physics of music ebook

THE PHYSICS OF MUSIC ANDMUSICAL INSTRUMENTS

DAVID R. LAPP, FELLOWWRIGHT CENTER FOR INNOVATIVE SCIENCE EDUCATION

TUFTS UNIVERSITYMEDFORD, MASSACHUSETTS

f1 f3 f5 f7

TABLE OF CONTENTSIntroduction 1

Chapter 1: Waves and Sound 5Wave Nomenclature 7Sound Waves 8ACTIVITY: Orchestral Sound 15Wave Interference 18ACTIVITY: Wave Interference 19

Chapter 2: Resonance 20Introduction to Musical Instruments 25Wave Impedance 26

Chapter 3: Modes, overtones, and harmonics 27ACTIVITY: Interpreting Musical Instrument Power Spectra 34Beginning to Think About Musical Scales 37Beats 38

Chapter 4: Musical Scales 40ACTIVITY: Consonance 44The Pythagorean Scale 45The Just Intonation Scale 47The Equal Temperament Scale 50A Critical Comparison of Scales 52ACTIVITY: Create a Musical Scale 55ACTIVITY: Evaluating Important Musical Scales 57

Chapter 5: Stringed Instruments 61Sound Production in Stringed Instruments 65INVESTIGATION: The Guitar 66PROJECT: Building a Three Stringed Guitar 70

Chapter 6: Wind Instruments 72The Mechanical Reed 73Lip and Air Reeds 74Open Pipes 75Closed Pipes 76The End Effect 78Changing Pitch 79More About Brass Instruments 79More about Woodwind instruments 81INVESTIGATION: The Nose flute 83INVESTIGATION: The Sound Pipe 86INVESTIGATION: The Toy Flute 89INVESTIGATION: The Trumpet 91PROJECT: Building a Set of PVC Panpipes 96

Chapter 7: Percussion Instruments 97Bars or Pipes With Both Ends Free 97Bars or Pipes With One End Free 99Toward a Harmonic Idiophone 100INVESTIGATION: The Harmonica 102INVESTIGATION: The Music Box Action 107PROJECT: Building a Copper Pipe Xylophone 110

References 111

1Everything is determined by forces over which we have nocontrol. It is determined for the insects as well as for the star.Human beings, vegetables, or cosmic dust we all dance to amysterious tune, intoned in the distance by an invisible piper. Albert Einstein

INTRODUCTIONHIS MANUAL COVERS the physics of waves, sound, music, andmusical instruments at a level designed for high school physics. However,it is also a resource for those teaching and learning waves and sound frommiddle school through college, at a mathematical or conceptual level. Themathematics required for full access to the material is algebra (to include

logarithms), although each concept presented has a full conceptual foundation thatwill be useful to those with even a very weak background in math.

Solomon proclaimed that there is nothing newunder the Sun and of the writing of books there is noend. Conscious of this, I have tried to producesomething that is not simply a rehash of what hasalready been done elsewhere. In the list of references Ihave indicated a number of very good sources, someclassics that all other writers of musical acousticbooks refer to and some newer and more accessibleworks. From these, I have synthesized what I believeto be the most useful and appropriate material for thehigh school aged student who has neither abackground in waves nor in music, but who desires afirm foundation in both. Most books written on thetopic of musical acoustics tend to be either verytheoretical or very cookbook style. The theoreticalones provide for little student interaction other thansome end of the chapter questions and problems. Theones I term cookbook style provide instructions forbuilding musical instruments with little or noexplanation of the physics behind the construction.This curriculum attempts to not only marry the bestideas from both types of books, but to includepedagogical aids not found in other books.

This manual is available as both a paper hardcopy as well as an e-book on CD-ROM. The CD-ROM version contains hyperlinks to interestingwebsites related to music and musical instruments. Italso contains hyperlinks throughout the text to soundfiles that demonstrate many concepts being developed.

MODES OF PRESENTATIONAs the student reads through the text, he or she

will encounter a number of different presentationmodes. Some are color-coded. The following is a keyto the colors used throughout the text:

Pale green boxes cover tables and figuresthat are important reference material.

Notes Frequencyinterval (cents)

Ci 0D 204E 408F 498G 702A 906B 1110Cf 1200

Table 2.8: Pythagoreanscale interval ratios

Light yellow boxes highlight derivedequations in their final form, which will be used forfuture calculations.

f1 =

Tm

2L

T

IIINNNTTTRRROOODDDUUUCCCTTTIIIOOONNN

2

Tan boxes show step-by-step examples formaking calculations or reasoning through questions.

ExampleIf the sound intensity of a screaming baby were

110-2 Wm 2

at 2.5 m away, what would it be at6.0 m away?The distance from the source of sound is greater by afactor of

6.02.5 = 2.4 . So the sound intensity is decreased

by

1(2.4)2

= 0.174 . The new sound intensity is:

(110-2 Wm 2

)(0.174) =

1.74 10-2 Wm 2

Gray boxes throughout the text indicatestopping places in the reading where students areasked, Do you get it? The boxes are meant toreinforce student understanding with basic recallquestions about the immediately preceding text. Thesecan be used to begin a discussion of the reading witha class of students.

Do you get it? (4)A solo trumpet in an orchestra produces a soundintensity level of 84 dB. Fifteen more trumpets jointhe first. How many decibels are produced?

In addition to the Do you get it? boxes,which are meant to be fairly easy questions doneindividually by students as they read through the text,there are three additional interactions students willencounter: Activities , Investigations, andProjects. Activities more difficult than the Do youget it? boxes and are designed to be done eitherindividually or with a partner. They either require ahigher level of conceptual understanding or draw onmore than one idea. Investigations are harder still anddraw on more than an entire section within the text.Designed for two or more students, each onephotographically exposes the students to a particularmusical instrument that they must thoroughly

consider. Investigations are labs really, often requiringstudents to make measurements directly on thephotographs. Solutions to the Do you get it?boxes, Activities, and Investigations are provided inan appendix on the CD-ROM. Finally, projectsprovide students with some background for buildingmusical instruments, but they leave the type ofmusical scale to be used as well as the key theinstrument will be based on largely up to the student.

PHYSICS AND MUSIC?

Without music life would be amistake. Friedrich Nietzsche

With even a quick look around most schoolcampuses, it is easy to see that students enjoy music.Ears are sometimes hard to find, covered byheadphones connected to radios or portable CDplayers. And the music flowing from them has thepower to inspire, to entertain, and to even mentallytransport the listener to a different place. A closerlook reveals that much of the life of a student eitherrevolves around or is at least strongly influenced bymusic. The radio is the first thing to go on in themorning and the last to go off at night (if it goes offat all). T-shirts with logos and tour schedules of


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popular bands are the artifacts of many teens mostcoveted event the concert. But the school bellringing for the first class of the day always bringswith it a stiff dose of reality.

H. L. Mencken writes, School days, I believe,are the unhappiest in the whole span of humanexistence. They are full of dull, unintelligible tasks,new and unpleasant ordinances, brutal violations ofcommon sense and common decency. This maypaint too bleak a picture of the typical studentsexperience, but its a reminder that what is taughtoften lacks meaning and relevance. When I think backto my own high school experience in science, I findthat there are some classes for which I have nomemory. Im a bit shocked, but I realize that it wouldbe possible to spend 180 hours in a science classroomand have little or no memory of the experience if theclassroom experience werelifeless or disconnectedfrom the reality of my life.Middle school and highschool students are a toughaudience. They want to beentertained but theydont have to be. What theyreally need is relevance.They want to see directconnections and immediateapplications. This is thereason for organizing anintroduction to the physicsof waves and sound aroundthe theme of music andmusical instruments.

Its not a stretch either.Both music and musicalinstruments are intimatelyconnected to the physics ofwaves and sound. To fullyappreciate what occurs in amusical instrument when itmakes music or to

understand the rationale for the development of themusical scales one needs a broad foundation in mostelements of wave and sound theory. With that said,the approach here will be to understand music andmusical instruments first, and to study the physics ofwaves and sound as needed to push the understandingof the music concepts. The goal however is a deeperunderstanding of the physics of waves and sound thanwhat would be achieved with a more traditionalapproach.

SOUND, MUSIC, AND NOISEDo you like music? No, I guess a better question

is, what kind of music do you like? I dont thinkanyone dislikes music. However, some parentsconsider their childrens music to be just noise.Likewise, if the kids had only their parents music tolisten to many would avoid it in the same way theyavoid noise. Perhaps thats a good place to start then the contrast between music and noise. Is there anobjective, physical difference between music andnoise, or is it simply an arbitrary judgment?

