www.SolutionManual.info The Physics of Musical Instruments Second Edition Neville H Fletcher Thomas D Rossing The Physics of Musical Instruments Second Edition With 485 Illustrations www.SolutionManual.info ~Springer Thomas D Rossing Department of Physics Northern Illinois University DeKalb, IL 60115 USA Neville H Fletcher Research School of Physical Sciences and Engineering Australian National University Canberra, A.C.T 0200 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 978-0-387-21603-4 (eBook) ISBN 978-1-4419-3120-7 DOI 10.1007/978-0-387-21603-4 Music - Acoustics and physics Musical instruments Construction I Rossing, Thomas D., 1929- II Title ML3805.F58 1998 97-35360 784.19'01'53-dc21 ISBN 978-1-4419-3120-7 Printed on acid-free paper © 1998 Springer Science+Business Media New York Originally published by Springer Science+Business 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 written 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, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden 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 Corrected 5th printing, 2005 springeronline.com SPIN 11340768 Preface When we wrote the first edition of this book, we directed our presentation 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 www.SolutionManual.info 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 civilization 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 musical 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 "improvements" 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 typesthe 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 reflecting the gradually changing mental image of the ideal instrument of that type The role of acoustical science in this context is an interesting one Centuries 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 scientist to point the way forward unless the problem or the opportunity has been identified adequately by the performer or the maker Only rarely 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 unravelled 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 reasonably 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 tradition, 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 community 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 originators 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 reasonable grasp of physics and who is not frightened by a little mathematics There are fine books in plenty about the history of particular musical instruments, 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- www.SolutionManual.info 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 contributed to this book Special thanks are due to Joanna Daly and Barbara Sullivan, who typed much of the manuscript and especially to Virginia Plemons, 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 publications, 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 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 Nonlinearity Appendix References www.SolutionManual.info Continuous Systems in One Dimension: Strings and Bars 2.1 2.2 2.3 2.4 2.5 Linear Array of Oscillators Transverse Wave Equation for a String General Solution of the Wave Equation: Traveling Waves Reflection at Fixed and Free Ends Simple Harmonic Solutions to the Wave Equation 10 11 13 18 21 23 25 26 28 29 32 34 34 36 37 38 39 xi Name Index Waller, M D., 79, 80, 82 Wang, 1., 130, 379 Warburton, G B., 83, 84, 87, 693 Ward, W D., 389 Warlimont, H., 686, 731, 732 Webster, A G., 207 Weibel, E S., 207 Weinreich, G., 121, 125, 252, 280, 301, 303, 304, 305, 310, 317, 333, 349, 385, 408 Weyer, R D., 344, 347 Wijnands, A P J., 405 Williams, A., 667 Williams, J., 260 Wilson, T A., 423, 486, 504 Wogram, K., 374, 375, 377, 378, 379, 380, 386, 532, 731 Wolf, D., 561 741 Wolfe, J., 223 Woodhouse, J., 65, 88, 278, 281, 283, 313, 665, 720, 723, 724 Worman, W E., 423, 486 Wornum, R., 352 Yongle, 703 Yoshikawa, S., 408, 456, 513, 515, 524, 531 Young, F J., 433 Young, R W., 390, 391 Zener, C., 713, 714 