Hearing: An Introduction to Psychological and Physiological Acoustics (6th Edition) PDF

 HEARING An Introduction to Psychological and Physiological Acoustics  HEARING An Introduction to Psychological and Physiological Acoustics SIXTH EDITION STANLEY A. GELFAND, PhD Professor Department of Linguistics and...

 HEARING An Introduction to Psychological and Physiological Acoustics  HEARING An Introduction to Psychological and Physiological Acoustics SIXTH EDITION STANLEY A. GELFAND, PhD Professor Department of Linguistics and Communication Disorders Queen’s College of the City University of New York Flushing, New York and PhD Program in Speech-Language-Hearing Sciences and AuD Program Graduate Center of the City University of New York New York, New York CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-4987-7542-7 (Pack-Hardback and eBook) This book contains information obtained from authentic and highly regarded sources. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identifi- cation and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Gelfand, Stanley A., 1948- author. Title: Hearing : an introduction to psychological and physiological acoustics / Stanley A. Gelfand. Description: Sixth edition. | Boca Raton : CRC Press, 2018. | Includes bibliographical references. Identifiers: LCCN 2017021446 (print) | LCCN 2017022201 (ebook) | ISBN 9781315154718 (eBook General) | ISBN 9781498775434 (eBook PDF) | ISBN 9781351650755 (eBook ePub) | ISBN 9781351641234 (eBook Mobipocket) | ISBN 9781498775427 (hardback : alk. paper) Subjects: | MESH: Hearing--physiology | Psychoacoustics Classification: LCC QP461 (ebook) | LCC QP461 (print) | NLM WV 272 | DDC 612.8/5--dc23 LC record available at https://lccn.loc.gov/2017021446 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Janice In Loving Memory Contents Preface xiii 1 Physical concepts 1 Physical quantities 1 Decibel notation 6 Harmonic motion and sound 8 Combining waves 15 Complex waves 17 Filters 20 Standing waves 21 Impedance 22 References 24 2 Anatomy 27 Gross anatomy and overview 27 Temporal bone 29 Outer ear 32 Pinna 32 Ear canal 32 Eardrum 33 Middle ear 33 Ossicular chain 35 Intratympanic muscles 36 Inner ear 37 Osseous and membranous labyrinths 37 Inner ear fluids 37 Vestibular organs 39 Cochlea 39 Hair cells 44 Innervation 48 Efferent innervation of the hair cells 50 Central auditory pathways 51 Ascending auditory pathways 52 Cochlear nuclei 52 Superior olivary complex 53 Lateral lemniscus 54 Inferior colliculus 54 Medial geniculate body 54 vii viii Contents Cortex 55 Descending auditory pathways 57 Olivocochlear bundle 58 Middle ear muscle reflex 59 References 59 3 Conductive mechanism 69 Outer ear 69 Pinna 69 Ear canal 69 Middle ear 70 Middle ear transformer mechanism 72 Area ratio 72 Curved-membrane mechanism 73 Ossicular lever 74 Middle ear response 76 Bone conduction 77 The acoustic reflex 80 Reflex parameters 81 Middle ear muscle theories 89 References 90 4 Cochlear mechanisms and processes 95 Action of sensory receptors 95 Classical theories of hearing 96 Classical resonance theory 96 Traveling wave theory 97 Classical temporal theories 97 Place-volley theory 98 The traveling wave 98 Hair cell activation 100 Mechanoelectrical transduction 103 Cochlear electrical potentials 104 Resting potentials 105 Receptor potentials 105 Cochlear microphonics 105 Distribution of the cochlear microphonic 108 Summating potentials 111 Cochlear tuning and frequency selectivity 112 Nonlinearity in the cochlea 118 Active processes and the cochlear amplifier 120 Otoacoustic emissions 122 References 127 5 Auditory nerve 137 Frequency coding 138 Tuning curves 138 Firing patterns 139 Responses to clicks 139 Responses to tones and tonal complexes 141 Two-tone suppression 144 Intensity coding 146 Speech coding 150  Contents ix Whole-nerve action potentials 151 References 154 6 Auditory pathways 159 Responses of the auditory nervous system 159 Binaural and related responses 161 Brainstem 161 Superior olivary complex 161 Inferior colliculus 162 Coding of binaural differences 163 Medial geniculate body 164 Cortex 164 Tonotopic organization 165 Cochlear nuclei 165 Superior olivary complex 167 Lateral lemniscus 167 Inferior colliculus 167 Medial geniculate body 168 Cortex 169 Auditory evoked potentials 170 Frequency-following response 173 Complex auditory brainstem response 173 Auditory steady-state response 174 Effects of cortical ablation 175 Medial olivocochlear reflex 177 References 181 7 Psychoacoustic methods 197 Scales of measurement 197 Measurement methods 198 Classical methods of measurement 198 Method of limits 198 Method of adjustment 201 Method of constant stimuli 201 Forced-choice methods 203 Adaptive procedures 203 Bekesy’s tracking method 203 Simple up-down or staircase method 204 Parameter estimation by sequential testing 205 Block up-down methods 206 Transformed up-down or staircase procedures 207 Modifications, other procedures, and comparisons 209 Direct scaling 209 Ratio estimation and production 210 Magnitude estimation and production 210 Cross-modality matches 211 Category rating scales 211 References 211 8 Signal detection theory 217 Factors affecting responses 217 Psychophysical methods in sdt 222 Yes/no methods 222 x Contents Two-interval and N-interval f­ orced-­choice methods 222 Confidence rating methods 222 Some implications of sdt 223 References 223 9 Auditory sensitivity 225 Absolute sensitivity 225 Minimum audible levels 225 Threshold microstructure 227 Upper limits of hearing 227 Reference levels 227 Hearing level 230 Effects of duration 230 Differential sensitivity 232 Intensity discrimination 233 Frequency discrimination 235 Profile analysis 236 Temporal resolution 238 Temporal discrimination 240 Stimulus uncertainty 241 Temporary threshold shift 242 Appendix 9.1 244 Appendix 9.2 244 References 244 10 Masking 251 Nature of masking 252 Frequency selectivity 255 Psychoacoustic tuning curves 259 Comodulation masking release 262 Overshoot 263 Temporal masking 264 Central masking 266 Informational masking 267 References 269 11 Loudness 275 Loudness level 275 Loudness scaling and loudness functions 277 Loudness and distance 280 Loudness and bandwidth 280 Temporal integration of loudness 281 Loudness adaptation 283 Induced loudness reduction 285 Binaural loudness summation 285 Annoyance 287 References 288 12 Pitch and timbre 295 Mel scales of pitch 295 Musical pitch 297 Consonance and dissonance 300 Pitch and intensity 301  Contents xi Beats, harmonics, and combination tones 301 Pitch of complex sounds 303 Missing fundamental and periodicity pitch 306 Pitch shift of the missing fundamental 308 Place/spectral theories 309 Temporal theories 309 Timbre 311 References 315 13 Binaural and spatial hearing 321 Binaural fusion 321 Binaural beats 322 Binaural summation 322 Differential sensitivity 323 Auditory scene analysis 324 Directional hearing 325 Localization 325 Head movements 330 Lateralization 331 Virtual auditory space localization 334 Minimum audible angle 334 Minimum audible movement angle 336 Directional hearing in infants 336 Species differences in directional hearing 336 Auditory distance perception 337 Precedence effect 338 Masking level differences 342 References 346 14 Speech and its perception 357 Speech sounds: Production and perception 358 Vowels 358 Consonants 360 Dichotic listening and cerebral lateralization 364 Categorical perception 365 The speech module 366 Power of speech sounds 367 Speech intelligibility 368 Audibility: Speech level and signal-to-noise ratio 369 Frequency 369 Amplitude distortion 370 Interruptions and temporal distortion 370 Masking and reverberation 372 Nonacoustic considerations 373 Speech perception theories and approaches 373 Models implicating the production system 374 Motor theory 374 Direct realist theory 374 Analysis-by-synthesis 374 General auditory approaches 374 Fuzzy logical model of perception 374 xii Contents Word recognition models 375 Prototype and exemplar models 375 Logogen model 375 Cohort model 375 Trace model 