After I saw the movie 8 Mile, the semi-autobiographical story of the famous rapper Eminem,I recommended it to many people but not to mymother. She would have hated it. To her, his music isjust noise. However, if she hears an old Buddy Hollysong, her toes start tapping and shes ready to dance.But the music of both of these men would beconsidered unpleasant by my late grandmother whoseemed to live for the music she heard on theLawrence Welk Show. I can appreciate all threeartists at some level, but if you ask me, nothing

beats a little Bob Dylan. Itsobviously not easy to definethe difference between noiseand music. Certainly there isthe presence of rhythm in thesounds we call music. At amore sophisticated level thereis the presence of tones thatcombine with other tones inan orderly and ... pleasingway. Noise is often associatedwith very loud and gratingsounds chaotic sounds whichdont sound good together orare somehow unpleasant tolisten to. Most would agreethat the jackhammer tearingup a portion of the street isnoise and the sound comingfrom the local marching bandis music. But the distinctionis much more subtle than that.If music consists of soundswith rhythmic tones of certainfrequencies then thejackhammer might be


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considered a musical instrument. After all, itpummels the street with a very regular frequency. Andif noise consists of loud sounds, which are unpleasantto listen to, then the cymbals used to punctuate theperformance of the marching band might beconsidered noise. I suppose we all define the pointwhere music becomes noise a bit differently. Perhapsits based on what we listen to most or on thegeneration we grow up in or make a break from.But we need to be careful about being cavalier as Iwas just now when I talked about pleasing sounds.John Bertles of Bash the Trash (a company dedicatedto the construction and performance of musicalinstruments from recycled materials:http://www.bashthetrash.com/ ) was quick to cautionme when I used the word pleasing to describemusical sound. Music that is pleasing to one personmay not be pleasing to others. Bertles uses thedefinition of intent rather than pleasing whendiscussing musical sound. He gives the example of anumber of cars all blaring their horns chaotically atan intersection. The sound would be considered noiseto most anyone. But the reason for the noise is not somuch the origin of the sound, but the lack of intentto organize the sounds of the horns. If someone at theintersection were to direct the car horns to beep atparticular times and for specific periods, the noisewould evolve into something more closely related tomusic. And no one can dispute that whether itsEminem, Buddy Holly, Lawrence Welk, or BobDylan, they all create(d) their particular recordedsounds with intent.

There are two means of refuge fromthe miseries of life: music and cats. Albert Schweitzer

BEGINNING TO DEFINE MUSICMusic makes us feel good, it whisks us back in

time to incidents and people from our lives; it rescuesus from monotony and stress. Its tempo and pace jivewith the natural rhythm of our psyche.

The simplest musical sound is some type ofrhythmical beating. The enormous popularity of thestage show Stomp http://www.stomponline.com/ andthe large screen Omnimax movie, Pulsehttp://www.Pulsethemovie.com/ gives evidence for

the vast appreciation of this type of music. Definingthe very earliest music and still prominent in manycultures, this musical sound stresses beat overmelody, and may in fact include no melody at all.One of the reasons for the popularity of rhythm-onlymusic is that anyone can immediately play it at somelevel, even with no training. Kids do it all the time,naturally. The fact that I often catch myselfspontaneously tapping my foot to an unknown beator lie in bed just a bit longer listening contentedly tomy heartbeat is a testament to the close connectionbetween life and rhythm.

Another aspect of music is associated with moreor less pure tones sounds with a constant pitch.Whistle very gently and it sounds like a flute playinga single note. But that single note is hardly a song,and certainly not a melody or harmony. No, to makethe single tone of your whistle into a musical soundyou would have to vary it in some way. So you couldchange the way you hold your mouth and whistleagain, this time at a different pitch. Going back andforth between these two tones would produce acadence that others might consider musical. Youcould whistle one pitch in one second pulses for threeseconds and follow that with a one second pulse ofthe other pitch. Repeating this pattern over and overwould make your tune more interesting. And youcould add more pitches for even more sophistication.You could even have a friend whistle with you, butwith a different pitch that sounded good with the oneyou were whistling.

If you wanted to move beyond whistling tomaking music with other physical systems, youcould bang on a length of wood or pluck a taut fiberor blow across an open bamboo tube. Adding morepieces with different lengths (and tensions, in the caseof the fiber) would give additional tones. To add morecomplexity you could play your instrument alongwith other musicians. It might be nice to have thesound of your instrument combine nicely with thesound of other instruments and have the ability toplay the tunes that others come up with. But to dothis, you would have to agree on a collection ofcommon pitches. There exist several combinations ofcommon pitches. These are the musical scales.

Here we have to stop and describe what the pitchof a sound is and also discuss the variouscharacteristics of sound. Since sound is a type ofwave, its additionally necessary to go even furtherback and introduce the idea of a wave.

5It is only by introducing the young to great literature, dramaand music, and to the excitement of great science that we opento them the possibilities that lie within the human spirit enable them to see visions and dream dreams. Eric Anderson

CHAPTER 1WAVES AND SOUND

SK MOST PEOPLEto define what a waveis and they get amental image of awave at the beach.

And if you really press them foran answer, they're at a loss. Theycan describe what a wave lookslike, but they can't tell you what itis. What do you think?

If you want to get energy from one placeto another you can do it by transferring itwith some chunk of matter. For example, ifyou want to break a piece of glass, you don'thave to physically make contact with ityourself. You could throw a rock and theenergy you put into the rock would travel onthe rock until it gets to the glass. Or if apolice officer wants to subdue a criminal, he doesn'thave to go up and hit him. He can send the energy tothe criminal via a bullet. But there's another way totransfer energy and it doesn't involve a transfer ofmatter. The other way to transfer energy is by using awave. A wave is a transfer of energy without atransfer of matter . If you and a friend hold onto bothends of a rope you can get energy down to her simplyby moving the rope back and forth. Although therope has some motion, it isn't actually transferred toher, only the energy is transferred.

A tsunami (tidal wave generally caused by anearthquake) hit Papua New Guinea in the summer of1998. A magnitude 7 earthquake 12 miles offshoresent energy in this 23-foot tsunami that killedthousands of people. Most people don't realize thatthe energy in a wave is proportional to the square ofthe amplitude (height) of the wave. That means that ifyou compare the energy of a 4-foot wave that youmight surf on to the tsunami, even though thetsunami is only about 6 times the height, it wouldhave 62 or 36 times more energy. A 100-foot tsunami

(like the one that hit the coast of the East Indies inAugust 1883) would have 252 or 625 times moreenergy than the four-foot wave.

Streaming through the place you're in right nowis a multitude of waves known as electromagneticwaves. Their wavelengths vary from so small thatmillions would fit into a millimeter, to miles long.Theyre all here, but you miss most of them. Theonly ones you're sensitive to are a small group thatstimulates the retinas of your eyes (visible light) anda small group that you detect as heat (infrared). Theothers are totally undetectable. But theyre there.

WAVES, SOUND, AND THE EARAnother type of wave is a sound wave. As small

in energy as the tsunami is large, we usually need anear to detect these. Our ears are incredibly awesomereceptors for sound waves. The threshold of hearing issomewhere around

110-12Watts /meter 2 . Tounderstand this, consider a very dim 1-watt night-light. Now imagine that there were a whole lot morepeople on the planet than there are now about 100

AFigure 1.1: Most people think of the ocean whenasked to define or describe a wave. The recurringtumult is memorable to anyone who has spent t imeat the beach or been out in the surf. But waves occurmost places and in many different forms, transferringenergy without transferring matter.

WWWAAAVVVEEESSS AAANNNDDD SSSOOOUUUNNNDDD

6

times more. Assume therewas a global populationof 1 trillion (that's amillion, million) people.If you split the light fromthat dim bulb equallybetween all those people,each would hold a radiantpower of

110-12Watts.Finally, let's say that oneof those people spread thatpower over an area of onesquare meter. At thissmallest of perceptiblesound intensities, theeardrum vibrates lessdistance than the diameterof a hydrogen atom! Well,it's so small an amount ofpower that you can hardlyconceive of it, but if youhave pretty good hearing,you could detect a soundwave with that smallamount of power. That'snot all. You could alsodetect a sound wave athousand times morepowerful, a million timesmore powerful, and even abillion times morepowerful. And, that's before it even starts to getpainful!

I have a vivid fifth grade memory of my goodfriend, Norman. Norman was blind and the first andonly blind person I ever knew well. I sat next to himin fifth grade and watched amazed as he banged awayon his Braille typewriter. I would ask him questionsabout what he thought colors looked like and if hecould explain the difference between light and dark.He would try to educate me about music beyond top-40 Pop, because he appreciated and knew a lot aboutjazz. But when it came to recess, we parted and wentour separate ways me to the playground and him tothe wall outside the classroom. No one played withNorman. He couldnt see and so there was nothing forhim. About once a day I would look over at Normanfrom high up on a jungle gym of bars and he wouldbe smacking one of those rubbery creepy crawlersagainst the wall. He would do it all recess everyrecess. I still marvel at how much Norman could get

out of a simple sound. He didnt have sight so he hadto compensate with his other senses. He got so muchout of what I would have considered a very simplisticsound. For him the world of sound was rich anddiverse and full. When I think of sound, I alwaysthink first of Norman. Hes helped me to look moredeeply and to understand how sophisticated the worldof sound really is.