Zhao, H., 604 Zheng Darui, 704 Zick, M., 389 Zuckermann, W J., 340 Zumpe, J., 352 www.SolutionManual.info Subject Index A-weighting, 161 absorption, of sound waves, 165-167 accordion, 413 acoustic impedance of infinite pipe, 191 measurement, 222-223 admittance mechanical, 20, 32 of violin bridge, 302 air attenuation coefficient, 166 speed of sound, 158 wave impedance, 159 air damping, of string, 53-54 air jet acoustic disturbance of, 509-511 acoustic drive by, 511-516 generator admittance, 517-521 geometry, 512 harmonic generation, 521-525 regeneration mechanism, 516-521 rigorous treatment, 521-522 turbulence noise, 528-529, 531 turbulent, 508 velocity profile, 505, 508, 509 wave growth on, 505, 506, 508 wave speed on, 504-505, 506, 507, 508 waves on, 503-509 air loading of membrane, 75-76 Green function method, 588-590 on kettledrums, 585 piston approximation, 587-588 air modes of cello, 318 of harp, 340 of violin, 286 alembas, 643, 645 aluminum, elastic properties, 625 "American" organ, 413 amplitude, complex, 6-7, 29 analog network, for simple oscillator, 30-32 antiresonance, 32, 119 Arundo donax, 725 attack transient, of oscillator, 22 attenuation, of sound waves, 165-167 attenuation coefficient of air, 166 in pipes, 196 bagpipes, 498-500 balalaika, 267 Balinese drum, 618, 619 bamboo, 725 banya, 610 bars bending waves, 58-60 driving point impedance, 97-98 end conditions, 6~3 longitudinal waves in, 56-57 moqe frequencies, 62-63, 64 thick, 63-64 torsional vibrations, 66-67 bass drums, 583, 599, 601-602 concert, 599 decay times, 602 743 744 Subject Index bass drums (cont.), mode frequencies, 601-602 bass handbells, 699 bassbar, 297 bassoon, 420, 480, 494-495, 725, 726 blowing pressure, 482, 483 cutoff frequency, 476 dimensions, 464 early, 489 modern, 488 reed, 402, 490, 725-726 batter head, 599, 601, 602-606 bayan, 610 beats, 9-10, 22 bell lyra, 624 bell plates, 665 bells See also handbells ancient Chinese two-tone, 70Q-702 bronze, 701 clappers, 676, 699-700 extensional modes, 678-679 history, 675 major-third and minor-third, 679, 685-686, 687, 688 materials for, 731-732 mode frequencies, 676, 677 modes of vibration, 676-678 radiation from, 186, 697-698 scaling, 688, 690-691 sound decay, 686 strike note, 682-685 temple, 703-705 tubular, 95-96 tuning and temperament, 681-682 tuning distribution among, 683 wall thickness, 691 warble, 686, 688 bending stiffness of membrane, 585-587 Bernoulli effect, 402, 403, 409, 493, 494 Bessel functions, 74, 75, 78, 169, 184, 214, 439, 585 Bessel horn, 432-433, 440, 462 bhaya, 610 blowing pressure recorder, 536, 537 threshold, 409, 410 woodwind instruments, 481-482 bongo drums, 583, 620 boobams, 583 boundary condition, hinged, 585 boundary layers, 166, 194 bow, 31Q-311 digital, 280 finite width, 283 reciprocal, 316-317 speed and force, 278-280 brass, 728, 730-731 elastic properties, 625 brass instruments blowing pressure, 456 cutoff frequency, 474-476 directivity, 439-440 dynamicrange,457 flow waveform, 446-448 frequency domain analysis, 451-453 history, 429-431 horn profiles, 431-433 input impedance, 435, 436, 449 lip motion in, 445 materials for, 730-731 mouthpiece, 433-437, 438-439 mouthpiece pressure waveform, 446, 447, 448-449 mutes, 453-455 performance technique, 455-458 power input, 457 power output, 457 shock waves, 448 slides and valves, 440-442 spectrum, 438, 439, 453, 455 system analysis, 434-435 system diagram, 434 time domain analysis, 451 transfer function, 439 transients, 450-453, 456 breathing mode, 679, 718 bridge cello, 298 motion of, 146 one-sided, 268 string coupling by, 12Q-124 www.SolutionManual.info Subject Index violin, 297-300 bronze, 728, 731-732 bronze bells, 701 bugle, 431 bulk modulus, of air, 156, 158 cane, 725 synthetic, 727 capped reeds, 497-500 carillon, 675 scaling, 690 carry head, 599, 601, 602 cast iron and steel, 732 Catgut Acoustical Society, 322 cavity(ies) in guitar, 248-251 impedance of, 228 mouth, 408, 409, 410, 415 normal modes in, 167-169 saxophone mouthpiece, 497 spherical, 169 cavity