Essentials of Audiology Fourth Edition PDF

Essentials of Audiology Fourth Edition Stanley A. Gelfand, PhD Professor Department of Linguistics and Communication Disorders Queens College of The City University of New York Flushing, New York PhD Program in Speech-Language-Hearing Sciences and AuD Program Graduate Center of The City Univers...

Essentials of Audiology Fourth Edition Stanley A. Gelfand, PhD Professor Department of Linguistics and Communication Disorders Queens College of The City University of New York Flushing, New York PhD Program in Speech-Language-Hearing Sciences and AuD Program Graduate Center of The City University of New York New York, New York 290 illustrations Thieme New York Stuttgart Delhi Rio de Janeiro Executive Editor: Anne Sydor Important note: Medicine is an ever-changing science undergoing Managing Editor: Elizabeth Palumbo continual development. Research and clinical experience are con- Director, Editorial Services: Mary Jo Casey tinually expanding our knowledge, in particular our knowledge of Editorial Assistant: Natascha Morris proper treatment and drug therapy. Insofar as this book mentions Production Editor: Kenneth L. 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This applies in par- ticular to photostat reproduction, copying, mimeographing or Also available as an eBook: duplication of any kind, translating, preparation of microfilms, eISBN 978-1-60406-862-7 and electronic data processing and storage.  To Janice In loving memory Contents Preface................................................................................................................................................................................. ix â ‡1 Acoustics and Sound Measurement..............................................................................................................................1 â ‡2 Anatomy and Physiology of the Auditory System.................................................................................................. 30 â ‡3 Measurement Principles and the Nature of Hearing............................................................................................. 70 â ‡4 The Audiometer and Test Environment.................................................................................................................... 91 â ‡5 Pure Tone Audiometry.................................................................................................................................................108 â ‡6 Auditory System and Related Disorders.................................................................................................................136 â ‡7 Acoustic Immittance Assessment.............................................................................................................................182 â ‡8 Speech Audiometry.......................................................................................................................................................215 â ‡9 Clinical Masking.............................................................................................................................................................248 10 Behavioral Tests for Audiological Diagnosis...........................................................................................................273 11 Physiological Methods in Audiology........................................................................................................................302 12 Assessment of Infants and Children.........................................................................................................................329 13 Audiological Screening.................................................................................................................................................348 14 Nonorganic Hearing Loss.............................................................................................................................................372 15 Audiological Management I........................................................................................................................................390 16 Audiological Management II......................................................................................................................................406 17 Effects of Noise and Hearing Conservation............................................................................................................456 Appendixes......................................................................................................................................................................497 Subject Index..................................................................................................................................................................513 Author Index...................................................................................................................................................................527 vii Preface What is audiology? Audiology is the clinical profession disorders, and being able to understand audiological that deals with hearing and balance disorders. It reports. Speech-language pathologists often find is also the scientific study of normal and abnormal themselves working hand-in-hand with their audition and related areas in the broadest sense. audiological colleagues. They also need to perform What is an audiologist? An audiologist is a certain audiological procedures themselves when practitioner of audiology as a clinical profession. these fall within the speech-language pathology scope Audiologists are principally concerned with the of practice, especially when screening is involved; identification, evaluation and management of patients and they regularly make interpretations and referrals with auditory and balance disorders, as well as with that are of audiological relevance. Moreover, speech- the prevention of hearing impairment. The scope language pathologists often work with patients who of audiological practice also includes such diverse have hearing losses and auditory processing disorders areas as the evaluation of the vestibular system, noise directly and on an ongoing basis. They frequently assessment, and hearing conservation, as well as the must explain the nature and management of auditory physiological monitoring of various neurological disorders to family members, teachers and other functions during surgical procedures. As a result, professionals. This is especially true in school settings audiologists possess a broad scope of knowledge and and long-term care facilities. What’s more, cochlear skills, and often have interests in common with a implant and other multidisciplinary programs are variety of other disciplines such as speech-language enhancing the scope and depth of interactions among pathology; speech and hearing science; education of speech-language pathologists and audiologists, the deaf and hearing impaired; engineering; acoustics; and are making a knowledge and understanding industrial hygiene; musicology; medicine; physiology; of audiology all the more important for budding psychology; linguistics; and vocational counseling. speech-language pathologists. With considerations Much like many other scholarly professions, like these in mind, I hope that students who become pursuing a career in audiology involves a rigorous speech-language pathologists will find this text course of doctoral education and training. Most useful as a reference source long after their audiology audiologists earn the Doctor of Audiology (Au.D.) courses have been completed. (Of course, I do admit degree, while others with research and academic hoping that at least a few speech-language pathology as well as clinical interests obtain a Ph.D. Some students will be attracted to a career in audiology by pursue both. In addition, those qualified to practice what they read here.) audiology are usually certificated by the American This textbook attempts to provide a comprehensive Speech-Language-Hearing Association (ASHA) and/ overview of audiology at the