Experiential Approach to Sound, Music, and Psychoacoustics

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Heller Why You Hear What You Hear An Experiential Journey through Sound, Music, and Psychoacoustics P R I N C E T O N U N I V E R S I T Y P R E S S Princeton and Oxford Copyright c 2013 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Heller, Eric Johnson. Why you hear what you hear : an experiential approach to sound, music, and psychoacoustics / Eric J. Heller. p. cm. Includes bibliographical references and index.

ISBN 978-0-691-14859-5 (hardback : alk. paper) 1. Hearing. 2. Sound–Transmission–Measurement. 3. Psychoacoustics. I. Title. QP461.H395 2012 612.8 5–dc23 2011053479 British Library Cataloging-in-Publication Data is available This book has been composed in Minion Pro and Myriad Pro Printed on acid-free paper. Typeset by S R Nova Pvt Ltd, Bangalore, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 To Sharl Contents Preface xix How to Use This Book xxiii Acknowledgments xxvii I Sound Itself 1 1 How Sound Propagates 3 1.1 Push and Pushback: Impedance 6 What Is Impedance, Really?

8 Antire ection Strategies 9 Impedance and the Violin 10 Bullwhip—The High Art of Impedance Matching 11 Impedance Mismatches Are Not Always Bad 11 Impedance of Masses and Springs Together 12 De ning and Measuring Impedance 12 1.2 Impedance of Air 13 1.3 Propagation of Sound in Pipes 16 Re ection of Sound at a Closed End 17 Re ection of Sound at an Open End 17 Re ection of Sound at the Junction of Different-diameter Pipes 19 2 Wave Phenomenology 21 2.1 Relation between Speed, Frequency, and Wavelength 21 2.2 Falloff with Distance from the Source 23 Loudness Falloff with Distance 24 Ripple Simulation 25 2.3 Measuring the Speed of Sound 26 Box 2.1 Father Marin Mersenne 27 vii viii Contents 2.4 Interference and Superposition 27 Active Noise Cancellation—Deliberate Destructive Interference 29 2.5 Re ection 29 Shiny and Matte 30 2.6 Refraction 32 2.7 Diffraction 34 Diffraction at an Edge 35 Brush with the Law of Similarity 36 Active Noise Reduction of Diffracted Sound 37 2.8 Schlieren Photography 38 2.9 Ray Tracing 39 Corner (Retro-) Re ector 40 Box 2.2 The SOFAR Channel 43 2.10 Measures of Sound Power 44 Box 2.3 How Big?

47 II Analyzing Sound 49 3 Sound and Sinusoids 51 3.1 The Atom of Sound 52 Building a Sine Wave 52 3.2 Sinusoidal Vibration 54 The Velocity 55 The Tuning Fork 56 The Sound of a Sinusoid 58 3.3 The Pendulum 58 3.4 The Double Tuning Fork 59 3.5 Microscopes for Vibration 62 3.6 Spying on Conversations 64 3.7 Fourier Decomposition 64 3.8 Power Spectra 66 3.9 Periodic Functions 68 3.10 Aperiodic Signals and Vibrations 69 4 The Power of Autocorrelation 71 4.1 Obtaining Autocorrelation Functions 74 Box 4.1 Autocorrelation Example: Temperature in Fairbanks 72 4.2 Autocorrelation and Power for a Sum of Sinusoids 74 Getting the Autocorrelation 74 Computing the Power Spectrum 76 4.3 Autocorrelation for Any Signal 76 Computing the Autocorrelation 77 Autocorrelation by Color 77 Contents ix 4.4 Power Spectrum from a General Autocorrelation 79 Power Spectrum by Color 81 The Wiener-Khinchin Theorem 82 4.5 The Uncertainty Principle 82 4.6 Autocorrelation and the Chorus Effect 85 4.7 Noise and Autocorrelation 87 Autocorrelation and Fast Echoes 87 Masking Signals with Noise 87 Box 4.2 Famous Fourier Transform Pairs 88 5 Sonograms 89 5.1 What Is a Sonogram?

89 5.2 Choosing Sonogram Parameters 91 6 Capturing and Re-creating Sound 93 6.1 Galileo—The First Recording?

93 6.2 Phonautograph—Sound Trace 95 6.3 Microphones and Loudspeakers 97 6.4 Sound Reproduction Fidelity 98 The Problem of Head Movement and Visual Concordance 99 The Edison Diamond Disc Phonograph 99 6.5 Digital Recording and Playback 100 6.6 Impulse Response and the Re-creation of a Soundspace 103 III Making Sound 105 7 Sources of Sound 107 7.1 Ampli cation without Active Ampli ers 108 Walls as Passive Ampli ers 109 Reactive versus Resistive Forces 110 7.2 The Method of Images 111 The 30-degree Wedge 112 7.3 The Horn 114 S .afı̄ al-Dı̄n Gets It Right in the Thirteenth Century 114 Low-frequency Piston Source 116 Monopole Source in a Pipe 117 Horns as Impedance Control 117 The Mouth of the Horn 118 The Shape of the Horn 118 Box 7.1 The Exponential Horn 119 Speaking Trumpets and Ear Trumpets 120 Box 7.2 Horns through the Ages 121 7.4 The Siren 125 Software Siren 127 7.5 Reciprocity of Sound Propagation 128 Box 4 x Contents 7.6 Law of Similarity 130 7.7 Dipole Sources 131 Dipoles as Acoustical Short-circuiting 132 Dipoles as Destructive Interference 132 Example Dipole Sources 133 Relative Phase of Loudspeakers 134 Simulations of a Dipole Source 135 Baf ing a Dipole 136 7.8 Tuning Fork—A Quadrupole Source 137 7.9 Supersonic Sources 138 Lightning and Thunder 142 7.10 Sound Launched by Surfaces 142 Sound Launched by a Baf ed Piston 143 Building Up Larger Pistons from Small Ones 144 Force Goes in Phase with Velocity for Larger Pistons 145 7.11 Sound Launched by Surface-bending Waves 146 Supersonic versus Subsonic Surface Waves 148 The Critical Frequency 149 Sound Radiation Pattern from Surface Waves 150 Box 7.3 Seneca Guns and Cookie Cutters 153 7.12 Soundboards and Surface Sound Generation 158 Box 7.4 The SST That Never Was 159 7.13 Thermophones—Sound without Vibration 161 Box 7.5 Sound That Won't Leave 162 7.14 The (Many) Other Sources of Sound 163 The 95 dB Sun Chips Bag 163 8 Making a Stretched String 165 8.1 Single Bead 167 Tension and Force 167 The Motion of the Bead 168 8.2 Two Beads 169 Box 8.1 Working with Loaded String 169 The Sinusoid Reigns Supreme 170 8.3 Three Beads 171 8.4 Combining Modes 172 8.5 More Beads 172 The Sound and Spectrum of a Pluck 173 Box 8.2 Spectrum for a Large Number of Beads 176 8.6 Putting Shape and Time Together 178 8.7 Combining Modes 179 8.8 Traveling Waves on the String 180 Standing versus Traveling Waves 181 Fourier Again 181 Ends and Boundaries 181 Contents xi Box 8.3 Experiment with Loaded String 182 Periodic or Not?