modes, in violin, 292 cello, 318-319, 324 bridge, 298 directivity, 308, 309 wolf note, 312 cellulose, 719 cembalo, 340 chalumeau, 467, 486 characteristic· impedance, of string, 51 chimes, 95-96, 641-642 mode frequencies, 641, 642, 643 ch'in, 334 Chinese gongs, 149 Chinese two-tone bell, mode frequencies, 701, 702 Chladni patterns, 80, 82, 89, 91, 291, 611, 612, 666 Chladni's law, 79, 68Q 681 modified form of, 650 chordophone, 711 cittern, 267 clappers, bells, 676, 699-700 clarinet, 419, 423, 424, 468, 486-491 acoustic power, 491 blowing pressure, 482, 483, 491 745 cutoff frequency, 476 dimensions, 464 early, 489 history, 486-489 input impedance, 473 key system, 487-489 modern, 488 multiphonics, 491 reed, 402, 490, 726 spectrum, 490 clarsach, 338 clavichord, 348 construction, 347-348 decay of sound, 349-350 radiated power, 349 soundboard, 349, 350 coincidence frequency, 188, 305, 698 composite materials, 726-727 concertina, 413 conga drums, 583, 618-619, 620 copper, 729 elastic properties, 625 cor anglais, 494 cornet, 449 cornett, 430 coupling, of vibrators, 102-125 coupling strength, 105-107 creep, 717 critical frequency, 698 for higher pipe modes, 192 for violin radiativity, 305-306 crotales, 663, 664 mode frequencies, 663, 664 crystallites, 715 cupro-nickel alloy, 730 cutoff frequency brass instruments, 474-476 finger hole, 476, 485 flute, 548 woodwind instruments, 476 cymbals, 150, 649-656 chaotic vibration, 655-656 decay times, 653 mode frequencies, 650-652 modes of vibration, 65Q-652 spectrum, 654-655 striking modes, 653-654 transients, 652-655 746 Subject Index damping air, 53-54 internal, 54-55, 732 of oscillations, 11-13 radiation, 731, 732 steel pans, 670 of string, 53-56 by supports, 55-56 of torsional waves, 67 viscous, 716 wall, 193-196 in wood, 722-723 dan tranh, 334 decay time, 12, 19, 22 of piano, 391, 392, 393 of piano sound, 383-387 of string, 54-56 degeneracy, on square membrane, 72-73 didjeridu, 429, 458 dipole source, 172-173 directional index, for pipe radiation, 201, 203 directional tone color, 308 directionality of brass instruments, 439-440 of woodwind instruments, 480-481 dispersion of bending waves, 59, 77 in stiff strings, 65-66 dispersion curves, 666 double bass, 318-319, 326 doublet effect, 601 drum heads, synthetic, 727 drums, 94-95 history, 583 membrane, 583 modern, 583 dry friction, 716 Duffing oscillator, 137-139, 656 dugga, 610 dulcimer, 332-334 elasticity adiabatic, 157 isothermal, 157 electric analogs, 227-228 electrical circuits, coupled, 111-115 end correction at finger hole, 465-466 flanged pipe, 199 flute embouchure hole, 540, 541 and jet drive, 514 open pipe, 560 pipe mouth, 560 at recorder mouth, 535 unfl.anged pipe, 20Q-201 energy of harmonic motion, 1Q-11 of vibrating string, 40 equal temperament, 567 fatigue, 717 finger hole impedance, 472-473 in woodwinds, 466-469 fingering chart, recorder, 534 finite element analysis, 130-131 flexed plates, 665-667 flexural wave speed, 698 Fliigge thick-bar theory, 641, 642 flute alto, 539 Baroque, 537, 539 blowing pressure, 545 Boehm, 538, 539 directionality, 544 embouchure hole, 541-542 harmonic generation, 521-525 head joint, 539-543 history, 503, 537-538 materials, 543-544 metal, 729-·730 mode transitions, 525-528 performance technique, 544-548 spectrum, 546-548 forced vibration, of two-mass system, 107-111, 115-116 Fourier analysis, 40, 41-44 Fourier transform, 224, 714 French horn, 442, 458 www.SolutionManual.info ear, human frequency response, 155, 162 structure, 155 edge tones, 521 efficiency, of loudspeakers, 163 elastic relaxation, 715 Subject Index frequency, wave velocity and, 588 frequency domain, 425, 451-453 frets, in guitar, 263-264 friction bow and string, 47-50 dry, 716 gamelan chimes, 646 gamelan instruments, 645 gekkin, 267 German silver, 730 glass, elastic properties, 625 glockenspiel, 623-624 mode frequencies, 624, 626 gold, 728 gold alloys, 730 Golden Ratio, 562 gongs, 660-663 pitch glide, 661, 663 shapes, 661 spectrum, 660-661, 662 graphite fibers, 727 Green