introductory level; or the American Academy of Audiology (AAA), as well including such topics as acoustics, anatomy and as possessing professional licenses from the states in physiology, sound perception, auditory disorders which they practice. and the nature of hearing impairment, methods of So who is this book for? Introductory audiology is measurement, screening, clinical assessment and an essential and fundamental aspect of the education clinical management. It is intended to serve as the core of all students who are interested in the two related text for undergraduate students in speech, language professions of Speech-Language Pathology and and hearing, and might also serve the needs of Audiology. This book is primarily intended to serve beginning-level graduate students who need to learn or as a comprehensive introductory text for students review the fundamentals of audiology. It is anticipated who are preparing to enter both of these fields. that the material will be covered in a one-, two- or As such, it tries to address the needs of two rather three-term undergraduate sequence, depending on different groups of students. Those planning a career the organization of the communication sciences and in audiology need a broad overview of the field and disorders curriculum at a given college or university. a firm understanding of its many basic principles so For example, the first three chapters are often used that they have a solid foundation for futures as doctors the text for an undergraduate hearing science course, of audiology in clinical practice. The audiological while selections from the other chapters might be needs of future speech-language pathologists are used in one or two audiology courses. just as important, and go well beyond knowing the With these considerations in mind, I have tried auditory implications of speech, language and related to prepare a textbook that is extensive enough for ix x Preface professors pick and choose material that provides and the history of audiology has been omitted. The use the right depth and scope of coverage for a particular of gender-specific pronouns (he, she, him, her, etc.)— course. For example, text readings can be assigned to originally undertaken with great trepidation—was cover clinical masking at almost any level from simple very well received in all three prior editions, and has to complex by selecting various sections of Chapter 9. been continued in this one. Its purpose is to maximize It is unlikely that all of that chapter will be assigned clarity for the benefit of the reader. The alternative in a single undergraduate class. However, the material would have resulted in longer phrases and a more is there if needed for further study, to provide the formal style, and would have detracted from the goal groundwork for a term-paper or independent study of providing the student with text that is maximally report, or for future reference. I have tried to provide reader-friendly (or at least minimally unfriendly). relatively extensive reference lists are provided for This style also makes the material considerably easier similar reasons. to follow by using different genders for the clinician This fourth edition was undertaken to provide the and patient when describing clinical procedures beginning student with an up-to-date coverage of and interactions. Gender fairness is maintained by a field that is steadily developing, as well as to take referring to both genders in both roles more-or-less advantage of accumulating experience to improve equally throughout the text. upon what is included and how it is presented. This book would not exist without the influence of Many developments and changes have taken place many very special people. I am particularly grateful since the third edition was published. Some of them to my colleagues and students in the Department of are in areas of rapidly unfolding development like Linguistics and Communication Disorders at Queens cochlear implants and related technologies and in College, and in the Au.D. Program and the Ph.D. electrophysiological assessment. But most of them Program in Speech Language and Hearing Sciences are the slow, methodical and often subtle—albeit at the City University of New York Graduate Center. important—advances that unfold over time in an I would also like to express my appreciation to the active clinical science. Others changes reflect the extraordinary, talented and dedicated professionals at influence of systematic reviews, changes in guidelines, Thieme Medical Publishers, who have been so helpful, expert position papers, standards and regulations cooperative and supportive throughout the process which affect clinical practices and technical matters. of preparing this book, in spite of its demanding and Of course, there are always a few developments that dyslexic author. With sincere apologies to anyone surfaced the day after the prior edition was printed—a inadvertently omitted, I would like to extend my frustration to the textbook author, but the kind of heartfelt thanks to the following individuals for thing that makes audiology such an exciting and their influence, insights, advice, encouragement, interesting field. assistance, support, and friendship: Moe Bergman, As with the prior editions, this one was influenced Arthur Boothroyd, Lauren Calandruccio, Kenny by the input graciously provided by of many Chumbley, Joseph Danto, Becky Dille, Lillian and audiologists involved in clinical practice, research, Sol Gelfand, Irving Hochberg, Gertrude and Oscar and teaching and student supervision. In addition, Katzen, Arlene Kraat, William Lamsbeck, Harry Levitt, considerable attention was given to the comments John Lutolf, Dave Mason, Maurice Miller, Natascha and insights of students who were taking or recently Morris, Elizabeth Palumbo, Neil Piper, John Preece, completed introductory audiology courses, including Brian Scanlan, Teresa Schwander, Shlomo Silman, those who used the third edition of this text as well Carol Silverman, Anne Sydor, Helen and Harris Topel, as other books. The content and especially the style of Barbara Weinstein, and Mark Weiss. the text were substantially influenced by their advice. Finally, my greatest appreciation is expressed to As a result of their insights, the current edition retains Janice, whose memory will always be a blessing and a writing style that has been kept as conversational inspiration; and to my wonderful children, Michael, and informal as possible; and only classroom-proven Joshua and Erin, and Jessica and Robert for their love, examples and drawings are included. Similarly, clinical encouragement, support, insight and unimaginable masking, acoustic immittance and screening have been patience. kept in separate chapters; the material on audiological management continues to be spread over two chapters; Stanley A. Gelfand 1 Acoustics and Sound Measurement We begin our study of audiology by reviewing the force and work are called newtons and joules in the nature of sound because, after all, sound is what we MKS system, and dynes and ergs in the cgs system, hear. The science of sound is called acoustics, which is respectively. We will emphasize the use of MKS units a branch of physics, and relies on several basic physi- because this is the internationally accepted standard cal principles. Many useful sources are available for in the scientific community, known as the Systeme students wishing to pursue the areas covered in this Internationale (SI). Equivalent cgs values will often chapter in greater detail (e.g., Beranek 1986; Gelfand be given as well because the audiology profession 2010; Hewitt 1974; Kinsler, Frey, Coppens, & Sand- has traditionally worked in cgs units, and the death ers 1982; Peterson & Gross 1972; Sears, Zemansky, of old habits is slow and labored. These quantities are & Young 1982). summarized with equivalent values in MKS and cgs units in Table 1.1. In addition, the correspondence between scientific notation and conventional num- bers, and the meanings of prefixes used to describe ⌀ Physical Quantities the sizes of metric