Principles of Vibration and Sound

335 Pages·2004·9.63 MB·English

by  Thomas D. Rossing

Principles of Vibration and Sound
Second Edition
Springer Science+Business Media, LLC
Thomas D. Rossing
Neville H. Fletcher
Principles of Vibration
and Sound
Second Edition
With 182 Illustrations
Springer
Thomas D. Rossing Neville H. Fletcher
Physics Department Department ofPhysical Sciences
Northem Illinois University Research School ofPhysical
DeKalb, IL 60015, USA Sciences and Engineering
Australian National University
Canberra, ACT 0200 Australia
Library of Congress Cataloging-in-Publieation Data
Rossing, Thomas, D., 1929-
Prineiples of vibration and sound / Thomas D. Rossing, NeviIle H.
Fletcher. - 2nd ed.
p. em.
lncludes bibliographical references and index.
ISBN 978-1-4419-2343-1 ISBN 978-1-4757-3822-3 (eBook)
DOI 10.1007/978-1-4757-3822-3
1. Acoustical engineering. 2. Vibration. L Fleteher, Neville H.
(Neville Homer) IL Title
TA365.R672003
534-dc21 2003054413
ISBN 978-1-4419-2343-1 Printed on acid-free paper.
© 2004, 1995 Springer Scienee+Business Media New York
Originally published by Springer-VerlagNew York, lnc. in 2004
Softeover reprint of the hardcover 2nd edition 2004
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Preface to the Second Edition
The first edition of this book presented the principles of vibration and sound
with only a little discussion of applications of these principles. During the
past eight years, our own experience, as well as that of other teachers who
used it as a textbook, has indicated that students would benefit from more
discussion of applications. In this edition we have revised some of the mate
rial in the first nine chapters, but more importantly we have added four new
chapters dealing with applications, including microphones, loudspeakers,
and other transducers; acoustics of concert halls and studios; sound and
noise outdoors; and underwater sound. Of course we could have selected
many additional applications of vibration and sound, but that would have
led to a book with too much material for the average acoustics course in
physics and engineering departments. We think there is now ample material
in the book so that instructors may select the applications of particular in
terest and omit the others without loss of continuity. We have continued to
stress concepts over detailed theory, as seems most appropriate for an in
troductory course.
We appreciate the comments we have received from users, students, and
teachers alike, and we continue to welcome feedback.
September 2003 Thomas D. Rossing
Neville H. Fletcher
Preface to the First Edition
Some years ago we set out to write a detailed book about the basic physics
of musical instruments. There have been many admirable books published
about the history of the development of musical instruments, about their
construction as a master craft, and about their employment in musical perfor
mance; several excellent books have treated the acoustics of musical instru
ments in a semiquantitative way; but none to our knowledge had then at
tempted to assemble the hard acoustic information available in the research
literature and to make it available to a wider readership. Our book The
Physics of Musical Instruments, published by Springer-Verlag in 1991 and
subsequently reprinted several times with only minor corrections, was the
outcome of our labor.
Because it was our aim to make our discussion of musical instruments as
complete and rigorous as possible, our book began with a careful introduction
to vibrating and radiating systems important in that field. We treated simple
linear oscillators, both in isolation and coupled together, and extended that
to a discussion of some aspects of driven and autonomous nonlinear oscilla
tors. Because musical instruments are necessarily extended structures, we then
went on to discuss the vibrations of strings, bars, membranes, plates, and
shells, paying particular attention to the mode structures and characteristic
frequencies, for it is these that are musically important. The generation and
propagation of acoustic waves in air is of obvious importance, and this too
received fairly thorough discussion, at least in relation to those parts of the
subject relevant to our major concern. Wind instruments, of course, consist
of pipes and horns, and the propagation of waves in these structures, their
normal modes, and their radiation properties were all carefully treated, again
in the musical instrument context. The first third of our book thus presented
a broad, but admittedly somewhat eclectic, treatment of the basic subject
matter of vibrations and acoustics.
In response to several suggestions, the publishers have decided to issue this
first section of The Physics ofM usical Instruments as a separate book, suitable
for use as a text in standard courses in vibrations and acoustics. We will not