What about when more than one wave is presentin the same place? For example, how is it that youcan be at a symphony and make out the sounds ofindividual instruments while they all play togetherand also hear and understand a message beingwhispered to you at the same time you detectsomeone coughing five rows back? How do the soundwaves combine to give you the totality as well as theindividuality of each of the sounds in a room? Theseare some of the questions we will answer as wecontinue to pursue an understanding of music andmusical instruments.

Figure 1.2: The ear is an astonishing receptor for sound waves. A tthe smallest of perceptible sound intensities, the eardrum vibratesless distance than the diameter of a hydrogen atom! If the energy i na single 1-watt night-light were converted to acoustical energy anddivided up into equal portions for every person in the world, i twould still be audible to the person with normal hearing.


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TWO TYPES OF WAVESWaves come in two basic types, depending on

their type of vibration. Imagine laying a long slinkyon the ground and shaking it back and forth. Youwould make what is called a transverse wave (seeFigure 1.3). A transverse wave is one in which themedium vibrates at right angles to the direction theenergy moves. If instead, you pushed forward andpulled backward on the slinky you would make acompressional wave (see Figure 1.4).Compressional waves are also known as longitudinalwaves. A compressional wave is one in which themedium vibrates in the same direction as themovement of energy in the wave

Certain terms and ideas related to waves are easierto visualize with transverse waves, so lets start bythinking about the transverse wave you could makewith a slinky. Imagine taking a snapshot of the wavefrom the ceiling. It would look like Figure 1.5. Somewave vocabulary can be taken directly from thediagram. Other vocabulary must be taken from amental image of the wave in motion:

CREST: The topmost point of the wave medium orgreatest positive distance from the rest position.

TROUGH: The bottommost point of the wavemedium or greatest negative distance from the restposition.

WAVELENGTH (l ): The distance from crest toadjacent crest or from trough to adjacent trough orfrom any point on the wave medium to the adjacentcorresponding point on the wave medium.

AMPLITUDE (A): The distance from the restposition to either the crest or the trough. Theamplitude is related to the energy of the wave. As the

energy grows, so does the amplitude. This makessense if you think about making a more energeticslinky wave. Youd have to swing it with moreintensity, generating larger amplitudes. Therelationship is not linear though. The energy isactually proportional to the square of the amplitude.So a wave with amplitude twice as large actually hasfour times more energy and one with amplitude threetimes larger actually has nine times more energy.

The rest of the vocabulary requires getting amental picture of the wave being generated. Imagineyour foot about halfway down the distance of theslinkys stretch. Lets say that three wavelengths passyour foot each second.

Figure 1.4: A compressional wave moves t othe right while the medium vibrates in thesame direction, right to left.

fi

fi

Figure 1.3: A transverse wave moves to theright while the medium vibrates at rightangles, up and down.

Amplitude

Amplitude

Wavelength

Wavelength

Crest

Trough

Restposition

Figure 1.5: Wave vocabulary

vocabulary

David Lapp


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Figure 1.6: The energy of the vibrating tuning forkviolently splashes water from the bowl. In the air, theenergy of the tuning fork is transmitted through the air andto the ears.

FREQUENCY (f): The number of wavelengths topass a point per second. In this case the frequencywould be 3 per second. Wave frequency is oftenspoken of as waves per second, pulses per second,or cycles per second. However, the SI unit forfrequency is the Hertz (Hz). 1 Hz = 1 per second, soin the case of this illustration, f = 3 Hz.

PERIOD (T): The time it takes for one fullwavelength to pass a certain point. If you see threewavelengths pass your foot every second, then thetime for each wavelength to pass is

13 of a second.

Period and frequency are reciprocals of each other:

T = 1f

and

f = 1T

SPEED (v) Average speed is always a ratio ofdistance to time,

v = d / t . In the case of wave speed,an appropriate distance would be wavelength, l . Thecorresponding time would then haveto be the period, T. So the wavespeed becomes:

v = lt

or

v = lf

SOUND WAVESIf a tree falls in the forest and

theres no one there to hear it, does itmake a sound? Its a commonquestion that usually evokes aphilosophical response. I could argueyes or no convincingly. You willtoo later on. Most people have avery strong opinion one way or theother. The problem is that theiropinion is usually not based on aclear understanding of what sound is.

I think one of the easiest waysto understand sound is to look atsomething that has a simplemechanism for making sound. Thinkabout how a tuning fork makessound. Striking one of the forkscauses you to immediately hear atone. The tuning fork begins to act

somewhat like a playground swing. The playgroundswing, the tuning fork, and most physical systemswill act to restore themselves if they are stressed fromtheir natural state. The natural state for the swing,is to hang straight down. If you push it or pull it andthen let go, it moves back towards the position ofhanging straight down. However, since its movingwhen it reaches that point, it actually overshoots and,in effect, stresses itself. This causes another attemptto restore itself and the movement continues back andforth until friction and air resistance have removed allthe original energy of the push or pull. The same istrue for the tuning fork. Its just that the movement(amplitude) is so much smaller that you cant visiblysee it. But if you touched the fork you could feel it.Indeed, every time the fork moves back and forth itsmacks the air in its way. That smack creates a smallcompression of air molecules that travels from thatpoint as a compressional wave. When it reaches yourear, it vibrates your eardrum with the same frequencyas the frequency of the motion of the tuning fork.You mentally process this vibration as a specifictone.


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A sound wave is nothing more than acompressional wave caused by vibrations . Next timeyou have a chance, gently feel the surface of a speakercone (see Figure 1.7). The vibrations you feel withyour fingers are the same vibrations disturbing theair. These vibrations eventually relay to your ears themessage that is being broadcast. So, if a tree falls inthe forest and theres no one there to hear it, does itmake a sound? Well yes, it will certainly causevibrations in the air and ground when it strikes theground. And no, if theres no one there tomentally translate the vibrations into tones, thenthere can be no true sound. You decide. Maybe it is aphilosophical question after all.

CHARACTERIZING SOUNDAll sound waves are compressional waves caused

by vibrations, but the music from a symphony variesconsiderably from both a babys cry and the whisperof a confidant. All sound waves can be characterizedby their speed, by their pitch, by their loudness,and by their quality or timbre.

The speed of sound is fastest in solids (almost6000 m/s in steel), slower in liquids (almost1500 m/s in water), and slowest in gases. Wenormally listen to sounds in air, so well look at thespeed of sound in air most carefully. In air, soundtravels at:

v = 331 ms + 0.6m / sC

Temperature

The part to the right of the + sign is thetemperature factor. It shows that the speed of soundincreases by 0.6 m/s for every temperature increase of1C. So, at 0 C, sound travels at 331 m/s (about 740

mph). But at room temperature (about 20C)sound travels at:

v = 331 ms

+ 0.6 m / sC

20C( )

=

343 ms

( This is the speed you shouldassume if no temperature is given).

It was one of aviations greatestaccomplishments when Chuck Yeager, onOct. 14, 1947, flew his X-1 jet at Mach1.06, exceeding the speed of sound by 6%.Regardless, this is a snails pace compared tothe speed of light. Sound travels through airat about a million times slower than light,which is the reason why we hear sound

echoes but dont see light echoes. Its also the reasonwe see the lightning before we hear the thunder. Thelightning-thunder effect is often noticed in bigstadiums. If youre far away from a baseball playerwhos up to bat, you can clearly see the ball hitbefore you hear the crack of the bat. You can considerthat the light recording the event reaches your eyesvirtually instantly. So if the sound takes half a secondmore time than the light, youre half the distancesound travels in one second (165 meters) from thebatter. Next time youre in a thunderstorm use thismethod to estimate how far away lightning isstriking. Click here for a demonstration of the effectof echoes.

Do you get it? (1.1)Explain why some people put their ears on railroadtracks in order to hear oncoming trains

Figure 1.7: The front of a speaker cone faces upwardwith several pieces of orange paper lying on top of i t .Sound is generated when an electric signal causes thespeaker cone to move in and out, pushing on the airand creating a compressional wave. The ear can detectthese waves. Here these vibrations can be seen as theycause the little bits of paper to dance on the surface o fthe speaker cone.


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Do you get it? (1.2)The echo of a ship's foghorn, reflected from the cliffon a nearby island, is heard 5.0 s after the horn issounded. How far away is the cliff?

