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 Physics Department Northem Illinois University DeKalb, IL 60015, USA

Library of Congress Cataloging-in-Publieation Data Rossing, Thomas, D., 1929-

Neville H. Fletcher Department ofPhysical Sciences Research School ofPhysical

Sciences and Engineering Australian National University Canberra, ACT 0200 Australia

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 mate­rial 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 in­terest and omit the others without loss of continuity. We have continued to stress concepts over detailed theory, as seems most appropriate for an in­troductory 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 perfor­mance; several excellent books have treated the acoustics of musical instru­ments in a semiquantitative way; but none to our knowledge had then at­tempted 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 oscilla­tors. 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 of Musical 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 princi­ples rather than on practical applications and with a moderately mathematical approach. It must therefore be left to supplementary reading to fill in fasci­nating and important material on such topics as physical acoustics, micro­phones, 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 re­lating to musical applications.

To make the book more useful in general courses in acoustics and vibra­tions, 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 Preface to the First Edition

PART I Vibrating Systems

CHAPTER I Free and Forced Vibrations of Simple Systems

1.1. Simple Harmonic Motion in One Dimension 1.2. Complex Amplitudes 1.3. Superposition of Two Harmonic Motions in One Dimension 1.4. Energy 1.5. Damped Oscillations 1.6. Other Simple Vibrating Systems 1.7. Forced Oscillations 1.8. Transient Response of an Oscillator 1.9. Two-Dimensional Harmonic Oscillator 1.10. Graphical Representations of Vibrations: Lissajous Figures 1.11. Normal Modes of Two-Mass Systems 1.12. Nonlinear Vibrations of a Simple System

APPENDIX A.l. Alternative Ways of Expressing Harmonic Motion A.2. Equivalent Electrical Circuit for a Simple Oscillator

References

CHAPTER2

Continuous Systems in One Dimension: Strings and Bars

2.1. Linear Array of Oscillators 2.2. Transverse Wave Equation for a String 2.3. General Solution of the Wave Equation: Traveling Waves 2.4. Reflection at Fixed and Free Ends

v vii

1

3

4 6 7

10 10 12 16 20 22 23 25 26

29 30 32

33

33 35 36 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

3.1. Wave Equation for a Rectangular Membrane 3.2. Square Membranes: Degeneracy 3.3. Circular Membranes 3.4. Real Membranes: Stiffness and Air Loading 3.5. Waves in a Thin Plate 3.6. Circular Plates 3.7. Elliptical Plates 3.8. Rectangular Plates 3.9. Square Plates 3.10. Square and Rectangular Plates with Clamped Edges 3 .11. Rectangular Wood Plates 3.12. Bending Stiffness in a Membrane 3.13. Shallow Spherical Shells 3.14. Nonlinear Vibrations in Plates and Shallow Shells 3.15. Driving Point Impedance

References

CHAPTER4

Coupled Vibrating Systems

4.1. Coupling Between Two Identical Vibrators 4.2. Normal Modes 4.3. Weak and Strong Coupling 4.4. Forced Vibrations 4.5. Coupled Electrical Circuits 4.6. Forced Vibration of a Two-Mass System 4. 7. Systems with Many Masses

65

65 68 69 70 71 72 74 75 78 81 82 85 86 88 89 92

95

95 96 98

100 103 107 109

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 A.2. Modal Analysis A.3. Finite Element Analysis

References

CHAPTERS Nonlinear Systems

5.1. A General Method of Solution 5.2: Illustrative Examples 5.3. The Self-Excited Oscillator 5.4. Multimode Systems 5.5. Mode Locking in Self-Excited Systems

References

PART II Sound Waves

CHAPTER6 Sound Waves in Air

6.1. Plane Waves 6.2. Spherical Waves 6.3. Sound Pressure Level and Intensity 6.4. Reflection and Transmission 6.5. Absorption 6.6. Normal Modes in Cavities

References

CHAPTER 7

Sound Radiation

7.1. Simple Multipole Sources 7.2. Pairs of Point Sources 7.3. Arrays of Point Sources 7.4. Radiation from a Spherical Source 7.5. Line Sources 7.6. Radiation from a Plane Source in a Baffle 7.7. Unbaffied Radiators 7.8. Radiation from Large Plates

References

117 121 122 123

125

126 128 130 131 133 135

137

139

139 143 145 147 151 153 156

157

157 160 162 164 166 167 170 172 174

xii Contents

CHAPTERS Pipes and Horns

8.1. Infinite Cylindrical Pipes 8.2. Wall Losses 8.3. Finite Cylindrical Pipes 8.4. Radiation from a Pipe 8.5. Impedance Curves 8.6. Horns 8.7. Finite Conical and Exponential Horns 8.8. Bessel Horns 8.9. Compound Horns 8.10. Perturbations 8.11. Numerical Calculations 8.12. The Time Domain

References

CHAPTER9 Acoustic Systems

9.1. Low-Frequency Components and Systems 9.2. High-Frequency Components and Systems 9.3. Finite Horns 9.4. Coupled Mechanical Components 9.5. Multi-Port Systems 9.6. Conclusion

References

CHAPTER 10 Microphones and Other Transducers

1 0.1. Microphone Principles 10.2. Omnidirectional Condenser Microphones 10.3. Directional Condenser Microphones 10.4. Studio Condenser Microphones 10.5. Piezoelectric Microphones 10.6. Dynamic Microphones 10.7. Ribbon Microphones 10.8. Electrical Circuitry 10.9. Loudspeakers 10.10. Dynamic Loudspeakers 10.11. Horn Loudspeakers 10.12. Hydrophones 10.13. Ultrasonic Transducers 10.14. Force Transducers and Accelerometers

References

175

175 178 181 186 186 189 194 197 199 201 203 203 206

209

209 216 219 222 224 226 227

229

229 232 236 237 239 240 240 241 242 244 246 246 247 248 250

Contents xiii

CHAPTER 11

Sound in Concert Halls and Studios 251

11.1. Spatial Dependence of the Sound Field 251 11.2. Time Dependence of the Sound Field 253 11.3. Sound Fields in Real Rooms 258 11.4. What Makes Good Acoustics? 262 11.5. Measuring Sound Absorption Coefficients 263 11.6. Standing Waves and Normal Modes 266 11.7. Small Rooms and Studios 267 11.8. Sound Diffusers and Absorbers 269 11.9. Rooms for Worship 271 11.10. Classrooms 271 11.11. Walls and Noise Barriers 272

References and Suggested Reading 275

CHAPTER12

Sound and Noise Outdoors

12.1. Sound Propagation in the Atmosphere 12.2. Effect of the Ground 12.3. Effect of Refraction 12.4. Diffraction and Sound Barriers 12.5. Atmospheric Turbulence 12.6. Motor Vehicle Noise 12.7. Railroad Noise 12.8. Aircraft Noise 12.9. Summary of Factors at Various Distances

References

CHAPTER 13 Underwater Sound

13.1. Underwater Sound Propagation 13.2. Underwater Waveguides 13.3. Sonar 13.4. Noise 13.5. Bubbles in Water

References

Selected Bibliography Problems Answers to Selected Problems Name Index Subject Index

277

277 280 283 287 288 289 290 291 292 293

294

294 296 297 300 302 307

309 311 321 323 325