function, 141-142, 224 Green function method, air loading of membrane, 588-590 group I modes, bells, 676 guitar body resonances, 251-253 construction, 239, 240 electric, 262-263 finger strokes, 255, 256 fret placement, 263 history, 239 mode frequencies, 248 new family, 26D-261 new materials, 261-262 sound quality, 258-260 string, 241-245 three-oscillator model, 25D-251 top-plate modes, 245-248 two-oscillator model, 248-250 handbells, 675 See also bells bass, 699 clappers, 676, 699-700 English tuning, 694-695 Malmark, 697 modes of vibration, 691, 693-694 747 scaling, 696-697 sound decay, 695-696 timbre and tuning, 694-695 warble, 696 hardness, 717 hardwoods, 723, 725 harmonic balance, 425 harmonic generation in brass instruments, 442-449 in reeds, 418-422, 422-426 harmonic motion alternative expression, 29-30 energy, 1D-11 simple, 4-6 superposition, 7-10 harmonica, 413, 415, 416 harmonium, 413, 414 harp construction, 336-338 Erard mechanism, 338 soundboard, 338-340 harpsichord construction, 340-343 decay time, 346-347 design parameters, 343-346 mechanism, 341-342 radiated power, 346-34 soundboard, 341, 343-344 string scaling, 344-346, 348 string vibration, 147 head joint, of recorder, 535-536 hearing, human, 155, 162 heat conduction, 714 Heisenberg uncertainty principle, 653 Helmholtz motion, 46-48, 50, 65, 275-278, 285 Helmholtz resonance, 232, 288 in guitars, 247 Helmholtz resonator, 13-14, 228-230, 323, 435 Hertz's law, 640 holographic interferometry, 290, 295, 296, 323 of cymbal modes, 650 of drum shell, 603-604, 605 of guitar top plate, 254, 255 of handbell modes, 693-694, 695 of steel pan, 671 748 Subject Index Hooke's Law, 712 horn Bessel, 213-216, 432-433, 440, 462 catenoidal, 210 compound, 216-218 conical, 21Q-213, 216, 420, 463 curved, 22Q-222 cylindrical, 463 equation, 207, 208 exponential, 210 hyperboloid, 206 mouthpiece, 433 numerical calculations, 220 Salmon, 209 in time domain, 223-227 horn function, 208 idiophone, 623, 711 impedance characteristic See characteristic impedance driving point, 96 mechanical, 19, 21 impedance coefficients for conical horn, 232 definition, 231 for horn, 231 for pipe, 231 impedance head, 96, 97 impulse response, 480 measurement, 226-227 Indian drums, 609-615 mode frequencies, 611-614 spectrum, 612, 613, 614, 615 Indonesian drums, 618 inharmonicity and mode locking, 143 in piano strings, 388-390 input impedance calculation, 471-472 of clarinet, 473 of conical horn, 21Q-212, 463 of cylindrical horn, 463 measurement, 47G-471 of oboe, 473, 475 of open cone, 211 of pipe, 198, 202-205 of recorder, 533 of stopped cone, 212 of woodwind instruments, 470-476 intensity definition, 162 of plane wave, 163 of spherical wave, 163 intensity level, definition, 162 internal damping, 732 iron, 728 Japanese drums, 615-618 mode frequencies, 616, 617 Javanese drums, 618 jegogan, 645, 646 jet See air jet jingles, 621 just tuning, 681 kendang ciblon, 618 kendang gending, 618 kettledrums (tympani), 91, 583, 584-599 air loading, 585 history, 584 inharmonic modes, 584-585 kettle of, 584, 599 Ludwig, 591, 593 mode frequencies, 584, 590, 591-595 modern, 584 normal strike point, 595 pitch, 595 radiation, 185, 598-599, 600 sound decay, 598-599 spectrum, 595-599 theoretical analysis, 585 volume, 591, 593, 594, 595 koto, 334-336, 339, 727 kotudumi, 616, 617, 618 krumhorn, 467, 498 www.SolutionManual.info larynx, in wind instrument playing, 458 Latin American drums, 618-620 spectrum, 619, 620 lead, 729 limit cycle, 140 limiting spectrum, of woodwind instruments, 484-486, 487 Subject Index lip valve, 402, 403, 408-409, 410, 411,417-418,422,423 admittance, 437 nonlinearity, 442-449 Lissajous figures, 25-26, 27, 47, 282 longitudinal modes, in piano strings, 387-388 longitudinal vibrations, 56-57 longitudinal waves, in plate, 76 LTAS {long-term average spectrum), 308, 575, 576 lute, 264-265, 266 mallets, percussion, 639-641 mandolin, 267 marimba, 624, 626-636 arch in underside of bars, 626 decay time, 634 mallets, 639-641 mode frequencies, 626-630 oscillograms of