375 Shortlist model 376 Neighborhood activation model 376 Speech intelligibility and acoustical measurements 376 Clear speech 378 References 379 Index 391 Preface This is the sixth edition of a textbook intended In addition to reflecting advances in the field, to provide beginning graduate students with an the sixth edition of Hearing has been strongly introduction to the sciences of hearing, as well as influenced by extensive comments and sugges- to provide an overview of the field for more expe- tions from both colleagues and graduate students. rienced readers. This has resulted in updates, changes, and addi- The need for a current text of this type has tions to the material as well as several new and been expanded by the advent of the professional revised figures; but every effort has been made to doctorate in audiology, the AuD in addition to maintain the fundamental characteristics of the those in PhD programs in the speech and hear- prior editions wherever possible. These include the ing sciences. However, an interest in hearing is by basic approach, structure, format, and the general no means limited to audiologists and speech and (and often irregular) depth of coverage, the provi- hearing scientists. It includes readers with widely sion of references at the end of each chapter, and diverse academic backgrounds, such as psycholo- the provision of liberal references to other sources gists, speech-language pathologists, physicians, for further study. As one might expect, the hard- deaf educators, industrial hygienists, linguists est decisions involved choosing material that could and engineers, among others. The result is a frus- be streamlined, replaced, or omitted, keeping the trating dilemma in which a text will likely be too original orientation and flavor of the book, and basic for some of its intended readers and too avoiding a “state-of-the-art” treatise. advanced for others. Thus, the idea is to provide a It is doubtful that all of the material covered in volume sufficiently detailed to serve as a core text this text would be addressed in a single one-semester for graduate students with a primary interest in course, nor that it would be the only source used. It hearing, while at the same time avoiding a reli- is more likely that this book would be used as a core ance on scientific or mathematical backgrounds text for a two-course sequence dealing with psy- not shared by those with different kinds of aca- chological and physiological acoustics, along with demic experiences. appropriately selected readings from the research Hearing science is an exciting area of study literature and state-of-the-art books. Suggested because of its broad, interdisciplinary scope, and readings are provided in context throughout the even more because it is vital and dynamic. Research text to provide a firm foundation for further study. continuously provides new information to expand My heartfelt appreciation is expressed to the on the old and also causes us to rethink what was numerous colleagues and students who provided once well established. The reader (particularly the me with valuable suggestions that have been incor- beginning student) is reminded that new findings porated into this and prior editions. I am especially occasionally disprove the “laws” of the past. Thus, indebted to my current and former colleagues and this textbook should be treated as a first step; it is students in the Department of Linguistics and by no means the final word. Communication Disorders at Queens College, xiii xiv Preface the PhD Program in Speech-Language-Hearing Lauren Calandruccio, Joseph Danto, Lillian and Sciences, and the AuD Program at the City Sol Gelfand, Irving Hochberg, Gertrude and Oscar University of New York Graduate Center, and at Katzen, Arlene Kraat, Linda Leggio, John Lutolf, the East Orange Veterans Affairs Medical Center. Grace McInnes, Maurice Miller, Neil Piper, Teresa Thank you all for being continuous examples of Schwander, Stanley Schwartz, Shlomo Silman, excellence and for your valued friendships. I am Carol Silverman, Helen and Harris Topel, Robert also grateful to the talented and dedicated staff of Vago, Barbara Weinstein, and Mark Weiss. Very CRC Press/Taylor & Francis Group, who contrib- special gratitude is expressed to Harry Levitt, who uted so much to this book and graciously arranged will always be my professor. for the preparation of the indices and the proof- Finally, my deepest gratitude goes to Janice, reading of the final page proofs. the love of my life, whose memory will always be At the risk of inadvertently omitting several, a blessing and inspiration; and to my wonderful I would like to thank the following people for children, Michael, Joshua, and Erin, and Jessica their advice, inspiration, influence, and support, and Robert for their love, support, confidence, and which have taken forms too numerous to mention: unparalleled patience. Cherry Allen, Nick Barber, Sandra Beberman, Moe Bergman, Arthur Boothroyd, Miranda Bromage, Stanley A. Gelfand  1 Physical concepts This book is concerned with hearing, and what basic quantities (and other derived quantities), and we hear is sound. Thus, both intuition and reason include such phenomena as velocity, force, and make it clear that a basic understanding of the work. If a quantity can be described completely nature of sound is prerequisite to an understand- in terms of just its magnitude (size), then it is a ing of audition. The study of sound is acoustics. ­scalar. Length is a good example of a scalar. On the An understanding of acoustics, in turn, rests upon other hand, a quantity is a vector if it needs to be knowing several fundamental physical principles. described by both its magnitude and its direction. This is so because acoustics is, after all, the physics For example, if a body moves 1 m from point x l of sound. We will therefore begin by reviewing a to point x2, then we say that it has been displaced. number of physical principles so that the following Here, the scalar quantity of length becomes the chapters can proceed without the constant need for vector quantity of displacement when both magni- the distracting insertions of basic definitions and tude and direction are involved. A derived quantity concepts. The material in this chapter is intended is a vector if any of its components is a vector. For to be a review of principles that were previously example, force is a vector because it involves the learned. Therefore, the review will be rapid and components of mass (a scalar) and acceleration (a somewhat cursory, and the reader may wish to con- vector). The distinction between scalars and vectors sult the American National Standard addressing is not just some esoteric concept. One must be able acoustical terminology and a physics or acoustics to distinguish between scalars and vectors because textbook for a broader coverage of these topics (e.g., they are manipulated differently in calculations. Pearce and David, 1958; van Bergeijk et al., 1960; The basic quantities may be more or less appreci- Peterson and Gross, 1972; Beranek, 1986; Kinsler ated in terms of one’s personal experience, and are et al., 1999; Speaks, 1999; Everest, 2000; Rossing expressed in terms of conventionally agreed upon et al., 2002; Hewitt, 2005; Young and Freedman, units. These units are values that are measurable 2007),* as well as the American National Standard and repeatable. The unit of time (t) is the s­ econd addressing acoustical terminology (ANSI, 2004). (s), the unit of length (L) is the meter (m), and the unit of mass (M) is the kilogram (kg). There PHYSICAL QUANTITIES is a common misconception that mass and weight are synonymous. This is actually untrue. Mass is Physical quantities may be thought of as being related to the density of a body, which is the same basic or derived, and as either scalars or vectors. for that body no matter where it is located. On The basic quantities of concern here are time, the other hand, an object’s weight is related to the length (distance), and mass. The derived quanti- force of gravity upon it, so that weight changes as a ties are the results of various combinations of the function of gravitational attraction. It is common knowledge that an object weighs more on earth * While no longer in print, the interested student may than it would on the moon, and that it weighs more be able to find the classical books by Pearce and David at sea level than it would in a high-flying airplane. (1958), van Bergeijk et al. (1960), and Peterson and In each of these cases, the mass of the body is the Gross (1972) in some libraries. same in spite of the fact that its weight is different. 