183 8.9 The Imperfect String 184 Weighted String 184 Real Strings 185 8.10 Membranes as Stretched Bead- lament Systems 185 8.11 A Metal Chair 187 8.12 Decomposing Complex Vibrations 187 Mersenne and Sauveur 188 9 Resonance Rules 191 9.1 Resonance and Constructive Interference 192 Proximity Resonance Revisited 192 Equivalent Viewpoints 192 Generalizing Proximity Resonance to Any Constructive Addition 193 Box 9.1 Echoes from Atoms 195 9.2 De nition of Driven Resonance 196 Remote versus Local Sources: Reciprocity 197 Multiple Sources 198 Autonomous Systems 199 Box 9.2 Resonance and the Divine Harmony 199 10 Damped and Driven Oscillation 202 10.1 Friction and Work 202 10.2 Friction and Decay 203 Kicked Damped Oscillator 204 10.3 Quality Factor Q 204 Equivalent De nitions of Q 204 10.4 Driving the Oscillator 207 Frequency of the Driven System 209 10.5 Resonance 209 Phase of the Drive: Reactive versus Resistive Force 209 Power near Resonance 211 10.6 Impedance and Forced Oscillation 212 Power, Impedance, and Admittance 213 Oscillator versus Wave Resonance 214 Driving a String 215 10.7 Coupling of Two or More Oscillators 216 Pure Modes 216 Two Coupled Pendula of Different Frequency 218 10.8 Tug-of-War: Resonance versus Damping 221 A Physical Model 223 11 Impulse Response 225 11.1 Impulse and Power 226 Five Easy Cases 226 Power and Echo 229 xii Contents 11.2 Average Power Theorem 231 Caveat for Proximity Resonance 232 11.3 Sculpting a Power Spectrum 232 Echo, Resonance, and Q 234 The Pop of a Cork and Its Echoes 235 Sculpting Principle for Any Signal 236 12 Impulse and Power for Complex Systems 239 12.1 Mode Density 239 12.2 Strength of Isolated Resonances 240 12.3 Impulse and Power Spectrum in an Open Wedge 241 12.4 High-Q Resonances: From Isolated to Densely Packed 245 12.5 Schroeder Frequency 246 Power Fluctuations above the Schroeder Frequency 247 Statistics of the Fluctuations 247 Statistics of the Wedge Spectrum 249 12.6 Is a Piano Soundboard Resonant?

250 Reverberant, Not Resonant 251 Foiling Short-circuiting 253 13 Helmholtz Resonators 255 13.1 How Helmholtz Resonators Work 255 Box 13.1 Deriving the Helmholtz Mode Frequency 257 The Ocarina: Size but Not Shape 257 13.2 Helmholtz Resonators and the Law of Similarity 258 Higher Modes 260 Ad Hominem Resonators 260 13.3 Phase and Power 261 Preresonance 262 Postresonance 262 On Resonance 263 13.4 Resonance and Short-circuiting of Pairs of Resonators 264 13.5 Helmholtz Resonance Ampli cation of Sound 266 Resonance and Reciprocity 266 13.6 Helmholtz Resonators at Work 266 Resonators as Transducers for Sound 267 Ported Loudspeakers 268 Box 13.2 Sound Enhancement in Ancient Greece?

268 Sound Attenuation 270 Helmholtz Bass Traps 271 Your Automobile as a Helmholtz Resonator 272 14 Sound Generation by Vortices and Turbulence 273 14.1 Vortex Streets 273 Föppl Vortices 274 Wagging, Shedding, and Sound Generation 274 Contents xiii 14.2 Resonant Vortex Shedding 276 Entrainment 277 Aeolian Harps Big and Small 278 14.3 Reynolds Number 278 14.4 Edge Tones 279 14.5 Whistling—Ring and Slit Vortices 281 Instability and Sensitivity 281 14.6 What Is Happening in a Lip Whistle?

281 Box 14.1 Experiment: Second Formant Resonance 284 14.7 Sound from Turbulence 285 Jet Noise 285 Turbulence: Fricatives and Speech 286 Box 14.2 Experiment: Speech Turbulence 287 14.8 Other Sources of Noise 287 Noise from Tires 288 15 Membranes and Shells 289 15.1 Networks of Strings 289 15.2 Stretched Membranes 290 Box 15.1 Paul Falstad's Stretched Membrane Applets 290 15.3 Vibrations of Plates and Shells 292 15.4 Chladni and the Era of Modern Acoustics 292 Box 15.2 Chladni and Napoleon 295 15.5 Baf ing and Acoustic Short-circuiting 296 15.6 Bowing a Metal Plate 297 15.7 Belleplates 298 15.8 Kettle Drums 299 IV Musical Instruments 303 16 Wind Instruments 305 16.1 Propagation of Sound in Pipes—Continued 305 Resonance in Tubes—Colored Echoes 306 Wall Losses 307 Box 16.1 Experiment: Resonance Frequencies and Wall Losses in a Tube 308 16.2 Frequencies of Tube Modes 309 Cylindrical Bore Tubes 309 The Conical Bore 312 The Inside-out Implosion 312 16.3 The Trumpet 315 Partials versus Resonances 315 Shaping the Trumpet's Timbre and Playing Qualities 316 The Mouthpiece Does Triple Duty 317 xiv Contents The Bell Does Triple Duty 320 Box 16.2 Gatekeeper Resonance Effect 320 The Trouble with Treble Boost 322 Box 16.3 The Horn Function 322 The Battle between Resonance and Wall Friction 325 Power in the Upper Partials—Up or Down When a Bell Is Added?