bar, 627, 630 radiation, 634 resonance frequency, 635 resonators, 633-636 tuning bars, 627-633 materials, 711-733 acoustical properties, 718 anelastic behavior and damping, 713-716 for bells, 731-732 for brass instruments, 73Q-731 composite, 726-727 linear elastic properties, 712-713 mechanical properties, 712-717 nonlinear properties, 716-717 membrane air loading on, 75-76 bending stiffness, 585-587 circular, 73-74 effect of stiffness, 91 rectangular, 70-72 square, 72-73 membranophone, 711 metal for flutes, 543-544 for musical instruments, 728 for pipes, 728-729 metal flute, 729-730 749 metal plates, 665 metallophones, 645 mixtures, 564-566 mobility, 20, 32, 118, 119 modal analysis, 128-130 of violin, 289-291 modal wave numbers, 586 mode frequencies of bars, 62-63, 64, 81 of bass drums, 601-602 of bells, 676, 677 of cello, 320, 321 of chimes, 641, 642, 643 of Chinese two-tone bell, 701, 702 for circular plates, 78-80 of coupled electrical circuits, 112-113 of crotales, 663, 664 of cymbals, 650-652 for elliptical plate, 80 of glockenspiel, 624, 626 of hard sound board, 339 of Indian drums, 611-614 of Japanese drums, 616, 617 of kettledrums, 584, 590, 591-595 of marimba, 626-630 for rectangular membrane, 72 for rectangular plates, 80, 83, 86, 87 for shells, 94, 148, 149 of snare drums, 603, 604 for square plate, 84 of tom-toms, 606, 608 of triangles, 643, 644 of vibes, 639 for wood plates, 89 mode locking, 143-144, 479 mode shapes for guitar plates, 245-248 for koto, 335 for rectangular plate, 81-82 for square plate, 84, 85 for violin body, 288 for wood plates, 88, 90 modes cavity, 247 for circular plate, 79 extensional, in shells, 92 750 Subject Index modes (cont.), nonextensional, in shells, 92 for square plate, 87 monopole source, 172, 173 mouth, 408, 409, 410, 415 mouthpiece, 433-437 influence on spectrum, 438-439 resonance, 435-436 mrdanga, 61Q-611 multimode systems, 14Q-144 multiphonics, 144, 424, 479 multipole expansion, of violin radiativity, 303-305 multi pole source, 171-174 musical saw, 665-667 mutations, 566 mute, violin, 299 Mylar, 584 natural frequency, network analogs, 227-232 Neumann-Green function, 589 new violin family, 322-326 nickel silver, 730 noise, aerodynamic, 528-529, 530 nonlinear oscillator, 133-150 forced, 136-139 general solution, 134-139 nonlinearity of brass instruments, 442-449 of jet drive, 521-525 of mechanical system, 28-29 of piano hammers, 366-367 in plates and shells, 148-150 of pressure-controlled valves, 422-424 in strings, 144-148 normal modes in cavities, 167-169 of coupled masses, 34-36 of coupled vibrators, 103-105 and Green functions, 141-142 of membrane, 72, 73 of string, 52-53 of two-mass system, 26-28 in violin cavity, 169 Norway spruce, 721 Nyquist plot, 21, 96, 129, 410 o-kawa, 617, 618 oak, 722 oboe,420, 473 acoustic power, 494 blowing pressure, 482, 483, 494 bore, 493 cutoff frequency, 476 dimensions, 464 early, 489 history, 491-492 input impedance, 473, 475 modern, 488 reed, 402, 490, 494, 725-726 spectrum, 494 oboe d'amore, 494 organ action, 555-556 design, 553-556 history, 552 mechanical action, 557 pipe ranks, 557-559 portative, 553-556 positive, 553-556 tonal architecture, 577-578 tuning and temperament, 566-568 organ flue pipes attack transients, 569 characteristic, 563-564 diapason, 563 flutes, 563-564 materials, 571-573 mixtures, 564-566 mouth correction, 559-561 mutations, 566 radiation, 568-569 scaling, 561-563 strings, 564 timbre, 56Q-561 voicing, 570 organ pipes drive by jet, 511-516, 516-521 harmonic generation, 521-525 mode transitions, 525-528 reed, 402, 411, 413, 417, 421-422 transients, 525, 526 organ reed pipes, 573-575 Oriental bells, 700-702 www.SolutionManual.info o-daiko, 615-616 Subject Index oscillations damped, 11-13 forced, 18-21, 107-111 longitudinal, 16-18 transverse, 16-18 oscillators analog circuit, 3Q-32 array of, 34-36 coupled, 102-125 Duffing, 137-139 electrical, 15-16, 3Q-32 many-mass, 116-117 mass and spring, 16-18, 23-28, 31 nonlinear See nonlinear oscillator one-dimensional, self-excited, 