Ernst Chladni's Researches in Acoustics

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Ernst Chladni's Researches in Acoustics

Overview

Ernst Chladni, an amateur musician and inventor of musical instruments, studied the production of sound by vibrating solid objects, particularly solid plates. Using sand sprinkled on the plates, he was able to discover the nodal lines associated with the different modes of vibration. The intricate patterns of curved lines, called Chladni figures, generated much popular interest and stimulated mathematical research that would have important implications for the physical sciences and engineering.

Background

The ancient Greeks understood that sounds originated in the vibrations of solid bodies. The Greek mathematician Pythagoras (c. 580-500 b.c.) and his disciples knew that strings (under equal tension) with lengths in the ratio of small whole numbers produced combinations of sounds pleasing to the ear. Around the year 1700 the French mathematician Joseph Sauveur (1653-1716) demonstrated that in its simplest vibrations the string would exhibit nodes (points of no movement) and antinodes (points of maximum movement) at equally spaced locations. For the lowest frequency motion, nodes only occurred at the ends, where the string was fastened to its supports. The overtone, an octave higher in pitch, had an additional node at the midpoint; the second overtone, two nodes, and so on.

A mathematical theory of the vibrating string was developed by the Swiss mathematician Johann Bernoulli (1667-1748) as one of the first applications on the new mechanics of Isaac Newton (1642-1727). A full understanding of the vibrating string would come with the work of the French mathematician Joseph Fourier (1768-1827), who provided the mathematical framework needed to understand the general case of vibration as a combination, or superposition, of the individual simple modes.

But strings are not the only source of musical sound. Relatively pure tones can be obtained from a membrane stretched tightly across a cylindrical frame, from the air column in a flute or an organ pipe, and from bells or flat plates such as are used in the modern xylophone. The vibrations of a drumhead were explained by the Swiss mathematician Leonhard Euler (1707-1783), and later independently by the French mathematical physicist Simeon Denis Poisson (1781-1840). The study of sound generation by wind instruments was taken up by the Swiss mathematician Daniel Bernoulli (1700-1782), Euler, and the Italian-French mathematician Joseph Louis Lagrange (1736-1813). Euler also began the study of the vibrations of bells and solid rods but did not progress far, finding the fourth order differential equations required beyond the capacity of the mathematical techniques of the time.

Ernst Florenz Friedrich Chladni (1756-1827) undertook the study of vibrating plates by experiment. Chladni was born in Germany, the son of a lawyer who forced him to study law against his wishes. On his father's death, he abandoned the law for the study of acoustics and the invention of musical instruments. By 1787 he had published his first experimental study. Much as Sauveur had demonstrated the existence of nodes or points of no motion in the different modes of vibration of a string, Chladni discovered nodal lines or curves on the two-dimensional vibrating surface. To excite the different modes of vibration he would draw a violin bow across the edge of the plate at different locations. The plate would be clamped at one location. To visualize the nodal lines and curves he sprinkled sand on the surface of the vibrating plate. The sand would only aggregate at locations where there was no motion. The resulting figures soon became known as Chladni figures. By changing the location of the clamp and the placement of the bow, many different figures could be obtained. Chladni also conducted investigations of the velocity of sound in different solid media.

Chladni spent much of his time touring through Europe, giving demonstrations of his figures and also of two new keyboard instruments he had invented that were based on the glass harmonica. When he visited the French Academy of Sciences in 1708, the Emperor Napoleon himself was present and was so intrigued by Chladni's intricate sand figures that he persuaded the Academy to issue a cash prize to the scientist or mathematician who could provide an explanation for what was observed.

The challenge was accepted by Sophie Germain (1776-1831), a remarkable French woman mathematician who had taught herself mathematics from lecture notes borrowed from male students attending the Ecole Polytechnique. In 1811 she submitted a memoir on the subject under a pseudonym to the academy but it was rejected. In 1913 a second entry of hers received honorable mention. In 1816 a paper submitted in her own name in which Germain solved the problem by applying the calculus of variations to a fourth order differential equation finally won the grand prize. Germain immediately gained world renown for this work, but died in 1831 before she could accept the honorary doctoral degree that would have been bestowed on her by the University of Göttingen.

Chladni's figures also stimulated further experimental work. The French physicist Felix Savart (1791-1841) extended Chladni's work using powders such as lycopodium, with much smaller particles than Chladni's sand. A much more extensive study was reported by the great English physicist Michael Faraday (1791-1867) in 1831.

Impact

The histories of music, the visual arts, science, and mathematics have been interconnected at many points. Early Greek notions of proportion entered into both music and geometry as well as into sculpture and architecture. The notion of a harmonious relationship based on numbers played an important role in early astronomical speculation. Innovations in musical instruments often followed discoveries in acoustics. Chladni's interest in music certainly played a role in motivating his discoveries and in popularizing them.

Chladni's researches presented a challenge to the mathematicians and physicists interested in the mathematical description of vibrating bodies. Compared to the description of the vibrating string, which involves a differential equation of second degree in one spatial dimension, the case of the vibrating plate requires a fourth-degree equation in two spatial dimensions. Fully understanding the mathematics of the vibrating plate would require not only the work of Germain, but further major contributions by the French civil engineer Claude Louis Navier (1785-1836), Poisson, and French Mathematician Augustin Louis Cauchy (1787-1857).

The understanding of the vibrations of solid plates that grew out of the researches of Chladni, Germain, and others has had important implications for science and technology. Many large structures include elements that are, to a first approximation, rigid flat plates. A very important design consideration in building a bridge, a skyscraper, or an airplane wing or fuselage is avoiding frequencies of vibration that might become excited by the wind or other disturbances. If the amplitude of such vibrations became large enough, the results could be unpleasant or dangerous for people near these structures. Special considerations apply to the design of auditoriums and speaker systems. The mathematics of Germain and Poisson has also been extended to Earth's crust and is used by seismologists studying wave motion.

The use of a visual characterization of a phenomenon in the absence of an adequate and usable mathematical theory is a recurrent theme in the history of physics. In Chladni's case the mathematical techniques were not yet available. Faraday was certainly acquainted with Chladni's results when he introduced the notion of lines of force to represent the electric and magnetic fields. As in Chladni's case, the required mathematics—the calculus of vector fields—had not been fully developed. In a much more recent example, the American theoretical physicist Richard Feynman (1918-1988) introduced a set of diagrams to represent the interactions between elementary problems. While the mathematical formalism underlying the use of Feynman diagrams has been established, the thinking and communication among particle physicists is mainly in terms of the diagrams rather than the equations.

By the mid-eighteenth century public demonstrations of scientific principles like those held by Chladni were becoming increasingly common. The demonstration of electrostatic principles and static electricity machines were becoming popular both in Europe and in colonial America. They would become increasingly more important in the nineteenth century, with the founding of the Royal Institution in London and similar activities in other countries.

Chladni's demonstrations also attracted some attention from the new writers of what would come to be called science fiction. In a short story, "The Atoms of Chladni," by J. D. Whelpley, published in Harper's New Monthly Magazine in 1859, a mentally unbalanced scientist builds a device based on Chladni's experiments that can record conversations. The magazine was almost certainly one that would have been sold by the young American inventor, Thomas Edison (1847-1931), as a 12-year-old newsboy on the Grand Trunk Railroad. While there is no solid evidence that Edison had read the story, it is a tantalizing speculation that Chladni's experiments might have, perhaps unconsciously, played a role in Edison's invention of the phonograph.

DONALD R. FRANCESCHETTI