327 The Lip Reed 330 Understanding Nonlinearities: Benade's Water Trumpet 332 Playing the Resonances on a Trumpet 334 Other Factors: Vocal Tract 336 Valves and Intonation 336 The Natural Trumpet 336 16.4 The Transverse Flute 337 Impedance of a Flute 337 The Flute Cork 338 The Golden Flute 340 16.5 The Clarinet 341 Register Holes 342 Toneholes 343 16.6 The Saxophone 345 The Saxophone Mouthpiece 346 16.7 Blown Closed versus Blown Open 346 Blown Closed 347 Blown Open 348 16.8 The Importance of Vocal Tract Resonances to Wind Instruments 349 Tract Resonances and Playability 349 Bending Down 350 17 Voice 352 17.1 Tubes That Change Diameter or Shape 352 Constriction Yielding a Helmholtz Resonator 355 17.2 The Source: Vocal Folds 356 17.3 Formants 358 Getting Q for Your Vocal Tract 359 17.4 Sayonara Source- lter Model 360 17.5 Formants and Vowels 361 17.6 Formant Tuning in Singing 362 Singer's Formant 362 17.7 Multiphonics—Playing Two Notes at Once 365 17.8 The Speaking Trumpet (Megaphone) Revisited 367 17.9 Helium and SF6 Voice 369 17.10 Vocal Disguise, Mimicry, and Gender Switching 370 17.11 Fricatives and Other Sounds 372 17.12 Organ Pipe—Vox Humana 372 18 Violin 374 18.1 Bowing, Stick-slip, and the Helmholtz Wave 375 Contents xv The Helmholtz Kink Wave 376 Nonlinear Cooperative Resonance 378 Inharmonic Strings 380 18.2 The Bridge and the Bridge Hill 380 Impulse on the Front Plate 383 18.3 Science and the Violin 384 18.4 Sound Radiation Patterns from a Violin 385 18.5 Strad or Bust?

386 18.6 The Helmholtz Air Mode 388 18.7 The Wolf 389 18.8 Summary of the Violin 390 18.9 Nondestructive Modi cations 390 Breakdown of the Helmholtz Wave 391 19 Piano 392 19.1 The Railsback Curve 393 19.2 Three Strings per Key 395 19.3 The Hammer 396 Where Should the Hammer Hit the String?

397 Shape, Mass, and Texture 398 19.4 Digital Piano 398 20 Hybrid Musical Instruments 400 20.1 Stroh Violin 400 20.2 Aeolian Harp 401 20.3 Tromba Marina 403 20.4 Instruments Based on Near- eld Capture (NFC) 403 The Marimba 404 20.5 Applying the NFC Mechanism 408 Savart's Cup and Resonator 409 Helmholtz Resonator Enhancement of a Tuning Fork 409 Wind Chimes and the Javanese Angklung 410 Other Hybrid and Unusual Musical Instruments 412 V Psychoacoustics and Music 413 21 Mechanisms of Hearing 415 21.1 Anatomy of the Hearing System 416 21.2 Outer Ear: Direction Detection 417 Repetition Resonances and Antiresonances (Peaks and Notches) 418 21.3 Middle Ear: Masterpiece of Impedance Transduction 419 Lever Action 420 21.4 Inner Ear: Masterpiece of Detection 422 Initial Frequency Sorting 422 Transduction to Nerve Impulses 424 Ampli cation and Sharpening 424 Sending Data to the Auditory Cortex 425 xvi Contents 21.5 The Bionic Ear 426 Box 21.1 Resonance and the Ear 428 22 Loudness 431 22.1 Fechner's (Weber's) Law 431 22.2 Equal Loudness Curves 432 22.3 Masking 434 22.4 Measuring Loudness 435 23 Pitch Perception 437 23.1 Overview 437 23.2 Pitch Is Not Partial 438 23.3 Pitch Is Not Periodicity 440 23.4 Pitched Battles 440 23.5 The Siren 442 23.6 Ohm's Law 443 23.7 Seebeck's Mistake 444 23.8 Ohm's Blunder 444 23.9 Helmholtz Falls Short 445 23.10 A Dramatic Residue Pitch Effect 447 Truth or Illusion?

449 23.11 Autocorrelation and Pitch 449 23.12 A Simple Formula for Pitch 450 23.13 Examples: Autocorrelation and Pitch 453 23.14 Seebeck's Pitch Experiments 456 The Marquee Effect 458 23.15 Shepard Tones 459 Shepard Tones and Autocorrelation 461 23.16 Chimes: Pitch without a Partial 463 The Hosanna Bell in Freiburg 464 Pitch of a Kettle Drum 465 23.17 Repetition Pitch 466 Huygens at Chantilly 467 Temple of Kukulkan, Chichén Itzá 468 Ground Re ections 469 23.18 Quantifying Frequency 472 Cents 472 Just Noticeable Difference (JND) 473 Time or Place?

473 23.19 Pitch Class, the Octave Ambiguity, and Perfect Pitch 475 23.20 Parsing and Persistence: Analytic versus Synthetic Hearing 476 23.21 Deutsch's Octave Illusion 477 Pitch and Loudness 478 23.22 An Extended De nition of Pitch 478 Contents xvii 24 Timbre 480 24.1 Timbre and Phase 480 Shape Depends on Phase 480 Ohm-Helmholtz Phase Law 481 Rationale for Insensitivity to Relative Phase of Harmonic Partials 482 24.2 Amplitude and Timbre Beats 483 Generalizing the Concept of Beats 484 24.3 Waveform Beats and the Phase Law 484 24.4 The Perception of Waveform Beats 487 24.5 A Dramatic Phase Sensitivity 488 24.6 Timbre and Context 489 Box 24.1 Helmholtz's and Koenig's Ingenious Tests of the Ohm-Helmholtz Phase Law 490 24.7 Timbre, Loudness, and Shock Waves 492 25 Phantom Tones 493 25.1 Lies and Illusions 493 25.2 Sounds That Aren't There 495 Hearing Phantom Tones 495 25.3 How and Where Do Phantom Tones Arise?

496 Mechanical Causes 496 Neural Causes and the Auditory Cortex 497 25.4 Beat Tones 499 Phantom Loudness Beat Tones 499 Examples of Beat Tones 500 25.5 Nonlinear Harmonic Generation 501 Box 25.1 Experiment in Nonlinear Harmonic Generation 502 Box 25.2 Rudolph Koenig 503 26 Dissonance and Temperament 505 26.1 Critical Bands 507 Autodissonance 508 26.2 Figuring Dissonance 510 26.3 Helmholtz Theory of Consonance and Dissonance 512 Trouble with 7 and 11?

515 26.4 The Impossible Perfection of Pythagoras 516 The Perfect Fifth as the Basis for a Musical Scale 516 Another Path to a Musical Scale 518 Pythagorean Just Intonation 519 26.5 The Pythagorean Comma 520 26.6 The Circular Musical Scale and the Circle of Fifths 522 The Wolf Fifth 523 26.7 The Modern Solution: Equal Temperament 524 The Barbershop Seventh—Just versus Equal 526 26.8 Stretched Scales and Partials—Extreme Tests of Dissonance Theory 527 26.9 Downshifting Chopin 528 xviii Contents VI Soundspaces 531 27 Modern Architectural Acoustics 533 27.1 Rooms as Resonant Spaces 533 Why Do Surfaces Absorb Sound? 536 Coloring Sound with Walls 537 27.2 W. C.