139-140 superposition, 7-10 transient response, 21-23 two-dimensional, 23-28 two-mass, 107-111 Van der Pol, 139-140 palladium, 730 panpipes, 529-531 parameters, slowly varying, 137 parametric amplification, 146 paulownia, 334 pendulums, 15, 105, 106 coupled, 102-103 pentangles, 645 pentatonic scale, 335 performance technique brass instruments, 455-458 woodwind instruments, 481-483 period, pernambuco, 310 perturbation, of horn profile, 218-220 pewter, 572-573, 728 phase, phase velocity, in pipes, 194, 195, 196 phasor, 6, phon, definition, 162 p'i-p'a, 266, 267 piano action, 354-362 751 attack, 394, 396 bridge, 374-375 construction, 353-362 decay time, 391, 392, 393 electric, 396 grand, 354, 379-383 hammer contact time, 367, 369, 371, 372 hammer impact, 44-46 hammer voicing, 372 hammers, 366-374 history, 352-353 inharmonicity, 363-364, 388-390 key dynamics, 357-362 longitudinal string modes, 387-388 pedals, 362, 386, 392 radiation, 392, 394, 395 radiation from soundboard, 185-186 sound decay, 383-387 sound radiation, 376-379 soundboard, 374-385, 386 spectrum, 367, 368-369, 370, 390-392 string scaling, 387 string stiffness, 65 strings, 353, 362-366, 387-388 timbre, 39Q-392 tuning, 388-390 upright, 354-357, 375-376 piccolo, 538-539 pipe organ See organ pipes See also organ pipes admittance, 520 finite, 196-201 higher modes in, 191-193 metal, 728-729 sounding frequency, 521 wall losses in, 193-196 waves in, 19Q-206 wood, 724 piston radiation from, 182 radiation load, 183 piston approximation for air loading, 587-588 pitch glide in gongs, 148-150, 661, 663 752 Subject Index pitch glide (cont.), in tom-toms, 608 plastic deformation, 716 plastic strings, 727 plastics, 726-727 woodwind instruments and, 726 plates circular, 78-80 driving point impedance, 97-98 elliptical, 80 nonlinearity, 148-150 radiation from, 184-185, 186-189 rectangular, 80-83, 85-90 square, 83-87 wood, 88-90 platinum, 730 plywood, 726 Poisson coupling, 83, 85 polarization, of string vibration, 146, 147 power, in spherical wave, 163 precession, of string vibration, 146, 147 psaltery, 331-332 pulsed video holography of cymbals, 652, 653 Pythagorean comma, 567 from large plates, 186-189 from line source, 181 by marimba, 634 from monopole source, 172 by organ flue pipes, 568-569 from pairs of point sources, 174-176 by piano, 376-379, 392, 394, 395 by piano soundboard, 185-186, 376-379 from pipe, 201-202 from piston, 182 from plane source in batHe, 181-184 from plates, 179, 184-185, 186-189 from quadrupole source, 174 by soundboard, 184-185 from spherical source, 179-180 by strings, 181 from unbaffied source, 185-186 from vibrating string, 181 by violin, 301-310 radiativity, 301, 303, 304, 305 radius of gyration, 58, 59 rayl, 159 Rayleigh thin-bar theory, 641, 642 reciprocity, 231 recorder, 533 blowing pressure, 536, 537 bore, 532-535 fingering chart, 534 history, 531-532 input impedance, 533 performance technique, 536-537 / reed generator See valves, pressure-controlled reed valve See valves, pressure-controlled reeds bassoon, 490,495,725-726 clarinet, 490, 726 free, 413-415 oboe, 490,494,725-726 saxophone, 726 wood for, 725-726 woodwind See woodwind reed www.SolutionManual.info Q, 13, 19, 22, 722 quadrupole source, 173-174 quality factor, 13, 19, 22 radiation damping, 731, 732 radiation impedance, 228 radiation load on open pipe, 198 on piston, 183 on sphere, 180 radiation of sound from arrays of point sources, 176-179 by bells, 186, 697-698 by brass instruments, 437-440 from dipole source, 173 from flexing plate, 184-185 from guitar, 256-258 by kettledrums, 185, 598-599, 600 Subject Index reflection at pipe end, 197 of sound waves, 163-165 reflection coefficient, 164 reflection function, in pipe, 224 register hole, woodwind instruments, 468, 469 relaxation times, 715-716 resonance(s) of conical horn, 211 of guitar body, 251-253, 257, 258-260 of lute body, 265 mouthpiece, 435-436 in nonlinear oscillator, 138-139 in simple oscillator, 19 in violin body, 286 resonance wood, 720-723 resonators, marimba, 