units are shown for convenience and ready reference in Table 1.2 and Table 1.3. The basic physical quantities are mass, time, and Quantities may be scalars or vectors. A scalar can length (or distance). All other physical quantities be fully described by its magnitude (amount or size), are derived by combining these three basic ones, but a vector has both direction and magnitude. For as well as other derived quantities, in a variety of example, length is a scalar because an object that is ways. The principal basic and derived quantities are one meter long is always one meter long. However, summarized in Table 1.1. These basic quantities are we are dealing with a vector when we measure the expressed in terms of conventional units that are distance between two coins that are one meter apart measurable and repeatable. The unit of mass (M) is because their relationship has both magnitude and the kilogram (kg) or the gram (g); the unit of length direction (from point x1 to point x2). This quantity is (L) is the meter (m) or the centimeter (cm); and the called displacement (d). Derived quantities will be unit of time (t) is the second (s). Mass is not really vectors if they have one or more components that synonymous with weight even though we express are vectors; for example, velocity is a vector because its magnitude in kilograms. The mass of a body is it is derived from displacement, and acceleration is related to its density, but its weight is related to the a vector because it involves velocity. We distinguish force of gravity. If two objects are the same size, the between scalars and vectors because they are han- one with greater density will weigh more. However, dled differently when calculations are being made. even though an object’s mass would be identical on Velocity Everyone knows that “55 miles per the earth and the moon, it would weigh less on the hour” refers to the speed of a car that causes it to moon, where there is less gravity. travel a distance of 55 miles in a one-hour period When we express mass in kilograms and length of time. This is an example of velocity (v), which is in meters, we are using the meter–kilogram-second equal to the amount of displacement (d) that occurs or MKS system. Expressing mass in grams and length over time (t): in centimeters constitutes the centimeter-gram-sec- ond or cgs system. These two systems also have dif- d ferent derived quantities. For example, the units of v= t 1 2 1â … Acoustics and Sound Measurement Table 1.1â … Principal physical quantities Quantity Formula MKS (Sl) units cgs units Comments Mass (M) M kilogram (kg) gram (g) 1 kg = 103 g Time (t) t second (s) s Area (A) A m2 cm2 1 m2 = 104 cm2 Displacement (d) d meter (m) centimeter (cm) 1 m = 102 cm Velocity (v) v = d/t m/s cm/s 1 m/s = 102 cm/s Acceleration (a) a = v/t m/s2 cm/s2 1 m/s2 = 102 cm/s2 Force (F) F = Ma kg m/s2 g cm/s2 1 N = 105 dyne = Mv/t newton (N) dyne Pressure (p) p = F/A N/m2 dyne/cm2 microbar 2 × 10−5 N/m2 or 20 μPa pascal (Pa) (μbar) (reference value) 2 × 10−4 dyne/cm2 or µbar (reference value) Work (W) W = Fd N m dyne cm 1 J = 107 erg joule (J) erg Power (P) P = W/t joule/s erg/s 1 w = 1 J/s â ›= Fd/t watt (w) watt (w) â ›= 107 erg/s â ›= Fv Intensity (I) I = P/A w/m2 w/cm2 10−12 w/m2 (reference value) 10−16 w/cm2 (reference value) Displacement is measured in meters and time is The term instantaneous velocity describes the measured in seconds (sec); thus, velocity is expressed velocity of a body at a particular moment in time. For in meters per second (m/s). Velocity is the vector the math-minded, it refers to the velocity when the equivalent of speed because it is based on displace- displacement and time between one point and the ment, which has both magnitude and direction. When next one approach zero, that is, the derivative of dis- we take a trip we usually figure out the distance trav- placement with respect to time: eled by making a mental note of the starting odometer reading and then subtracting it from the odometer dx v= reading at the destination (e.g., if we start at 10,422 dt miles and arrive at 10,443 miles, then the distance Acceleration Driving experience has taught us must have been 10,443 – 10,422 = 21 miles). We do all that a car increases its speed to get onto a high- the same thing to calculate the time it took to make way, slows down when exiting, and also slows down the trip (e.g., if we left at 1:30 and arrived at 2:10, then while making a turn. “Speeding up” and “slowing the trip must have taken 2:10 – 1:30 = 40 minutes). down” mean that the velocity is changing over time. Physical calculations involve the same straightforward The change of velocity over time is acceleration (a). approach. When an object is displaced, it starts at Suppose a body is moving between two points. Its point x1 and time t1 and arrives at point x2 and time t2. velocity at the first point is v1, and the time at that Its average velocity is simply the distance traveled (x2 – point is t1. Similarly, its velocity at the second point x1) divided by the time it took to make the trip (t2 – t1): is v2 and the time at that point is t2. Average accelera- x2 − x1 v= t 2 − t1  1â … Acoustics and Sound Measurement 3 Table 1.2â … Expressing numbers in standard notation tion is the difference between the two velocities (v2 and scientific notation – v1) divided by the time interval (t2 – t1): Standard notation Scientific notation v2 − v1 a= 0.000001 10−6 t 2 − t1 0.00001 10−5 In more general terms, acceleration is written simply as 0.0001 10−4 v 0.001 10–3 a= t 0.01 10–2 Because velocity is the same as displacement divided 0.1 10–1 by time, we can replace v with d/t, so that 1 100 d /t a= 10 101 t which can be simplified to 100 102 103 d 1,000 a= t2 10,000 104 Consequently, acceleration is expressed in units of 100,000 105 meters per second squared (m/s2) in the MKS system. 1,000,000 106 When measurements are made in cgs units, acceler- ation is expressed in centimeters per second squared 3600 3.6 × 103 (cm/s2). Acceleration at a given moment is called instanta- 0.036 3.6 × 10–2 neous acceleration, and quantitatively oriented read- 0.0002 2 × 10−4 ers should note it is equal to the derivative of velocity with respect to time, or 0.00002 2 × 10−5 dv a= dt Table 1.3â … Examples of prefixes used to express metric units Multiply by Prefix Symbol Definition Standard notation Scientific notation micro µ millionths 1/1,000,000 or 0.000001 10−6 milli m thousanths 1/1000 or 0.001 10–3 cent c hundredths 1/100 or 0.01 10–2 deci d tenths 1/10 or 0.1 10–1 deka da tens 10 101 hecto h hundreds 100 102 kilo k thousands 1000 103 mega M millions 1,000,000 106 4 1â … Acoustics and Sound Measurement Because velocity is the first derivative of displace- Many different forces are usually acting upon an ment, we find that acceleration is the second deriva- object at the same time. Hence, the force we have tive of displacement: been referring to so far is actually the net or resultant force, that is, the “bottom line” effect of all the forces d2 x that act upon an object. If a force of 3 N is pushing an a= dt 2 object toward the right and a second force of 8 N is also pushing that object toward the right, then the net force Force An object that is sitting still will not move would be 3 + 8 = 11 N toward the right. In other words, unless some outside influence causes it to do so, and if two forces push a body in the same direction, then an object that is moving will continue moving at the net force would be the sum of those two forces. the same speed unless some outside influence does Conversely, if a 4 N force pushes an object toward the something to change it. This commonsense state- right at the same time that a 9 N force pushes it toward ment is Newton’s first law of motion. It describes the the left, then the net force is 9 – 4 = 5 N toward the left. attribute of inertia, which is the property of mass to Thus, if two forces push an object in opposite direc- continue doing what it is already doing. The “outside tions, then the net force is the difference between the influence” that makes a stationary object move, or