Sabine and Architectural Acoustics 537 The Right Questions 538 Decay of Reverberations 539 Box 27.1 Sabine's Experiments 540 27.3 Understanding T60 540 Box 27.2 Deriving the Sabine Reverberation Formula 542 Rectangular Rooms and the Law of Similarity 545 Strength G 546 The Problem of Low Frequencies 548 27.4 Diffusion by Walls 548 27.5 Special Shapes 550 Box 27.3 Acoustics of the Mormon Tabernacle 551 27.6 Auditory Scene 551 27.7 The Precedence Effect 552 Electronic Enhancement in Concert Halls 553 27.8 Blind Navigation in Spaces 554 27.9 Frequency Response of Rooms and Concert Halls 555 Power Spectrum and Mode Density 555 Point-to-point Frequency-dependent Transmission 556 27.10 Reverberation Timeline 559 27.11 Best Hall Acoustics 560 27.12 Acoustical Triumphs and Disasters 560 Boston Symphony Hall 561 Philharmonic Hall, New York 561 Munich Philharmonic 563 28 Sound Outdoors 564 28.1 The Battle of Gaines Farm 564 28.2 Long-range Sound Propagation in the Atmosphere 565 Upwind versus Downwind 567 28.3 Scintillating Sound 569 28.4 Echoes 571 The Mystery of the Harmonic Echo 572 Flaws in Rayleigh's Arguments 574 Sir William Henry Bragg Gets into the Act 575 Bibliography 579 Index 583 Preface No book about vision and visual art is devoid of diagrams and reproduc- tions, yet books about sound and music are traditionally mute.

It has been possible to print images in books for centuries, but conveying sound has historically been much more dif cult. The situation started to change when the Laboratory of Psychophysics of Harvard University (active from 1940 to 1972) under Professor Stanley Smith Stevens produced and recorded 20 demonstrations on psycho- acoustics, plus an explanatory booklet. Later Houtsma, Rossing, and Wage- naars created a set of improved demonstrations on a CD illustrating many important psychoacoustic phenomena. Available now on the Internet, their work has been recommended listening by many texts.

This was a good beginning, but new technology has made it possible and relatively easy to do far more. This book is integrated with many example sound les and interactive applets that generate and analyze sound. They are available on the book's website, whyyouhearwhatyouhear.com. If a picture is worth 1000 words, so too is a sound le. Sounds and effects created and analyzed on the y with well-conceived applets are worth 10,000 words. Computer animation, Java, MAX patches, Mathematica applets, sound processing and analysis tools (such as Audacity) not to mention the World Wide Web, all ow into crisp display screens and high- delity headphones—at little or no expense.

Any book on sound and acoustics that doesn't take advantage of these technological miracles is missing a huge opportunity. The many excellent books of the past, no matter how good they otherwise are, cannot provide the reader with the rsthand interactive knowledge and listening experience we integrate into this book. Yet we hope to have given new life to some parts of these older classics, by providing interactive examples illustrating some of their major lessons. If nothing had evolved in the last 20 years, it would be quite pre- sumptuous to offer a conceptually higher level book about acoustics to xix xx Preface the nonspecialist.

But things have evolved: anyone with a laptop has a fully portable sound laboratory and recording studio that might have cost hundreds of thousands of dollars not so long ago. Now it is possible to achieve true understanding by showing and doing, at one's own desk or anywhere a personal computer is taken. We seize this new opportunity to actually explain sound to the nonspecialist, rather than to present descrip- tions or mnemonics received from on high. This approach certainly puts more demands on the reader, but the reward is an intuitive understanding previously reserved for the best sound engineers and acousticians. In spite of its long history, acoustics is still wide open to discovery.

The level of this book is only a step away from original research, and many times we point the way to something that needs further investigation. With the approach we take here and the new tools available, readers can experience the sense of discovery that scientists crave. New phenomena or interesting variants on known effects can be exposed using the tools and point of view provided here. You will certainly learn much about your own hearing, including whether it is "normal" and whether you have special abilities or tendencies, such as the ability to listen analytically rather than holistically to complex tones.

Musical instruments are understood through representative cases that focus on the way these instruments actually work. We trust the reader to extrapolate from trumpet to trombone, from violin to viola. This focus enriches the understanding of the important physical effects at play and explains rather than describes the instrument.

Coupled resonators, Fourier analysis, autocorrelation, impulse response, impedance mismatch and re ection at open tube ends and toneholes, wall losses, phase of drives near resonance, and launching of sound by accelerating surfaces all help explain the effects of a mouthpiece, bell, violin body, the phase of the lip buzzing on a trumpet, the bending of notes on a sax, and so on. We do not shy from controversy; indeed, we welcome it and even try to stir some up from time to time. Nothing could be a better learning experience for practitioners or students than to participate in spirited debate.

It gives us practice in applying the principles and demonstrates to students that their own struggles are not so distant from those at the research frontier. This book grew out of years of teaching The Physics of Music and Sound, rst a core curriculum course and then a general education course after Harvard switched to that system. Originally designed and taught by Professor John Huth and myself, the course was never intended primarily as an excuse to teach physics to nonscience undergraduates; rather, our rst love and our rst intent was to really understand sound and the mechanisms that generate it and receive it.

It is always a challenge to arrange a linear path through a multidi- mensional subject. Rather than adopting a "the rewards will come later" approach, we seed many of the applications as early as possible as we Preface xxi introduce the principles. This does mean that not all the relevant material about pianos—for example, piano soundboards—is actually in the chapter on pianos. There is a signi cant component of spiral learning: we are never nished with the topic of resonance, for example. Most universities have general education requirements that help to en- sure a liberal education.

For humanities students, these requirements used to mean enrolling in Rocks and Stars or Physics for Poets classes, often with predictable results. These courses are now evolving into more interesting and relevant ones, as professors are discarding the "eat your spinach" approach in favor of engagement and relevance. Case in point: Physics for Poets has become Physics for Future Presidents. Poets don't need much physics, or at least they don't think they do; modern presidents do. Figure P.1 Illustration of the divine monochord, in the book De Musica Mundana, by Robert Flood. Notice the hand of God tuning the monochord.

The connection between length proportions on a string and pleasing musical intervals is attributed to Pythagoras. According to the legend, Pythagoras as early as 600 BCE used a monochord, a stretched string over a resonator, to connect intervals like the octave and the fth with length ratios of 2:1 and 3:2, respectively. This reinforced deep mysticism about the fundamental connection between small whole numbers and the clockwork of the heavens. It is said that Pythagoras' followers believed only he could hear the music of the spheres, the divine harmonies of small integers governing the motion of the planets and the heavens.