conceal the fact that, had we set out to write such a book in the first place, its
viii Preface to the First Edition
content would probably have been rather different. But the subject matter of
acoustics is so wide and the possible manners of approach so various that we
believe the academic community may welcome this view of the subject. It is
an unashamedly basic book, with emphasis on fundamental dynamical princi
ples rather than on practical applications and with a moderately mathematical
approach. It must therefore be left to supplementary reading to fill in fasci
nating and important material on such topics as physical acoustics, micro
phones, loudspeakers, architectural acoustics, and auditory physiology. Even
for musical instruments the interested reader is referred to our complete book.
The references in the text are similarly eclectic, with emphasis on those re
lating to musical applications.
To make the book more useful in general courses in acoustics and vibra
tions, we have added several new sections and one new chapter-on network
analogs for acoustic systems. We have also included some problems at the end
of each chapter to assist with the use of the book in a teaching environment.
January 1994 Neville H. Fletcher
Thomas D. Rossing
Contents
Preface to the Second Edition v
Preface to the First Edition vii
PART I
Vibrating Systems 1
CHAPTER I
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 10
1.6. Other Simple Vibrating Systems 12
1.7. Forced Oscillations 16
1.8. Transient Response of an Oscillator 20
1.9. Two-Dimensional Harmonic Oscillator 22
1.10. Graphical Representations of Vibrations: Lissajous Figures 23
1.11. Normal Modes of Two-Mass Systems 25
1.12. Nonlinear Vibrations of a Simple System 26
APPENDIX
A.l. Alternative Ways of Expressing Harmonic Motion 29
A.2. Equivalent Electrical Circuit for a Simple Oscillator 30
References 32
CHAPTER2
Continuous Systems in One Dimension: Strings and Bars 33
2.1. Linear Array of Oscillators 33
2.2. Transverse Wave Equation for a String 35
2.3. General Solution of the Wave Equation: Traveling Waves 36
2.4. Reflection at Fixed and Free Ends 36
x Contents
2.5. Simple Harmonic Solutions to the Wave Equation 37
2.6. Standing Waves 38
2. 7. Energy of a Vibrating String 39
2.8. Plucked String: Time and Frequenyy Analyses 39
2.9. Struck String 42
2.10. Bowed String 45
2.11. Driven String Impedance 47
2.12. Motion of the End Supports 48
2.13. Damping 50
2.14. Longitudinal Vibrations of a String or Thin Bar 53
2.15. Bending Waves in a Bar 54
2.16. Bars with Fixed and Free Ends 57
2.17. Vibrations of Thick Bars: Rotary Inertia and Shear Deformation 60
2.18. Vibrations of a Stiff String 61
2.19. Dispersion in Stiff and Loaded Strings: Cutoff Frequency 61
2.20. Torsional Vibrations of a Bar 63
References 64
CHAPTER3
Two-Dimensional Systems: Membranes and Plates 65
3.1. Wave Equation for a Rectangular Membrane 65
3.2. Square Membranes: Degeneracy 68
3.3. Circular Membranes 69
3.4. Real Membranes: Stiffness and Air Loading 70
3.5. Waves in a Thin Plate 71
3.6. Circular Plates 72
3.7. Elliptical Plates 74
3.8. Rectangular Plates 75
3.9. Square Plates 78
3.10. Square and Rectangular Plates with Clamped Edges 81
3. 11. Rectangular Wood Plates 82
3.12. Bending Stiffness in a Membrane 85
3.13. Shallow Spherical Shells 86
3.14. Nonlinear Vibrations in Plates and Shallow Shells 88
3.15. Driving Point Impedance 89
References 92
CHAPTER4
Coupled Vibrating Systems 95
4.1. Coupling Between Two Identical Vibrators 95
4.2. Normal Modes 96
4.3. Weak and Strong Coupling 98
4.4. Forced Vibrations 100
4.5. Coupled Electrical Circuits 103
4.6. Forced Vibration of a Two-Mass System 107
4. 7. Systems with Many Masses 109
Contents xi
4.8. Graphical Representation of Frequency Response Functions 110
4.9. Vibrating String Coupled to a Soundboard 112
4.10. Two Strings Coupled by a Bridge 114
APPENDIX
A.1. Structural Dynamics and Frequency Response Functions 117
A.2. Modal Analysis 121
A.3. Finite Element Analysis 122
References 123
CHAPTERS
Nonlinear Systems 125
5.1. A General Method of Solution 126
5.2: Illustrative Examples 128
5.3. The Self-Excited Oscillator 130
5.4. Multimode Systems 131
5.5. Mode Locking in Self-Excited Systems 133
References 135
PART II
Sound Waves 137
CHAPTER6
Sound Waves in Air 139
6.1. Plane Waves 139
6.2. Spherical Waves 143
6.3. Sound Pressure Level and Intensity 145
6.4. Reflection and Transmission 147
6.5. Absorption 151
6.6. Normal Modes in Cavities 153
References 156
CHAPTER 7
Sound Radiation 157
7.1. Simple Multipole Sources 157
7.2. Pairs of Point Sources 160
7.3. Arrays of Point Sources 162
7.4. Radiation from a Spherical Source 164
7.5. Line Sources 166
7.6. Radiation from a Plane Source in a Baffle 167
7.7. Unbaffied Radiators 170
7.8. Radiation from Large Plates 172
References 174