Chapter 2 Further Reading and Resources

Classic Texts

1. Rayleigh, Lord (John William Strutt). The Theory of Sound. (Vols. 1 and 2, Macmillan, 1877 and 1878; Dover reprint, 1945.)

Lord Rayleigh's two-volume masterwork remains one of the most comprehensive classical treatments of acoustic theory ever written. Volume 1, Chapter VI covers the vibrating string in exhaustive mathematical detail, deriving the wave equation, the normal modes, and the effects of real-world complications (stiffness, damping, loading). The mathematics is rigorous 19th-century physics — some calculus is required — but the physical reasoning is impeccable. For readers who want to understand the rigorous mathematics underlying Chapter 2, this is the primary source. The Dover reprint is inexpensive and widely available.

2. French, A.P. Vibrations and Waves. (MIT Introductory Physics Series, W.W. Norton, 1971.)

French's undergraduate textbook covers mechanical vibrations and waves with exactly the mathematical level appropriate for this textbook's more technically inclined readers. Chapters 4–7 develop the standing wave theory from first principles, treating the string with care and making the connection to Fourier analysis explicit. The treatment of normal modes and superposition is particularly clear. A standard reference in first-year university physics courses, this book makes an excellent companion to Chapters 2 and 3.

3. Rossing, Thomas D. The Science of String Instruments. (Springer, 2010.)

An edited volume with chapters by leading experts on the acoustics of specific string instruments: bowed strings (violin, viola, cello, bass), plucked strings (guitar, harp, lute, banjo), keyboard strings (piano, harpsichord, clavichord), and struck strings (hammered dulcimer, cimbalom). Each chapter combines physical theory with experimental measurements on real instruments. The chapter on piano acoustics is particularly detailed and directly relevant to Case Study 2.1. Requires calculus for the technical sections but the introductory material in each chapter is readable without advanced mathematics.

Modern Acoustics Research

4. Gough, Colin E. "Science and the Stradivarius." Physics World 13(4), April 2000, pp. 27–33.

A highly readable review article on the acoustics of violins and what makes great instruments great. Gough covers the vibrating string, the bridge, the body modes, and the radiation of sound from the violin with clarity and physical precision. This article is available through university library databases and provides excellent background for Chapter 3's Stradivarius case study.

5. Young, Dana A. "Inharmonicity of Plain Wire Piano Strings." Journal of the Acoustical Society of America 24(3), 1952, pp. 267–273.

The classic technical paper establishing the theory of piano string inharmonicity quantitatively. Young derives the formula for inharmonic overtone frequencies in real strings with bending stiffness and compares theory to measurement. Requires undergraduate physics mathematics, but the results and their musical implications are discussed in terms accessible to the general reader. A historical landmark in piano acoustics.

6. McIntyre, M.E., Schumacher, R.T., and Woodhouse, J. "On the Oscillations of Musical Instruments." Journal of the Acoustical Society of America 74(5), 1983, pp. 1325–1345.

The authoritative mathematical treatment of the stick-slip mechanism of bowed string oscillation. McIntyre, Schumacher, and Woodhouse developed a computer model of the bowing process that accurately predicts the Helmholtz motion of a bowed string (the sawtooth-like displacement pattern). This paper is the foundation of modern bowed-string acoustics. Requires significant mathematical facility, but the physical explanations in the introduction and conclusion sections are accessible.**

On Quantum Mechanics and Musical Strings

7. Feynman, Richard P., Leighton, Robert B., and Sands, Matthew. The Feynman Lectures on Physics, Vol. III: Quantum Mechanics. (Addison-Wesley, 1965; freely available online at feynmanlectures.caltech.edu.)

Feynman's legendary undergraduate physics lectures. Volume III covers quantum mechanics with characteristic physical insight and minimal mathematical formalism. Chapter 1's introduction to quantum behavior and Chapter 3's treatment of the position-momentum uncertainty principle are essential for understanding the quantum parallels drawn in this chapter. The Feynman Lectures are freely available online at the link above — one of the great gifts to science education.

8. Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. (W.W. Norton, 1999.)

The definitive popular account of string theory for the general reader. Greene explains the basic ideas of string theory — vibrating strings, extra dimensions, supersymmetry, M-theory — with remarkable clarity and appropriate musical analogies. Chapters 6–9 are most directly relevant to Case Study 2.2. Greene's treatment of the limitations and open questions in string theory is honest, though his enthusiasm for the theory as the likely correct framework reflects his own position in a field where consensus is lacking.

9. Woit, Peter. Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. (Basic Books, 2006.)

A vigorous critique of string theory from a mathematical physicist who argues that the theory makes no testable predictions and has diverted resources from more promising directions. Reading this book alongside Greene's The Elegant Universe gives a more complete picture of the genuine scientific debate about string theory. Chapters 1–5 provide excellent background on the history of particle physics that led to string theory; the later chapters argue the specific scientific objections.

Online Resources

10. Walter Lewin's MIT OpenCourseWare lectures on "Vibrations and Waves" (8.03).

MIT Professor Walter Lewin (now retired) recorded his undergraduate physics lectures, including a complete course on vibrations and waves that is freely available through MIT OpenCourseWare and YouTube. The lectures on standing waves on strings and the connection to musical instruments are particularly relevant to this chapter. Lewin's theatrical style and use of physical demonstrations make these lectures unusually engaging. Search for "MIT 8.03 Lewin" to find the complete course.

11. Hyperphysics "String Instruments" module (hyperphysics.phy-astr.gsu.edu).

Hyperphysics is a free physics reference site maintained by Georgia State University. The string instruments module provides interactive calculations for string frequencies, mode diagrams, and harmonic series, with links to adjacent topics. Useful for verifying calculations from this chapter and exploring numerical examples.

12. The Physics of Music and Musical Instruments (online resource by David Lapp, Tufts University).

An extensive, well-organized online text available freely at various physics education repositories. Lapp's treatment of string vibration, standing waves, and the harmonic series at the introductory level complements this chapter's treatment. Includes laboratory activities that can be done with simple materials, many of which align with the "Try It Yourself" activities in this textbook.