In 1618, English physician and mystic Robert Fludd wrote De Musica Mundana, which included a compelling illustration of the divine monochord ( gure P.1), elevating the monochord to the governing engine of the universe. This idea of a "vibratory universe" has not died away. If you Google that phrase, you will get many websites physicists think of as crackpot; the mysticism side of this idea is as strong as ever. In fact, the vibrating plates of Chladni became enormously popular around 1800. These are taken up in section 15.4 and seem to have provided a segue between the scienti c and the mystical that has lasted to this day.

It is well-known that Hans Christian Ørsted, the discoverer of electromagnetism and an unassailably brilliant scientist, took off in a mystical direction for quite a while after he saw and heard Chladni plate vibrations. The vibratory universe idea has not been entirely left to mystics, how- ever. Indeed, I cannot think of any aspect of the physical universe that is not vibratory at some level. Quantum mechanics teaches us that matter is actually made of waves, which have the usual properties of wavelength and frequency; the evidence of this is abundantly clear, but of course we can't go into it here. Light, microwaves, radio waves, and so on exhibit obvious vibratory wavelike properties.

Cosmologists tell us that the whole universe is still vibrating in various modes as a remnant of the Big Bang. Even the most modern and abstruse corner of theoretical physics, string theory, supposes that the different particles found in nature are distinct vibratory modes of tiny stringlike objects. I am not a mystic, but I do believe the universe is vibratory.

xxii Preface The science of acoustics Earth sciences Engineering Life sciences Arts O c e a n o g r a p h y U n d e r w a t e r s o u n d Physics of Earth and atmosphere Seismic waves Sound in the atmosphere Medicine Bioacoustics P h y s i o l o g y H e a r i n g P s y c h o lo g y E le c t r ic a l a n d c h e m ic a l Ele ctr oa co ust ics S o n i c a n d u l t r a s o n i c e n g i n e e r i n g Shock and vibration Noise Room and theater acoustics M usical scales and instrum ents C o m m u n i c a t i o n Psychoacousti cs M echanical Architectural Visual arts M usic S p e e c h Fundamental physical acoustics Mechanical radiation in all material media Phonons Figure P.2 A gure by R.

Bruce Lindsay showing the range and breadth of the eld of acoustics. Subjects treated extensively in this book are highlighted in darker blue; subjects partially treated are shown in lighter blue. Adapted from R. Bruce Lindsay, Acoustics: Historical and Philosophical Development, Dowden, Hutchinson, and Ross, Stroudsburg, PA, 1973. This universality is another reason for studying sound, the most acces- sible of all vibrational and wavelike manifestations, for in doing so you are studying the clockwork of the universe. Perhaps this is simply a less poetic way of expressing the idea of music of the spheres, which so captivated Pythagoras and those after him. R.

Bruce Lindsay, the late professor at Brown University, understood the universality of acoustics in a more practical way. In the introduction to his marvelous book of reprints of some of the seminal works and papers on acoustics in the past few thousand years, Acoustics: Historical and Philosophical Development, Professor Lindsay created a graphic that makes clear the vast range of applications of acoustics and some relations among them. A modi ed version is shown in gure P.2. Topics that are key to this book are highlighted in dark blue; some related areas that we touch upon are shown in lighter blue.

How to Use This Book The book was written with a wide range of interests and musical/acoustical backgrounds in mind, from neophyte to professional. Students, musicians, sound engineers, psychologists, phonetics and audiology professionals, and anyone wanting or needing to know more about sound and music generation and perception can expect to emerge with a real understanding of sound, because the real story is told. Not much prior technical sophis- tication is demanded, yet teachers, musicians, acoustical engineers, and scientists will recognize a fresh perspective and hopefully be entertained on almost every page.

The book is designed so that students of the subject are not hindered by the subtext for the insider. Rather, students are presented the truth and pretty much the whole truth at the minimum possible level of technical sophistication. There is too much material here for a one-semester undergraduate course. The book makes various pathways through parts of the subject possible. An instructor can steer a course (several are suggested here) through the material, con dent that curious students with an interest in something not speci cally covered in class can nd it in this book.

The website paired with the book, whyyouhearwhatyouhear.com, is an essential addition to the package and a multifaceted resource. The book is heavily cross-referenced to help smooth the way for creative pathways through the material. Many of the chapters and parts are mostly self-contained—some are almost books in themselves. This is partly a consequence of the spiral learning approach, so that concepts introduced earlier keep reappearing, not just mentioned in passing but brought up anew in a context that enriches understanding. These facts make it quite possible and even recommended to read the book on a "need-to-know" basis.

For example, if you play the violin, start with that chapter, and follow all the cross-references to the violin from other chapters. If you nd you are fascinated by the bridge hill resonance and the reason for its existence, you might nd yourself reading the chapters devoted mainly to the concept xxiii xxiv How to Use This Book of resonance and impulse response. Before long, you might put a little piece of putty on a violin string to see what happens (you'll be surprised). To understand the drastic result, you might end up reading about the Helmholtz wave, harmonic vibrations on a string, and stick-slip motion of bow and string.

Next you might buy a cheap, tiny accelerometer (there's one in every smart phone), attach it to your violin and then to your laptop, and start making measurements on your own violin. Free sound capture software will record and analyze all the data you need. Who cares if you read the whole book? You're on your way to acoustical discovery. The psychophysics chapters are another good place to start; they are rather self-contained in some respects, but de nitely enriched by all the material before and after if you choose to explore further. For example, dive into the chorus effect (section 4.6), and branch out from there. You'll read about autocorrelation.

Now you have a reason to know what autocorrelation is, in order to understand how a chorus can have a de nite pitch even through every singer is a little bit off pitch, or doing vibrato, and so on. Musically inclined readers might want to start with psychophysics and especially pitch perception, moving into the theory of dissonance and the chapters on systems of musical scales, nishing with chapters on musical instruments and the acoustics of musical spaces. Forays into other cross-referenced sections of the book would be required for the best understanding, but reading the whole book would not be required.

If singing, phonetics, and voice are a special interest, it is possible to start in chapter 17, backtracking to sound in tubes and sound from turbulence as the topics arise. A more conventional and "safe" approach for a college class would be to introduce qualitative ideas in part I (chapters 1 and 2), further develop the language and analysis tools in part II, and then introduce resonance through the effects of walls and horns, jumping over the more technical aspects of resonance and impulse response, and then proceeding directly to musical instruments (part IV) or psychoacoustics (part V), backtracking cross-references where necessary.