633-636 reverberation time, 166 Reynolds number, 528 ring-driven modes, bells, 676 rooms, normal, modes in, 168-169 roughness effects, 718 rule of eighteen, 263 sackbut, 440 Saxhorn, 449 saxophone, 420,496-497 blowing pressure, 482, 483 dimensions, 464 reed, 402, 726 scaling, of organ pipe ranks, 561-563 scattering, of sound waves, 165 self-excited oscillator, 139 140 self-sounder, 623 serpent, 430 shakuhachi, 530, 531 shawm, 491, 498 shell-driven modes, bells, 676 shells cylindrical, 94-96 extensional modes, 92 mode frequencies, 94 nonextensional modes, 92 nonlinearity, 148-150 shallow, 93-94 vibration of, 92-96 753 shock waves, in brass instruments, 448 side drums See snare drums silver, 728, 729 730 simple harmonic motion, 4-6 singer's formant, 315 sitar, 266, 268-269 snare drums, 583, 602-606 action, 604, 606 coupling,602 mode frequencies, 603, 604 spectrum, 604, 606, 607 snare head, 602-606 velocity, 604, 606 soh, 334 sound radiation of See radiation of sound speed of See speed of sound sound decay, 716 sound pressure level, definition, 161 sound waves absorption of, 165-167 attenuation of, 165-167 reflection of, 163-165 scattering of, 165 transmission of, 165 in tubes, 157-158 soundboard clavichord, 349, 350 harp, 338-340 harpsichord, 341, 343-344 piano, 374-385, 386 radiation from, 184-185 string coupling, 119-120 soundpost, 295-297 source dipole, 172-173 monopole, 172, 173 multipole, 171-174 quadrupole, 173-174 specific acoustic impedance of air, 159 definition, 159 spectrum of brass instruments, 438, 439, 453, 455 of clarinet, 490 of cymbals, 654-655 754 Subject Index spectrum (cont.), of flute, 546-548 of gongs, 660-661, 662 of Indian drums, 612, 613, 614, 615 of kettledrums, 595-599 of Latin American drums, 619, 620 of oboe, 494 of organ pipes, 575-577 of piano, 367, 368-369, 370, 39Q-392 of plucked string, 4Q-44 of snare drums, 604, 606, 607 of steel pans, 668-669, 670 of struck string, 45-46 of triangles, 643, 644 of violin, 307, 313-316 of woodwind instruments, 484-486,487 speed of sound in air, 158 temperature variation, 158 spotted metal, 729 spring air, 13 nonlinear, 28-29 spruce, 722 steel, elastic properties, 625 steel drum set, 668 steel drums, tuned, 667 steel pans, 667-672 construction and tuning, 672 damping, 670 harmonics, 672 mechanical coupling between note areas, 669-672 spectrum, 668-669, 670 steel plates, 665 stress behavior, 714 strike note, bells, 682-685 strings bowed,46-50, 275-285 clavichord, 348-349 coupled, 385-386 coupled by bridge, 12Q-124, 125 coupled to soundboard, 119-120 dispersion by stiffness, 65-66 driven, 5Q-52 longitudinal motion in bowing, 281-282 longitudinal waves in, 56-57 low-pitched tones from, 285 missing mode generation, 145-146 nonlinearity, 144-148 physical properties, 283-284 plastic, 727 plucked, 40-44, 241-245 radiation from, 181 rotation of polarization, 146-147 stiff, 64-66 stiffness, 280-281, 363 struck, 44-46, 367-374 torsional motion in bowing, 281-282 torsional waves in, 67 vibration direction in guitar, 253-255 waves on, 36-56 Strouhal number, 505 structural dynamics, 125-128 Struve function, 183 supports, string, motion of, 52-53 surface density, of air load, 587 surface tension of membrane, 602 swinging modes, bells, 677 www.SolutionManual.info tabla, 583, 610 left-handed, 610 tam-tams, 150, 655, 656-660 construction, 656-657 modes of vibration, 657-658 nonlinear mode coupling, 658-660 timbre, 659-660 tambourine, 62Q-621 tambura, 266, 268-269 tawa tawa gong, 660, 662 temperature, effect on pitch, 470 temple bells, 703-705 tenor drum, 583 thermal conduction, 714 threshold pressure, 409, 410, 414 timbales, 620 timbre, of organ pipes, 575-577 time domain, 426, 451, 480 Subject Index formalism, 223-227 Timoshenko beam model, 632, 633 tin, 729 tom-toms, 583, 606, 608-609 mode frequencies, 606, 608 onset and decay of sound, 608-609, 610 pitch glide, 608 tone-hole lattice, 476, 484 tracheid cells, 719, 720 transfer function, 130 transient response, of oscillator, 21-23 transients in brass instruments, 