The pitch of sound is the same as thefrequency of a sound wave. With no hearing losses ordefects, the ear can detect wave frequencies between 20Hz and 20,000 Hz. (Sounds below 20 Hz areclassified as subsonic; those over 20,000 Hz areultrasonic). However, many older people, loud concertattendees, and soldiers with live combat experiencelose their ability to hear higher frequencies. The goodnews is that the bulk of most conversation takesplace well below 2,000 Hz. The average frequencyrange of the human voice is 120 Hz to approximately1,100 Hz (although a babys shrill cry is generally2,000 - 3,000 Hz which is close to the frequencyrange of greatest sensitivity hmm, interesting).Even telephone frequencies are limited to below3,400 Hz. But the bad news is that the formation ofmany consonants is a complex combination of veryhigh frequency pitches. So to the person with highfrequency hearing loss, the words key, pee, and teasound the same. You find people with these hearinglosses either lip reading or understanding aconversation by the context as well as the actualrecognition of words. Neil Bauman, a hearing expertat www.hearinglosshelp.com , offers the followinginformation:

Vowels are clustered around thefrequencies between about 300 and 750 Hz.Some consonants are very low frequency,such as j, z, and v at about 250 Hz. Otherssuch as k, t, f, s, and th are high frequencysounds and occur between 3,000 and 8,000Hz. All of these consonants are voiceless onesand are produced by air hissing from aroundthe teeth. Most people, as they age, beginlosing their hearing at the highest frequenciesfirst and progress downwards. Thus, theabove consonant sounds are the first to belost. As a result, it is most difficult to

distinguish between similar sounding wordsthat use these letters. Furthermore, thevowels are generally loud (they use about95% of the voice energy to produce). Theconsonants are left with only 5% to go aroundfor all of them. But it is mostly the consonantsthat give speech its intelligibility. That is whymany older people will say, I can hear peopletalking. I just can't understand what they aresaying.

One important concept in music is the octave a doubling in frequency. For example, 40 Hz is oneoctave higher than 20 Hz. The ear is sensitive over afrequency range of about 10 octaves: 20 Hz 40 Hz 80 Hz 160 Hz 320 Hz 640 Hz 1,280 Hz 2,560 Hz 5,120 Hz 10,240 Hz 20,480 Hz. And within that range it can discriminatebetween thousands of differences in sound frequency.Below about 1,000 Hz the Just NoticeableDifference (JND) in frequency is about 1 Hz (at theloudness at which most music is played), but thisrises sharply beyond 1,000 Hz. At 2,000 the JND isabout 2 Hz and at 4,000 Hz the JND is about 10 Hz.(A JND of 1 Hz at 500 Hz means that if you wereasked to listen alternately to tones of 500 Hz and501 Hz, the two could be distinguished as twodifferent frequencies, rather than the same). It isinteresting to compare the ears frequency perceptionto that of the eye. From red to violet, the frequency oflight less than doubles, meaning that the eye is onlysensitive over about one octave, and its ability todiscriminate between different colors is only about125. The ear is truly an amazing receptor, not onlyits frequency range, but also in its ability toaccommodate sounds with vastly different loudness.

The loudness of sound is related to theamplitude of the sound wave. Most people have somerecognition of the decibel (dB) scale. They might beable to tell you that 0 dB is the threshold of hearingand that the sound on the runway next to anaccelerating jet is about 140 dB. However, mostpeople dont realize that the decibel scale is alogarithmic scale. This means that for every increaseof 10 dB the sound intensity increases by a factor often. So going from 60 dB to 70 dB is a ten-foldincrease, and 60 dB to 80 dB is a hundred-foldincrease. This is amazing to me. It means that we canhear sound intensities over 14 orders of magnitude.This means that the 140 dB jet on the runway has aloudness of 1014 times greater than threshold. 1014 is100,000,000,000,000 thats 100 trillion! It meansour ears can measure loudness over a phenomenallylarge range. Imagine having a measuring cup thatcould accurately measure both a teaspoon and 100trillion teaspoons (about 10 billion gallons). The earis an amazing receptor! However, our perception isskewed a bit. A ten-fold increase in loudness doesnt


11

sound ten times louder. It may only sound twice asloud. Thats why when cheering competitions aredone at school rallies, students are not very excited bythe measure of difference in loudness between afreshmen class (95 dB) and a senior class (105 dB).The difference is only 10 dB. It sounds perhaps twiceas loud, but its really 10 times louder. (Those lungsand confidence grow a lot in three years!)

Do you get it? (1.3)When a passenger at the airport moves from inside awaiting area to outside near an airplane the decibellevel goes from 85 dB to 115 dB. By what factor hasthe sound intensity actually gone up?

The quality of sound or timbre is thesubtlest of all its descriptors. A trumpet and a violincould play exactly the same note, with the same pitchand loudness, and if your eyes were closed you couldeasily distinguish between the two. The difference inthe sounds has to do with their quality or timbre.The existence of supplementary tones, combined withthe basic tones, doesnt change the basic pitch, butgives a special flavor to the sound being produced.Sound quality is the characteristic that gives theidentity to the sound being produced.

DETAILS ABOUT DECIBELSIt was mentioned earlier that the sensitivity of

the human ear is phenomenally keen. The thresholdof hearing (what a young perfect ear could hear) is

110-12Watts /meter 2 . This way of expressing soundwave amplitude is referred to as Sound Intensity (I).It is not to be confused with Sound Intensity Level(L), measured in decibels (dB). The reason whyloudness is routinely represented in decibels ratherthan

Watts /meter 2 is primarily because the earsdont hear linearly. That is, if the sound intensitydoubles, it doesnt sound twice as loud. It doesntreally sound twice as loud until the Sound Intensity isabout ten times greater. (This is a very roughapproximation that depends on the frequency of thesound as well as the reference intensity.) If the soundintensity were used to measure loudness, the scalewould have to span 14 orders of magnitude. That

means that if we defined the faintest sound as 1, wewould have to use a scale that went up to100,000,000,000,000 (about the loudest sounds youever hear. The decibel scale is much more compact(0 dB 140 dB for the same range) and it is moreclosely linked to our ears perception of loudness.You can think of the sound intensity as a physicalmeasure of sound wave amplitude and sound intensitylevel as its psychological measure.

The equation that relates sound intensity to soundintensity level is:

L = 10 log I 2I1

L The number of decibels I2 is greater than I1I2 The higher sound intensity being comparedI1 The lower sound intensity being compared

Remember, I is measured in

Watts /meter 2 . It islike the raw power of the sound. The L in thisequation is what the decibel difference is betweenthese two. In normal use, I1 is the threshold ofhearing,

110-12Watts /meter 2 . This means that thedecibel difference is with respect to the faintest soundthat can be heard. So when you hear that a busyintersection is 80 dB, or a whisper is 20 dB, or a classcheer is 105 dB, it always assumes that thecomparison is to the threshold of hearing, 0 dB. (80dB means 80 dB greater than threshold). Dontassume that 0 dB is no sound or total silence. This issimply the faintest possible sound a human withperfect hearing can hear. Table 1.1 provides decibellevels for common sounds.

If you make I2 twice as large as I1, then

DL @ 3dB . If you make I2 ten times as large as I1,then

DL = 10dB . These are good reference numbers totuck away:

Double Sound Intensity @ +3dB

10 Sound Intensity = +10dB

Click here to listen to a sound intensity level reducedby 6 dB per step over ten steps. Click here to listento a sound intensity level reduced by 3 dB per stepover ten steps.


12

Source of soundSoundIntensityLevel (dB)

SoundIntensity

Wm2

Threshold of hearing 0

110-12Breathing 20

110-10Whispering 40

110-8Talking softly 60

110-6Loud conversation 80

110-4Yelling 100

110-2Loud Concert 120

1Jet takeoff 140

100

Table 1.1 Decibel levels for typical sounds

FREQUENCY RESPONSE OVER THEAUDIBLE RANGE

We hear lower frequencies as low pitches andhigher frequencies as high pitches. However, oursensitivity varies tremendously over the audiblerange. For example, a 50 Hz sound must be 43 dBbefore it is perceived to be as loud as a 4,000 Hzsound at 2 dB. (4,000 Hz is the approximatefrequency of greatest sensitivity for humans with nohearing loss.) In this case, we require the 50 Hz soundto have 13,000 times the actual intensity of the4,000 Hz sound in order to have the same perceivedintensity! Table 1.2 illustrates this phenomenon ofsound intensity level versus frequency. The lastcolumn puts the relative intensity of 4,000 Hzarbitrarily at 1 for easy comparison with sensitivity atother frequencies.

If you are using the CD version of thiscurriculum you can try the following demonstration,which illustrates the response of the human ear tofrequencies within the audible range Click here tocalibrate the sound on your computer and then clickhere for the demonstration.

Frequency(Hz)

SoundIntensityLevel (dB)

SoundIntensity

Wm 2( )

RelativeSoundIntensity

50 43

2.0 10-8 13,000100 30

1.0 10-9 625200 19

7.9 10-11 49500 11

1.310-11 8.11,000 10

1.0 10-11 6.32,000 8

6.310-12 3.93,000 3

2.0 10-12 1.34,000 2

1.6 10-12 15,000 7

5.0 10-12 3.16,000 8

6.310-12 3.97,000 11

1.310-11 8.18,000 20

1.0 10-10 62.59,000 22

1.6 10-10 10014,000 31

1.310-9 810

Table 1.2: Sound intensity and soundintensity level required to perceivesounds at different frequencies to beequally loud. A comparison of relativesound intensities arbitrarily assigns4,000 Hz the value of 1.

ExampleThe muffler on a car rusts out and the decibel levelincreases from 91 dB to 113 dB. How many timeslouder is the leaky muffler?