two opposing forces, and it causes the object to accel- causes a moving object to change its speed or direc- erate in the direction of the greater force. If two equal tion, is called force (F). Notice that force causes the forces push in opposite directions, then the net force is moving object to change velocity or the motionless zero. Because the net force is zero it will not cause the object to move, which is also a change in velocity motion of the object to change. The situation in which (from zero to some amount). Recall that a change of net force is zero is called equilibrium. In this case, a velocity is acceleration. Hence, force is that influence moving object will continue moving and an object that (conceptually a “push” or “pull”) that causes a mass is at rest (i.e., not moving) will continue to remain still. to be accelerated. In effect, the amount of “push” or Friction When an object is moving in the real “pull” needed depends on how much mass you want world, it tends to slow down and eventually comes to to influence and the amount of acceleration you are a halt. This happens because anything that is moving trying to produce. In other words, force is equal to in the real world is always in contact with other bod- the product of mass times acceleration: ies or mediums. The sliding of one body on the other constitutes a force that opposes the motion, called F = Ma resistance or friction. The opposing force of friction or resistance Since acceleration is velocity over time (v/t), we can depends on two parameters. The first factor is that also specify force in the form the amount of friction depends on the nature of the materials that are sliding on one another. Simply Mv stated, the amount of friction between two given F= t objects is greater for “rough” materials than for “smooth” or “slick” ones, and is expressed as a quan- The quantity Mv is called momentum, so we may tity called the coefficient of friction. The second factor also say that force equals momentum over time. that determines how much friction occurs is easily The amount of force is measured in kg m/s2 appreciated by rubbing the palms of your hands back because force is equal to the product of mass (mea- and forth on each other. First rub slowly and then sured in kg) and acceleration (measured in m/s2). The more rapidly. The rubbing will produce heat, which unit of force is the newton (N), where one newton is occurs because friction causes some of the mechani- the amount of force needed to cause a 1 kg mass to cal energy to be converted into heat. This notion will be accelerated by 1 m/s2; hence, 1 N = 1 kg 1 m/s2. be revisited later, but for now we will use the amount (This might seem very technical, but it really simpli- of heat as an indicator of the amount of resistance. fies matters; after all, it is easier to say “one newton” Your hands become hotter when they are rubbed than “one kg m/s2.”). It would take a 2 N force to together more quickly. This illustrates the notion cause a 1 kg mass to be accelerated by 2 m/s2, or a that the amount of friction depends on the velocity 2 kg mass to be accelerated by 1 m/s2. A 4 N force is of motion. In quantitative terms, needed to accelerate a 2 kg mass by 2 m/s2, and a 63 N force is needed to accelerate a 9 kg mass by 7 m/s2. F = Rv In the cgs system, the unit of force is called the dyne, which is the force needed to accelerate a 1 g mass by where F is the force of friction, R is the coefficient of 1 cm/s2; that is, 1 dyne = 1 g cm/s2. It takes 105 dynes friction between the materials, and v is the velocity to equal 1 N. of the motion.  1â … Acoustics and Sound Measurement 5 Elasticity and restoring force It takes some The unit of pressure is the Pascal (Pa), so that 1 Pa effort (an outside force) to compress or expand a = 1 N/m2. In the cgs system, pressure is measured in spring; and the compressed or expanded spring will dynes per square centimeter (dynes/cm2), occasion- bounce back to its original shape after it is released. ally referred to as microbars (μbars). Compressing or expanding the spring is an example Work and energy As a physical concept, work of deforming an object. The spring bouncing back (W) occurs when the force applied to a body results to its prior shape is an example of elasticity. More in its displacement, and the amount of work is equal formally, we can say that elasticity is the property to the product of the force and the displacement, or whereby a deformed object returns to its original form. Notice the distinction between deformation W = Fd and elasticity. A rubber band and saltwater taffy can both be stretched (deformed), but only the rubber Because force is measured in newtons and dis- band bounces back. In other words, what makes a placement is measured in meters, work itself is rubber band elastic is not that it stretches, but rather quantified in newton-meters (N m). For example, that it bounces back. The more readily a deformed if a force of 2 N displaces a body by 3 m, then the object returns to its original form, the more elastic amount of work is 2 × 3 = 6 N. There can only be (or stiff) it is. work if there is displacement. There cannot be work We know from common experiences, such as if there is no displacement (i.e., if d = 0) because work using simple exercise equipment, that it is relatively is the product of force and displacement, and zero easy to begin compressing a spring (e.g., a “grip exer- times anything is zero. The MKS unit of work is the ciser”), but that it gets progressively harder to con- joule (J). One joule is the amount of work that occurs tinue compressing it. Similarly, it is easier to begin when one newton of force effects one meter of dis- expanding a spring (e.g., pulling apart the springs on placement, or 1 J = 1 N m. In the cgs system, the unit a “chest exerciser”) than it is to continue expanding of work is called the erg, where 1 erg = 1 dyne cm. it. In other words, the more a spring-like material (an One joule corresponds to 107 ergs. elastic element) is deformed, the more it opposes the Energy is usually defined as the capability to applied force. The force that opposes the deforma- do work. The energy of a body at rest is potential tion of an elastic or spring-like material is known as energy and the energy of an object that is in motion the restoring force. If we think of deformation in is kinetic energy. The total energy of a body is the terms of how far the spring has been compressed or sum of its potential energy plus its kinetic energy, expanded from its original position, we could also say and work corresponds to the exchange between that the restoring force increases with displacement. these two forms of energy. In other words, energy is Quantitatively, then, restoring force (FR) depends on not consumed when work is accomplished; it is con- the stiffness (S) of the material and the amount of its verted from one form to the other. This principle is displacement as follows: illustrated by the simple example of a swinging pen- dulum. The pendulum’s potential energy is greatest FR = Sd when it reaches the extreme of its swing, where its motion is momentarily zero. On the other hand, the Pressure Very few people can push a straight pin pendulum’s kinetic energy is greatest when it passes into a piece of wood, yet almost anyone can push a through the midpoint of its swing because this is thumbtack into the same piece of wood. This is pos- where it is moving the fastest. Between these two sible because a thumbtack is really a simple machine extremes, energy is being converted from potential that concentrates the amount of force being exerted to kinetic as the pendulum speeds up (on each down over a larger area (the head) down to a very tiny area swing), and from kinetic to potential as the pendu- (the point). In other words, force is affected by the lum slows down (on each up swing). size of the area over which it is applied in a way that Power The rate at which work is done is called constitutes a new quantity. This quantity, which is power (P), so that power can be defined as work equal to force divided by