1 2 Physical concepts A brief word is appropriate at this stage regard- moment in time. Instantaneous velocity reflects ing the availability of several different systems of the speed at some point in time when the displace- units. When we express length in meters and mass ment and time between that point and the next one in kilograms we are using the units of the Système approaches zero. Thus, students with a background International d’Unités, referred to as the SI or the in mathematics will recognize that instantaneous MKS system. Here, MKS stands for meters, kilo- velocity is equal to the derivative of displacement grams, and seconds. An alternative scheme using with respect to time, or smaller metric units coexists with MKS, which is the cgs system (for centimeters, grams, and sec- dx onds), as does the English system of weights and v= (1.3) dt measures. Table 1.1 presents a number of the major basic and derived physical quantities we will deal with, their units, and their conversion factors.* As common experience verifies, a fixed speed is Velocity (v) is the speed at which an object is rarely maintained over time. Rather, an object may moving, and is derived from the basic quantities of speed up or slow down over time. Such a change displacement (which we have seen is a vector form of velocity over time is acceleration (a). Suppose of length) and time. On average, velocity is the we are concerned with the average acceleration of distance traveled divided by the amount of time it a body moving between two points. The velocity takes to get from the starting point to the destina- of the body at the first point is v1 and the time as it tion. Thus, if an object leaves point x l at time t1 and passes that point is t1. Similarly, its velocity at the arrives at x 2 at time t2, then we can compute the second point and the time when it passes this point average velocity as are, respectively, v2 and t2. The average accelera- tion is the difference between these two velocities ( x 2 − x1 ) divided by the time interval involved: v= (1.1) ( t 2 − t1 ) (v 2 − v1 ) a= (1.4) ( t 2 − t1 ) If we call (x 2 − x l) displacement (x) and (t2 − t l) time (t), then, in general: or, in general: x v= (1.2) v t a= (1.5) t Because displacement (x) is measured in meters and time (t) in seconds, velocity is expressed in If we recall that velocity corresponds to displace- meters per second (m/s). ment divided by time (Equation 1.2), we can sub- In contrast to average velocity as just defined, stitute x/t for v, so that instantaneous velocity is used when we are con- cerned with the speed of a moving body at a specific x t x (1.6) a= = t t2 * Students with a penchant for trivia will be delighted to know the following details. (1) One second is the Therefore, acceleration is expressed in units of time needed to complete 9,192,631,770 cycles of radi- meters per second squared (m/s2) or centimeters ation of cesium-133 atoms in an atomic clock (for an per second squared (cm/s2). interesting and informative discussion, see Finkleman The acceleration of a body at a given moment is et al., 2011). (2) The reference value for 1 kg of mass called its instantaneous acceleration, which is the is that of a cylinder of platinum–iridium alloy kept in derivative of velocity with respect to time, or the International Bureau of Weights and Measures in France. (3) One meter is 1,650,763.73 times the wave- dv length of orange-red light emitted by krypton-86 a= (1.7) under certain conditions. dt  Physical quantities 3 Table 1.1 Principal physical quantities Quantity Formula SI (MKS) units cgs units Equivalent values Time (t) t second (s) s Mass (M) M kilogram (kg) gram (g) 1 kg = 1000 g Displacement (x) x meter (m) centimeter (cm) 1 m = 100 cm Area (A) A m2 cm2 1 m2 = 104 cm2 Velocity (v) v = x/t m/s cm/s 1 m/s = 100 cm/s Acceleration (a) a = v /t m/s2 cm/s2 1 m/s2 = 100 cm/s2 = x /t 2 Force (F) F = Ma newton (N) dyne (d) 1 N = 105 d = Mv/t kg ∙ m/s2 g ∙ cm/s2 Work (w) w = Fx joule (J) erg 1 J = 107 erg N∙m d ∙ cm Power (P) P = w /t watt (W) watt (W) 1 W = 1 J/s = 107 erg/s = Fx /t = Fv Intensity (I) I = P/A W/m2 W/cm2 Reference values: 10−12 W/m2 or 10−16 W/cm2 Pressure (p) p = F/A pascal (Pa) microbar (µbar) Reference values: N/m2 d/cm2 2 × 10−5 N/m2 (µPa) or 2 × 10−4 d/cm2 (µbar)a a The reference value for sound pressure in cgs units is often written as 0.0002 dynes/cm2. Recalling that velocity is the first derivative of that is, to change its speed or direction. The amount displacement (Equation 1.3), and substituting, we of force is equal to the product of mass times accel- find that acceleration is the second derivative of eration (Newton’s second law of motion): displacement: F = Ma (1.9) d x 2 a= (1.8) dt 2 Recall that acceleration corresponds to velocity over time (Equation 1.5). Substituting v/t for a Common experience and Newton’s first law of (acceleration) reveals that force can also be defined motion tell us that if an object is not moving (is at in the form rest), then it will tend to remain at rest, and that if an object is moving in some direction at a given Mv F= (1.10) speed, that it will tend to continue doing so. This t phenomenon is inertia, which is the property of mass to continue doing what it is already doing. where Mv is the property of momentum. Stated An outside influence is needed in order to make a in this manner, force is equal to momentum over stationary object move, or to change the speed or time. direction of a moving object. That is, a force (F) Because force is the product of mass and accel- is needed to overcome the body’s inertia. Because eration, the amount of force is measured in kg ∙ m/ a change in speed is acceleration, we may say that s2. The unit of force is the newton (N), which is the force is that which causes a mass to be accelerated, force needed to cause a 1-kg mass to be accelerated 4 Physical concepts by 1 kg ∙ m/s2 (i.e., 1 N = kg ∙ m/s2). It would thus The opposing force of friction depends on take a 2-N force to cause a 2-kg mass to be accel- two factors. Differing amounts of friction occur erated by 1 m/s2, or a 1-kg mass to be accelerated depending upon what is sliding on what. The mag- by 2 kg ∙ m/s2. Similarly, the force required to nitude of friction between two given materials is accelerate a 6-kg mass by 3 m/s2 would be 18 N. called the coefficient of friction. Although the The unit of force in cgs units is the dyne, where 1 details of this quantity are beyond current inter- dyne = 1 g ∙ cm/s2 and 105 dynes = 1 N. est, it is easily understood that the coefficient of Actually, many forces tend to act upon a given friction is greater for “rough” materials than for body at the same time. Therefore, the force referred “smooth” or “slick” ones. to in Equations 1.9 and 1.10 is actually the resultant The second factor affecting the force of fric- or net force, which is the net effect of all forces act- tion is easily demonstrated by an experiment the ing upon the object. The concept of net force is clari- reader can do by rubbing the palms of his hands fied by a few simple examples: If two forces are both back and forth on one another. First rub slowly pushing on a body in the same direction, then the and then rapidly. Not surprisingly, the rubbing net force would be the sum of these two forces. (For will produce heat. The temperature rise is due to example, consider a force of 2 N