Individual or class projects could be assigned as forays into the chapters on impulse response or architectural acoustics, for example. A casual reader will nd much material of historical and human interest, including the culture of acoustics and waves. Fascinating characters like Ernst Chladni and Sophie Germain enliven the subject, as do scienti c curiosities, matters of importance to society, and so on. For example, why was Moodus, Connecticut, named by the Indians for sonic booms long before settlers arrived?

What could cause pieces of sod weighing several tons and resembling cookies from a giant cookie cutter to wind up 75 feet from the hole they left behind, as has happened in several places in the world? How do whales communicate over thousands of miles by diving down almost a kilometer? Why is it that you can easily be heard when you How to Use This Book xxv shout downwind, but you can't hear what anyone says when they shout upwind back to you? On which side of a busy highway would you prefer to live? These and many other stories and examples are to be found within the pages of this book.

The chapters can be read and appreciated without the use of a computer to download and play sound les, run demonstrations, and measure and analyze sound, but it is highly recommended that you get interactive to best assimilate the subject matter. Descriptions, screen shots, and the like are provided, but nothing beats the hands-on, ears-open experience of trying and testing the concepts for yourself. Some experiments, like pitch or phantom tone perception, are done on yourself. Your perceptions may differ from the norm, and with the ears-open approach you will nd yourself listening for and able to hear new aspects of sound.

If you are a performer, you will become aware of new aspects of sound that you may be able to control. Verso Filler Page z z How Sound Propagates Sound takes place when bodies strike the air, ... by its being moved in a corresponding manner; the air being contracted and expanded and overtaken, and again struck by the impulses of the breath and the strings, for when air falls upon and strikes the air which is next to it, the air is carried forward with an impetus, and that which is contiguous to the rst is carried onward; so that the same voice spreads every way as far as the motion of the air takes place.

—Aristotle (384–322 BCE), Treatise on Sound and Hearing More than two thousand years ago, Aristotle correctly declared that sound consists of the propagation of air pressure variations. Even to the casual observer, sound is plainly revealed to be a conse- quence of vibrating or pulsating objects in contact with air. Surfaces feel a force from all the molecules colliding with them; every molecule bouncing off the surface gives it a tiny shove.

A bounce is a change of velocity and thus an acceleration, imparting a force F on the molecule (and an equal and opposite force acting on the surface) according to Sir Isaac Newton's law F = ma, where m is the mass of the accelerated molecule, and a is the acceleration. Fluctuations of pressure above (condensations) and below (rarefactions) the average pressure, arriving at the surface as sound, cause a very small increase or decrease in the number of collisions per second, and a cor- responding tiny but measurable change of force on the surface.

These uctuations above and below the ambient pressure are called the pressure amplitude δP, where the total pressure is P = P0 + δP, and P0 is the ambient pressure. Usually only the amplitudes matter to us; it is changes in pressure that we hear, not the ambient pressure. We (and other animals) however are spectacularly sensitive to these changes; a pressure uctuation of just a few parts in a billion (a few billionths of an atmosphere) is enough for us to hear if it happens fast enough. 3 4 Part I Sound Itself As small as it is, the tympanum is huge on the molecular scale.

There are so many molecules colliding with it every millisecond (roughly 1023 — that's 1 followed by 23 zeros) that they average out and give a nearly steady pressure, amounting to about 14 lb of force on every square inch. Air pressure is usually measured in kilopascals (1 kPa = 0.145 pounds per square inch, or psi). Sea-level air pressure is about 100 kPa, or 14.5 psi. The tympanum membrane, which separates the middle and outer ear, normally has equal air pressure on both sides, so there is no net force on it, except for tiny uctuations. Aristotle could not have known that air is a seething mass of molecules crashing into one another.

More than a billion collisions are suffered by every molecule every second at sea level and room temperature. In spite of all the collisions, air is mostly empty space: the molecules occupy only about one part in 5000 of the available volume. Think of 10 bumper cars in an area the size of a football eld. You might think that this was a relatively safe, low density of cars—unless each car was traveling at thousands of kilometers per hour. There would be many col- lisions every second. Between collisions, molecules speed along a straight path at typically half a kilometer per second, managing to travel only a tenth of the length of a typical bacterium before suffering another collision.

The density and speed of air molecules are in this way suf cient to explain atmospheric air pressure and the speed of sound. Individually, the air molecules (mostly diatomic nitrogen and oxygen) act like drunken messengers ying and colliding every which way. Nonetheless, these col- lisions can collectively communicate even slight uctuations in pressure to neighboring collections of molecules, which in turn pass them on to their neighbors, leading to sound propagation.

Air molecules are usually not traveling directly along the path of the sound wave; the information that there is higher or lower pressure somewhere propagates no faster than the average speed of molecules along a given direction. The typical 500 meter/second (m/s) molecule is traveling either in the wrong direction or only 300 to 400 m/s along the direction of propagation of the sound. Thus the effective speed with which the morass of mole- cules communicates pressure variations is less than their average speed of 500 m/s. The measured speed of sound in air is about 343 m/s at room temperature.

The "seething mass of molecules" picture explains why the speed of sound is insensitive to pressure, since pressure hardly affects the speed of individual molecules. They crash into each other more often at high pressure, but between collisions they travel at a speed that depends only on the temperature, not the pressure. The speed of sound on Mount Everest is nearly the same as at sea level, if the temperatures are the same. The average speed of molecules is proportional to the square root of the temperature, and inversely proportional to the square root of the mass of the molecules in the gas.

Chapter 1 How Sound Propagates 5 "Helium voice," the Donald Duck–like sound when someone speaking has just inhaled a puff of helium, is the result of the much higher speed of sound in helium than in air. Helium has a mass of four atomic units; air has an average mass of about 29 atomic units and 29/4 2.7. The speed of sound in helium, 972 m/s, is about 2.8 times that of air, at 343 m/s. Another harmless gas (except that like helium, it displaces oxygen and can be lethal if breathed for more than a short time), sulfur hexa ouride, SF6, is much heavier at 146 atomic units and should have a speed of 343 × 29/146 = 153 m/s; the measured value is 150, less than half the speed of sound in air.

"SF6 voice" is even more astonishing in its effect than helium voice, and in the opposite direction. (However, the nature of and reasons for the changes in the sound of the voice using helium and SF6 will be explained in section 17.9. In spite of impressions, the gases do not change the pitch of the voice!) The energy needed to make audible sound is very small. You can shout for a year, and the energy produced that winds up as sound would not be enough to boil a cup of water. A full orchestra playing loudly produces only about enough sound energy to power a weak lightbulb. An orchestral crescendo might bathe a listener in sound pressure uctuations of about 1 pascal (1 Pa).