area (A), is called pressure divided by time, (p), so W P= F t p= A The unit of power is called the watt (w). One unit Because force is measured in newtons and area of power corresponds to one unit of work divided is measured in square meters in MKS units, pressure by one unit of time. Hence, one watt is equal to one is measured in newtons per square meter (N/m2). joule divided by one second, or 1 w = 1 J/s. Power 6 1â … Acoustics and Sound Measurement is also expressed in watts in the cgs system, where work is measured in ergs. Since 1 J = 107 erg, we can g in also say that 1 w = 107 erg/s. wid en a e Power can also be expressed in other terms. For ver urc d o so ide from example, because W = Fd, we can substitute Fd for W i v is d ance er t in the power formula, to arrive at t o ow dis f p ith w n a ou re am ce a Fd Fin ite surfa P= t source We know that v = d/t, so we can substitute v for d/t Proportionately less power falls on the same unit and rewrite this formula as of area at increasing distances from source P = Fv In other words, power is also equal to force times velocity. Intensity Consider a hypothetical demonstration in which one tablespoonful of oil is placed on the sur- face of a still pond. At that instant the entire amount Fig.€1.1â … Intensity (power divided by area) decreases with dis- tance from the sound source because a fixed amount of power of oil will occupy the space of a tablespoon. As time is spread over an increasing area, represented by the thinning passes, the oil spreads out over an expanding area on of the lines. Proportionately less power falls on the same unit the surface of the pond, and it therefore also thins area (represented by the lighter shading of the ovals) with out so that much less than all the oil will occupy the increasing distance from the source. space of a tablespoon. The wider the oil spreads the more it thins out, and the proportion of the oil cov- ering any given area gets smaller and smaller, even though the total amount of oil is the same. Clearly, there is a difference between the amount of oil, per is distributed over area. Specifically, intensity is equal se, and the concentration of the oil as it is distributed to power per unit area, or power divided by area, or across (i.e., divided by) the surface area of the pond. An analogous phenomenon occurs with sound. It P I= is common knowledge that sound radiates outward in A every direction from its source, constituting a sphere that gets bigger and bigger with increasing distance Because power is measured in watts and area is mea- from the source, as illustrated by the concentric cir- sured in square meters in the MKS system, inten- cles in Fig.€1.1. Let us imagine that the sound source sity is expressed in watts per square meter (w/m2). Intensity is expressed in watts per square centimeter is a tiny pulsating object (at the center of the concen- (w/cm2) in the cgs system. tric circles in the figure), and that it produces a finite Intensity decreases with increasing distance from amount of power, analogous to the fixed amount of a sound source according to a rule called the inverse oil in the prior example. Consequently, the sound square law. It states that the amount of intensity power will be divided over the ever-expanding sur- drops by 1 over the square of the change in distance. face as distance increases from the source, analogous Two examples are illustrated in Fig.€1.2. Frame a to the thinning out of the widening oil slick. This shows that when the distance from a loudspeaker is notion is represented in the figure by the thinning of doubled from 5 m to 10 m, the amount of intensity the lines at greater distances from the source. Sup- at 10 m will be one quarter of what it was at 5 m pose we measure how much power registers on a (because 1/22 = 1/4). Similarly, frame b shows that certain fixed amount of surface area (e.g., a square tripling the distance from 5 m to 15 m causes the inch). As a result, a progressively smaller propor- intensity to fall to one ninth of its value at the closer tion of the original power falls onto a square inch as point because 1/32 = 1/9. the distance from the source increases, represented An important relationship to be aware of is that in the figure by the lighter shading of the same-size power is equal to pressure squared, ovals at increasing distances from the source. The examples just described reveal that a new quantity, called intensity (I), develops when power P = p2  1â … Acoustics and Sound Measurement 7 include a playground swing, a pendulum, the floor- Intensity boards under a washing machine, a guitar string, a tuning fork prong, and air molecules. The vibration is Distance usually called sound when it is transferred from air particle to air particle (we will see how this happens (a) a 1/22 = 1/4 later). The vibration of air particles might have a sim- ple pattern such as the tone produced by a tuning 0m 5m 10 m 15 m fork, or a very complex pattern such as the din heard in a school cafeteria. Most naturally occurring sounds Doubling of distance are very complex, but the easiest way to understand sound is to concentrate on the simplest ones. Simple Harmonic Motion b (b) 1/32 = 1/9 A vibrating tuning fork is illustrated in Fig.€1.3. The initial force that was applied by striking the tuning 0m 5m 10 m 15 m fork is represented by the green arrow in frame 1. The progression of the drawings represents the motion of Tripling of distance the prongs at selected points in time after the fork has been activated. The two prongs vibrate as mirror Fig.€1.2â … Illustrations of the inverse square law. (a) Doubling images of each other, so that we can describe what of distance: The intensity at 10 m away from a loudspeaker is is happening in terms of just one prong. The insert one quarter of its intensity at 5 m because 1/22 = 1/4. (b) Tri- highlights the motion of the right prong. Here the pling of distance: The intensity at 15 m away from the sources center position is where the prong would be at rest. is one ninth of its intensity at 5 m because 1/32 = 1/9. When the fork is struck the prong is forced inward as shown by arrow a. After reaching the leftmost posi- tion it bounces back (arrow b), accelerating along the way. The rapidly moving prong overshoots the center and continues rightward (arrow c). It slows and pressure is equal to the square root of power, down along the way until it stops for an instant at the extreme right, where it reverses direction again and p= P starts moving toward the left (arrow d) at an ever- increasing speed. It overshoots the center again, and as before, the prong now follows arrow a, slowing In addition, intensity is proportional to pressure down until it stops momentarily at the extreme left. squared, Here it reverses direction again and repeats the same process over and over again. One complete round trip I ∝ p2 (or replication) of an oscillating motion is called a cycle. The number of cycles that occur in one second and pressure is proportional to the square root of is called frequency. intensity, This form of motion occurs when a force is applied to an object having the properties of inertia p∝ I and elasticity. Due to its elasticity, the deformation of the fork caused by the applied force is opposed by a restoring force. In the figure the initial leftward This simple relationship makes it easy to convert force is opposed by a restoring force in the opposite between sound intensity and sound pressure. direction, that is, toward the right. The rightward restoring force increases as the prong is pushed pro- gressively toward the left. As a result, the movement ⌀ The Nature of Sound of the prong slows down and eventually stops. Under the influence of its elasticity the prong now reverses Sound is often defined as a form of vibration that direction and starts moving rightward. As the restor- propagates through the air in the form of a wave. ing force brings the prong back toward the center, we Vibration is nothing more than the to-and-fro must also consider its mass. Because the prong has motion (oscillation) of an object. Some examples mass, inertia causes it to accelerate as it moves back 8 1â … Acoustics and Sound Measurement pletely. The dying out of vibrations over time is called force damping, and it occurs due to resistance or fric- tion. Resistance occurs because the vibrating prong is always in contact with the surrounding air. As a 1 result, there will be friction between the oscillating metal and the surrounding air molecules. This fric- tion causes some of the mechanical energy that has been supporting the motion of the tuning fork to be converted into heat. In turn, the energy that has been 2 a d converted into heat is no longer available to main- b c tain the vibration of the tuning fork. Consequently, the sizes of the oscillations dissipate and eventually die out altogether. A diagram summarizing the concepts just 3 described is shown in Fig.