The Theory of Sound, Volume One

John William Strutt, known as Lord Rayleigh, was a mathematician, a mathematical physicist and something of an experimental physicist as well. He was a pupil of E. J. Routh, a famous coach for the Tripos exam at Cambridge, and later served a five-year term as Cavendish professor of physics at Cambridge. James Clerk Maxwell, whom he followed in that position, had been a friend and mentor. Rayleigh was a very prolific scientist with broad interests. Some indications of this are the number of terms associated with his name: the Rayleigh quotient (used for the estimation of eigenvalues), Rayleigh scattering (associated with light and other electromagnetic radiation), the Rayleigh limit of optical resolution, Rayleigh waves (acoustic waves that travel along the surfaces of solids) and many more. He was also the co-discoverer of the gas argon and for this won the Nobel Prize in Physics in 1904.

Rayleigh began working on the volumes reviewed here in 1872 while he was recovering from rheumatic fever. They were eventually published in 1877. Sometime during that period Rayleigh had assumed his father’s title and moved to the family estate. Before taking the Cavendish professorship he had set up his own private laboratory at his father’s estate and he became very adept at making good measurements with rather marginal equipment. At the time Cambridge had no physical laboratory of its own. Apparently Rayleigh was motivated by Helmholtz’s work “On the Sensations of Tone” to pursue his interest in acoustics more deeply.

The two volumes of his work on the theory of sound break naturally into two parts. The first volume, after some introductory material, concentrates on the vibrations of dynamical systems, especially those that result in acoustic radiation. Thus its primary goal is to describe how sound is generated. The second volume is devoted to the study of acoustic radiation through air and water. More simply, it describes how sound propagates.

Rayleigh’s style is comfortably informal and in that respect somewhat at odds with his contemporaries. He begins with the simplest possible case — oscillations of a system with one degree of freedom — and each aspect of theory is accompanied by an experimental illustration. He follows the simple case with a general theory of vibrations for a system with n degrees of freedom. This leads to a complicated system of differential equations that he proceeds to solve with an approach using maximum potential and kinetic energies that has become known as the Rayleigh-Ritz method. Throughout his work Rayleigh was especially fond of using energy and virtual work to set up the relevant differential equations.

In his study of vibrations Rayleigh considers strings, bars, flat and curved plates. Along the way he examines how violin and piano strings move and also has a charming bit on how the tones arise and change in church bells. He also considers electrical vibrations in the context of alternating current motivated in part by the telephone, a relatively new invention at the time he was writing.

The second volume’s work on the transmission of sound treats a more difficult subject. A compressional disturbance in a fluid (air or water, for example) is a very complex thing. Much of the difficulty arises because ordinary fluids of interest are never ideal, so the art is to identify good approximations that give physically useful results. In this second part Rayleigh begins with the transmission of sound in the air (in the open atmosphere, in tubes, in chambers, and when resonance occurs as in organ pipes), then discusses the propagation of sound in water and in solids. He concludes by looking at the ear and the perception of sound.

Rayleigh’s is a work of mathematical physics. His mathematical analysis is careful and thorough, but he clearly felt no need to be particularly formal. He did require that his theories made sense physically, were mathematically consistent, and — as much as possible — agreed with measurements.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.