Sea-level air pressure is 100,000 Pa, so the crescendo loud enough to damage your hearing, if it lingered too long, is varying the pressure by just 0.001%. Clearly, a very delicate detection system is at work. We will nd in chapter 21 that human hearing depends on a few thousand single-molecule links between cochlear hair cells. At the extreme—loud sound near the threshold of pain—the air pressure variations are over a million times bigger than the threshold of hearing, or about a 0.03% pressure variation, 30 Pa or so. This still seems small, and yet is almost immediately damaging!

This sound level corresponds to a power arriving at the ear 10,000,000,000,000 (1013 ) times larger than that which produces the softest sound we can hear. (The power increases as the square of the pressure variations.) The dynamic range of our hearing is truly remarkable. Why should you buy a 600-watt (W) ampli er for your loudspeakers if a full orchestra normally produces just a watt of power, 40 or 50 W at the loudest? The answer is that to reproduce sound, rather large forces must be exerted on the speaker cone to get it to vibrate in a prescribed way. The conversion ef ciency from motion of a loudspeaker cone to sound is very low.

The cones are moved with electric currents in coils near magnets, wasting considerable energy. Imagine all the effort you would expend waving your hands back and forth 1000 times. Only a tiny fraction of that energy would go into pushing air around; most of the energy expended would go into working against yourself, so to speak: internal friction, stopping your arms with one set of muscles after starting them swinging with another, working against gravity, and so on. So it is with a loudspeaker. For that matter, musicians can work up a sweat playing an instrument, all to produce well under a tenth of a watt of sound.

6 Part I Sound Itself 1.1 Push and Pushback: Impedance We need to develop a better intuitive foundation for sound propagation. The "drunken messenger" picture explains the speed of sound but applies on the molecular scale, too small to give a good sense of wave phenomena such as re ection, diffraction, refraction, and so on. For example, much of the sound traveling down a tube re ects from its open end, reversing direction rather than exiting to freedom. Why doesn't the sound just leave the tube? Why is the re ected wave a rarefaction (pressure low relative to ambient) if the incident wave approaching the end of the tube was a compression (pressure high relative to ambient)?

Why does sound of high-enough frequency (the frequency is the number of wave crests traveling by per second), on the other hand, mostly escape the tube without re ecting? There are not many references that provide a foundation for a comprehensive understanding of these sorts of phenomena; those written for engineers and physicists all too often derive equations and formulas but skimp on the intuition. Imagine dividing air into small cells. Each cell is large on the molecular scale; they are packed one next to the other.

The size of the cell is determined by the wavelength of the sound involved (there needs to be at least several cells per wavelength) and the details of any obstacles, sound sources, and so on. If we can understand how the cells communicate with each other, are pushed by and then push back on neighboring cells, we can understand propagation, re ection, diffraction, and even refraction of sound. This is our rst glance at a powerful engineer's trick, wherein the properties of complex objects are lumped into a few well- chosen summarizing properties.

These have vastly less information than the original system, but enough to carry the essential physics, and lead more easily to the correct conclusions. To understand impedance in air, we begin by considering solid elastic bodies, such as pucks on an air hockey table or coins on a slick surface. We need to understand such things in any case, because usually before air is set in motion, something more massive, like a string or a block of wood, is set in motion. Each puck or coin is a lumped object—we ignore the details of atomic or molecular structure inside, but keep essentials such as size, density, and elasticity, just as we will for air cells when we return to them.

The essentials are used to build a theory of what happens when adjacent lumps interact. You may have noticed, for example, that in a head-on collision between two pennies, one initially at rest, the moving penny stops dead in its tracks, and the second one picks up where the rst one left off. (The demonstration does not work well with quarters or coins having serrated, gearlike edges colliding with other such coins. Presumably the serrations cause a rather nonideal collision, gnashing of the gears, chattering, and so on.) Chapter 1 How Sound Propagates 7 Complete transfer of momentum does not happen when a nickel collides with a penny at rest, nor when a penny collides with a nickel at rest.

The energy of the rst, moving coin is only partially transferred to the second. If we make a line of coins, each coin becomes an agent of transfer of energy from left to right, if the rst coin was traveling in that direction. A coin that is much heavier or lighter than its neighbors will impede the transfer of energy. Two nickels are impedance matched; the stationary nickel gives as good as it gets, stopping the moving nickel dead. A penny and a nickel are impedance mismatched; a penny does not exert as much force back on the colliding nickel as another nickel would and does not decelerate the nickel all the way to zero speed. The nickel continues on its way, albeit more slowly.

Only in the case of equal masses does the energy get completely transferred from one coin to the other; this is clear since for the head-on nickel-penny and penny-nickel collisions, both coins remain in motion and that movement carries energy. If you line up 5 or 6 pennies perfectly on a slick surface and hit the end of the row head-on with another penny, you will notice the row stays intact, with the projectile penny adding to the row and the last penny popping off at the opposite end.

The impedance matching works all the way down the penny chain, each penny for an instant carrying the momentum, giving as good as it got on its left, and then almost instantly giving and getting forces on its right that stop it cold and give the momentum to the next penny. Put a nickel in the chain of pennies and the rst penny will rebound from the row; the last will still pop off the end but with less energy than before. All the energy of the rst coin is not transferred down the chain; rather, part of the energy has been re ected and part transmitted, because of the impedance mismatch, which can be blamed on the interloping nickel. The situation is depicted in gure 1.1.

The impedance of the untethered coins is proportional to their mass. Two untethered objects of equal mass, therefore, indeed have the same impedance. The bigger the impedance mismatch, the more energy is re ected and the less transmitted. The formula for the fraction of energy Figure 1.1 In the top row, a penny collides head-on with a row of ve pennies, resulting in the expulsion of the last penny in the row with the same speed as the rst penny had. The masses are all the same and the chain of pennies is impedance matched, resulting in 100% transfer of the energy from the rst penny to the last, except for friction.

In the bottom row, the presence of the nickel replacing one of the pennies causes a mismatched impedance, with some of the energy re ected back toward the rst penny, causing it to rebound; only part of the energy reaches the last penny. 8 Part I Sound Itself R that the moving mass m1 retains in a head-on collision with a stationary mass m2 is R = (m1 m2)2 (m1 + m2)2 . (1.1) If one coin weighs one-tenth as much as the rst, say, m1 = 1, m2 = 10, R will be 92 /112 = 81/121, which means 67% of the energy gets re ected on one bounce, and 33% transmitted. What Is Impedance, Really?

Roughly speaking, impedance, which we symbolize with the letter Z, measures the response of a body to a force—in fact, the force applied divided by the velocity attained (Z force ÷ velocity). A heavy object moves slower than a light one after the same force is applied starting at rest, so impedance is high for a heavy object, low for a light one. This is still a rough de nition, since in the measurement of Z, the force is taken to vary sinusoidally (see chapter 3), and the velocity, while also sinusoidal, may lag or lead the force. We will consider these complications later.