€1.4. The curve in the fig- ure represents the tuning fork’s motion. The amount of displacement of the tuning fork prong around its resting (or center) position is represented by dis- tance above and below the horizontal line. These events are occurring over time, which is represented 4 by horizontal distance (from left to right). The ini- tial displacement of the prong due to the original applied force is represented by the dotted segment of the curve. Inertial forces due to the prong’s mass and elastic restoring forces due to the elasticity of 5 the prong are represented by labeled arrows. Damp- ing of the oscillations due to friction is shown by Fig.€1.3â … After being struck, a tuning fork vibrates or oscil- the decline in the displacement of the curve as time lates with a simple pattern that repeats itself over time. One replication (cycle) of this motion is illustrated going from goes on. The curve in this diagram is an example of frames 1 to 5. The arrows in the insert highlight the motion a waveform, which is a graph that shows displace- of one of the prongs. ment (or another measure of magnitude) as a func- tion of time. Sound Waves toward its center resting position. In fact, the prong is moving at its maximum speed as it passes through Tuning fork vibrations produce sound because the the resting position. The force of inertia causes the oscillations of the prongs are transmitted to the sur- prong to overshoot the center and continue moving rounding air particles. When the tuning fork prong rightward. The deformation process begins again moves to the right, it displaces air molecules to its once the prong overshoots its resting position. As a right in the same direction. These molecules are thus result, opposing elastic restoring forces start building displaced to the right of their own resting positions. up again, now in the leftward direction. Just as before, Displacing air molecules toward the right pushes the increasing (leftward) restoring force eventually them closer to the air particles to their right. The overcomes the rightward inertial force, thereby stop- pressure that exists among air molecules that are not ping the prong’s displacement at the rightmost point, being disturbed by a driving force (like the tuning and causing a reversal in the direction of its move- fork) is known as ambient or atmospheric pressure. ment. Hence, the same course of events is repeated We can say that the rightward motion of the tuning again, this time in the leftward direction; then right- fork prong exerts a force on the air molecules that ward, then leftward, etc., over and over again. This pushes them together relative to their undisturbed, kind of vibration is called simple harmonic motion resting situation. In other words, forcing the air mol- (SHM) because the oscillations repeat themselves at ecules together causes an increase in air pressure the same rate over and over again. relative to the ambient pressure that existed among We know from experience that the oscillations the undisturbed molecules. This state of positive air just described do not continue forever. Instead, they pressure is called compression. The amount of com- dissipate over time and eventually die out com- pression increases as the prong continues displacing  1â … Acoustics and Sound Measurement 9 that are associated with simple harmonic motion are called pure tones. Let us consider one of the air molecules that has already been set into harmonic motion by the tuning fork. This air particle now vibrates to-and-fro in the same direction that was originally imposed by the vibrating prong. When this particle moves toward its right it will cause a similar displacement of the par- ticle that is located there. The subsequent leftward motion is also transmitted to the next particle, etc. Thus, the oscillations of one air particle are transmit- Fig.€1.4â … Diagrammatic representation of tuning fork oscilla- ted to the molecule next to it. The second particle is tions over time. Vertical displacement represents the amount therefore set into oscillation, which in turn initiates of the tuning fork prong displacement around its resting posi- oscillation of the next one, and so forth down the tion. Distance from left to right represents the progression of line. In other words, each particle vibrates back and time. (From Gelfand 2010, courtesy of Informa.) forth around its own resting point, and causes suc- cessive molecules to vibrate back and forth around their own resting points, as shown schematically in Fig.€1.5. Notice that each molecule vibrates “in place” around its own average position; it is the vibratory the air molecules rightward. A maximum amount of pattern that is transmitted through the air. positive pressure occurs when the prong and air mol- This propagation of vibratory motion from par- ecules reach their greatest rightward displacement. ticle to particle constitutes the sound wave. This The tuning fork prong then reverses direction, wave appears as alternating compressions and overshoots its resting position, and proceeds to its rarefactions radiating from the sound source in leftmost position. The compressed air molecules all directions, as already suggested in Fig.€1.1. The reverse direction along with the prong. This occurs transmission of particle motion along with the because air is an elastic medium, so the particles compressed to the right develop a leftward restoring force. Small as they are, air particles do have mass. Therefore, inertia causes the rebounding air particles to overshoot their resting positions and to continue toward their extreme leftward positions. As the particles move leftward, the amount of compres- sion decreases and is momentarily zero as they pass through their resting positions. As they continue to move to the left of their resting positions, the parti- cles are now becoming increasingly farther from the molecules to their right (compared with when they are in their resting positions). We now say that the air particles are rarefied compared with their resting states, so that the air pressure is now below atmo- spheric pressure. This state of lower than ambient pressure is called rarefaction. When the air particles reach the leftmost position they are maximally rar- efied, which means that the pressure is maximally negative. At this point, the restoring force instigates a rightward movement of the air molecules. This movement is enhanced by the push of the tuning fork prongs that have also reversed direction. The air molecules now accelerate rightward (so that the Fig.€1.5â … Sound is initiated by transmitting the vibratory pat- amount of rarefaction decreases), overshoot their tern of the sound source to nearby air particles, and then the resting positions, and continue to the right, and so vibratory pattern is passed from particle to particle as a wave. on. The tuning fork vibrations have now been trans- Notice how it is the pattern of vibration that is being transmit- mitted to the surrounding particles, which are now ted, whereas each particle oscillates around its own average also oscillating in simple harmonic motion. Sounds location. 10 1â … Acoustics and Sound Measurement resulting variations in air pressure with distance Wave propagation from the source are represented in Fig.