Nineteenth-Century Advances in the Mathematical Theory and Understanding of Sound

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Nineteenth-Century Advances in the Mathematical Theory and Understanding of Sound

Overview

The nineteenth century saw the development of the mathematical techniques and experimental methods needed to understand the vibrations of objects and the motion of sound waves in air and other media. These developments were summarized and integrated by John Strutt (1842-1919), better known as Lord Rayleigh, in his The Theory of Sound, first published in 1877. Raleigh's text remains one of the fundamental reference works for acoustical scientists and engineers.

Background

The history of the science of acoustics is somewhat unusual in that, unlike the case of heat or projectile motion, the essential nature of sound has been correctly understood since the time of the ancient Greeks. They identified the origin of sound with the vibration of bodies and understood it to be transmitted through the air in some fashion. The fact that vibrating strings under tension produced harmonious sounds if their lengths were in simple numerical ratios was known to the disciples of Greek mathematician Pythagoras (580?-500? B.C.).

The development of musical instruments, begun in antiquity, advanced rapidly in the Renaissance and subsequent baroque and classical periods. Scientific understanding of sound lagged behind, however, for lack of a means of recording sound and the mathematics needed to describe its origin and movement. Galileo (1564-1642) included a discussion of vibrating bodies in his 1638 Discourses Concerning Two New Sciences, relating vibration to the behavior of the pendulum. Robert Hooke (1635-1703), who discovered the fundamental law of elasticity, tried to study the vibration of sound sources. Isaac Newton (1642-1727) included a somewhat incorrect discussion of the motion of sound waves in his Principia Mathematica, published in 1687.

The description of the vibrating string drew the attention of some of the best mathematicians of the seventeenth and eighteenth centuries, including Jakob Bernoulli (1654-1705), Jean d'Alembert (1717-1783), Leonhard Euler (1707-1783), and Joseph Louis Lagrange (1736-1813). The description of vibration required generalizing the calculus, which had been invented by Newton, to describe the rates of change of quantities in time, to a calculus of more than one variable, which could describe quantities that were functions of both time and position. The necessary methodology was provided by French engineer Jean Baptiste Fourier (1768-1830) in his Analytical Theory of Heat, published in 1822. Although Fourier concerned himself primarily with the flow of heat through matter, the mathematics of heat flow presented the same problems as that of sound, and he was able to show how any solution of the corresponding equations defined over a region of space could be expressed as a sum of sine- and cosine-like functions, each having its own characteristic variation with time.

Hermann Helmholtz (1821-1894) was a military surgeon who became professor of physiology at the University of Königsberg and later professor of physics at the University of Berlin. His early career was devoted to a study of the physical senses and the nervous system. He was the first to measure the speed of a nerve impulse. His interest turned to acoustics in 1853, apparently after spotting some mathematical errors in a paper on the subject. He identified the cochlea as the organ of hearing and proposed a plausible mechanism for hearing whereby sounds are detected by the resonance of parts of this structure with the incoming sound wave. He then explained how the quality of musical tones resulted from there being a combination of different frequencies. His book summarizing this work, On the Sensations of Tone, was first published in 1863.

Helmholtz's book stimulated interest in acoustics on the part of numerous physicists, among them Strutt, who would soon inherit the title and estate of his father, becoming the third Baron Rayleigh. Apparently, Strutt read Helmholtz's work upon the suggestion of Professor Donkin, an astronomer at Oxford University who recommended that he should acquire a reading knowledge of German. Throughout his career Lord Rayleigh worked in several areas of physics, publishing over 400 scientific papers, 128 of them dealing with acoustical phenomena. He began writing his two-volume treatise The Theory of Sound while recuperating from an illness in 1872, and saw its publication in 1877, with a revised and enlarged edition incorporating many new developments in 1894.

Rayleigh's treatise is largely a work of synthesis, integrating his own discoveries with those of others. It deals both with the origin of sound and its transmission through the air and other media. As Professor R. B. Lindsay wrote in the preface to the first American edition, "Lord Rayleigh appeared on the acoustical scene when the time was precisely ripe for a synthesis of experimental phenomena, much of which was, however too idealized for practical application." Much of Rayleigh's book deals with approximation methods for dealing with vibrations and wave motion in situations in which the equations of motion can not be solved in any simple mathematical form. Rayleigh's careful organization, clarity of expression, and balance between mathematical elegance and the pragmatic realities of laboratory investigation assured the book's value to generations of future physicists.

Impact

Modern acoustics is the study of the generation, transmission, and detection of sound waves travelling through gaseous, liquid, or solid media, including the reflection and refraction of sound waves at the boundaries between different media. Rayleigh's treatise provides a compact and thorough summary of the first two areas, generation and transmission, and in its second edition makes a connection with the emerging technology of the telephone.

The need for understanding the fundamentals of sound production, transmission, and detection was to grow rapidly following the appearance of Rayleigh's treatise in 1877, the same year that Thomas Edison (1847-1931) demonstrated the first phonograph. The previous year, a United States Patent had been issued to Alexander Graham Bell (1847-1922) for a "device to transmit speech sounds over electric wires." Fourteen years later, Carnegie Hall in New York City opened for its first concert, a building considered nearly perfect for musical performances without electronic amplification due to its special acoustic design. With the invention of the vacuum tube triode by the American Lee De Forest (1873-1961) in 1907 it became possible to amplify sounds to levels audible throughout a large area, the acoustics of loudspeakers and ever-larger auditoriums became matters of immediate practical concern. The first motion picture with sound was demonstrated in 1926.

The use of ultrasound, or sound waves traveling above the range of human hearing, has come to be very important in many areas of technology. Military applications of ultrasound generation and detection were developed during the First World War. SONAR (sound navigation and ranging) was needed to track submarines under water. Depth sounders, which bounce sound waves from the water bottom, are an important byproduct for civilian use. Ultrasound is routinely used in industry to nondestructively test for defects in materials. It has also proven its value in medical diagnosis, as a portable means of imaging soft tissues without the radiation hazard inherent in x-rays. Expectant mothers are now routinely examined by ultrasound to check the health of the developing fetus. Frequently they receive a videotape or computer-generated image to take with them, and sometimes these are carried and displayed like baby photos by the proud parents to be.

With the invention of the transistor by John Bardeen (1908-1991), Walter Brattain (1902-1987), and William Shockley (1910-1989) in 1947, it became possible to build very small sound amplification systems, systems small enough to be carried about by the hearing impaired. Since the 1950s portable hearing aids have gotten smaller and smaller with many modern aids fitting entirely within the ear canal.

Acoustical techniques are also employed in seismology, the study of vibrations of the earth, and particularly of earthquakes. Seismic waves are typically subsonic, or lower in frequency than the range of human hearing. An earthquake gives rise to several different types of acoustic waves that arrive at points in the Earth's surface at different times. At first, the acoustic data from identified earthquakes was used to refine the model of the Earth's inner structure. Currently seismic data is used to locate and profile new earthquakes. Seismic monitoring is also used to identify belowground explosions of nuclear weapons.

The mathematical analysis of sound waves can be used in modern electronic technology to synthesize both speech and musical sounds. Thus the approach begun by Helmholtz and Rayleigh has, in a sense, come full circle to produce computers that can speak and listen, as well as the keyboards and synthesizers that are ubiquitous in contemporary music.

DONALD R. FRANCESCHETTI