that is pushing an the conversion of the mechanical energy into heat object toward the north, and a second force of 5 N as a result of the friction, and will be addressed that is also pushing that object in the same direc- again in another context. For the moment, we will tion. The net force would be 2 N + 5 N, or 7 N and accept the amount of heat as an indicator of the the direction of acceleration would be to the north.) amount of friction. Note that the hands become Alternatively, if two forces are pushing on the same hotter when they are rubbed together more rap- body but in opposite directions, then the net force is idly. Thus, the amount of friction is due not only to the difference between the two, and the object will the coefficient of friction (R) between the materials be accelerated in the direction of the greater force. involved (here, the palms of the hands), but also to (Suppose, for example, that a 2-N force is pushing an the velocity (v) of the motion. Stated as a formula, object toward the east and that a 5-N force is simul- the force of friction (F) is thus taneously pushing it toward the west. The net force would be 5 N – 2 N, or 3 N which would cause the F = Rv (1.11) body to accelerate toward the west.) If two equal forces push in opposite directions, A compressed spring will bounce back to its then net force would be zero, in which case there original shape once released. This property of a would be no change in the motion of the object. deformed object to return to its original form is This situation is called equilibrium. Thus, under called elasticity. The more elastic or stiff an object, conditions of equilibrium, if a body is already mov- the more readily it returns to its original form after ing, it will continue in motion, and if it is already being deformed. Suppose one is trying to compress at rest, it will remain still. That is, of course, what a coil spring. It becomes increasingly more diffi- Newton’s first law of motion tells us. cult to continue squeezing the spring as it becomes Experience, however, tells us that a moving more and more compressed. Stated differently, the object in the real world tends to slow down and will more the spring is being deformed, the more it eventually come to a halt. This occurs, for example, opposes the applied force. The force that opposes when a driver shifts to “neutral” and allows his car the deformation of a spring-like material is called to coast on a level roadway. Is this a violation of the the restoring force. laws of physics? Clearly, the answer is no. The rea- As the example just cited suggests, the restoring son is that in the real world a moving body is con- force depends on two factors; the elastic modulus stantly in contact with other objects or mediums. of the object’s material and the degree to which The sliding of one body against the other consti- the object is displaced. An elastic modulus is the tutes a force opposing the motion, called friction ratio of stress to strain. Stress (s) is the ratio of the or resistance. For example, the coasting automo- applied force (F) to the area (A) of an elastic object bile is in contact with the surrounding air and the over which it is exerted, or roadway; moreover, its internal parts are also mov- ing one upon the other. s = F/A (1.12)  Physical quantities 5 The resulting relative displacement or change in accomplished. The rate at which work is done is dimensions of the material subjected to the stress power (P) and is equal to work divided by time, is called strain. Of particular interest is Young’s modulus, which is the ratio of compressive stress P = w /t (1.15) to compressive strain. Hooke’s law states that stress and strain are proportional within the elastic in joules per second (J/s). The watt (W) is the unit limits of the material, which is equivalent to stat- of power, and 1 W is equal to 1 J/s. In the cgs sys- ing that a material’s elastic modulus is a constant tem, the watt is equal to 107 ergs/s. within these limits. Thus, the restoring force (F) of Recalling that w = Fx, then Equation 1.15 may an elastic material that opposes an applied force is be rewritten as F = Sx (1.13) P = Fx /t (1.16) where S is the stiffness constant of the material and If we now substitute v for x/t (based on Equation x is the amount of displacement. 1.2), we find that The concept of “work” in physics is decidedly more specific than its general meaning in daily P = Fv (1.17) life. In the physical sense, work (w) is done when the application of a force to a body results in its Thus, power is equal to the product of force and displacement. The amount of work is therefore the velocity. product of the force applied and the resultant dis- The amount of power per unit of area is called placement, or intensity (I). In formal terms, w = Fx (1.14) I = P/A (1.18) Thus, work can be accomplished only when where I is intensity, P is power, and A is area. there is displacement: If the displacement is zero, Therefore, intensity is measured in watts per then the product of force and displacement will square meter (W/m2) in SI units, or in watts per also be zero no matter how great the force. Work square centimeter (W/cm2) in cgs units. Because is quantified in Newton-meters (N ∙ m); and the of the difference in the scale of the area units in unit of work is the joule (J). Specifically, one joule the MKS and cgs systems, we find that 10−12 W/m2 (1 J) is equal to 1 N ∙ m. In the cgs system, work corresponds to 10−16 W/cm2. This apparently pecu- is expressed in ergs, where 1 erg corresponds to 1 liar choice of equivalent values is being provided dyne-centimeter (1 d ∙ cm). because they represent the amount of intensity The capability to do work is called energy. The required to just barely hear a sound. energy of an object in motion is called kinetic An understanding of intensity will be better energy and the energy of a body at rest is its poten- appreciated if one considers the following. Using tial energy. Total energy is the body’s kinetic for the moment the common knowledge idea of energy plus its potential energy. Work corresponds what sound is, imagine that a sound source is a tiny to the change in the body’s kinetic energy. The pulsating sphere. This point source of sound will energy is not consumed, but rather is converted produce a sound wave that will radiate outward in from one form to the other. Consider, for exam- every direction, so that the propagating wave may ple, a pendulum that is swinging back and forth. be conceived of as a sphere of ever-increasing size. Its kinetic energy is greatest when it is moving the Thus, as distance from the point source increases, fastest, which is when it passes through the mid- the power of the sound will have to be divided over point of its swing. On the other hand, its potential the ever-expanding surface. Suppose now that we energy is greatest at the instant that it reaches the measure how much power registers on a one-unit extreme of its swing, when its speed is zero. area of this surface at various distances from the We are concerned not only with the amount source. As the overall size of the sphere is get- of work, but also with how fast it is being ting larger with distance from the source, so this 6 Physical concepts one-unit sample must represent an ever-decreasing another form, called decibels (dB), which make the proportion of the total surface area. Therefore, less values both palatable and rationally meaningful. power “falls” onto the same area as the distance One may conceive of the decibel as basically from the source increases. It follows that the mag- involving two characteristics, namely ratios and log­ nitude of the sound appreciated by a listener would arithms. First, the value of a quantity is expressed become less and less with increasing distance from in relation to some meaningful baseline value in a sound source. the form of a ratio. Because it makes sense to use The intensity of a sound decreases with distance the softest sound one can hear as our baseline, we