€1.6. Most people are more familiar with the kinds of waves Oscillations of individual particles that develop on the surface of a pond when a peb- ble is dropped into the water. These are transverse waves because the particles are moving at right Longitudinal representation angles to the direction that the wave is propagat- ing. That is, the water particles oscillate up and Wavelength down (vertically) even though the wave moves out Transverse horizontally from the spot where the pebble hit the representation water. This principle can be demonstrated by float- ing a cork in a pool, and then dropping a pebble in Wavelength the water to start a wave. The floating cork reflects Amplitude the motions of the water particles. The wave will move out horizontally, but the floating cork bobs Distance up and down (vertically) at right angles to the Fig.€1.6â … Longitudinal and transverse representations of a wave. In contrast, sound waves are longitudinal sound wave. Wavelength (λ) is the distance covered by one waves because each air particle oscillates in the replication (cycle) of a wave, and is most easily visualized as same direction in which the wave is propagating the distance from one peak to the next. (Fig.€1.6). Although sound waves are longitudi- nal, it is more convenient to draw them with a transverse representation, as in Fig.€1.6. In such a diagram the vertical dimension represents some consider one cycle to have 360°. Since 45/360 = ⅛, measure of the size of the signal (e.g., displace- a phase angle (θ) of 45° is the same as one eighth ment, pressure, etc.), and left to right distance of the way around a circle or one eighth of the way represents time (or distance). For example, the into a sine wave. Returning to the circle, the vertical waveform in Fig.€1.6 shows the amount of positive displacement from the horizontal to the point where pressure (compression) above the baseline, nega- r intersects the circle is represented by a vertical line tive pressure (rarefaction) below the baseline, and labeled d. This vertical line corresponds to the dis- distance horizontally going from left to right. placement of point b on the sine wave, where the displacement of the air particle is represented by the height of the point above the baseline. Notice that we The Sinusoidal Function now have a right triangle in the circle, where r is the Simple harmonic motion is also known as sinusoidal motion, and has a waveform that is called a sinu- soidal wave or a sine wave. Let us see why. Fig.€1.7 shows one cycle of a sine wave in the center, sur- rounded by circles labeled to correspond to points on the wave. Each circle shows a horizontal line corre- sponding to the horizontal baseline on the sine wave, as well as a radius line (r) that will move around the circle at a fixed speed, much like a clock hand but in a counterclockwise direction. Point a on the waveform in the center of the figure can be viewed as the “starting point” of the cycle. The displacement here is zero because this point is on the horizontal line. The radius appears as shown in circle b when it reaches 45° of rotation, which corresponds to point b on the sine wave. The angle between the radius and the horizontal is called the phase angle (θ) and is a handy way to tell loca- tion going around the circle and on the sine wave. In other words, we consider one cycle (one “round trip” of oscillation) to be the same as going around Fig.€1.7â … Sinusoidal motion (θ, phase angle; d, displacement). a circle one time. Just as a circle has 360°, we also (Adapted from Gelfand 2010, courtesy of Informa.)  1â … Acoustics and Sound Measurement 11 hypotenuse, θ is an angle, and d is the leg opposite 180° is covered in ½ second; 90° takes ¼ second; that angle. Recall from high school math that the sine 270° takes ¾ second; etc. Hence, the phase angle of an angle equals the length of the opposite leg over also reflects the elapsed time from the onset of rota- the length of the hypotenuse. Here, sin θ = d/r. If we tion. This is why the horizontal axis in Fig.€1.8 can be conveniently assume that the length of r is 1, then labeled in terms of phase. As such, the phase of the displacement d becomes the sine of angle θ, which wave at each of the points indicated in Fig.€1.7 is 0° happens to be 0.707. In other words, displacement is at a, 45° at b, 90° at c, 135° at d, 180° at e, 225° at f, determined by the sine of the phase angle, and dis- 270° at g, 315° at h, and 360° at i, which is also 0° for placement at any point on the sine wave corresponds the next cycle. to the sine of θ. This is why it is called a sine wave. Phase is often used to express relationships The peak labeled c on the sine wave corresponds between two waves that are displaced relative to to circle c, where the rotating radius has reached the each other, as in Fig.€1.8. Each frame in the figure straight-up position. We are now one fourth of the shows two waves that are identical to each other way into the wave and one fourth of the way around except that they do not line up exactly along the hor- the circle. Here, θ = 90° and the displacement is the izontal (time) axis. The top panel shows two waves largest it can be (notice that d = r on the circle). Con- that are 45° apart. Here, the wave represented by tinuing the counterclockwise rotation of r causes the thicker line is at 45° at the same time that the the amount of displacement from the horizontal to other wave (shown by the thinner line) is at 0°. The decrease, exemplified by point d on the sine wave phase displacement is highlighted by the shaded and circle d, where θ is 135°. The oscillating air parti- area and the dotted vertical guideline in the figure. cle has already reversed direction and is now moving This is analogous to two radii that are always 45° back toward the resting position. When it reaches apart as they move around a circle. In other words, the resting position there is again no displacement, these two waves are 45° apart or out-of-phase. The as shown by point e on the sine wave and by the fact second panel shows the two waves displaced from that r is now superimposed on the horizontal at θ = each another by 90°, so that one wave is at 90° when 180° in circle e. Notice that 180° is one half of the other one is at 0°. Hence, these waves are 90° out- 360° round trip, so we are now halfway around the of-phase, analogous to two radii that are always 90° circle and halfway into the cycle of SHM. In addition, apart as they move around a circle. The third panel displacement is zero at this location (180°). shows two waves that are 180° out-of-phase, Here, Continuing the rotation of r places it in the lower one wave it at its 90° (positive) peak at the same time left quadrant of circle f, corresponding to point f on that the other one is at its 270° (negative) peak, which the wave, where θ = 225°. The oscillating particle has is analogous to two radii that are always 180° apart overshot its resting position and the displacement is as they move around a circle. Notice that these two now increasing in the other direction, so that we are otherwise identical waves are exact mirror images of in the rarefaction part of the wave. Hence, displace- each other when they are 180° out-of-phase, just as ment is now drawn in the negative direction, indi- the two radii are always pointing in opposite direc- cating rarefaction. The largest negative displacement tions. The last example in the bottom panel shows is reached at point g on the wave, where θ = 270°, the two waves 270° out-of-phase. corresponding to circle g, in which r points straight down. The air particle now begins moving in the posi- Parameters of Sound Waves tive direction again on its way back toward the rest- ing position. At point h and circle h the displacement We already know that a cycle is one complete repli- in the negative direction has become smaller as the cation of a vibratory pattern. For example, two cycles rotating radius passes through the point where θ = are shown for each sine wave in the upper frame of 315° (point h on the wave and circle h). The air par- Fig.