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The Physics of Musical Instruments Second Edition Neville H. Fletcher Thomas D. Rossing The Physics of Musical Instruments Second Edition With 485 Illustrations ~Springer Neville H. Fletcher Thomas D. Rossing Research School of Physical Department of Physics Sciences and Engineering Northern Illinois University Australian National University DeKalb, IL 60115 Canberra, A.C.T. 0200 USA Australia Cover illustration: French hom. © The Viesti Collection, Inc. Library of Congress Cataloging-in-Publication Data Fletcher, Neville H. (Neville Homer) The physics of musical instruments I Neville H. Fletcher : Thomas D. Rossing. - 2nd ed. p. em. Includes bibliographical references (p. ) and index. ISBN 978-1-4419-3120-7 ISBN 978-0-387-21603-4 (eBook) DOI 10.1007/978-0-387-21603-4 1. Music - Acoustics and physics. 2. Musical instruments - Construction. I. Rossing, Thomas D., 1929- II. Title. ML3805.F58 1998 784.19'01'53-dc21 97-35360 ISBN 978-1-4419-3120-7 Printed on acid-free paper. © 1998 Springer Science+B usiness Media New York Originally published by Springer Science+B usiness Media, Inc. in 1998 Softcover reprint of the hardcover 2nd edition 1998 All rights reserved. This work may not be translated or copied in whole or in part without the writ ten permission of the publisher Springer Science+ Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, com puter software, or by similar or dissimilar methodology now known or hereafter developed is for bidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 Corrected 5th printing, 2005. SPIN 11340768 springeronline.com Preface When we wrote the first edition of this book, we directed our presenta tion to the reader with a compelling interest in musical instruments who has "a reasonable grasp of physics and who is not frightened by a little mathematics." We are delighted to find how many such people there are. The opportunity afforded by the preparation of this second edition has allowed us to bring our discussion up to date by including those new insights that have arisen from the work of many dedicated researchers over the past decade. We have also taken the opportunity to revise our presentation of some aspects of the subject to make it more general and, we hope, more immediately accessible. We have, of course, corrected any errors that have come to our attention, and we express our thanks to those friends who pointed out such defects in the early printings of the first edition. We hope that this book will continue to serve as a guide, both to those undertaking research in the field and to those who simply have a deep interest in the subject. June 1991 N.H.F and T.D.R. v Preface to the First Edition The history of musical instruments is nearly as old as the history of civiliza tion itself, and the aesthetic principles upon which judgments of musical quality are based are intimately connected with the whole culture within which the instruments have evolved. An educated modern Western player or listener can make critical judgments about particular instruments or particular performances but, to be valid, those judgments must be made within the appropriate cultural context. The compass of our book is much less sweeping than the first paragraph might imply, and indeed our discussion is primarily confined to Western musical instruments in current use, but even here we must take account of centuries of tradition. A musical instrument is designed and built for the playing of music of a particular type and, conversely, music is written to be performed on particular instruments. There is no such thing as an "ideal" instrument, even in concept, and indeed the unbounded possibilities of modern digital sound-synthesis really require the composer or performer to define a whole set of instruments if the result is to have any musical coherence. Thus, for example, the sound and response of a violin are judged against a mental image of a perfect violin built up from experience of violins playing music written for them over the centuries. A new instrument may be richer in sound quality and superior in responsiveness, but if it does not fit that image, then it is not a better violin. This set of mental criteria has developed, through the interaction of musi cal instruments makers, performers, composers, and listeners, over several centuries for most musical instruments now in use. The very features of particular instruments that might be considered as acoustic defects have become their subtle distinguishing characteristics, and technical "improve ments" that have not preserved those features have not survived. There are, of course, cases in which revolutionary new features have prevailed over tradition, but these have resulted in almost new instrument types the violin and cello in place of the viols, the Boehm flute in place of its baroque ancestor, and the saxophone in place of the taragato. Fortunately, perhaps, such profound changes are rare, and most instruments of today vii viii Preface to the First Edition have evolved quite slowly, with minor tonal or technical improvements re flecting the gradually changing mental image of the ideal instrument of that type. The role of acoustical science in this context is an interesting one. Cen turies of tradition have developed great skill and understanding among the makers of musical instruments, and they are often aware of subtleties that are undetected by modern acoustical instrumentation for lack of precise technical criteria for their recognition. It is difficult, therefore, for a scien tist to point the way forward unless the problem or the opportunity has been identified adequately by the performer or the maker. Only rarely do all these skills come together in a single person. The first and major role