With this notion of Z (force applied ÷ velocity attained), it is possible to see why matched impedance leads to complete energy transfer between two bodies. According to one of Newton's laws, they experience equal and opposite force as they collide or interact, and what velocity is lost by one is gained by the other—just the ticket if you want to transfer energy from one place to another, or from one thing to another. One coin stops and the other takes off with the same velocity. The utility of impedance is to help determine the transfer of energy be- tween bodies. Matched impedance means ef cient energy transfer; unequal impedances mean rejection or re ection of energy.

Ideally, impedance can be determined for any part of an object, such as a block of metal or a section of pipe with air in it. If two such objects are joined somehow, an impedance mismatch (if any) can be calculated, and the transduction (transfer) of energy from one part to another can then be determined. As an example, suppose two strings of different density are tied together. We will see in chapter 8 that waves travel down a uniform string quite readily, with a velocity c = T/ρ, where T is the tension (a force) along the string, and ρ is the density (mass per unit length) of the string.

The two parts tied together have the same tension, since tension is communicated all along the string, but they have different density, and thus different wave speeds c. They also have different impedances. The impedance of transverse oscillations of a stretched string is Z = Tρ. (1.2) Chapter 1 How Sound Propagates 9 Given the densities ρ1 and ρ2 of the two string segments, we can easily calculate the re ection and transmission of energy at their junction using formulas 1.3 and 1.4 given below. Antire ection Strategies Suppose we insert a third coin between two mismatched coins, one more massive than the other.

The middle coin should be of some intermediate mass, to make the mismatches of adjacent coins less severe. It is not dif cult to show that taking the mass of the middle coin to be the geometric mean of the two original coins (that is, m = m1m2) is optimal. The transmission with the intermediate coin in place in the 1:10 impedance mismatch considered earlier then works out to 53% from the rst to the last coin; an improvement over the previous 33%. We would do even better with more intermediate coins selected to further reduce the adjacent impedance mismatches. Abrupt changes in impedance at a boundary between two objects or regions lead to low transmission of energy across the boundary.

Like the nickel in a line of pennies, regions with different impedance push back too much or too little. Suppose we have a system of one impedance Z1 on the left side connected to a second system on the right with a different impedance Z2. The sudden change of impedance at the interface causes a fraction of energy R to be re ected: R = (Z1 Z2)2 (Z1 + Z2)2 . (1.3) Thus equation 1.1 generalizes to more general types of impedance, including (as we shall see) restoring force and friction.

The transmitted energy is T = 4 (Z1 Z2) (Z1 + Z2)2 , (1.4) and the re ected and transmitted fractions sum to one: R+T = 1—that is, what is not transmitted is re ected.1 Impedance matching plays a role in many domains. In the preceding ex- ample, the coins were a "medium" for the propagation of the translational energy possessed by the rst coin. Light is similar: it propagates nicely through transparent media, such as air and glass, but these do not have the same impedance.

The impedance (called refractive index in the case of light) has a mismatch passing from air to glass, with the result that some 1The impedances are in fact complex numbers, so we have R = ("Z1 Z2"2/"Z1 + Z2"2) and T = 4[Re(Z 1 Z2)/"Z1 + Z2"2], where Re denotes the real part of the variables within the parenthesis, and " · · · "2 is the absolute value squared of · · · . 10 Part I Sound Itself light will re ect at the interface, whether it is coming from air to glass or vice versa. If a coating can be found with intermediate impedance, it can break up the impedance mismatch into two smaller steps, with the result that less light will be re ected and more transmitted.

This is the principle of antire ection-coated eyeglasses and camera lenses. The coating works better for some colors (wavelengths) than others; this explains the color sheen often seen on coated optics. As an example of the importance of impedance to sound and music, consider a violin. The body of a violin is much heavier and stiffer than a string and has a much higher impedance. Both impedances vary with frequency too. The body needs to tap into the energy of the string in order to make sound. (Vibrating strings by themselves are almost silent—this will be made clear in the following chapters; see especially the discussion of dipole sources—for example, section 7.7).

Hypothetically the string could be attached directly to the body, but there are several problems with this (see gure 1.2). The directly connected string may not set the correct body vibrations into play. Worse, there is a large impedance mismatch between string and body, preventing the string from imparting enough of its energy to the violin. (Note: We don't want the transfer of energy from string to body to be too ef cient either, lest the string dump its energy too fast.) Figure 1.2 What would this sound like? A string is attached directly to a violin body at one spot (no bridge) and to a rigid wall at another. It is bowed in the usual way.

Impedance and the Violin Air has a refractive index nair of about 1, and glass can be nglass = 1.5 or so. The refractive index is essentially impedance; the formula for the fraction of light re ected is R = (n1 n2)2 (n1 + n2)2 . (1.5) This is a 4% re ection of light for air–glass, for each surface, and there are always at least two surfaces and sometimes many more, as in expensive camera lenses. By adding an optimal single coating, with the geometric mean refractive index nair nglass , we can get this down to a 2% re ection. Multiple coatings can do even better.

Can something be inserted between string and violin body to lessen the impedance mismatch, thus allowing the energy to take two smaller steps, instead of one large one? While we are at it, can we sweeten the sound by modulating the impedance (and ultimately the loudness of the instrument) according to frequency? The answer is yes: this is the job of the bridge, as we discuss in chapter 18. The bridge is the "intermediate coin" that mediates the transfer of energy from string to body. Its impedance is cleverly tuned by choice of shape, size, and material to depend in a certain way on the frequency of vibration.

Chapter 1 How Sound Propagates 11 Figure 1.3 An Australian bullwhip can achieve supersonic speeds at the whip end, resulting in a loud crack heard some distance away. Courtesy Cgoodwin. Bullwhip—The High Art of Impedance Matching The bullwhip is a spectacular example of impedance matching ( gure 1.3). If most of the energy from the relatively heavy handle region can somehow be ef ciently transferred to a light string ("popper") at the other end, the popper will wind up moving very fast. Sudden impedance mismatches along the whip would re ect energy, so the bullwhip is gradually tapered and also carefully constructed so as to have no abrupt changes in density or stiffness.

The energy of a moving mass m due to its motion is E = 1 2 mv2 , where v is its velocity. A reasonable estimate is that the popper weighs 1/400th as much per centimeter of length as does the handle end. The energy per centimeter if the handle region weighs M kilograms per centimeter is E = 1/2MV2 , where M is the mass of a centimeter near the handle end, and V its initial velocity. If this gets transferred to the popper, then the same energy is now written E = 1/2mv2 , where m is the mass per centimeter of the popper, and v is the velocity of the popper. The ratio of the two velocities is v V = M m = 20…