€1.9, and four cycles are shown for each sine wave ticle is again passing through its resting position at in the lower frame. Each of the sine waves in this point i, having completed one round trip or 360° of figure is said to be periodic because it repeats itself rotation. Here, displacement is again zero. Having exactly over time. Sine waves are the simplest kind completed exactly one cycle, 360° corresponds to 0°, of periodic wave because simple harmonic motion is and circle i is the same one previously used as circle a. the simplest form of vibration. Later we will address Recall that r rotates around the circle at a fixed complex periodic waves. speed. Hence, how fast r is moving will determine The duration of one cycle is called its period. how many degrees are covered in a given amount The period is expressed in time (t) because it refers of time. For example, if one complete cycle of rota- to the amount of time that it takes to complete one tion takes 1 second, then 360° is covered in 1 second; cycle (i.e., how long it takes for one round trip). For 12 1â … Acoustics and Sound Measurement 45° Fig.€1.8â … Each panel shows two waves that are 45° identical in every way except they are displaced from one another in terms of phase, highlighted 0° by the shaded areas and the dotted vertical guidelines. Analogous examples of two radii moving around a circle are shown to the left of the waveforms. Top panel: Two waves that are 90° 45° out-of-phase, analogous to two radii that 90° are always 45° apart as they move around a circle. Second panel: Waves that are 90° out-of- phase, analogous to two radii moving around a 0° circle 90° apart. Third panel: Waves that are 180° out-of phase, analogous to two radii that are always 180° apart (pointing in opposite direc- tions) moving around a circle. Bottom panel: 180° Two waves (and analogous radii moving around a circle) that are 270° out-of-phase. 180° 0° 270° 0° 270° 0° example, a periodic wave that repeats itself every to complete one cycle, then frequency and period one hundredth of a second has a period of 1/100 sec- must be related in a very straightforward way. Let us ond, or t = 0.01 second. One hundredth of a second consider the three examples that were just used to is also 10 thousandths of a second (milliseconds), so illustrate the relationship of period and frequency: we could also say that the period of this wave is 10 A period of 0.01 second goes with a frequency (f) milliseconds. of 100 Hz. Similarly, a wave that repeats itself every one A period of 0.002 second goes with a frequency thousandth of a second has a period of 1 millisecond of 500 Hz. or 0.001 second; and the period would be 2 millisec- A period of 0.001 second goes with a frequency onds or 0.002 second if the duration of one cycle is of 1000 Hz. two thousandths of a second. The number of times a waveform repeats itself Now, notice the following relationships among in one second is its frequency (f), or the number of these numbers: cycles per second (cps). We could say that frequency 1/100 = 0.01 and 1/0.01 = 100. is the number of cycles that can fit into one second. 1/500 = 0.002 and 1/0.002 = 500. Frequency is expressed in units called hertz (Hz), 1/1000 = 0.001 and 1/0.001 = 1000. which means the same thing as cycles per second. For example, a wave that is repeated 100 times per In each case the period corresponds to 1 over the second has a frequency of 100 Hz; the frequency of a frequency, and the frequency corresponds to 1 over wave that has 500 cycles per second is 500 Hz; and a the period. In formal terms, frequency equals the 1000 Hz wave has 1000 cycles in one second. reciprocal of period, If frequency is the number of cycles that occur each second, and period is how much time it takes 1 f = t  1â … Acoustics and Sound Measurement 13 and period equals the reciprocal of frequency, 1 t= f Amplitude Each wave in the upper frame of Fig.€1.9 contains two cycles in 4 milliseconds, and each wave in the lower frame contains four cycles in the 4 millisec- onds. If two cycles in the upper frame last 4 millisec- onds, then the duration of one cycle is 2 milliseconds. Hence, the period of each wave in the upper frame is 2 milliseconds (t = 0.002 second), and the frequency is 1/0.002, or 500 Hz. Similarly, if four cycles last 4 0 1 2 3 4 milliseconds in the lower frame, then one cycle has Time (msec) a period of 1 millisecond (t = 0.001), and the fre- quency is 1/0.001, or 1000 Hz. Fig.€1.9 also illustrates differences in the ampli- tude between waves. Amplitude denotes size or magnitude, such as the amount of displacement, power, pressure, etc. The larger the amplitude at Amplitude some point along the horizontal (time) axis, the greater its distance from zero on the vertical axis. With respect to the figure, each frame shows one wave that has a smaller amplitude and an otherwise identical wave that has a larger amplitude. As illustrated in Fig.€1.10, the peak-to-peak amplitude of a wave is the total vertical distance 0 1 2 3 4 between its negative and positive peaks, and peak Time (msec) amplitude is the distance from the baseline to one peak. However, neither of these values reflects the Fig.€1.9â … Each frame shows two sine waves that have the overall, ongoing size of the wave because the ampli- same frequency but different amplitudes. Compared with the tude is constantly changing. At any instant an oscil- upper frame, twice as many cycles occur in the same amount lating particle may be at its most positive or most of time in the lower frame; thus the period is half as long and negative displacement from the resting position, or the frequency is twice as high. anywhere between these two extremes, including Fig.€1.10â … Peak, peak-to-peak, and root-mean-square +1 (RMS) amplitude. 0.707 0.707 Peak RMS Peak-to-peak Amplitude –1 Root mean square (RMS) amplitude is 0.707 of peak amplitude. 14 1â … Acoustics and Sound Measurement the resting position itself, where the displacement other words, low frequencies have long wavelengths is zero. The magnitude of a sound at a given instant and high frequencies have short wavelengths. (instantaneous amplitude) is applicable only for that moment, and will be different at the next moment. Yet we are usually interested in a kind of Complex Waves “overall average” amplitude that reveals the magni- tude of a sound wave throughout its cycles. A simple When two or more pure tones are combined, the average of the positive and negative instantaneous result is called a complex wave. Complex waves may amplitudes will not work because it will always be contain any number of frequencies from as few as equal to zero. A different kind of overall measure is two up to an infinite number of them. Complex peri- therefore used, called the root-mean-square (RMS) odic waves have waveforms that repeat themselves amplitude. Even though measurement instruments over time. If the waveform does not repeat itself over provide us with RMS amplitudes automatically, we time, then it is an aperiodic wave. can understand RMS by briefly reviewing the steps that would be used to calculate it manually: (1) All Combining Sinusoids of the positive and negative values on the wave are squared, so that all values are positive (or zero for The manner in which waveforms combine into more values on the resting position itself). (2) A mean complex waveforms involves algebraically add- (average) is calculated for the squared values. (3) ing the amplitudes of the two waves at every point This average of the squared values is then rescaled along the horizontal (time) axis. Consider two sine back to the “right size” by taking its square root. This waves that are to be added. Imagine that they are is the RMS value. The RMS amplitude is numerically drawn one above the other on a piece of graph paper equal to 70.7% of (or 0.707 times) the peak amplitude so that the gridlines can be used to identify similar (or 0.354 times the peak-to-peak amplitude). Even moments in time (horizontally) for the two waves, though these values technically apply only to sinu- and their amplitudes