from the source according to an orderly rule as long use the intensity or pressure of the softest audible as there are no reflections, in which case a free field sound as our reference value. is said to exist. Under these conditions, increasing As introduced earlier, the reference sound the distance (D) from a sound source causes the intensity is 10−12 W/m2 and the equivalent ref- intensity to decrease to an amount equal to 1 over erence sound pressure is 2 × 10−5 N/m2. Recall the square of the change in distance (1/D2). This also that the equivalent corresponding values in principle is known as the inverse-square law. In cgs units are 10−16 W/cm2 for sound intensity and effect, the inverse square law says that doubling 2 × 10−4 dynes/cm2 for sound pressure. The appro- the distance from the sound source (e.g., from 1 priate reference value becomes the denominator to 2 m) causes the intensity to drop to 1/22 or 1/4 of our ratio and the absolute intensity or pressure of the original intensity. Similarly, tripling the dis- of the sound in question becomes the numerator. tance causes the intensity to fall to 1/32, or 1/9 of Thus, instead of talking about a sound having an the prior value; four times the distance results in absolute intensity of 10−10 W/m2, we express its 1/42, or 1/16 of the intensity; and a tenfold increase intensity relatively in terms of how it relates to our in distance causes the intensity to fall 1/102, or reference, as the ratio: 1/100 of the starting value. Just as power divided by area yields intensity, so (10−10 W/m2 ) force (F) divided by area yields a value called pres- (10−12 W/m2 ) sure (p): p = F/A (1.19) which reduces to simply 102. This intensity ratio is then replaced with its common logarithm. The reason is that the linear distance between numbers so that pressure is measured in N/m2 or in dynes/ having the same ratio relationship between them cm2. The unit of pressure is called the pascal (Pa), (say, 2:1) becomes wider when the absolute magni- where 1 Pa = 1 N/m2. As for intensity, the softest tudes of the numbers become larger. For example, audible sound can also be expressed in terms of its the distance between the numbers in each of the pressure, for which 2 × 10−5 N/m2 and 2 × 10−4 following pairs increases appreciably as the size dynes/cm2 are equivalent values. of the numbers becomes larger, even though they all involve the same 2:1 ratio: 1:2, 10:20, 100:200, DECIBEL NOTATION and 1000:2000. The logarithmic conversion is used because equal ratios are represented as equal dis- The range of magnitudes we concern ourselves tances on a logarithmic scale. with in hearing is enormous. As we shall discuss The decibel is a relative entity. This means that in Chapter 9, the sound pressure of the loudest the decibel in and of itself is a dimensionless quan- sound that we can tolerate is on the order of 10 mil- tity, and is meaningless without knowledge of the lion times greater than that of the softest audible reference value, which constitutes the denomina- sound. One can immediately imagine the cumber- tor of the ratio. Because of this, it is necessary to some task that would be involved if we were to deal make the reference value explicit when the magni- with such an immense range of numbers on a lin- tude of a sound is expressed in decibel form. This is ear scale. The problems involved with and related accomplished by stating that the magnitude of the to such a wide range of values make it desirable to sound is whatever number of decibels with respect transform the absolute physical magnitudes into to the reference quantity. Moreover, it is common  Decibel notation 7 practice to add the word “level” to the original pressure level. Here, we must be aware that inten- quantity when dealing with dB values. Intensity sity is proportional to pressure squared: expressed in decibels is called intensity level (IL) and sound pressure in decibels is called sound I ∝ p2 (1.22) pressure level (SPL). The reference values indi- cated above are generally assumed when decibels and are expressed as dB IL or dB SPL. For example, one might say that the intensity level of a sound is “50 dB re: 10−12 W/m2” or “50 dB IL.” p∝ I (1.23) The general formula for the decibel is expressed in terms of power as As a result, converting the dB IL formula into the equivalent equation for dB SPL involves replac- P ing the intensity values with the squares of the cor- PL dB = 10 ⋅ log   (1.20) responding pressure values. Therefore,  P0   p2  SPL dB = 10 ⋅ log  2  (1.24) where P is the power of the sound being mea-  p0  sured, P0 is the reference power to which the for- mer is being compared, and PL is the power level. where p is the measured sound pressure and p0 is Acoustical measurements are, however, typically the reference sound pressure (2 × 10−5 N/m2). This made in terms of intensity or sound pressure. The formula may be simplified to applicable formula for decibels of intensity level is thus:  p 2 SPL dB = 10 ⋅ log   (1.25)  p0  I IL dB = 10 ⋅ log   (1.21)  I0  Because the logarithm of a number squared cor- responds to two times the logarithm of that num- where I is the intensity (in W/m2) of the sound in ber (log x = 2 ∙ log x), the square may be removed question, and I0 is the reference intensity, or 10−12 to result in W/m2. Continuing with the example introduced above, where the value of I is 10−10 W/m2, we thus p find that SPL dB = 10 ⋅ 2 ⋅ log   (1.26)  p0   10−10 W/m2  IL dB = 10 ⋅ log  −12   10 W/m2  Therefore, the simplified formula for decibels of SPL becomes = 10 ⋅ log 102 = 10×2 p SPL dB = 20 ⋅ log   (1.27) = 20 dB re: 10−12 W/m2  p0  In other words, an intensity of 10−10 W/m2 where the value of 20 (instead of 10) is due to having ­corresponds to an intensity level of 20 dB re: 10−12 removed the square from the earlier described ver- W/m2, or 20 dB IL. sion of the formula. One cannot take the intensity Sound intensity measurements are important ratio from the IL formula and simply insert it into and useful, and are preferred in certain situations. the SPL formula, or vice versa. The square root of (See Rasmussen for a review of this topic.) the intensity ratio yields the corresponding pressure However, most acoustical measurements involved ratio, which must then be placed into the SPL equa- in hearing are made in terms of sound pres- tion. Failure to use the proper terms will result in sure, and are thus expressed in decibels of sound an erroneous doubling of the value in dB SPL. 8 Physical concepts By way of example, a sound pressure of 2 × 10−4 reference sound pressure of 2 × 10−5 N/m2. Notice N/m2 corresponds to an SPL of 20 dB (re: 2 × 10−5 that 0 dB does not mean “no sound.” Rather, 0 dB N/m2), which may be calculated as follows: implies that the quantity being measured is equal to the reference quantity. Negative decibel values  2×10−4 N/m2  indicate that the measured magnitude is smaller SPL dB = 20 ⋅ log    2×10−5 N/m2  than the reference quantity. Recall that sound intensity drops with distance = 20 ⋅ log 101 from the sound source according to the inverse- = 20×1 square law. However, we want to know the effect of = 20 dB re: 10−5 N/m2 the inverse-square law in terms of decibels of sound pressure level because sound is usually expressed in these terms. To address this, we must first remem- What would happen if the intensity (or pres- ber that pressure is proportional to the square root sure) in question were the same as the reference of intensity. Hence, pressure decreases according intensity (or pressure)? In other words, what is the to the inverse of the distance change (1/D) instead dB value of the reference itself? In terms of inten- of the inverse of the square of the distance change sity, the answer to this question may be found by (1/D2). In effect, the inverse-square law for inten- simply using 10−12 W/m2 as both the numerator (I) sity becomes an inverse-distance law when we are and denominator (I0) in the dB formula; thus, dealing with pressure. Let us assume a doubling as the distance change, because