are not inherent in timpani tone. The sound spectra created by timpani are not generated by vibrating columns of air or vibrating strings, but rather by vibrating circular membranes. Vibrating circular membranes do not vibrate with a harmonic series yet they do have an overtone series, it is just not harmonic. Unlike strings or columns of air, which vibrate in one dimension, vibrating circular membranes vibrate in two dimensions simultaneously. Furthermore, the fundamental of a vibrating circular membrane is not very resonant and doesn’t produce a pleasant sound.

A vibrating circular membrane: Modes
(0,1) (0,2) (1,1) (2,1)

The theory of how vibrating membranes function has been of interest to the scientific community for well over two centuries. In the second half of the eighteenth century the noted Swiss mathematician and physicist Leonhard Euler (1707-1783) presented a treatise titled “De motu vibratorio tympanorum” (On the motion of vibrations in drums) to the Berlin Academy on January 22, 1761 and to the Petersburg Academy on May 17, 1762. It was later published in Novi Commentarii academiae scientiarum Petropolitanae in 1766. This is the first known treatise expressly devoted to the science of a vibrating membrane. Later on in the century, the lesser known Italian scientist Giordano Ricatti (1709-1790) published his treatise “Delle vibrationi del tambour” (vibration of the drum) in the Saggi scientifici e letterari dell’Academia di Padova in 1786. According to Peter Zimmerman in his 1996 paper “Zur Theorie der schwingenden Membran bei Leonhard Euler und Giordano Riccati: Erfindung, Nacherfindung, Fama“, these two treatises paved the way for our mathematical understanding of how a vibrating circular membrane functions.1

Perhaps the most visible contribution to the understanding of vibrating circular membranes came from the German physicist and musician, Ernst Florens Friedrich Chladni (1756–1827). One of Chladni’s best-known achievements was inventing a technique to show the various modes of vibration on a mechanical surface now know as the Chladni patterns. Chladni’s technique was first published in 1787 in his book, “Entdeckungen über die Theorie des Klanges” (Discoveries in the Theory of Sound).2 These patterns are still in use today to illustrate the vibrational modes of a mechanical surface.

In the middle of the nineteenth century the German scientist and physicist Herman von Helmholtz worked out an equation for studying physical problems involving both space and time. This equation became know as the Helmholtz Equation and was applied to any problem that dealt with basic shapes. The Helmholtz equation was solved for the circular membrane by the German mathematician Alfred Clebsch (1833-1872)in 1862.3

Chapter two will introduce the theory of how circular vibrating membranes function and how the various modes of vibration contribute to the sound of timpani.