The brute force way to do this problem would be tostart by using the decibel equation to calculate thesound intensity both before and after the muffler rustsout. Then you could calculate the ratio of the two.Its easier though to recognize that the decibeldifference is 22 dB and use that number in the decibelequation to find the ratio of the sound intensitiesdirectly:

L = 10 log I 2I1

22dB = 10 log I 2I1

fi 2.2 = log I 2I1

Notice I just dropped the dB unit. Its not a real unit,just kind of a placeholder unit so that we dont haveto say, The one sound is 22 more than the othersound. and have a strange feeling of 22 what?

102.2 = I 2I1

fi I 2 = 102.2 I1 fi

I2 = 158I1

So the muffler is actually 158 times louder thanbefore it rusted out.


13

Do you get it? (1.4)A solo trumpet in an orchestra produces a soundintensity level of 84 dB. Fifteen more trumpets jointhe first. How many decibels are produced?

Do you get it? (1.5)What would it mean for a sound to have a soundintensity level of -10 dB

Another factor that affects the intensity of thesound you hear is how close you are to the sound.Obviously a whisper, barely detected at one metercould not be heard across a football field. Heres theway to think about it. The power of a particularsound goes out in all directions. At a meter awayfrom the source of sound, that power has to cover anarea equal to the area of a sphere (4pir2) with a radiusof one meter. That area is 4pi m2. At two meters awaythe same power now covers an area of4pi(2 m)2 = 16pi m2, or four times as much area. Atthree meters away the same power now covers an areaof 4pi(3 m)2 = 36pi m2, or nine times as much area.So compared to the intensity at one meter, theintensity at two meters will be only one-quarter asmuch and the intensity at three meters only one-ninthas much. The sound intensity follows an inversesquare law, meaning that by whatever factor thedistance from the source of sound changes, theintensity will change by the square of the reciprocalof that factor.

ExampleIf the sound intensity of a screaming baby were

110-2 Wm 2

at 2.5 m away, what would it be at6.0 m away?The distance from the source of sound is greater by afactor of

6.02.5 = 2.4 . So the sound intensity is decreased

by

12.4 2( )

= 0.174 . The new sound intensity is:

110-2 Wm 2( ) 0.174( ) =

1.74 10-3 Wm 2

Another way to look at this is to first considerthat the total power output of a source of sound is itssound intensity in

Watts /meter 2 multiplied by thearea of the sphere that the sound has reached. So, forexample, the baby in the problem above creates asound intensity of

110-2W /m2 at 2.5 m away.This means that the total power put out by the babyis:

Power = Intensity sphere area

fi P = 110-2 Wm2

4p 2.5m( )2[ ] = 0.785 W

Now lets calculate the power output from theinformation at 6.0 m away:

P = 1.74 10-3 Wm2

4p (6.0m)2[ ] = .785 W

Its the same of course, because the power outputdepends on the baby, not the position of the observer.This means we can always equate the power outputsthat are measured at different locations:

P1 = P2 fi I1( ) 4pr12( ) = I2( ) 4pr22( )

fi

I1r12 = I 2r2

2


14

Do you get it? (1.6)Its a good idea to make sure that you keep thechronic sound youre exposed to down under 80 dB. Ifyou were working 1.0 m from a machine that createda sound intensity level of 92 dB, how far would youneed move away to hear only 80 dB? (Hint: rememberto compare sound intensities and not sound intensitylevels.)


15

ACTIVITYORCHESTRAL SOUND

You can hear plenty of sound in a concert hallwhere an orchestra is playing. Each instrumentvibrates in its own particular way, producing theunique sound associated with it. The acoustical powercoming from these instruments originates with themusician. It is the energy of a finger thumping on apiano key and the energy of the puff of air across thereed of the clarinet and the energy of the slam ofcymbals against each other that causes theinstruments vibration. Most people are surprised tolearn that only about 1% of the power put into theinstrument by the musician actually leads to thesound wave coming from the instrument. But, as youknow, the ear is a phenomenally sensitive receptor ofacoustical power and needs very little power to bestimulated to perception. Indeed, the entire orchestraplaying at once would be loud to the ear, but actuallygenerate less power than a 75-watt light bulb! Anorchestra with 75 performers has an acoustic power ofabout 67 watts. To determine the sound intensity

level at 10 m, we could start by finding the soundintensity at 10 m:

I = 67W4p 10m( )2

= 0.056 Wm2

.

The sound intensity level would then be:

L = 10 log0.056 W

m2

110-12 Wm2

= 107.5dB .

Table 1.3 lists the sound intensity level of variousinstruments in an orchestra as heard from 10 m away.You can use these decibel levels to answer thequestions throughout this activity.

OrchestralInstrument

Sound IntensityLevel (dB )

Violin (at its quietest) 34.8Clarinet 76.0Trumpet 83.9Cymbals 98.8Bass drum (at itsloudest)

103

Table 1.3: Sound intensity l eve l s(measured at 10 m away) for variousmusical instruments

1. How many clarinets would it take to equal the acoustic power of a pair of cymbals?


16

2. If you had to replace the total acoustic power of a bass drum with a single light bulb, what wattage wouldyou choose?

3. How far would you have to be from the violin (when its at its quietest) in order to barely detect its sound?

4. If the sound emerges from the trumpet at 0.5 m from the trumpet players ear, how many decibels does heexperience during his trumpet solo?


17

5. If the orchestra conductor wanted to produce the sound intensity of the entire orchestra, but use only violins(at their quietest) to produce the sound, how many would need to be used? How about if it were to be donewith bass drums (at their loudest?

6. The conductor has become concerned about the high decibel level and wants to make sure he does notexperience more than 100 dB. How far away from the orchestra must he stand?

7. If he doesnt use the one bass drum in the orchestra, how far away does he need to stand?


18

WAVE INTERFERENCETS INTRIGUING THAT at a lively partywith everyone talking at once, you can hearthe totality of the noise in the room andthen alternately distinguish and concentrate onthe conversation of one person. That isbecause of an interesting phenomenon of

waves called superposition. Wave superpositionoccurs when two or more waves move through thesame space in a wave medium together. There are twoimportant aspects of this wave superposition. One isthat each wave maint ains its own identity and isunaffected by any other waves in the same space . Thisis why you can pick out an individual conversationamong all the voices in the region of your ear. Thesecond aspect is that when two or more waves are inthe same medium, t he overall amplitude at any pointon the medium is simply the sum of the individualwave amplitudes at that point . Figure 1.8 illustratesboth of these aspects. In the top scene, two wave

pulses move toward each other. In the second scenethe two pulses have reached the same spot in themedium and the combined amplitude is just the sumof the two. In the last scene, the two wave pulsesmove away from each other, clearly unchanged bytheir meeting in the second scene.

When it comes to music, the idea of interferenceis exceptionally important. Musical sounds are oftenconstant frequencies held for a sustained period. Soundwaves interfere in the same way other waves, butwhen the sound waves are musical sounds (sustainedconstant pitches), the resulting superposition cansound either pleasant (consonant) or unpleasant(dissonant). Musical scales consist of notes (pitches),which when played together, sound consonant. Welluse the idea of sound wave interference when webegin to look for ways to avoid dissonance in thebuilding of musical scales.

I

Figure 1.8: Wave superposition. Note in the middle drawing that the wave shape is simply thearithmetic sum of the amplitudes of each wave. Note also in the bottom drawing that the twowaves have the same shape and amplitude as they had before encountering each other.


19

ACTIVITYWAVE INTERFERENCE

In each of the following two cases, the wave pulses are moving toward each other. Assume that each wave pulsemoves one graph grid for each new graph. Draw the shape the medium would have in each of the blank graphsbelow.

20

Music should strike fire from the heart of man, and bringtears from the eyes of woman. Ludwig Van Beethoven

CHAPTER 2RESONANCE, STANDING WAVES,

AND MUSICAL INSTRUMENTSHE NEXT STEP in pursuing the physics of music and musicalinstruments is to understand how physical systems can be made to vibratein frequencies that correspond to the notes of the musical scales. Butbefore the vibrations of physical systems can be understood, a diversionto the behavior of waves must be made once again. The phenomena of

resonance and standing waves occur in the structures of all musical instruments. Itis the means by which all musical instruments make their music.