this is the most use- 10−12 W/m2  ful relationship. We can now calculate the size of IL dB = 10 ⋅ log  −12  (1.28) 10 W/m2  the decrease in decibels between a point at some distance from the sound source (Dl, e.g., 1 m) and a Because anything divided by itself equals 1, and point at twice the distance (D2, e.g., 2 m) as follows: the logarithm of 1 is 0, this equation reduces to Level drop in SPL = 20 ⋅ log(D2 /D1 ) IL dB = 10 ⋅ log 1 = 20 ⋅ log(2/1) = 10×0 = 20 ⋅ log 2 = 0 dB re: 10−12 W /m2 = 20×0.3 = 6 dB Hence, 0 dB IL is the intensity level of the refer- ence intensity. Just as 0 dB IL indicates the inten- In other words, the inverse-square law causes sity level of the reference intensity, so 0 dB SPL the sound pressure level to decrease by 6 dB when- similarly implies that the measured sound pressure ever the distance from the sound source is doubled. corresponds to that of the reference For example, if the sound pressure level is 60 dB at  2×10−5 N/m2  1 m from the source, then it will be 60 − 6 = 54 SPL dB = 20 ⋅ log   (1.29) dB when the distance is doubled to 2 m, and  2×10−5 N/m2  54 − 6 = 48 dB when the distance is doubled again from 2 to 4 m. Just as we saw in the previous example, this equation is solved simply as follows: HARMONIC MOTION AND SOUND SPL dB = 20 ⋅ log 1 What is sound? It is convenient to answer this ques- = 20×0 tion with a formally stated sweeping generality. For example, one might say that sound is a form of = 0 dB re: 10−5 N/m2 vibration that propagates through a medium (such as air) in the form of a wave. Although this state- In other words, 0 dB SPL indicates that the pres- ment is correct and straightforward, it can also be sure of the sound in question corresponds to the uncomfortably vague and perplexing. This is so  Harmonic motion and sound 9 because it assumes knowledge of definitions and center (C), the rapidly moving prong overshoots concepts that are used in a very precise way, but this point. It now continues rightward (arrow 3), which are familiar to most people only as “gut-level” slowing down along the way until it comes to a generalities. As a result, we must address the under- halt at point R (right). It now reverses direction lying concepts and develop a functional vocabulary and begins moving leftward (arrow 4) at an ever- of physical terms that will not only make the gen- increasing speed, so that it again overshoots the eral definition of sound meaningful, but will also center. Now, again following arrow 1, the prong allow the reader to appreciate its nature. slows down until it reaches a halt at L, where it Vibration is the to-and-fro motion of a body, reverses direction and repeats the process. which could be anything from a guitar string to The course of events just described is the result the floorboards under the family refrigerator, or a of applying a force to an object having the proper- molecule of air. Moreover, the motion may have a ties of elasticity and inertia (mass). The initial force very simple pattern as produced by a tuning fork, to the tuning fork displaces the prong. Because the or an extremely complex one such as one might tuning fork possesses the property of elasticity, the hear at lunchtime in an elementary school caf- deformation caused by the applied force is opposed eteria. Even though few sounds are as simple as by a restoring force in the opposite direction. In that produced by a vibrating tuning fork, such an the case of the single prong in Figure 1.2, the initial example provides what is needed to understand the force toward the left is opposed by a restoring force nature of sound. toward the right. As the prong is pushed farther Figure 1.1 shows an artist’s conceptualization to the left, the magnitude of the restoring force of a vibrating tuning fork at different moments of increases relative to the initially applied force. As its vibration pattern. The heavy arrow facing the a result, the prong’s movement is slowed down, prong to the reader’s right in Figure 1.1a repre- brought to a halt at point L, and reversed in direc- sents the effect of applying an initial force to the tion. Now, under the influence of its elasticity, the fork, such as by striking it against a hard surface. prong starts moving rightward. Here, we must The progression of the pictures in the figure from consider the mass of the prong. (a) through (e) represents the movements of the As the restoring force brings the prong back prongs as time proceeds from the moment that the toward its resting position (C), the inertial force of outside force is applied. its mass causes it to increase in speed, or acceler- Even though both prongs vibrate as mirror ate. When the prong passes through the resting images of one another, it is convenient to consider position, it is actually moving fastest. Here, inertia just one of them for the time being. Figure 1.2 high- does not permit the moving mass (prong) to simply lights the right prong’s motion after being struck. stop, so instead it overshoots the center and con- Point C (center) is simply the position of the prong tinues its rightward movement under the force of at rest. Upon being hit (as in Figure 1.1a) the prong is pushed, as shown by arrow 1, to point L (left). L C R The prong then bounces back (arrow 2), picking 1 4 up speed along the way. Instead of stopping at the 2 3 (a) (b) (c) (d) (e) Figure 1.1 Striking a tuning fork (indicated by the heavy arrow) results in a pattern of movement that repeats itself over time. One complete cycle Figure 1.2 Movements toward the right (R) of these movements is represented from frames and left (L) of the center (C) resting position of (a) through (e). Note that the two prongs move as a single tuning fork prong. The numbers and mirror images of one another. arrows refer to the text. 10 Physical concepts its inertia. However, the prong’s movement is now The events and forces just described are summa- resulting in deformation of the metal again once rized in Figure 1.3, where the tuning fork’s motion it passes through the resting position. Elasticity is represented by the curve. This curve represents therefore comes into play with the buildup of an the displacement to the right and left of the center opposing (now leftward) restoring force. As before, (resting) position as the distance above and below the restoring force eventually equals the applied the horizontal line, respectively. Horizontal dis- (now inertial) force, thus halting the fork’s dis- tance from left to right represents the progression placement at point R and reversing the direction of of time. The initial dotted line represents its initial its movement. Here, the course of events described displacement due to the applied force. The elastic above again comes into play (except that the direc- restoring forces and inertial forces of the prong’s tion is leftward), with the prong building up speed mass are represented by arrows. Finally, damping again and overshooting the center (C) position as a is shown by the reduction in the displacement of result of inertia. The process will continue over and the curve from center as time goes on. over again until it dies out over time, seemingly “of The type of vibration just described is called its own accord.” simple harmonic motion (SHM) because the to- Clearly, the dying out of the tuning fork’s vibra- and-fro movements repeat themselves at the same tions does not occur by some mystical influence. rate over and over again. We will discuss the nature On the contrary, it is due to resistance. The vibrat- of SHM in greater detail below with respect to the ing prong is always in contact with the air around motion of air particles in the sound wave. it. As a result, there will be friction between the The tuning fork serves as a sound source by vibrating metal and the surrounding air particles. transferring its vibration to the motion of the sur- The friction causes some of the mechanical energy rounding air particles (Figure 1.4). (We will again involved in the movement of the tuning fork to