of acoustics is therefore to try to understand all the details of sound production by traditional instruments. This is a really major program, and indeed it is only within the past few decades that we have achieved even a reasonable understanding of the basic mechanisms determining tone quality in most instruments. In some cases even major features of the sounding mechanism itself have only recently been unrav elled. This is an intellectual exercise of great fascination, and most of our book is devoted to it. Our understanding of a particular area will be rea sonably complete only when we know the physical causes of the differences between a fine instrument and one judged to be of mediocre quality. Only then may we hope that science can come to the help of music in moving the design or performance of contemporary instruments closer to the present ideal. This book is a record of the work of very many people who have studied the physics of musical instruments. Most of them, following a long tradi tion, have done so as a labor of love, in time snatched from scientific or technical work in a field of more immediate practical importance. The com munity of those involved is a world-wide and friendly one in which ideas are freely exchanged, so that, while we have tried to give credit to the origi nators wherever possible, there will undoubtedly be errors of oversight. For these we apologize. We have also had to be selective, and many interesting topics have perforce been omitted. Again the choice is ours, and has been influenced by our own particular interests, though we have tried to give a reasonably balanced treatment of the whole field. The reader we had in mind in compiling this volume is one with a rea sonable grasp of physics and who is not frightened by a little mathematics. There are fine books in plenty about the history of particular musical in struments, lavishly illustrated with photographs and drawings, but there is virtually nothing outside the scientific journal literature that attempts to come to grips with the subject on a quantitative basis. We hope that we have remedied that lack. We have not avoided mathematics where precision is necessary or where hand-waving arguments are inadequate, but at the same time we have not pursued formalism for its own sake. Detailed phys- Preface to the First Edition ix ical explanation has always been our major objective. We hope that the like-minded reader will enjoy coming to grips with this fascinating subject. The authors owe a debt of gratitude to many colleagues who have con tributed to this book. Special thanks are due to Joanna Daly and Barbara Sullivan, who typed much of the manuscript and especially to Virginia Ple mons, who typed most of the final draft and prepared a substantial part of the artwork. Several colleagues assisted in the proofreading, including Rod Korte, Krista McDonald, David Brown, George Jelatis, and Brian Finn. We are grateful to David Peterson, Ted Mansell, and other careful readers who alerted us to errors in the first printing. Thanks are due to our many colleagues for allowing us to reprint figures and data from their publica tions, and to the musical instrument manufacturers that supplied us with photographs. Most of all, we thank our colleagues in the musical acoustics community for many valuable discussions through the years that led to our writing this book. December 1988 Neville H. Fletcher Thomas D. Rossing Contents Preface v Preface to the First Edition vii I. Vibrating Systems 1. Free and Forced Vibrations of Simple Systems 3 1.1. Simple Harmonic Motion in One Dimension 4 1.2. Complex Amplitudes 6 1.3. Superposition of Two Harmonic Motions in One Dimension 7 1.4. Energy 10 1.5. Damped Oscillations 11 1.6. Other Simple Vibrating Systems 13 1.7. Forced Oscillations 18 1.8. Transient Response of an Oscillator 21 1.9. Two-Dimensional Harmonic Oscillator 23 1.10. Graphical Representations of Vibrations: Lissajous Figures 25 1.11. Normal Modes of Two-Mass Systems 26 1.12. Nonlinearity 28 Appendix 29 References 32 2. Continuous Systems in One Dimension: Strings and Bars 34 2.1. Linear Array of Oscillators 34 2.2. Transverse Wave Equation for a String 36 2.3. General Solution of the Wave Equation: Traveling Waves 37 2.4. Reflection at Fixed and Free Ends 38 2.5. Simple Harmonic Solutions to the Wave Equation 39 xi xii Contents 2.6. Standing Waves 39 2.7. Energy of a Vibrating String 40 2.8. Plucked String: Time and Frequency Analyses 40 2.9. Struck String 44 2.10. Bowed String 46 2.11. Driven String: Impedance 50 2.12. Motion of the End Supports 52 2.13. Damping 53 2.14. Longitudinal Vibrations of a String or Thin Bar 56 2.15. Bending Waves in a Bar 58 2.16. Bars with Fixed and Free Ends 60 2.17. Vibrations of Thick Bars: Rotary Inertia and Shear Deformation 63 2.18. Vibrations of a Stiff String 64 2.19. Dispersion in Stiff and Loaded Strings: Cutoff Frequency 65 2.20. Torsional Vibrations of a Bar 66 References 68 3. Two-Dimensional Systems: Membranes, Plates, and Shells 70 3.1. Wave Equation for a Rectangular Membrane 70 3.2. Square Membranes: Degeneracy 72 3.3. Circular Membranes 73 3.4. Real Membranes: Stiffness and Air Loading 75 3.5. Waves in a Thin Plate 76 3.6. Circular Plates 78 3. 7. Elliptical Plates 80 3.8. Rectangular Plates 80 3.9. Square Plates 83 3.10. Square and Rectangular Plates with Clamped Edges 85 3.11. Rectangular Wood Plates 88 3.12. Bending Stiffness in a Membrane 91 3.13. Vibration of Shells 92 3.14. Driving Point Impedance 96 References 99 4. Coupled Vibrating Systems 102 4.1. Coupling Between Two Identical Vibrators 102 4.2. Normal Modes 103 4.3. Weak and Strong Coupling 105 4.4. Forced Vibrations 107 4.5. Coupled Electrical Circuits 111 4.6. Forced Vibration of a Two-Mass System 115 4.7. Systems with Many Masses 116