Why is it that eight-year-old boys have such anaversion to taking baths? I used to hate getting in thetub! It was perhaps my least favorite eight-year-oldactivity. The one part of the bathing ritual that madethe whole process at least tolerable was makingwaves! If I sloshed back and forth in the water withmy body at just the right frequency, I could createwaves that reached near tidal wave amplitudes. Toolow or too high a frequency and the waves would dieout. But it was actually easy to find the rightfrequency. It just felt right. I thought I had discoveredsomething that no one else knew. Then, 20 yearslater, in the pool at a Holiday Inn in New Jersey witha group of other physics teachers, I knew that mydiscovery was not unique. Lined up on one side of thepool and with one group movement, we heaved ourbodies forward. A water wave pulse moved forwardand struck the other side of the pool. Then it reflectedand we waited as it traveled back toward us, continuedpast us, and reflected again on the wall of the poolright behind us. Without a word, as the crest of thedoubly reflected wave reached us, we heaved ourbodies again. Now the wave pulse had greateramplitude. We had added to it and when it struck theother side, it splashed up on the concrete, drawing theamused and, in some cases, irritated attention of theother guests sunning themselves poolside. As a childpushes her friend on a playground swing at just righttime in order to grow the amplitude of the swingsmotion, we continued to drive the water waveamplitude increasingly larger with our rhythmicmotion. Those who were irritated before were nowcurious, and those who were amused before were nowcheering. The crowd was pleased and intrigued by

something they knew in their gut. Most had probablyrocked back and forth on the seat of a car at astoplight with just the right frequency to get theentire car visibly rocking. Many had probably had theexperience of grabbing a sturdy street sign andpushing and pulling on it at just the right times toget the sign to shake wildly. They had all experiencedthis phenomenon of resonance.

To understand resonance, think back to thediscussion of the playground swing and tuning forkrestoring themselves to their natural states after beingstressed out of those states. Recall that as the swingmoves through the bottom of its motion, itovershoots this natural state. The tuning fork doestoo, and both of the two will oscillate back and forthpast this point (at a natural frequency particular to thesystem) until the original energy of whatever stressedthem is dissipated. You can keep the swing movingand even increase its amplitude by pushing on it atthis same natural frequency. The same could be doneto the tuning fork if it were driven with an audiospeaker producing the forks natural frequency.Resonance occurs whenever a physical system isdriven at its natural frequency. Most physical systemshave many natural frequencies. The street sign, forexample can be made to shake wildly back and forthwith a low fundamental frequency (first mode)or with a higher frequency of vibration, in thesecond mode. Its easier to understand how musicalinstruments can be set into resonance by thinkingabout standing waves.

Standing waves occur whenever two waves withequal frequency and wavelength move through amedium so that the two perfectly reinforce each other.

T

SSSTTTAAANNNDDDIIINNNGGG WWWAAAVVVEEESSS

21

In order for this to occur the length of the mediummust be equal to some integer multiple of half thewavelength of the waves. Usually, one of the twowaves is the reflection of the other. When theyreinforce each other, it looks like the energy isstanding in specific locations on the wave medium,hence the name standing waves (see figure 2.1).There are parts of the wave medium that remainmotionless (the nodes) and parts where the wavemedium moves back and forth between maximumpositive and maximum negative amplitude (theantinodes).

Standing waves can occur in all wave mediumsand can be said to be present during any resonance.Perhaps youve heard someone in a restaurant rubbinga finger around the rim of a wineglass, causing it tosing. The singing is caused by a standing wave inthe glass that grows in amplitude until the pulsesagainst the air become audible. Soldiers are told tobreak step march when moving across small bridgesbecause the frequency of their march may be the

natural frequency of the bridge, creating and thenreinforcing a standing wave in the bridge and causingthe same kind of resonance as in the singing wineglass. A standing wave in a flat metal plate can becreated by driving it at one of its natural frequencies atthe center of the plate. Sand poured onto the platewill be unaffected by and collect at the nodes of thestanding wave, whereas sand at the antinodes will bebounced off, revealing an image of the standing wave(see figure 2.2).

Resonance caused the destruction of the TacomaNarrows Bridge on November 7, 1940 (see Figure2.3). Vortices created around its deck by 35 - 40 mphwinds resulted in a standing wave in the bridge deck.When its amplitude reached five feet, at about 10 am,the bridge was forced closed. The amplitude of motioneventually reached 28 feet. Most of the remains ofthis bridge lie at the bottom of the Narrows, makingit the largest man-made structure ever lost to sea andthe largest man-made reef.

Figure 2.1: A time-lapse view of a standing wave showing the nodes and antinodes present. It i sso named because it appears that the energy in the wave stands in certain places (the antinodes).Standing waves are formed when two waves of equal frequency and wavelength move through amedium and perfectly reinforce each other. In order for this to occur the length of the mediummust be equal to some integer multiple of half the wavelength of the waves. In the case of th i sstanding wave, the medium is two wavelengths long (4 half wavelengths).

ANTINODES

NODES


22

Figure 2.2: A flat square metal plate is driven in four of its resonance modes. These photographsshow each of the resulting complicated two-dimensional standing wave patterns. Sand poured ontothe top of the plate bounces off the antinodes, but settles into the nodes, allowing the standingwave to be viewed. The head of a drum, when beaten, and the body of a guitar, when played,exhibit similar behaviors.


23

The Tacoma Narrows Bridge on November 7 ,1940. Workers on the construction of thebridge had referred to it as Galloping Gertiebecause of the unusual oscillations that wereoften present. Note nodes at the towers and i nthe center of the deck. This is the secondmode.

Vortices around its deck caused by winds o f35 40 mph caused the bridge to begin risingand falling up to 5 feet, forcing the bridge t obe closed at about 10 am. The amplitude o fmotion eventually reached 28 feet.

The bridge had a secondary standing wave i nthe first mode with its one node on thecenterline of the bridge. The man in the photowisely walked along the node. Although thebridge was heaving wildly, the amplitude at thenode was zero, making it easy to navigate.

At 10:30 am the oscillations had f inallycaused the center span floor panel to fal lfrom the bridge. The rest of the breakupoccurred over the next 40 minutes.

Figure 2.3: The Tacoma Narrows Bridge collapse (used with permission from University of WashingtonSpecial Collection Manuscripts).


24

Since the collapse of the Tacoma NarrowsBridge, engineers have been especially conscious ofthe dangers of resonance caused by wind orearthquakes. Paul Doherty, a physicist at theExploratorium in San Francisco, had this to sayabout the engineering considerations now made whenbuilding or retrofitting large structures subject todestruction by resonance:

The real world natural frequency of largeobjects such as skyscrapers and bridges isdetermined via remote sensoraccelerometers. Buildings like the Trans-America Pyramid and the Golden GateBridge have remote accelerometers attachedto various parts of the structure. When windor an earthquake excites the building theaccelerometers can detect the ringing(resonant oscillation) of the structure. TheGolden Gate Bridge was detuned by havingmass added at various points so that a standingwave of a particular frequency would affectonly a small portion of the bridge. TheTacoma Narrows Bridge had the resonanceextend the entire length of the span and onlyhad a single traffic deck which could flex. TheGolden Gate Bridge was modified after theNarrows Bridge collapse. The underside ofthe deck had stiffeners added to dampentorsion of the roadbed and energy-absorbingstruts were incorporated. Then massadditions broke up the ability of the standingwave to travel across the main cablesbecause various sections were tuned todifferent oscillation frequencies. This is whysitting in your car waiting to pay the toll youcan feel your car move up a down when alarge truck goes by but the next large truckmay not give you the same movement. Thatfirst truck traveling at just the right speed mayexcite the section you are on while a truck ofdifferent mass or one not traveling the samespeed may not affect it much. As for thehorizontal motion caused by the wind, thesame differentiation of mass elements underthe roadbed keeps the whole bridge from goingresonant with olian oscillations.Just envision the classic Physics experiment

where different length pendulums are hungfrom a common horizontal support. Measuredperiodic moving of the support will make only

one pendulum swing depending on the periodof the applied motion. If all the pendulums hadthe same oscillation you could get quite amotion going with a small correctly timedforce application. Bridges and buildings nowrely on irregular distributions of mass to helpkeep the whole structure from moving as aunit that would result in destructive failure.Note also on the Golden Gate the secondarysuspension cable keepers (spacers) arelocated at slightly irregular intervals todetune them. As current structuralengineering progresses more modifications ofthe bridge will be done. The new super bridgein Japan has hydraulically movable weightsthat can act as active dampeners. What isearthquake (or wind) safe today will besubstandard in the future.

The collapse of the Tacoma Narrows Bridge isperhaps the most spectacular example of thedestruction caused by resonance, but everyday thereare boys and girls who grab hold of street sign polesand shake them gently intuitively at just the rightfrequency to get them violently swaying back andforth. Moreover, as mentioned previously, all musicalinstruments make their music by means of standingwaves.

To create its sound, the physical structure ofmusical instruments is set into a standing wavecondition. The connection with resonance can be seenwith a trumpet, for example. The buzzing lips of thetrumpet player create a sound wave by allowing aburst of air into the trumpet. This burst is largelyreflected back when it reaches the end of the trumpet.If the trumpet player removes his lips, the soundwave naturally reflects back and forth between thebeginning and the end of the trumpet, quickly dyingout as it leaks from each end. However, if the playerslips stay in contact with the mouthpiece, the reflectedburst of air can be reinforced with a new burst of airfrom the players lips. The process continues,creating a standing wave of growing amplitude.Eventually the amplitude reaches the point whereeven the small portion of the standing wave thatescapes the trumpet becomes audible. It is resonancebecause the trumpet player adds a kick of air at theprecise frequency (and therefore also the samewavelength) of the already present standing soundwave.