45Q-453, 456 in cymbals, 652-655 in organ flue pipes, 569 in organ pipes, 525, 526 in woodwind instruments, 483 transmission, of sound waves, 165 transmission coefficient, 164 transverse modes, coupling on strings, 146-147 transverse waves, on plate, 76-77 triangles, 642-645 mode frequencies, 643, 644 spectrum, 643, 644 tristimulus diagram, 450, 575, 576 trombone, 431, 440, 441 trumpet, 431, 441, 448 mouthpiece, 433 tsuzumi, 617 tubaphones, 645-646 tube walls, 718 tubular bells, 95-96, 641-642 tudumi, 617 tuning, of organs, 566-568 turbulence in air jet, 508 noise generation by, 528-529 turi-daiko, 616-617 twang, in strings, 145 twisting modes, bells, 677 tympani See kettledrums ud, 267 755 valves brass instrument, 441-442 French horn, 442 pressure-controlled acoustic admittance, 410-413 coupled to horn, 415-418 description, 401-403 large-amplitude behavior, 418-422 nonlinearity, 422-424 numerical simulation, 424-426 oscillation conditions, 407-408 playing frequency, 406-413 quasi-static model, 403-406 trumpet, 441 Vander Pol oscillator, 139-140 varnishes, wood, 723-724 vibes, 638-639 mode frequencies, 639 vibrato, 638-639 vibrations, torsional, 66-67 vibrato flute, 548 string instruments, 317-318 vibes, 638-639 violin, 124 viola, 318-319 wolf note, 312 violin air modes, 286 body resonances, 314 body vibrations, 285-297 cavity modes, 292 construction, 274-275 directivity, 301-310 free plate modes, 292-294 history, 272-274 new family, 322-326 radiation, 301-310 soundpost, 295-297 spectrum, 307 tonal quality, 313-316 top-plate modes, 88-89 transient response, 294-295 vibrato, 317-318 wolf note, 312-313 violin bow, 310-311 violin bridge, 297-300 viols, 319, 321-322 viscous damping, 716 756 Subject Index vocal tract, 410, 418, 458, 482-483, 491 voicing, of organ flue pipes, 570 vortices, 508, 521 warble bells, 686, 688 handbells, 696 warping, 717 wave equation, 588 bending waves, 58-60 one-dimensional, 36-40 for plate, 77 for rectangular membrane, 70-74 sound, 157 spherical, 160 for stiff membrane, 91 stiff string, 64 for string, 36-37 three-dimensional, 167 two-dimensional, 70-74 wave impedance of air, 159 definition, 159 for spherical wave, 161 wave numbers, modal, 586 wave speed, flexural, 698 wave velocity, frequency and, 588 waves bending, 58-60 longitudinal, 56-57 plane, 156-160 reflection, 38-39 sinuous, on jet, 504-509 spherical, 16Q-161 standing, 39-40 torsional vibrations, 66-67 transverse, on string, 36-37 traveling, 37-39 varicose, on jet, 504-509 Webster equation, 207, 208 wolf note, 312-313 wood, 719-726 bent, 723 cut, 722 damping in, 722-723 density, 721 elastic moduli, Sitka spruce, 88 elastic parameters, 721 elastic properties, 625 for pipe, 724 quarter-cut, 88 for reeds, 725-726 resonance, 720-723 varnishes, 723-724 for violin bows, 310-311 for violins, 274 for woodwind instruments, 724-725 woodwind instruments acoustic efficiency, 484 blowing pressure, 481-482 cutoff frequency, 476,485 directionality, 480-481 finger holes, 464-466, 468, 472 fingering chart, 467 fingering systems, 467-469 history, 461 horn shapes, 461-464 input impedance, 470-476 performance technique, 481-483 pitch, 469-470 plastics and, 726 reed See woodwind reed register hole, 468, 469 spectrum, 484-486, 487 system diagram, 477-4 79 transients, 483 vocal tract effects, 482-483 wood for, 724-725 woodwind reed, 402, 403, 407, 409, 411, 416, 477-480 double, 420-421 resonance frequency, 479-480 single, 418-420 work hardening, 717 Worman's theorem, 485 www.SolutionManual.info xylomarimba, 636 xylophone, 624, 636-638 decay times, 637 Young's modulus, 713, 714 zither, 331, 332 ... the Helmholtz resonator shown in Fig 1.7(b), the mass of air in the neck serves as the piston and the large volume of air V as the spring The mass of air in the neck and the spring constant of. .. The Physics of Musical Instruments Second Edition Neville H Fletcher Thomas D Rossing The Physics of Musical Instruments Second Edition With 485 Illustrations... 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