be concentrate on the activity to the right of the fork, converted into heat. The energy that has been con- remembering that a mirror image of this pattern verted into heat by friction is no longer available occurs to the left.) The rightward motion of the to support the to-and-fro movements of the tuning tuning fork prong displaces air molecules to its fork. Hence, the oscillations die out as continuing right in the same direction as the prong’s motion. friction causes more and more of the energy to be These molecules are thus displaced to the right of converted into heat. This reduction in the size of their resting positions, thereby being forced closer the oscillations due to resistance is called damping. and closer to the particles to their own right. In Left Time Displacement from center (C) rce Restoring Damping due to friction ied fo force Elasticity resting position Appl C Inertia Mass Right Figure 1.3 Conceptualized diagram graphing the to-and-fro movements of the tuning fork prong in Figure 1.2. Vertical distance represents the displacement of the prong from its center (C) or resting position. The dotted line represents the initial displacement of the prong as a result of some applied force. Arrows indicate the effects of restoring forces due to the fork’s elasticity, and the inertia due to its mass. The damping effect due to resistance (or friction) is shown by the decreasing displace- ment of the curve as time progresses, and is highlighted by the shaded triangles (and double-headed arrows) above and below the curve.  Harmonic motion and sound 11 L C R As the air molecules move left of their ambi- (a) ent positions, they are now at an increasingly greater distance from the molecules to their right than when they were in their resting positions. (b) Consequently, the air pressure is reduced below atmospheric pressure. This state is the opposite (c) of compression, and is called rarefaction. The air particles are maximally rarefied so that the pres- (d) sure is maximally negative when the molecules reach the leftmost position. Now, the restoring force yields a rightward movement of the air mol- (e) ecules, enhanced by the push of the tuning fork prong which has also reversed direction. The air molecules now accelerate rightward, overshoot Figure 1.4 Transmittal of the vibratory pattern their resting positions (when rarefaction and nega- from a tuning fork to the surrounding air par- tive pressure are zero), and continue rightward. ticles. Frames represent various phases of the Hence, the SHM of the tuning fork has been trans- tuning fork’s vibratory cycle. In each frame, the filled circle represents an air particle next to the mitted to the surrounding air, so that the air mol- prong as well as its position, and the unfilled ecules are now also under SHM. circle shows an air molecule adjacent to the first Consider now one of the air molecules set into one. The latter particle is shown only in its resting SHM by the influence of the tuning fork. This air position for illustrative purposes. Letters above molecule will vibrate back and forth in the same the filled circle highlight the relative positions of direction as that of the vibrating prong. When this the oscillating air particle (C, center [resting]; L, molecule moves rightward, it will cause a similar leftward; R, rightward). The line connecting the displacement of the particle to its own right. Thus, particle’s positions going from frames (a) through the SHM of the first air molecule is transmitted to (e) reveals a cycle of simple harmonic motion. the one next to it. The second one similarly initi- ates vibration of the one to its right, and so forth other words, the air pressure has been increased down the line. above its resting (ambient or atmospheric) pres- In other words, each molecule moves to-and-fro sure because the molecules are being compressed. around its own resting point, and causes successive This state is clearly identified by the term “com- molecules to vibrate back and forth around their pression.” The amount of compression (increased own resting points, as shown schematically by the air pressure) becomes greater as the tuning fork arrows marked “individual particles” in Figure 1.5. continues displacing the air molecules rightward; Notice in the figure that each molecule stays in its it reaches a maximum positive pressure when the prong and air molecules attain their greatest right- Transverse: λ ward amplitude. The prong will now reverse direction, over- shoot its resting position, and then proceed to its Longitudinal: extreme leftward position. The compressed air molecules will also reverse direction along with the prong. The reversal occurs because air is an elastic medium, so that the rightward compressed Individual particles: particles undergo a leftward restoring force. The rebounding air molecules accelerate due to mass effects, overshoot their resting position, and con- Wave propagation tinue to an extreme leftward position. The amount of compression decreases as the molecules travel Figure 1.5 Transverse and longitudinal repre- leftward, and falls to zero at the moment when the sentations of a sinusoidal wave illustrating points molecules pass through their resting positions. made in the text. 12 Physical concepts own general location and moves to-and-fro about as though they were transverse, as in the upper part this average position; and that it is the vibratory of Figure 1.5. Here, the dashed horizontal baseline pattern that is transmitted. represents the particle’s resting position (ambient This propagation of vibratory motion from par- pressure), distance above the baseline denotes com- ticle to particle constitutes the sound wave. This pression (positive pressure), and distance below the wave appears as alternating compressions and rar- baseline shows rarefaction (negative pressure). The efactions radiating from the sound source as the passage of time is represented by the distance from particles transmit their motions outward, and is left to right. Beginning at the resting position, the represented in Figure 1.5. air molecule is represented as having gone through The distance covered by one cycle of a propagat- one cycle (or complete repetition) of SHM at point ing wave is called its wavelength (λ). If we begin 1, two cycles at point 2, three complete cycles at where a given molecule is at the point of maxi- point 3, and four cycles at point 4. mum positive displacement (compression), then The curves in Figure 1.5 reveal that the wave- the wavelength would be the distance to the next form of SHM is a sinusoidal function, and is thus molecule, which is also at its point of maximum called a sinusoidal wave, also known as a sine compression. This is the distance between any two wave or a sinusoid. Figure 1.6 elucidates this con- successive positive peaks in the figure. (Needless to cept and also indicates a number of the character- say, such a measurement would be equally correct istics of sine waves. The center of the figure shows if made between identical points on any two suc- one complete cycle of SHM, going from points a cessive replications of the wave.) The wavelength of through i. The circles around the sine wave cor- a sound is inversely proportional to its frequency, respond to the various points on the wave, as as follows: indicated by corresponding letters. Circle (a) cor- responds to point a on the curve, which falls on the c baseline. This point corresponds to the particle’s λ= (1.30) resting position. f Circle (a) shows a horizontal radius (r) drawn from the center to the circumference on the right. where f is frequency and c is a constant represent- Imagine as well a second radius (r′) that will rotate ing the speed of sound. (The speed of sound in air around the circle in a counterclockwise direction. approximates 344 m/s at a temperature of 20°C.) The two radii are superimposed in circle (a) so that Similarly, frequency can be derived if one knows the angle between them is 0°. There is clearly no the wavelength, as: distance between these two superimposed lines. This situation corresponds to point a on the sine c wave at the center of the figure. Hence, point a may f= (1.31) λ be said to have an angle of 0°, and no displacement from the origin. This concept may appear quite Figure 1.5 reveals that the to-and-fro motions of vague at first, but it will become clear as the second each air molecule is in the same direction as that in radius (r′) rotates around the circle. which the overall wave is propagating. This kind of Let us assume that radius r′ is rotating coun- wave, which characterizes sound, is a longitudinal terclockwise at a fixed speed. When r′ has rotated wave. In contrast to longitudinal waves, most peo- 45°, it arrives in the position shown in circle (b). ple are more familiar with transverse waves, such Here, r′ is at an angle of 45° to r. We will call this as those that develop on the water’s surface when angle the phase angle (θ), which simply reflects the a pebble is dropped into a still pool. The latter are degree of rotation around the circle, or the number called transverse waves because the water particles of degrees into the sine wave at the correspond- vibrate up and down around their resting positions ing point b. We now drop a vertical line from the at right angles (transverse) to the horizontal propa- point where r′ intersects the circle down to r. We gation of the surface waves out from the spot where label this line d, representing the vertical distance the pebble hit the water. between r and the point where r′ intersects the Even though sound waves are longitudinal, it is circle. The length of this line corresponds to the more convenient to show them diagrammatically displacement of point b from the baseline of the  Harmonic motion and sound 13 (b) (c) (d) r′ r′ θ d θ r′ θ r r r θ = 45° θ = 90° θ = 135° sin θ =.707 sin θ = 1.0 sin θ =.707 c b d (a,i) (e) e a i θ r, r′ r′ r f h θ = 180° θ = 0°, 360° g sin θ = 0 sin θ = 0 0 45 90 135 180 225 270 315 360 Degrees 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π Radians (h) (g) (f) r r θ θ θ r r′ θ = 315° r′ θ = 270° r′ θ = 225° sin θ = –.707 sin θ = –.1 sin θ = –.707 Figure 1.6 The nature of sinusoidal motion (see text). sine wave (dotted line at [b]). We now see that point (e) by the fact that r and r′ constitute a single hori- b on the sine wave is 45° into the cycle of SHM, zontal line (diameter). Alternatively stated, r and at which the displacement of the air particle from r′ intersect the circle’s circumference at points that its resting position is represented by the height are 180° apart. Here, we have completed half of the of the point above the baseline. It should now be cycle of SHM, and the phase angle is 180° and the clear that the sine wave is related to the degrees of displacement from the baseline is again zero. rotation around a circle. The sine wave’s shape cor- Continuing rotation of r′ places its intersection responds to the sine of θ as r′ rotates around the with the circumference in the lower left quadrant circle, which is simply equal to d/r′. of the circle, as in circle (f). Now, θ is 225°, and the The positive peak of the sine wave at point c cor- particle has overshot and is moving away from its responds to circle (c), in which r′ has rotated to the resting position in the negative (rarefaction) direc- straight up position. It is now at a 90° angle to r, and tion. The vertical displacement from the baseline the distance (d) down to the horizontal radius (r) is is now downward or negative, indicating rarefac- greatest. Here, we have completed a quarter of the tion. The negative peak of the wave occurs at 270°, wave and an arc equal to a quarter of the circum- where displacement is maximum in the negative ference of the circle. Notice now that further coun- direction (point and circle [g]). terclockwise rotation of r′ results in decreasing the Circle (h) and point h show that the negative distance (d) down to the horizontal, as shown in displacement has become smaller as the rotating circle (d) and by the displacement of point d from radius passes 315° around the circle. The air parti- the baseline of the sine wave. Note also that θ is cle has reversed direction again and is now moving now 135°. Here, the air particle has reversed direc- toward its original position. At point i, the air par- tion and is now moving back toward the resting ticle has once again returned to its resting position, position. When the particle reaches the resting where displacement is again zero. This situation position (point [e]), it is again at no displacement. corresponds to having completed a 360° rotation, The zero displacement condition is shown in circle so that r and r′ are once again superimposed. Thus, 14 Physical concepts 360° corresponds to 0°, and circle (i) is one and the repetition of the wave. Thus, four cycles of a sinu- same with circle (a). We have now completed one soidal wave were shown in Figure 1.5 because it full cycle. depicts four complete repetitions of the waveform. Recall that r′ has been rotating at a fixed speed. Because the waveform is repeated over time, this It therefore follows that the number of degrees tra- sound is said to be periodic. In contrast, a wave- versed in a given amount of time is determined by form that does not repeat itself over time would be how fast r′ is moving. If one complete rotation takes called aperiodic. 1 s, then 360° is covered each second. It clearly fol- The amount of time that it takes to complete lows that if 360° takes 1 s, then 180° takes 0.5 s, 90° one cycle is called its period, denoted by the sym- takes 0.25 s, 270° takes 0.75 s, and so on. It should bol t (for time). For example, a periodic wave that now be apparent that the phase angle reflects the repeats itself every millisecond is said to have a elapsed time from the onset of rotation. Recall from period of 1 ms, or t = 1 ms or 0.001 s. The periods Figure 1.3 that the waveform shows how particle of the waveforms considered in hearing science are displacement varies as a function of time. We may overwhelmingly less than 1 s, typically in the mil- also speak of the horizontal axis in terms of phase, liseconds and even microseconds. However, there or the equivalent of the number of degrees of rota- are instances when longer periods are encountered. tion around a circle. Hence, the phase of the wave at The number of times a waveform repeats itself each of the labeled points in Figure 1.6 would be: 0° per unit of time is its frequency (f). The standard at (a), 45° at (b), 90° at (c), 135° at (d), 180° at (e), 225° unit of time is the second; thus, frequency is the at (f), 270° at (g), 315° at (h), and 360° at (i). With number of times that a wave repeats itself in a sec- an appreciation of phase, it should be apparent that ond, or the number of cycles per second (cps). By each set of otherwise identical waves in Figure 1.7 convention, the unit of cycles per second is the differs with respect to phase: (a) wave 2 is offset from hertz (Hz). Thus, a wave that is repeated 1000 wave 1 by 45°; (b) waves 3 and 4 are apart in phase times per second has a frequency of 1000 Hz, and by 90°; and (c) waves 5 and 6 are 180° out of phase. the frequency of a wave that repeats at 2500 cycles We may now proceed to define a number of per second is 2500 Hz. other fundamental aspects of sound waves. A If period is the time it takes to complete one cycle has already been defined as one complete cycle, and frequency is the number of cycles that occur each second, then it follows that period and (a) 1 2 frequency are intimately related. Consider a sine wave that is repeated 1000 times per second. By definition it has a frequency of 1000 Hz. Now, if exactly 1000 cycles take exactly 1 s, then each cycle must clearly have a duration of 1 ms, or 1/1000 s. 3 4 Similarly, each cycle of a 250-Hz tone must last (b) 1/250 s, or a period of 4 ms. Formally, then, fre- quency is the reciprocal of period, and period is the reciprocal of frequency: 1 5 6 f= (1.32) (c) t and 1