MMMUUUSSSIIICCCAAALLL IIINNNSSSTTTRRRUUUMMMEEENNNTTTSSS

25

INTRODUCTION TO MUSICAL INSTRUMENTSF YOU DONT play a musical instrument,its humbling to pick one up and try to createsomething that resembles a melody or tune. Itgives you a true appreciation for the musicianwho seems to become one with hisinstrument. But you certainly dont have to be

a musician to understand the physics of music or thephysics of musical instruments. All musicians createmusic by making standing waves in their instrument.The guitar player makes standing waves in the stringsof the guitar and the drummer does the same in theskin of the drumhead. The trumpet blower and fluteplayer both create standing waves in the column of air

within their instruments. Most kids have done thesame thing, producing a tone as they blow over thetop of a bottle. If you understand standing waves youcan understand the physics of musical instruments.Well investigate three classes of instruments:

Chordophones (strings) Aerophones (open and closed pipes) Idiophones (vibrating rigid bars and pipes)

Most musical instruments will fit into these threecategories. But to fully grasp the physics of thestanding waves within musical instruments andcorresponding music produced, an understanding ofwave impedance is necessary.

IFigure 2.4:Chordophones aremusical instrumentsin which a standingwave is initiallycreated in the stringsof the instruments.Guitars, violins, andpianos fall into thiscategory.

Figure 2.5:Aerophones aremusical instrumentsin which a standingwave is initiallycreated in the columnof air within theinstruments.Trumpets, flutes, andoboes fall into thiscategory.

Figure 2.6:Idiophones aremusical instrumentsin which a standingwave is initiallycreated in the physicalstructure of theinstruments.Xylophones,marimbas, and chimesfall into thiscategory.

MMMUUUSSSIIICCCAAALLL IIINNNSSSTTTRRRUUUMMMEEENNNTTTSSS

26

MUSICAL INSTRUMENTS ANDWAVE IMPEDANCE

The more rigid a medium is the less effect theforce of a wave will have on deforming the shape ofthe medium. Impedance is a measure of how muchforce must be applied to a medium by a wave in orderto move the medium by a given amount. When itcomes to standing waves in the body of a musicalinstrument, the most important aspect of impedancechanges is that they always cause reflections.Whenever a wave encounters a change in impedance,some or most of it will be reflected. This is easy tosee in the strings of a guitar. As a wave moves alongthe string and encounters the nut or bridge of theguitar, this new medium is much more rigid than thestring and the change in impedance causes most of thewave to be reflected right back down the string (goodthing, because the reflected wave is needed to createand sustain the standing wave condition). Its harderto see however when you consider the standing wave

of air moving through the inside of the tuba. How isthis wave reflected when it encounters the end of thetuba? The answer is that wave reflection occursregardless of how big the impedance change is orwhether the new impedance is greater or less. Thepercentage of reflection depends on how big thechange in impedance is. The bigger the impedancechange, the more reflection will occur. When thewave inside the tuba reaches the end, it is not asconstricted less rigid, so to speak. This slightchange in impedance is enough to cause a significantportion of the wave to reflect back into the tuba andthus participate and influence the continuedproduction of the standing wave. The part of the wavethat is not reflected is important too. The transmittedportion of the wave is the part that constitutes thesound produced by the musical instrument. Figure 2.7illustrates what happens when a wave encountersvarious changes in impedance in the medium throughwhich it is moving.

A .

B .

C .

D .

Before

Before

Before

Before

After

After

After

After

Figure 2.7: Wave pulse encountering medium with different impedanceA. Much greater impedance fi Inverted, large reflection.B. Slightly greater impedance fi Inverted, small reflection.C. Much smaller impedance fi Upright, large reflection.D. Slightly smaller impedance fi Upright, small reflection.

27

After silence that which comes nearest to expressing theinexpressible is music.Aldous Huxley

CHAPTER 3MODES, OVERTONES, AND HARMONICS

HEN A DESIRED note (tone) is played on a musical instrument,there are always additional tones produced at the same time. Tounderstand the rationale for the development of consonant musicalscales, we must first discuss the different modes of vibrationproduced by a musical instrument when it is being played. It doesnt

matter whether it is the plucked string of a guitar, or the thumped key on a piano,or the blown note on a flute, the sound produced by any musical instrument is farmore complex than the single frequency of the desired note. The frequency of thedesired note is known as the fundamental frequency, which is caused by the firstmode of vibration, but many higher modes of vibration always naturally occursimultaneously.

Higher modes are simply alternate, higherfrequency vibrations of the musical instrumentmedium. The frequencies of these higher modes areknown as overtones. So the second mode producesthe first overtone, the third mode produces the secondovertone, and so on. In percussion instruments, (likexylophones and marimbas) the overtones are notrelated to the fundamental frequency in a simple way,but in other instruments (like stringed and windinstruments) the overtones are related to thefundamental frequency harmonically.

When a musical instruments overtones areharmonic, there is a very simple relationship betweenthem and the fundamental frequency. Harmonics areovertones that happen to be simple integer multiplesof the fundamental frequency. So, for example, if astring is plucked and it produces a frequency of110 Hz, multiples of that 110 Hz will also occur atthe same time: 220 Hz, 330 Hz, 440 Hz, etc will allbe present, although not all with the same intensity.A musical instruments fundamental frequency and allof its overtones combine to produce that instrumentssound spectrum or power spectrum. Figure 3.1shows the sound spectrum from a flute playing thenote G4. The vertical line at the first peak indicates itsfrequency is just below 400 Hz. In the musical scaleused for this flute, G4 has a frequency of 392 Hz.Thus, this first peak is the desired G4 pitch. Thesecond and third peaks are also identified with verticallines and have frequencies of about 780 Hz and1,170 Hz (approximately double and triple the lowestfrequency of 392 Hz). This lowest frequency occurring

when the G4 note is played on the flute is caused bythe first mode of vibration. It is the fundamentalfrequency. The next two peaks are the simultaneouslypresent frequencies caused by the second and thirdmodes of vibration. That makes them the first twoovertones. Since these two frequencies are integermultiples of the lowest frequency, they are harmonicovertones.

When the frequencies of the overtones areharmonic the fundamental frequency and all theovertones can be classified as some order of harmonic.The fundamental frequency becomes the firstharmonic, the first overtone becomes the secondharmonic, the second overtone becomes the thirdharmonic, and so on. (This is usually confusing formost people at first. For a summary, see Table 3.1).Looking at the Figure 3.1, it is easy to see that thereare several other peaks that appear to be integermultiples of the fundamental frequency. These are allindeed higher harmonics. For this flute, the thirdharmonic is just about as prominent as thefundamental. The second harmonic is a bit less thaneither the first or the third. The fourth and higherharmonics are all less prominent and generally followa pattern of less prominence for higher harmonics(with the exception of the eighth and ninthharmonics). However, only the flute has thisparticular spectrum of sound. Other instrumentsplaying G4 would have overtones present in differentportions. You could say that the sound spectrum ofan instrument is its acoustical fingerprint.

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Mode ofvibration

Frequency name (forany type ofovertone)

Frequency name(for harmonicovertones)

First Fundamental First harmonicSecond First overtone Second harmonicThird Second overtone Third harmonicFourth Third overtone Fourth harmonic

Table 3.1: Names given to the frequenciesof different modes of vibration. Theovertones are harmonic if they areinteger multiples of the fundamentalfrequency.

TIMBRE, THE QUALITY OF SOUNDIf your eyes were closed, it would still be easy to

distinguish between a flute and a piano, even if bothwere playing the note G4. The difference in intensitiesof the various overtones produced gives eachinstrument a characteristic sound quality ortimbre (tam-brrr), even when they play the samenote. This ability to distinguish is true even betweenmusical instruments that are quite similar, like theclarinet and an oboe (both wind instruments usingphysical reeds). The contribution of total soundarising from the overtones varies from instrument toinstrument, from note to note on the same instrumentand even on the same note (if the player produces thatnote differently by blowing a bit harder, for example).In some cases, the power due to the overtones is lessprominent and the timbre has a very pure sound to it,

Figure 3.1: The sound spectrum of a flute shows the frequencies present when the G4 note i splayed. The first designated peak is the desired frequency of 392 Hz (the fundamentalfrequency). The next two designated peaks are the first and second overtones. Since these andall higher overtones are integer multiples of the fundamental frequency, they are harmonic.(Used with permission. This and other sound spectra can be found athttp://www.phys.unsw.edu.au/music/flute/modernB/G4.html )

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like in the flute or violin. In other instruments, likethe bassoon and the bagpipes, the overtonescontribute much more significantly to the powerspectrum, giving the timbre a more complex sound.

A CLOSER LOOK AT THEPRODUCTION OF OVERTONES

To begin to understand the reasons for theexistence of overtones, consider the guitar and how itcan be played. A guitar string is bound at both ends.If it vibrates in a standing wave condition, thosebound ends will necessarily have to be nodes. Whenplucked, the string can vibrate in any standing wavecondition in which the ends of the string are nodesand the length of the string is some integer number ofhalf wavelengths of the modes wavelength (seeFigure 3.2). To understand how the intensities ofthese modes might vary, consider plucking the stringat its center. Plucking it at its center will favor anantinode at that point on the string. But all the oddmodes of the vibrating string have antinodes at thecenter of the string, so all these modes will bestimulated. However, since the even modes all havenodes at the center of the string, these will generallybe weak or absent when the string is plucked at thecenter, forcing this point to be moving (see Figure3.2). Therefore, you would expect the guitars soundquality to change as its strings are plucked at differentlocations.

An additional (but more subtle) explanation forthe existence and intensity of overtones has to dowith how closely the waveform produced by themusical instrument compares to a simple sine wave.Weve already discussed what happens when a flexiblephysical system is forced from its position of greateststability (like when a pendulum is moved from itsrest position or when the tine of a garden rake ispulled back and then released). A force will occur thatattempts to restore the system to its former state. Inthe case of the pendulum, gravity directs thependulum back to its rest position. Even though theforce disappears when the pendulum reaches this restposition, its momentum causes it to overshoot. Nowa new force, acting in the opposite direction, againattempts to direct the pendulum back to its restposition. Again, it will overshoot. If it werent forsmall amounts of friction and some air resistance,this back and forth motion would continue forever. Inits simplest form, the amplitude vs. time graph ofthis motion would be a sine wave (see Figure 3.3).Any motion that produces this type of graph isknown as simple harmonic motion.

Not all oscillatory motions are simple harmonicmotion though. In the case of a musical instrument,its not generally possible to cause a physicalcomponent of the instrument (like a string or reed) tovibrate as simply as true simple harmonic motion.

Instead, the oscillatory motion will be a morecomplicated repeating waveform (Figure 3.4). Here isthe big idea. This more complicated waveform canalways be created by adding in various intensities ofwaves that are integer multiples of the fundamentalfrequency. Consider a hypothetical thumb piano tinevibrating at 523 Hz. The tine physically cannotproduce true simple harmonic motion when it isplucked. However, it is possible to combine a1,046 Hz waveform and a 1,569 Hz waveform withthe fundamental 523 Hz waveform, to create thewaveform produced by the actual vibrating tine. (The1,046 Hz and 1,569 Hz frequencies are integermultiples of the fundamental 523 Hz frequency and donot necessarily have the same intensity as thefundamental frequency). The total sound produced bythe tine would then have the fundamental frequency of523 Hz as well as its first two overtones.

mode 1

mode 2

mode 3

mode 4

mode 5

Figure 3.2: The first five modes of avibrating string. Each of these (and all otherhigher modes) meets the following criteria:The ends of the string are nodes and thelength of the string is some integer numberof half wavelengths of the modeswavelength. The red dashed line indicatesthe center of the string. Note that if thecenter of the string is plucked, forcing th i sspot to move, modes 2 and 4 (and all othereven modes) will be eliminated since theyrequire a node at this point. However, modes1, 3, and 5 (and all other odd modes) w i l lall be stimulated since they have antinodesat the center of the string.

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The remainder of this section providesvisualization for the timbre of a variety of musicalinstruments. Each figure provides two graphs for aparticular instrument. The first graph is a histogramof the power spectrum for that musical instrument.The first bar on the left is the power of thefundamental frequency, followed by the overtones. Becareful when comparing the relative strength of theovertones to the fundamental and to each other. Thescale of the vertical axis is logarithmic. The top ofthe graph represents 100% of the acoustic power ofthe instrument and the bottom of the graph represents

80 dB lower than full power (

10-8 less power). Thesecond graph is a superposition of the fundamentalwave together with all the overtone waves. Threecycles are shown in each case. Notice that wheregraphs for the same instrument are shown that thisacoustical fingerprint varies somewhat from note tonote the bassoon almost sounds like a differentinstrument when it goes from a very low note to avery high note.

You can click on the name of the instrument inthe caption to hear the sound that produced bothgraphs.

Figure 3.5: E5 played on the VIOLIN . Note the dominance of the fundamental frequency. Themost powerful overtone is the 1st, but only slightly more than 10% of the total power. Thehigher overtones produce much less power. Of the higher overtones, only the 2nd overtoneproduces more than 1% of the total power. The result is a very simple waveform and thecharacteristically pure sound associated with the violin.

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Figure 3.3: The sine wave pictured hereis the displacement vs. time graph of themotion of a physical system undergoingsimple harmonic motion.

Figure 3.4: The complicated waveformshown here is typical of that produced by amusical instrument. This waveform can beproduced by combining the waveform of thefundamental frequency with waveformshaving frequencies that are integermultiples of the fundamental frequency.


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Figure 3.6: F3 played on the CLARINET . Note that although all harmonics are present, the1st, 3rd, 5th, and 7th harmonics strongly dominate. They are very nearly equal to each other i npower. This strong presence of the lower odd harmonics gives evidence of the closed pipenature of the clarinet. The presence of these four harmonics in equal proportion (as well asthe relatively strong presence of the 8 th harmonic) creates a very complex waveform and theclarinets characteristically woody sound

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Figure 3.7: B4 played on the BAGPIPE . There is a strong prominence of odd harmonics(through the 7 th). This is a hint about the closed pipe nature of the bagpipes soundproduction. The 3rd harmonic ( 2nd overtone) is especially strong, exceeding all other modefrequencies in power. Not only is the third harmonic more powerful than the fundamentalfrequency, it appears to have more power than all the other modes combined. This gives thebagpipe its characteristic harshness.

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Figure 3.8: F2 played on the BASSOON . Note that the fundamental frequency produces farless than 10% of the total power of this low note. Its also interesting to note that the firstfive overtones not only produce more power than the fundamental, but they eachindividually produce more than 10% of the total power of this note. This weak fundamentalcombined with the dominance of the first five harmonics creates a very complex waveformand the foghorn-like sound of the bassoon when it produces this note.

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Figure 3.9: B4 played on the BASSOON . The power spectrum is much different for this highnote played on the same bassoon. Over two octaves higher than the note producing thepower spectrum in Figure 3.8, the first two harmonics dominate, producing almost all thepower. The third and fourth harmonics produce far less than 10% of the total power and a l lthe remaining higher harmonics produce less than 1% combined. The result is a s implewaveform with an almost flute-like sound.

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Figure 3.11: D5 played on the TRUMPET . The fundamental frequency and the first twoovertones dominate the power spectrum, with all three contributing over 10% of the totalpower. The next five harmonics all contribute above 1% of the total power. However, evenwith the much greater prominence of the 3rd through 7 th overtones, when compared to thebassoons B4 note, the waveform graphs of the trumpet D5 and the bassoon B4 aresurprisingly prominent.

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Figure 3.10: Bb3 played on the TRUMPET . Like the F2 note on the bassoon, the fundamentalfrequency here produces less than 10% of the total power of this low note. Also like thebassoons F2, the first five overtones here not only produce more power than thefundamental, but they each individually produce more than 10% of the total power of th i snote. There are significant similarities between the two power spectra, but the seemingsubtle differences on the power spectra graph become far more obvious when eithercomparing the two waveform graphs or listening to the notes.

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ACTIVITYINTERPRETING MUSICAL INSTRUMENT POWER SPECTRA

In this activity, you will view the power spectrum graphs for a pure tone as well as three musical instruments: aflute (open end), a panpipe (closed end), and a saxophone. The power spectra are presented randomly below. For eachpower spectrum you will be asked to:

Match the power spectrum to one of four waveform graphs. Make a general statement about the timbre of the sound (pure, complex, harsh, clarinet-like, etc.). Match the power spectrum to the correct musical instrument or to the pure tone.

1. a. Which of the waveforms shown at the end ofthis activity corresponds to this powerspectrum? Explain.

b. Describe the timbre of this sound. Explain.

c. Predict which, if any, of the musical instruments listed above produced this power spectrum. Explain.

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4. a. Which of the waveforms shown at the end ofthis activity corresponds to this power spectrum?Explain.



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BEGINNING TO THINK ABOUT MUSICAL SCALESOW THAT YOU know a bit aboutwaves and even more so about thesound waves that emerge frommusical instruments, youre ready tostart thinking about musical scales.In some cultures, the music is made

primarily with percussion instruments and certaincommon frequencies are far less important thanrhythm. But in most cultures, musical instrumentsthat produce sustained frequencies are more prominentthan percussion instruments. And, for theseinstruments to be played simultaneously, certainagreed upon frequencies must be adopted. How didmusicians come up with widely agreed upon commonfrequencies used in musical scales? How would youdo it? Take a moment to consider that question beforeyou move on. What aspects of the chosen frequencieswould be important in the development of your scale?

Use the space below to make some proposals forwhat you believe would be important as you begin tobuild a musical scale.

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Now show your proposals to a neighbor and have that person provide feedback concerning your ideas. Summarizethis feedback below.

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